Script | Spoken-Tutorial - User contributions [en]

Applications-of-GeoGebra/C3/Statistics-using-GeoGebra/English

2019-01-16T07:48:55Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Statistics using GeoGebra'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to perform:

'''One Variable Analysis''' to calculate different statistical parameters

'''Two Variable Regression Analysis''' to estimate best fit line

'''Multiple Variable Analysis''' to calculate different statistical parameters
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux '''OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Statistics
|-
| | '''Slide Number 5'''

'''Statistics'''

Data analysis and interpretation

'''Measures of central tendency'''

'''Measures of Dispersion'''

Comparing '''variability '''of data series

'''Additional material '''
| |

'''Statistics'''

Statistics deals with

Data analysis and interpretation

Measures of '''central tendency'''

Measures of '''Dispersion'''

Comparing '''variability''' of data series

Please refer to '''additional material''' provided along with this '''tutorial'''.
|-
| | '''Slide Number 6'''

'''Fish Feed '''

A fishery is testing four feed formulations on its fish: '''A, B, C''' and '''D'''

Length (mm)

Weight (lbs)

Girth (mm)

| | '''Fish Feed '''

Let us look at an example.

A fishery is testing four types of feed formulations on its fish: '''A, B, C''' and '''D'''.

Data to be collected after feeding the fish for 6 months are:

Length in millimeters

Weight in pounds

Girth in millimeters

Let us look at some of these data.
|-
| | '''Slide Number 7'''

'''Fish Feed Data'''

[[Image:]]
| | We will use these data for our '''analyses'''.

Please download the '''code file''', '''Fishery-data''', provided along with this '''tutorial'''.
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' tool and select '''Spreadsheet'''.
|-
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.

This will close these views.
|-
| | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''.

Show data in '''columns H''' and '''I'''.

Hold '''Ctrl''' key down and press '''C'''.
| | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''.

These are data for fish that have been fed formulation '''C'''.

Hold '''Control''' key down and press '''C'''.
|-
| | Click on '''Spreadsheet''' view in '''GeoGebra'''.
| | Click in the top of the '''Spreadsheet''' in '''GeoGebra'''.
|-
| | Press '''Ctrl''' key down and press '''V'''.
| | Hold '''Control''' key down and press '''V'''.

This will copy and paste the highlighted data from the '''code file''' into '''GeoGebra'''.
|-
| | Place the '''cursor''' on the first column header in '''Spreadsheet''' view.

Drag and adjust '''column A''''s width.
| | Place the '''cursor''' on the first column header in '''Spreadsheet''' view.

Drag and adjust '''column A''''s width.
|-
| | Right-click on '''column A''' heading of '''Length (mm)'''.

Select '''Object Properties'''.

Point to '''dialog box'''.
| | Right-click on '''column A''' heading of '''Length millimetres'''.

Select '''Object Properties'''.

A '''dialog box''' opens.
|-
| | Click on '''Text tab''' and change the name to '''Length (mm)-C'''.

Close the '''dialog box'''.

Similarly, add '''–C''' to '''Weight (lbs)'''.
| | Click on '''Text tab''' and change the name to '''Length millimetres''' hyphen '''C'''.

Close the '''dialog box'''.

Similarly, add '''hyphen C''' to '''Weight pounds'''.
|-
| | Adjust '''column B''' width.
| | Adjust '''column B''' width.
|-
| | Use '''mouse''' to drag and highlight first '''column A'''’s length data and label in '''GeoGebra'''.
| | Click on '''column A''' heading of '''Length millimetres C'''.

Drag to highlight length data in '''Spreadsheet''' view.
|-
| | Below the '''menubar''', click on '''One Variable Analysis''' tool.

Point to '''Data Source''' popup window.

Click on '''Analyze button'''.
| | Below the '''menubar''', click on '''One Variable Analysis''' tool.

A '''Data Source''' popup window appears.

Click on '''Analyze button'''.
|-
| | Point to '''Data Analysis''' window and '''histogram'''.
| | A '''Data Analysis''' window appears.

By default, a '''histogram''' is plotted.
|-
| | Drag the boundary to see the graph properly.
| | Drag the boundary to see the graph properly.
|-
| | Point to length on the '''x-axis''' and frequency on the '''y-axis'''.
| | The length is plotted on the '''x-axis'''.

The number of fish that are of a particular length, the '''frequency''', is plotted on the '''y-axis'''.
|-
| | Point to the '''display box''' above the graph containing the word '''Histogram'''.
| | Note the '''display box''' above the graph containing the word '''Histogram'''.
|-
| | In the '''display box''', click on the '''dropdown menu button'''.
| | In the '''display box''', click on the '''dropdown menu button''' to display the list of plots.
|-
| | Select '''Histogram'''.

Point to '''slider''' to the right of the display.
| | We will stay with the '''histogram''' option.

To the right of the '''dropdown menu''' is a '''slider'''.
|-
| | Drag the '''slider''' from left to right to go to 20.
| | Drag the '''slider''' from left to right to go to 20.
|-
| | Point to rectangles between minimum and maximum values of data.
| | The '''slider''' changes the number of rectangles between the minimum and maximum values of data.
|-
| | Click on '''Options''' button to the right of the '''slider'''.
| | Click on '''Options''' button to the right of the '''slider'''.
|-
| | Under '''Classes''', check '''Set Classes Manually'''.
| | Under '''Classes''', check '''Set Classes Manually check box'''.

This displays '''Start''' and '''Width text-boxes''' to the left of the '''Options button'''.
|-
| | Type 800 in the '''Start text-box''' and press '''Enter'''.

Show the value of 5 in the '''Width text-box'''.
| | As all the fish are over 800 mm long, type 800 in the '''Start text-box''' and press '''Enter'''.

We will stay with the default value of 5 for rectangle '''width'''.
|-
| | Uncheck '''Set Classes Manually'''.
| | Uncheck '''Set Classes Manually''' check box.
|-
| | Under '''Show''', uncheck '''Histogram''' check box.
| | Under '''Show''', uncheck '''Histogram''' check box to make it disappear.
|-
| | Scroll down and check '''Frequency Polygon''' to show it.
| | Scroll down and check '''Frequency Polygon''' to show it.
|-
| | Check '''Cumulative''' option as the '''Frequency Type'''.
| | Under '''Frequency Type''', check '''Cumulative''' option.
|-
| | Point to default '''Count''' selection.

Point to the '''cumulative frequency count'''.
| | The default '''Count''' selection shows the '''cumulative frequency count''' for the data.
|-
| | Drag '''slider''', bring it back to 20.
| | Drag the '''slider''' and note the effects on smoothness of the '''cumulative frequency count curve'''.

We will drag the '''slider''' back to 20.
|-
| | Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''.
| | Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''.
|-
| | Under '''Show''', check '''Histogram''' and uncheck '''Frequency Polygon''' > > click on '''Options''' button again to hide the window.
| | Under '''Show''', check '''Histogram''' and uncheck '''Frequency Polygon'''.

And click on '''Options''' button again to hide the window.
|-
| | Click on '''Show Data''' tool >> point to data highlighted in the '''Spreadsheet'''.
| | Above the '''Histogram text-box''', click on the third '''Show Data tool button'''.

This displays all the data highlighted in the '''Spreadsheet'''.
|-
| | Drag the boundary to see the data properly.
| | Drag the boundary to see the data properly.
|-
| | Click on '''Show Data''' tool again to hide the list.
| | Click on the '''Show Data''' tool again to hide the list.
|-
| | Click on '''Show 2nd Plot''' tool.
| | Above the '''Histogram text-box''', click on the last '''Show 2nd Plot tool button'''.
|-
| | Select '''histogram''' for top plot and '''box plot''' for bottom plot.
| | The same data are graphed in two vertically placed plots.

You can select plot types from the '''dropdown menu button''' above each plot.
|-
| | Click on '''Show Statistics''' tool.

Point to '''Statistics''' for both plots.
| | Above the '''Histogram text-box''', click on the second '''Show Statistics tool button'''.

'''Statistics''' for the plot appears as a '''panel''' in the middle.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | '''Slide Number 8'''

'''Box Plot'''

[[Image:]]
| | '''Box Plot'''

'''Box plot''' is a standardized way of showing data, based on the '''five number summary'''.
|-
| | Click and point to '''Median''', '''Min''', '''Max''', '''Q1''' and '''Q3''' values in the '''box plot'''.
| | Let us compare '''histogram''' and '''box plot'''.

In the '''box plot''', locate the '''Median''', '''Min''', '''Max''', '''Q1''' and '''Q3''' values.
|-
| | Click on the button next to '''Options button''' above the plot.
| | Above each plot, in the upper right corner, click on the '''button''' next to '''Options'''.

A '''dropdown menu''' appears with which you can copy each plot to '''Clipboard''' or '''export''' it as an '''image'''.
|-
| | Click on '''Show Statistics tool button''' to hide the data.
| | Click on '''Show Statistics tool button''' to hide the data.
|-
| | Close the '''Data Analysis''' window.
| | Close the '''Data Analysis''' window.
|-
| | '''Slide Number 9'''

'''Least Squares Linear Regression (LSLR)'''

Changing an independent variable '''x''' changes the dependent variable '''y'''.

'''LSLR''' predicts '''y''' based on '''x''' value.

'''LSRL (best fit line) y = b0 + b1x'''

'''Coefficient of determination R2'''
| | '''Least Squares Linear Regression (LSLR)'''

Changing an independent variable '''x''' changes the dependent variable '''y'''.

'''LSLR''' predicts '''y''' based on '''x''' value.

'''Least Squares Regression Line (LSRL)''' is also called the '''best fit line'''.

It is given by '''y = b0 + b1x'''.

'''b1''', the '''slope''', is the '''regression coefficient'''.

'''Coefficient of determination R squared'''

'''R squared''' ranges from 0 to 1.

The closer '''R squared''' is to 1, the better is the prediction of variance in '''y''' from '''x'''.
|-
| | Show length and weight data in the '''Spreadsheet''' in the '''GeoGebra'''.
| | Let us go back to the length and weight data in the '''Spreadsheet''' view in '''GeoGebra'''.
|-
| | Drag '''mouse''' to highlight all labels and data in the two '''columns'''.
| | Drag and select all the data in both '''columns'''.
|-
| | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool.
| | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool.
|-
| | Click '''Analyze button''' in the '''Data Source window''' that pops up.
| | In the '''Data Source window''' that pops up, click '''Analyze button'''.
|-
| | '''Data Analysis''' window appears.
| | A '''Data Analysis''' window appears with two plots.
|-
| | Show both plots.
| | By default, the upper plot is a '''Scatterplot''' and the lower a '''Residual plot'''.
|-
| | Click on '''Show Statistics''' tool to see '''Statistics'''.
| | Click on '''Show Statistics''' tool to see the '''Statistics'''.
|-
| | Drag the boundary to see them properly.
| | Drag the boundary to see them properly.
|-
| | Below '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''.
| | Below the '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''.
|-
| | Point to the red line that is drawn through some points.
| | Note the red line in the '''Scatterplot'''.
|-
| | Point to equation is given in red, '''y= 0.08x-48.39'''.
| | This is the '''best fit line''' that passes through as many points as possible.

Its equation is given in red at the bottom.
|-
| | Point to '''R2''' value of 0.7722.
| | This ''' R squared''' value indicates good fit between the model and the actual data.
|-
| | Select other '''regression models''' to see effects on '''R2'''.
| | Select other '''regression models''' to see effects on the '''R squared''' value.
|-
| | Point to the lower '''Residual Plot'''.
| | The lower plot is the '''Residual Plot'''.

'''Residuals '''are the differences between observed and predicted values of all points.
|-
| | Click on '''Switch Axes button'''.
| | Above the '''Statistics window''', click on the last '''Switch Axes button'''.
|-
| | Point to length now plotted on '''y-axis''' and weight on '''x-axis'''.
| | For the '''scatterplot''', length is now plotted along '''y-axis''' and weight along '''x-axis'''.
|-
| | Point to the '''best fit line''' and '''statistics'''.

Point to equation '''y= 9.91x + 684.3'''.
| | Observe that the '''best fit line''' and many '''statistics''' change.

Its equation is now '''y= 9.91x + 684.3'''.
|-
| | Point to '''r''', '''R2 '''and '''rho (ρ)'''.
| | The only statistics that remain the same are '''r''', '''R squared '''and '''rho (ρ)'''.

Note that '''r''' and '''rho''' are greater than 0.8, indicating positive '''correlation'''.

Weight increases as length increases for fish given '''feed C'''.

The relationship is strong and well predicted by the '''best fit lines'''.
|-
| | Again, click on '''Switch Axes button'''.
| | Again, click on '''Switch Axes button'''.
|-
| | Point to '''Symbolic Evaluation''' at the bottom.
| | At the bottom, in '''Symbolic Evaluation''', you can enter a value for '''x''' to get a prediction for '''y'''.
|-
| | Point at the line in the '''Scatterplot'''.
| | To get logical predictions, we will enter '''x''' values above the '''x-intercept'''.
|-
| | In '''Symbolic Evaluation''', type in a value for '''x''' and press '''Enter'''.
| | In '''Symbolic Evaluation''', in the text-box for '''x''', type 800 and press '''Enter'''.
|-
| | Point to '''y''' value appearing next to the '''display box'''.
| | Note that a '''y''' value appears next to the '''display box'''.

The '''x''' value was substituted in the '''best fit line''' equation to get the '''y''' value.
|-
| | Again, click on '''Show Statistics tool button'''.
| | Again, click on '''Show Statistics tool button'''.
|-
| | Close the '''Data Analysis window'''.
| | Close the '''Data Analysis window'''.
|-
| | Point to length and weight data in the '''Spreadsheet'''.
| | Let’s go back to the length and weight data in the '''Spreadsheet'''.
|-
| | Drag '''mouse''' to highlight all labels and data in the two '''columns'''.
| | In the '''Spreadsheet''', select all the data in both '''columns'''.
|-
| | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool.
| | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool.
|-
| | Click '''Analyze button''' in the '''Data Source window''' that pops up.
| | In the '''Data Source window''' that pops up, click '''Analyze button'''.
|-
| | Point to '''Box Plots''' in the window and to the cell numbers in each row.
| | '''Box Plots''' appear in the window.

They are for length and weight data.
|-
| | Click on '''Show Statistics''' tool.

Point to '''Statistics''' for both plots.
| | Above the plot, click on the second '''Show Statistics''' tool.

'''Statistics''' for both plots appear below.
|-
| | Place the '''cursor''' on the boundary between the plot and statistics.

When the arrow appears, drag the boundary to resize the windows.
| | Place the '''cursor''' on the boundary between the plot and statistics.

When the '''arrow''' appears, drag the boundary to resize the windows.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 10'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to perform:

'''One Variable Analysis''' to calculate different statistical parameters

'''Two Variable Regression Analysis''' to estimate best fit line

'''Multiple Variable Analysis''' to calculate different statistical parameters
|-
| | '''Slide Number 11'''

'''Assignment'''

Perform statistical analyses for weight and girth data

Is any of the oils absorbed more than the others?

[[Image:]]

| | '''Assignment'''

Perform statistical analyses for weight and girth data given in this '''tutorial'''

Four oils were used to deep fry chips. Amount of absorbed fat was measured for 6 chips fried in 4 oils. Is any of the oils absorbed more than the others?
|-
| | '''Slide Number 12'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 13'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 15'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

2019-01-15T05:35:36Z

Vidhya:

{|border=1
|| '''Visual Cue'''
|| '''Narration'''

|-
|| '''Slide Number 1'''

'''Title Slide'''
|| Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
|-
|| '''Slide Number 2'''

'''Learning Objectives'''
|| In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate:

'''Area Under a Curve (AUC)'''

Area bounded by two '''functions'''
|-
|| '''Slide Number 3'''

'''System Requirement'''
|| Here I am using:

'''Ubuntu Linux''' Operating System version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
|| '''Slide Number 4'''

'''Pre-requisites'''

[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
|| To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Integration

For relevant '''tutorials''', please visit our website.
|-
|| '''Slide Number 5'''

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''[a,b]''' above '''x-axis'''

'''a''' is lower limit, b is upper limit

<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>

Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
|| '''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''a, b''' above the '''x-axis'''.

'''a''' and '''b''' are called the lower and upper limits of the integral.

Integral of '''f of x''' from '''a''' to '''b''' with respect to '''x''' is the notation for this definite integral.

It is the area bounded by '''y''' equals '''f of x''', '''x''' equals '''a, x''' equals '''b''' and the '''x-axis'''.
|-
|| '''Slide Number 6'''

'''Calculation of a Definite Integral'''

Let us calculate the definite integral

<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
|| Let us calculate the definite integral of this function with respect to '''x'''.
|-
|| Open a new '''GeoGebra''' window.
|| Let us open a new '''GeoGebra''' window.
|-
|| Type '''g(x)= - 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.
|| In the '''input bar''', type the following line and press '''Enter'''.
|-
|| Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.
|| Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.
|-
|| Click on '''Slider''' tool and click in '''Graphics''' view.

Type '''n''' in the '''Name''' field.

Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''

Point to '''slider n''' in '''Graphics''' view.
|| Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.
It should range from 1 to 50 in increments of 1.
|-
|| Drag '''slider n''' to 5.
|| Drag the resulting '''slider n''' to 5.
|-
|| Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.
|| Under '''Point''', click on '''Point on Object''' and click at -1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.
|-
|| Cursor on the GeoGebra interface.
|| Let us look at a few ways to approximate '''area under the curve'''.

These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.

We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
|-
|| Type '''uppersum=Upp''' in the '''Input Bar'''.

Show option.

'''UpperSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles> )'''

Click on it.
|| In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.

The following option appears.
Click on it.
|-
|| Type '''g''' instead of highlighted '''<Function>'''.
|| Type '''g''' instead of highlighted '''<Function>'''.
|-
|| Press '''Tab''' to highlight '''<Start x-Value>'''.
|| Press '''Tab''' to highlight '''<Start x-Value>'''.
|-
|| Type '''x(A)'''.
|| Type '''x A in parentheses'''.
|-
|| Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
|| Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.

Press '''Enter'''.
|-
|| Point to five rectangles between '''x'''<nowiki= -1 and 2. </nowiki>
|| Note that five rectangles appear between '''x''' equals -1 and 2.
|-
|| Under '''Move Graphics View,''' click on '''Zoom In ''' >> click in '''Graphics''' view.
|| Under '''Move Graphics View,''' click on '''Zoom In ''' and click in '''Graphics''' view.
|-
|| Click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
|| Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
|-
|| '''Point''' to '''upper sum area under the curve (AUC).'''
|| The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
|-
|| Point to the rectangles extending above the curve.
|| It is an overestimation of the area under the curve.

This is because some portion of each rectangle extends above the curve.
|-
|| Drag the background to move the graph to the left.
|| Drag the background to move the graph to the left.
|-
||
|| Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.
|-
|| Type '''trapsum=Tra''' in the '''Input bar'''.
|| In the '''input bar''', type '''trapsum''' is equal to '''Capital T ra'''.
|-
|| Point to the menu that appears.
|| A menu with various options appears.
|-
|| Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> )'''.
|| Select the following option.
|-
||
|| We will type the same values as before and press '''Enter'''.
|-
|| In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Point to trapezoids.
|| In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Note the shape of the trapezoids.
|-
||
|| Let us now look at the integral as the area under the curve.
|-
|| Finally, type '''Int''' in the '''Input Bar'''.
|| Finally, in the '''input bar''', type '''capital I nt'''.
|-
|| '''Point''' to the menu with various options.
|| A menu with various options appears
|-
|| Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.
|| Select the following option.
|-
|| Enter '''g , x(A), x(B)'''
|| Again, we will enter the same values as before.
And Press '''Enter.'''
|-
|| In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
|| In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
|-
|| Point to the integrated''' AUC'''.
|| For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals -1 to 2.
|-
|| In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
|| In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
|-
|| Click on '''Text''' tool under '''Slider''' tool.
|| Under '''Slider''', click on '''Text'''.
|-
|| Click in '''Graphics''' view to open a '''text box'''.
|| Click in '''Graphics''' view to open a '''text box'''.
|-
|| In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
|| In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
|| Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
|| Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
|| Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

Click '''OK''' in the '''text box'''.
|| Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

In the '''text box''', click '''OK'''.
|-
|| Click on '''Move''' >> drag the '''text box''' in case you need to see it better.
|| Click on '''Move''' and drag the '''text box''' in case you need to see it better.
|-
|| Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
|| Now, click on the '''text box''', click on the '''Graphics''' panel and select '''bold''' to make the text bold.
|-
|| In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
|| In '''Algebra''' view, check '''a''', '''trapsum''' and '''uppersum''' to show all of them.
|-
|| Point to text box and to '''slider n'''.
|| Observe the values in the '''text box''' as you drag '''slider n'''.
|-
|| Point to '''Graphics''' view.
|| '''Trapsum''' is a better approximation of '''AUC''' at high '''n''' values.

'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.
|-
|| Open a new '''GeoGebra''' window.
|| Let us open a new '''GeoGebra''' window
|-
|| Cursor on GeoGebra interface.
|| We will look at the relationship between '''differentiation''' and '''integration'''.

Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.
|-
|| Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.
|| In the '''input bar''', type the following line and press '''Enter'''.
|-
|| Point to f of x.
|| Let us call '''integral''' of '''f of x capital F of x'''.
|-
|| Type '''F(x)= Integral(f)''' in the '''Input Bar''' >> '''Enter'''.
|| In the '''input bar''', type the following line and press '''Enter'''.
|-
|| Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.

Point to equation for '''F(x)=1/3 x3+ x2+x''' appears in '''Algebra''' view.
|| The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.

Its equation for '''capital F of x''' appears in '''Algebra''' view.

Confirm that this is the integral of '''f of x'''.
|-
|| Drag the boundary to see the equations properly.
|| Drag the boundary to see the equations properly.
|-
|| Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.
|| In the '''input bar''', type the following and press '''Enter'''.
|-
|| Point to '''F'(x)''' and '''f(x)'''.
|| Note that this graph coincides with '''f of x'''.

The equations for '''f of x''' and '''h of x''' are the same.

Thus, we can see that '''integration''' is the inverse process of '''differentiation'''.

Taking the derivative of an integral, gives back the original '''function'''.
|-
|| Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
|| Click on '''Point''' tool and create a point at '''1 comma 3'''.
|-
|| Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.
|| In the '''input bar''', type the following and press '''Enter'''.
|-
|| Click on '''Create Sliders''' in the window that pops up.
|| Click on '''Create Sliders''' in the window that pops up.
|-
|| Point to '''slider k'''.
|| A '''slider k''' appears.
|-
|| Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.

Close the '''Preferences''' window.
|| Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5.

Scroll right to set the '''Increment''' to 0.01.

Close the '''Preferences''' box.
|-
|| Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
|| In '''Algebra''' view, double-click on '''i of x''' and on '''Object Properties'''.
|-
|| Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
|| Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
|-
|| Drag '''k''' to make '''i(x)''' pass through point '''A'''.

Point to integral function '''(1/3)x3+x2+x+0.7'''.
|| Drag '''k''' to make '''i of x''' pass through point '''A'''.
|-
|| Drag the boundary to see '''i of x''' properly.
|| Drag the boundary to see '''i of x''' properly.
|-
|| Point to '''F(x)+0.7''': the curve and equation.
|| This function is '''capital F of x''' plus 0.7.
|-
|| '''Slide Number 7'''

'''Double Integrals'''

'''Double integrals''' can be used to find:

'''AUC''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z=f(x,y)'''
|| '''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''' directions

The volume under a surface '''z''' which is equal to '''f of x''' and '''y'''
|-
|| '''Slide Number 8'''

'''Double Integral-An Example'''

Let us find the area between parabola '''x=y2''' and the line '''y=x'''.

The '''limits''' are from '''(0,0)''' to '''(1,1)'''.

This area can be expressed as the '''double integral'''

'''=<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''

'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>'''
||'''Double Integral-An Example'''

Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.

The limits are from '''0 comma 0''' to '''1 comma 1'''.

This area can be expressed as the double integrals shown here.

Observe the limits and the order of the integrals in terms of the variables.

|-
||
|| Let us open a new '''GeoGebra''' window.

We will first express '''x''' in terms of '''y''', for both '''functions'''.
|-
|| In the '''input bar''', type '''x=y2''' >> press '''Enter'''.
|| In the '''input bar''', type '''x''' equals '''y caret''' 2 and press '''Enter'''.
|-
|| Next, in the '''input bar''', type '''y=x''' >> press '''Enter'''.
|| Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.
|-
|| Click on '''View''' tool and select '''CAS'''.
|| Click on '''View''' tool and select '''CAS'''.
|-
|| In '''Algebra''' view, click top right button to close '''Algebra''' view.
|| In '''Algebra''' view, click top right button to close '''Algebra''' view.
|-
|| Drag the boundary to make '''CAS''' view bigger.
|| Drag the boundary to make '''CAS''' view bigger.
|-
|| In '''CAS''' view, type '''Int''' in line 1.

Point to the menu that appears.
|| In '''CAS''' view, type '''Int capital I''' in line 1.

A menu with various options appears.
|-
|| Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
|| Scroll down.

Select the following option.
|-
|| Type '''y''' for the first '''function'''.
|| Type '''y''' for the first '''function'''.
|-
|| Press '''Tab''' >> type '''y^2''' for the second '''function'''.
|| Press '''Tab '''and type '''y caret 2''' for the second '''function'''.
|-
|| Press '''Tab''' >> type '''y''' as the '''variable'''.
|| Press '''Tab''' and type '''y''' as the '''variable'''.
|-
|| Press '''Tab''' >> type 0 and 1 as '''start''' and '''end values''' of '''y'''.
|| Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
|-
|| Press '''Enter'''.
|| Press '''Enter'''.
|-
|| Point to the value of 1/6 below the entry.

Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.
|| A value 1 divided by 6 appears below the entry.

This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.
|-
|| Let us now express '''y''' in terms of '''x''' for both '''functions'''.
|| Let us now express '''y''' in terms of '''x''' for both '''functions'''.
|-
|| In '''CAS''' view, type '''Int''' and observe the same menu as before.
|| In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.
|-
|| Cursor in '''CAS''' view.
|| Now, let us reverse the order of '''functions''' and '''limits'''.
|-
|| Type '''sqrt(x)''' for the first function and '''x''' for the second.
|| Type the following and press '''Enter'''.
|-
|| Point to the '''input bar'''.
|| You can also use the '''input bar''' instead of the '''CAS''' view.
|-
|| Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
|| Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
|-
|| Drag the boundaries to make '''CAS''' view smaller.
|| Drag the boundaries to make '''CAS''' view smaller.
|-
|| In the '''input bar''', type '''Int'''.

From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start Value''' and again press '''Tab''' to move to and type 1 as the '''End Value'''.

Press '''Enter'''.

This will also give you an area a of 0.17 or 1 divided by 6.
||In the '''input bar''', type '''Int capital I'''.

From menu, select the following option.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start x Value''' and again press '''Tab''' to move to and type 1 as the '''End x Value'''.

Press '''Enter'''.

This will also give you an area '''a''' of 0.17 or 1 divided by 6.
|-
||
|| Let us summarize.
|-
|| '''Slide Number 9'''

'''Summary'''
|| In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of:

'''Area Under a Curve''' ('''AUC''')

Area bounded by two '''functions'''
|-
|| '''Slide Number 10'''

'''Assignment'''

Calculate '''<math>{\int }_{0}^{0.5}f\left(x\right)dx</math>''' where '''f(x) = 1/(1-x)'''

Calculate '''<math>{\int }_{x\left(A\right)}^{x\left(B\right)}g\left(x\right)dx</math>''' and

'''<math>{\int }_{x\left(B\right)}^{x\left(C\right)}g\left(x\right)dx</math>'''

where '''g(x) = 0.5x3+2x2-x-3.75'''

'''A''', '''B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
|| As an '''assignment''':

Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.

Explain the results for '''g of x'''.
|-
|| '''Slide Number 11'''

'''Assignment'''

Calculate the area bounded by the following '''functions''':
'''y=4x-x2, y=x'''

'''x2+y2<nowiki>=9, y=3-x</nowiki>'''

'''y=1+x2, y=2x2'''
|| As another '''assignment''':

Calculate the shaded areas between these pairs of '''functions'''.
|-
|| '''Slide Number 12'''

'''About Spoken Tutorial project'''
|| The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
|| '''Slide Number 13'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials

gives certificates on passing online tests.

For more details, please write to us.
|-
|| '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
|| Please post your timed queries on this forum.
|-
|| '''Slide Number 15'''

'''Acknowledgement'''
|| '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
||
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Differentiation-using-GeoGebra/English

2019-01-11T13:09:17Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Differentiation using GeoGebra'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn how to use '''GeoGebra''' to:

Understand Differentiation

Draw graphs of derivative of functions

|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d

|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Differentiation

For relevant '''tutorials''', please visit our website.

|-
| | '''Slide Number 5'''

'''Differentiation: First Principles'''

[[Image:]]

'''f(x) = x2-x'''

'''f'(x)''' is derivative of '''f(x)'''

'''A (x, f(x)), B (x+j, f(x+j))'''
| |

Let us understand differentiation using '''first principles''' for the '''function f of x'''.

'''f of x''' is equal to '''x squared''' minus '''x'''

'''f prime x''' is the derivative of '''f of x'''.

Consider 2 points, '''A''' and '''B'''.

'''A''' is '''x''' comma '''f of x''' and '''B''' is '''x''' plus '''j''' comma '''f of x''' plus '''j'''
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Type '''f(x)=x^2-x''' in the '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line.
For the '''caret symbol''', hold the '''Shift''' key down and press 6.

Press '''Enter'''.
|-
| | Point to the equation in '''Algebra''' view.

Point to '''parabola''' in '''Graphics''' view.
| | Observe the equation and the parabolic graph of '''function f'''.
|-
| | Click on '''Point on Object''' tool >> click on the parabola at '''(2,2)'''.

Point to '''A''' at '''(2,2)'''.

Click on '''Point''' tool and click on '''(3,6)'''.

| | Clicking on the '''Point on Object''', create point A at 2 comma 2 and B at 3 comma 6.

|-
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
|-
| | Click on the '''Move''' tool.

Double click on the resulting '''line g''' and click on '''Object Properties'''.

Click on '''Color''' tab and select blue.

Click on '''Style''' tab and select '''dashed style'''.
| | As shown earlier in this series, make this line '''g '''blue and dashed.
|-
| | Click on '''Tangents''' tool under '''Perpendicular Line''' tool.
| | Under '''Perpendicular Line''', click on '''Tangents'''.
|-
| | Click on '''A''' and then on the '''parabola'''.
| | Click on '''A''' and then on the '''parabola'''.
|-
| | Point to '''tangent h''' at point '''A''' to the '''parabola'''.
| | This draws a '''tangent h''' at point '''A''' to the '''parabola'''.
|-
| | Double click on '''tangent h''' and click on '''Object Properties'''.

Under '''Color''' tab, select red.

Close the '''Preferences''' box.
| | Let us make '''tangent h''' a red line.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.

Point to point '''C'''.
| | Click on the '''Point''' tool and click anywhere in '''Graphics''' view.

This creates point '''C'''.
|-
| | Double click on point '''C''' in '''Algebra''' view and change its '''coordinates''' to '''(x(B),y(A))'''.

Point to '''C'''.
| | In '''Algebra''' view, double-click on '''C''' and change its '''coordinates''' to the following ones.
|-
| |
| | Now C has the same '''x coordinate''' as point '''B''' and the same '''y coordinate''' as point '''A'''.
|-
| | Under '''Line''', click on '''Segment''' and click on '''B '''and '''C''', and then on '''A''' and '''C'''.
| | Let us use the '''Segment''' tool to draw segments '''BC''' and '''AC.'''
|-
| | Right-click on '''AC''' >> Select '''Object Properties''' >> '''Color''' tab >> Purple

Click on '''Style''' tab >> select dashed line

Under '''Basic''' tab >> choose '''Name and Value''' >> '''Show Label''' check box.

Close the '''Preferences''' dialog box.
| | We will make '''AC''' and '''BC''' purple and dashed segments.
|-
| | With '''Move''' highlighted, drag '''B''' towards '''A''' on the '''parabola'''.
| | With '''Move''' highlighted, drag '''B''' towards '''A''' on the '''parabola'''.
|-
| | Point to the value of '''j''' (length of '''AC''') and lines '''g''' and '''h'''.
| | Observe lines '''g''' and '''h''' and the value of '''j''' (length of '''AC''').

As '''j''' approaches 0, points '''B'''and '''A''' begin to overlap.

Lines '''g''' and '''h''' also begin to overlap.
|-
| | Point to line '''g''', '''BC''' and '''AC'''.
| | Slope of line '''g''' is the ratio of length of '''BC''' to length of '''AC'''.
|-
| | Point to all the points on the parabola.
| | Derivative of the parabola is the slopes of tangents at all points on curve.
|-
| | Point to text-box that appears in '''GeoGebra''' window.

As '''B''' approaches '''A''', slope '''AB''' approaches slope of tangent at '''A'''.
| | As '''B''' approaches '''A''' on '''f of x''', slope of '''AB''' approaches the slope of tangent at '''A'''.
|-
| |
| | Now let us look at the '''Algebra''' behind these concepts.
|-
| | '''Slide Number 7'''

'''Differentiation: First Principles, the Algebra'''

'''f'(x) = lim_j→0 (length of Segment BC / length of Segment AC)'''

'''<nowiki>= lim_j→0 </nowiki>[(f(x+j) – f(x)]/[(x+j) – x]'''

Remember '''f(x) = x2-x, (x+j)2 = x2+2xj+j2'''

'''f'(x) = lim_j→0 [(x+j)2-(x+j)-(x2-x)]/(x+j-x)'''

| | Slope of line '''AB''' equals the ratio of the lengths of '''BC''' to '''AC'''.

Line '''AB''' becomes the tangent at point '''A''' as distance '''j''' between '''A''' and '''B''' approaches 0.

'''BC''' is the difference between '''y' coordinates''', '''f of x''' plus '''j''' and '''f of x''', for '''A''' and '''B'''.

'''AC''' is the difference between the '''x-coordinates''', '''x''' plus '''j''' and '''x'''.

Let us rewrite '''f of x''' plus '''j''' and '''f of x''' in terms of '''x squared''' minus '''x'''.

We will expand the terms in the numerator.
|-
| | '''Slide 8 The Algebra-Cont’d'''

'''f'(x) = lim_j→0 [x2+2xj+j2-x-j-x2+x]/j'''

'''<nowiki>= lim_j→0 [</nowiki>2xj+j2-j]/j = lim_j→0 [j(2x+j-1)]/j'''

'''<nowiki>= lim_j→0 [2x+j-1] = 2x-1</nowiki>'''

'''f'(x2-x) = 2x -1'''
| | After expanding the terms in the numerator, we will cancel out similar terms with opposite signs.

We will pull out '''j''' from the numerator, and cancel it.

Note that as '''j''' approaches 0, '''j''' can be ignored so that '''2x''' plus '''j''' minus 1 approaches '''2x''' minus 1.

As we know, derivative of '''x squared''' minus '''x''' is '''2x''' minus 1.

|-
| |
| | Let us look at derivative graphs for some '''functions'''.
|-
| | '''Slide Number 10'''

'''Differentiation of a Polynomial Function'''

Consider '''g(x)=5+12x-x3'''

'''Differentiation rules''':

'''<div>d(u±v)/dx = d(u)/dx ± d(v)/dx</div>'''

'''d(5+12x-x3)/dx = d(5)/dx + d(12x)/dx - d(x3)/dx <nowiki>= 0 + 12 - 3x</nowiki>2 <nowiki>= -3x</nowiki>2 +12'''

For '''g(x)=5+12x-x3, g'(x) = -3x2 +12'''
| |

Consider '''g of x'''.

Derivative '''g prime x''' is the sum and difference of derivatives of the individual components.

'''g prime x''' is calculated by applying these rules.
|-
| |
| | Let us differentiate '''g of x''' in '''GeoGebra'''.
|-
| | Open a new '''GeoGebra''' window.
| | Open a new '''GeoGebra''' window.
|-
| | Type '''g(x)=5+12x-x^3''' in '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out'''.

Click in '''Graphics''' view until you see '''function g'''.

| | As shown earlier in the series, zoom out to see '''function g''' properly.

|-
| | Right-click in '''Graphics''' view and select '''xAxis : yAxis''' option.

Select '''1:5'''.
| | Right-click in '''Graphics''' view and select '''xAxis''' is to '''yAxis''' option.

Select 1 is to 5.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' again.

Click in '''Graphics''' view to zoom out.
| | I will zoom out again.
|-
| | Click on '''Point on Object''' tool and click on the curve to create point '''A'''.

Click on '''Tangent''' under '''Perpendicular Line'''.

Click on point '''A''' and the curve.

| | As shown earlier, draw point '''A''' on curve '''g''' and a tangent '''f''' at this point.
|-
| | Click on '''Slope''' tool under '''Angle''' tool and on tangent line '''f'''.
| | Under '''Angle''', click on '''Slope''' and on tangent line '''f'''.
|-
| | Point to slope of '''line f''' at '''A''' appearing as '''m''' value in '''Graphics''' view.

| | Slope of line '''f'''at '''A''' appears as '''m''' value in '''Graphics''' view.
|-
| | Click on '''Point''' tool and in '''Graphics''' view to create point '''B'''.

Double click on point '''B''' in '''Algebra''' view and change '''coordinates''' to ('''x(A), m)'''.

Point to points '''A''' and '''B''' and slope ''' m''' of tangent line '''g'''.

| | Draw point '''B''' and change its '''coordinates''' to '''x A''' in parentheses comma '''m'''.
|-
| | Right-click on point '''B''' and select '''Trace On''' option.
| | Right-click on point '''B''' and select '''Trace On''' option
|-
| | Click on '''Move''' tool and move point '''A''' on curve.

Observe the curve traced by point '''B'''.
| | With '''Move''' tool highlighted, move point '''A''' on the curve.

Observe the curve traced by point '''B'''.
|-
| |
| | Let us check whether we have the correct '''derivative''' graph.
|-
| | Type '''Deri''' in '''input bar''' >> select '''Derivative( <Function> )''' >> Type '''g''' instead of highlighted '''<Function>''' >> press '''Enter'''
| | In the '''input bar''', type '''capital D e r i'''.

From the menu that appears, select '''Derivative Function''' option.

Type '''g''' to replace the highlighted word '''<Function>'''.

Press '''Enter'''.
|-
| |
| | Note the equation of '''g prime x''' in '''Algebra''' view.

Drag the boundary to see it properly
|-
| | Compare slide’s calculations with equation of '''g'(x)''' in '''Algebra''' view.
| | Compare the calculations in the previous slide with the equation of '''g prime x'''
|-
| |
| | Let us find the maxima and minima of the '''function g of x'''.
|-
| | Point to derivative curve '''g'(x)''' above the '''x-axis''' and to '''g(x)'''.
| | Derivative curve '''g prime x''' remains above the '''x-axis''' (is positive) as long as '''g of x''' is increasing.
|-
| | Point to derivative curve '''g'(x)''' below the '''x-axis''' and to '''g(x)'''.
| | '''g prime x''' remains below the '''x-axis''' (is negative) as long as '''g of x''' is decreasing.
|-
| | Point to derivative curve '''g'(x)''' intersecting '''x-axis''' at '''x = -2 '''and''' x = 2'''.
| | 2 and -2 are the values of '''x''' when '''g prime x''' equals 0.
|-
| |
| | Slope of the tangents at the corresponding points on '''g of x''' is 0.

These points on '''g of x''' are maxima or a minima.
|-
| |
Point to '''(-2,-11)''' and '''(2,21)'''.
| | Hence, for '''g of x,''' -2 comma -11 is the minimum and 2 comma 21 is the maximum.
|-
| | Point to minimum of '''g(x)''' and '''x=-3''' and '''x = -1'''.
| | In '''GeoGebra''', we can see that the minimum value of '''g of x''' lies between '''x''' equals -3 and '''x''' equals -1.
|-
| | In the '''input bar''', type '''Min'''.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' for '''Function'''.

Press '''Tab''' to go to the next argument.

Type -4 and -1 as '''Start''' and '''End x-Values'''.

Press '''Enter'''.
| | In the '''input bar''', type '''Min'''.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' for '''Function'''.

Press '''Tab''' to go to the next argument.

Type -4 and -1 as '''Start''' and '''End x-Values'''.

Press '''Enter'''.
|-
| | Point to minimum '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
| | We see the minimum on '''g of x'''.

Its '''co-ordinates''' are -2 comma -11 in '''Algebra''' view.
|-
| | In the '''input bar''', type '''Max'''.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''', 1 and 4 as the arguments.

Press '''Enter.'''
| | In the '''input bar''', type '''Max'''.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''', 1 and 4 as the arguments.

Press '''Enter.'''
|-
| | Point to maximum '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
| | We see the maximum on '''g of x''', 2 comma 21.
|-
| |
| | Finally, let us take a look at a practical application of differentiation.
|-
| | '''Slide Number 16'''

'''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard

We have to convert it into a box

Squares have to be cut from the four corners

What size squares should we cut out to get the maximum volume of the box?
| | '''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard.

We have to convert it into a box.

Squares have to be cut from the four corners.

What size squares should we cut out to get the maximum volume of the box?
|-
| | '''Slide Number 17'''

'''A Sketch of the Cardboard'''

Let’s draw the cardboard:

[[Image:]]

The volume function here is '''(24-2x)*(15-2x)*x''' cubic inches.
| | '''A Sketch of the Cardboard'''

Let us draw the cardboard:

This is the volume '''function''' here.

You could expand it into a '''cubic polynomial'''<nowiki>; but we will leave it as it is. </nowiki>

|-
| | Open a new '''GeoGebra''' window.

| | Open a new '''GeoGebra''' window.
|-
| | Type '''(24-2 x) (15-2 x) x''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Drag the boundary to see the equation properly in '''Algebra''' view.
| | Drag the boundary to see the equation properly in '''Algebra''' view.
|-
| | Right-click in '''Graphics''' view and set '''xAxis : yAxis''' to '''1:50'''.

Under '''Move Graphics View''', click on '''Zoom Out'''.

Click in '''Graphics''' view to see the '''function''' properly.

| | Right-click in '''Graphics''' view and set '''xAxis''' is to '''yAxis''' to 1 is to 50.

Now, zoom out to see the function properly.
|-
| | Point to the graph for this volume '''function''' in '''Graphics''' view.

Click in and drag the background to move '''Graphics''' view to see the maximum.

| | Observe the graph that is plotted for this volume '''function''' in '''Graphics''' view.

Drag the background to see the maximum.

|-
| | Point to the maximum on top of the broad peak and to '''x''' = 0 and '''x''' = 7.
| | Note that the maximum is on the top of a broad peak from '''x''' equals 0 to '''x''' equals 7.
|-
| | Point to both axes.
| | The length of the square side is plotted along the '''x-axis'''.

Volume of the box is plotted along the '''y-axis'''.
|-
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-value'''.

Instead of highlighted '''Function''', type '''f'''.

Press '''Tab''' to move and highlight '''Start x-Value''' and type 0.

Again, press '''Tab''' to move and highlight '''End x-Value''' and type 10.

Press '''Enter'''.
| | As before, let us find the maximum of this '''function'''.

|-
| |

Point to the maximum,''' A''', in '''Graphics''' view and its '''coordinates''' in '''Algebra''' view.

| | This maps the maximum, point '''A''', on the curve.

Its '''coordinates''' 3 comma 486 appear in '''Algebra''' view.

Thus, we have to cut out 3 inch squares from all corners.

This will give the maximum possible volume of 486 cubic inches for the cardboard box.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 19'''

'''Summary'''
| | In this tutorial, we have learnt how to use '''GeoGebra''' to:

Understand differentiation

Draw graphs of derivatives of '''functions'''
|-
| | '''Slide Number 16'''

'''Assignment'''

Draw graphs of derivatives of the following functions in '''GeoGebra''':

'''h(x)=ex'''

'''i(x)=ln(x)'''

'''j(x)=(5x3+3x-1)/(x-1)'''

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
| | As an assignment:

Draw graphs of derivatives of the following functions in '''GeoGebra'''.

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
|-
| | '''Slide Number 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

Conducts workshops using spoken tutorials and
Gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 20'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from''' IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Radioactive-Dating-Game/English

2019-01-09T12:54:52Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '''interactive PhET simulation.'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Radioactive Dating Game PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Radioactive Dating Game''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school physics and chemistry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''

| |

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Radioactive Dating Game simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.
| | To open the '''jar file''', open the '''terminal'''.
|-
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
|-
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
| | Type '''java space hyphen jar space radioactive-dating-game underscore en dot jar'''.
|-
| | Point to the '''browser''' address.
| | '''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Radioactive Dating Game simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to four '''screens''' in the '''interface'''.
| | The '''interface''' has four '''screens''':

'''Half Life'''

'''Decay Rates'''

'''Measurement'''

'''Dating Game'''
|-
| | Show the '''Half Life''' screen.
| | We are already looking at the '''Half Life''' screen.
|-
| | Point to the graph at the top of the screen.

Point to units of time along the '''x-axis'''.
| | At the top of the screen is an '''Isotope versus Time''' graph.

Pay attention to the units of time.
|-
| | Show '''Choose Isotope''' panel to the right.

Point to the three options in '''Choose Isotope''' panel.
| | On the right side of the screen, you see a '''Choose Isotope''' panel.

It has three options showing unstable nucleus decaying to stable nucleus.
|-
| | Point to the '''Bucket o’ Atoms''' in the '''simulation''' panel.

Point to the '''C-14''' atoms in the bucket and to the default selection of '''C-14'''.

Point to the “'''Add 10'''” button attached to the bottom of the bucket.
| | In the middle is the '''simulation''' panel containing a '''Bucket o’ Atoms'''.

Note that it contains '''C-14''' atoms as the default selection is '''C-14'''.

Attached to the bottom of the bucket is a '''button''' called “'''Add 10'''”.
|-
| | '''Slide Number 7'''

'''Stable and Unstable Nuclei'''

Electrostatic repulsion between protons in nucleus

Strong nuclear force is associated with binding energy

If binding energy is low, the nucleus is unstable and is radioactive

| |

'''Stable and Unstable Nuclei'''

The strong nuclear force overcomes this electrostatic repulsion between protons.

The energy associated with this force is the binding energy.

The lower the binding energy, the more unstable is the nucleus.

Such an unstable nucleus is said to be radioactive.
|-
| | Point to '''Play/Pause button''' and '''Step button''' next to it.
| | Below the '''simulation''' panel is a '''Play/Pause button''' and a '''Step button''' next to it.
|-
| | Point to blue '''Reset All Nuclei button''' in '''simulation''' panel.
| | In the '''simulation''' panel is a blue '''Reset All Nuclei button'''.

It lets you return to the start but with the selected isotope.
|-
| | Point to the white '''Reset All button''' below the right panel.
| | Below the right panel is a white '''Reset All button'''.

It resets the '''simulation''' in this '''screen''' to all the default settings.
|-
| | Point to the '''Isotope versus time''' graph.

Point to vertical red dashed line labeled '''Half Life''' near the 5000 year mark.
| | Observe the '''Isotope versus time''' graph.

There is a vertical red dashed line labeled '''Half Life''' near the 5000 year mark.

The '''half-life''' of '''C-14''' is 5730 years.
|-
| | Point to red '''C-14''' symbol above the blue '''N-14''' symbol along the '''y-axis'''.
| | Along the '''y-axis''', you can see the red '''C-14''' symbol above the blue '''N-14''' symbol.

'''C-14''' atoms will appear in the upper row and '''N-14''' atoms in the lower one.
|-
| | Point to the red circle to the left of the '''Isotope''' label.

Point to '''hash symbols''' on the left of the circle.
| | To the left of the '''Isotope''' label is a red circle.

Numbers of '''C-14''' and '''N-14''' atoms shown by '''hash symbols''' will appear to the left of the circle.
|-
| | '''Slide Number 8'''

'''Radioactive Decay'''
Radioactive Decay

Alpha decay

Beta decay

Gamma decay

'''Radioactive Decay'''

'''Radioactive Decay '''is the spontaneous conversion of an unstable nucleus into a stable nucleus.

It involves the release of subatomic particles and their energy as radiation.

It is of the following types:

'''Alpha decay'''

'''Beta decay'''

'''Gamma decay'''

|-
| | '''Slide Number 9'''

'''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
| | '''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| |
| | Let us get back to the '''simulation'''.
|-
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
|-
| | Point to the 10 '''C-14''' atoms added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
| | Ten '''C-14''' atoms have been added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
|-
| | Show the red '''C-14''' atoms flying across the graph in the upper row.

Show the blue '''N-14''' atoms in the lower row.
| | Observe the atoms moving across the graph in the two rows.

|-
| | Point to the circle changing to blue.
| | Note how the circle changes to blue as more '''N-14''' atoms form.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
| | Keep clicking on '''Step button''' to the right of '''Pause'''.

The circle is half red and half blue.
|-
| | Point to 5 blue '''N-14''' atoms on the left of the dashed half-life line.
| | Observe that there are 5 blue '''N-14''' atoms in the graph.
|-
| | Point to 5 '''C-14''' and '''N-14''' atoms in the '''simulation''' panel.
| | There are 5 '''C-14''' atoms and 5 '''N-14''' atoms in the '''simulation''' panel also.

This is the definition of '''half-life'''.

|-
| | Point to '''Add 10 button''' and to the bucket.
| | If you click again on '''Add 10''', another 10 '''C-14''' atoms will be added to the '''simulation''' panel.
|-
| |
| | Predict the number of '''C-14''' atoms remaining after different periods.
|-
| | Perform the same '''simulation''' for the other nuclei.
| | Perform the same '''simulation''' for the other nuclei.
|-
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

Show the '''interface'''.
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

The '''interface''' has a similar arrangement as the '''Half Life screen'''.

Please explore this '''screen''' in the same way.
|-
| | Click on the '''Measurement tab'''.
| | Now, let us click on the '''Measurement tab''' to go to that '''screen'''.
|-
| | Point to '''Tree''', the default selection under '''Choose an Object''' on the right.
| | In the right panel, under '''Choose an Object''', we will stay with '''Tree''', the default selection.
|-
| | Point to '''Carbon-14''', and '''Objects''' under '''Probe Type''' in top left.
| | In the top left, under '''Probe Type''', we will retain the default selections, '''Carbon-14''' and '''Objects'''.
|-
| | Click on '''Plant Tree button''' in the bottom right corner.

Point to the tree growing right where the '''probe''' is placed.
| | In the bottom right corner, click on '''Plant Tree button'''.

|-
| | Immediately click on the '''Pause button'''.
| | Immediately click on the '''Pause button'''.
|-
| | Point to 100% seen above '''Probe Type''' in the upper left corner.
| | Observe 100% appear above '''Probe Type''' in the upper left corner.
|-
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
| | Keep clicking on the '''Step button''' to the right of '''Pause''' to move the simulation along.
|-
| | Show '''% of C-14''' above the graph.
| | Above the graph, '''% of C-14''' is the default selection.
|-
| | Point to the white box below the graph.
| | The white box below the graph shows the number of years since the tree was planted.
|-
| | Point to the red line at the top of the graph.
| | The red line shows % of '''C-14''' remaining in the tree.
|-
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
|-
| | Point to the red line at the top of the graph.
| | Now the red line shows the '''C-14''' to '''C-12''' ratio in the tree.
|-
| | Click again on the % of '''C-14 radio button''' above the graph.
| | Click again on the % of '''C-14 radio button''' above the graph.
|-
| | Point to % in the top left, the tree and the white box below the graph.
| | Keep track of the % in the top left, the tree and the number of years below the graph.
|-
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
|-
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
|-
| | Point to 50% on '''y-axis''', to red line and to '''x co-ordinate'''.
| | Note the number of years after which you see 50% of '''C-14''' in the tree.
|-
| | Click on '''Rock''' and '''Uranium-238 radio buttons'''.
| | Click on '''Rock''' and '''Uranium-238 radio buttons'''.
|-
| | Click on '''Erupt Volcano''' and '''Cool rock buttons'''.

Measure '''U-238''' levels in the cooled volcanic rock.
| | Click on '''Erupt Volcano''' and '''Cool rock buttons'''.

Measure '''U-238''' levels in the cooled volcanic rock.
|-
| | Click on the '''Air radio button''' to compare isotope levels in objects to air levels.
| | Click on the '''Air radio button''' to compare isotope levels in objects to air levels.
|-
| | Click on the last '''Dating Game tab'''.

Show the '''interface'''.
| | Let us click on the last '''Dating Game tab''' to go to that '''screen'''.

We can measure levels of '''C-14, U-238''' or other '''custom nuclei''' in this '''screen'''.

We see objects on and below the ground on which we can place the probe to measure these levels.
|-
| | '''Slide Number 10'''

'''Radioactive Dating'''

Two isotopes of C: 12C and 14C

Both isotopes → CO2, living organisms

Death of organism, ratio and 14C fall
| |

'''Radioactive Dating'''

Carbon has two isotopes: '''C-12''' and '''C-14'''.

Both are converted to carbon dioxide and are taken in by living organisms.

When an organism dies, it no longer takes in any carbon.

So levels of '''C-14''' and ratio of '''C-14''' to '''C-12''' fall.
|-
| | '''Slide Number 11'''

'''Radioactive Dating-Cont’d'''

Radioactive dating, 14C:12C of samples vs recently dead specimens

Ur-Pb dating for rocks, artefacts etc
| |

'''Radioactive Dating-Continued'''

Radioactive dating compares C-14 C-12 ratio of samples to recently dead specimens.

It estimates how long the organism has been dead.

'''Uranium-lead dating''' is used for rocks, archaeological artefacts etc.
|-
| | Point to the graph on the top.
| | On the top, we see the graph like the ones in the previous screens.
|-
| | Point to these default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
| | Let us keep the following default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
|-
| | Drag the '''probe''' and place it on the animal skull on the ground, to the left.
| | We will drag the '''probe''' and place it on the animal skull on the ground, to the left.
|-
| | Point to the '''pop-up box''' next to the skull.

Show the '''text''', “'''Estimate age of Animal Skull'''”.

Show the empty box and “'''yrs'''” next to it.
| | Observe a '''pop-up box''' that appears next to the skull.

We see the text, “'''Estimate age of Animal Skull'''” and “'''years'''” next to the empty box below.
|-
| | Point to the '''Check Estimate button'''.
| | Below this is a '''Check Estimate button'''.
|-
| | Show 98.2% in the top left side, above '''Probe Type'''.
| | Observe that in the top left side, above '''Probe Type''', we see 98.2%.
|-
| | Drag the double-headed green arrow to the left.

Show 98.2% in the white box above the arrow.
| | Let us drag the double-headed green arrow above the graph.

In the white box above the arrow, % of '''C-14''' should be approximately 98.2%.
|-
| | Show '''t = 123 yrs''' in the white box below % of '''C-14'''.
| | Observe that '''t equals 123 yrs''' appears in the white box below % of '''C-14'''.
|-
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
|-
| | Point to green text-box with 123 years in its place with a green '''smiley''' face next to it.
| | The '''Estimate pop-up box''' disappears.

A green text-box with 123 years appears in its place with a green '''smiley''' face next to it.
|-
| |
| | We have successfully dated the animal skull by measuring the % of '''C-14''' remaining in it.
|-
| | '''Slide Number 12'''

'''Assignment'''

Estimate ages of all objects in '''Dating Game screen'''

Correlate age (years) with percentage of '''unstable''' nucleus

Correlate age (years) with depth at which object found

'''C-14''': animal remains; '''U-238''': rocks, objects
| |

As an '''assignment''',

Estimate ages of all the objects in the '''Dating Game screen'''.

Correlate age in years with percentage of '''unstable''' nucleus.

Correlate age in years with the depth at which the object is found.

Remember to use '''C-14''' for animal remains and '''U-238''' for rocks and other objects.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 13'''

'''Summary'''

We have demonstrated,

'''Radioactive Dating Game PhET simulation'''
| |

In this '''tutorial''', we have demonstrated how to use the '''Radioactive Dating Game PhET simulation'''.
|-
| | '''Slide Number 14'''

'''Summary'''

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
| |

Using this '''simulation''', we looked at:

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

2019-01-07T09:47:51Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate:

'''Area Under a Curve (AUC)'''

Area bounded by two '''functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Integration

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''[a,b]''' above '''x-axis'''

'''a''' is lower limit, b is upper limit

<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>

Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
| |

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''a b''' above the '''x-axis'''.

'''a''' and '''b''' are called the lower and upper limits of the integral.

Integral of '''f of x''' from '''a''' to '''b''' with respect to '''x''' is the notation for this definite integral.

It is the area bounded by '''y''' equals '''f of x, x''' equals '''a, x''' equals '''b''' and the '''x-axis'''.
|-
| | '''Slide Number 6'''

'''Calculation of a Definite Integral'''

Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
| |

Let us calculate the definite integral of this function with respect to '''x'''.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''g(x)= ‑ 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.
| | Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.
|-
| | Click on '''Slider''' tool and click in '''Graphics''' view.
| | Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.

It should range from 1 to 50 in increments of 1.
|-
| | Leave the '''Number''' radio button checked.
| |
|-
| | Type '''n''' in the '''Name''' field.
|-
| | Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''
|-
| | Point to '''slider n''' in '''Graphics''' view.

|-
| | Drag '''slider n''' to 5.
| | Drag the resulting '''slider n''' to 5.
|-
| | Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.
| | Under '''Point''', click on '''Point on Object''' and click at ‑1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.
|-
| |
| | Let us look at a few ways to approximate '''area under the curve'''.

These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.

We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
|-
| | Type '''uppersum=Upp''' in the '''Input Bar'''.

Show option.

'''UpperSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles> )'''

Click on it.
| | In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.

The following option appears.

Click on it.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Type '''g''' instead of highlighted '''<Function>'''.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
|-
| | Type '''x(A)'''.
| | Type '''x A in parentheses'''.
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
| | Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.

Press '''Enter'''.
|-
| | Point to five rectangles between '''x'''<nowiki= -1 and 2. </nowiki>
| | Note that five rectangles appear between '''x''' equals -1 and 2.
|-
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
|-
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
|-
| | '''Point''' to '''upper sum area under the curve (AUC).'''
| | The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
|-
| | Point to the rectangles extending above the curve.
| | It is an overestimation of the area under the curve.

This is because some portion of each rectangle extends above the curve.
|-
| | Drag the background to move the graph to the left.
| | Drag the background to move the graph to the left.
|-
| |
| | Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.
|-
| | Type '''trapsum=Tra''' in the '''Input bar'''.
| | In the '''input bar''', type '''trapsum''' is equal to '''Tra'''.
|-
| | Point to the menu that appears.
| | A menu with various options appears.
|-
| | Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
| | Select the following option.
|-
| |
| | We will type the same values as before and press '''Enter'''.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| |
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A)'''.
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.
| |
|-
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Point to trapezoids.
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Note the shape of the trapezoids.
|-
| |
| | Let us now look at the integral as the area under the curve.
|-
| | Finally, type '''Int''' in the '''Input Bar'''.
| | Finally, in the '''input bar''', type '''Int'''.
|-
| | '''Point''' to the menu with various options.
| | A menu with various options appears
|-
| | Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.
| | Select the following option.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Again, we will enter the same values as before.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A)'''.
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value'''.

Press '''Enter.'''
| |
|-
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
|-
| | Point to the integrated''' AUC'''.
| | For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals ‑1 to 2.
|-
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
|-
| | Click on '''Text''' tool under '''Slider''' tool.
| | Under '''Slider''', click on '''Text'''.
|-
| | Click in '''Graphics''' view to open a '''text box'''.
| | Click in '''Graphics''' view to open a '''text box'''.
|-
| | In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
| | In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
| | Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

Click '''OK''' in the '''text box'''.
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

In the '''text box''', click '''OK'''.
|-
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
|-
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
|-
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
|-
| | Point to text box and to '''slider n'''.
| | Observe the values in the '''text box''' as you drag '''slider n'''.
|-
| | Point to '''Graphics''' view.
| | '''Trapsum''' is a better approximation of '''AUC''' at high '''n''' values.

'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.

'''F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x2 + C'''
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window
|-
| |
| | We will look at the relationship between '''differentiation''' and '''integration'''.

Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.
|-
| | Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| |
| | Let us call '''integral''' of '''f of x capital F of x'''.
|-
| | Type '''F(x)=Integral(f)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.

Point to equation for '''F(x)=1/3 x3+ x2+x''' appears in '''Algebra''' view.
| | The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.

Its equation for '''capital F of x''' appears in '''Algebra''' view.

Confirm that this is the integral of '''f of x'''.
|-
| | Drag the boundary to see the equations properly.
| | Drag the boundary to see the equations properly.
|-
| | Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following and press '''Enter'''.
|-
| | Point to '''F'(x)''' and '''f(x)'''.
| | Note that this graph coincides with '''f of x'''.

The equations for '''f of x''' and '''h of x''' are the same.

Thus, we can see that '''integration''' is the inverse process of '''differentiation'''.

Taking the derivative of an integral, gives back the original '''function'''.
|-
| | Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
| | Click on '''Point''' tool and create a point at '''1 comma 3'''.
|-
| | Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following and press '''Enter'''.
|-
| | Click on '''Create Sliders''' in the window that pops up.
| | Click on '''Create Sliders''' in the window that pops up.
|-
| | Point to '''slider k'''.
| | A '''slider k''' appears.
|-
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.

Close the '''Preferences''' window.
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5.

Scroll right to set the '''Increment''' to 0.01.

Close the '''Preferences''' box.
|-
| | Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
| | In '''Algebra''' view. double-click on '''i of x''' and on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
|-
| | Drag '''k''' to make '''i(x)''' pass through point '''A'''.

Point to integral function '''(1/3)x3+x2+x+0.7'''.
| | Drag '''k''' to make '''i of x''' pass through point '''A'''.
|-
| | Drag the boundary to see '''i of x''' properly.
| | Drag the boundary to see '''i of x''' properly.
|-
| | Point to '''F(x)+0.7''': the curve and equation.
| | This function is '''capital F of x''' plus 0.7.
|-
| | '''Slide Number 7'''

'''Double Integrals'''

'''Double integrals''' can be used to find:

'''AUC''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z=f(x,y)'''
| | '''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z''' which is equal to '''f of x and y'''
|-
| | '''Slide Number 8'''

'''Double Integral-An Example'''

Let us find the area between parabola '''x=y2''' and the line '''y=x'''.

The '''limits''' are from '''(0,0)''' to '''(1,1)'''.

This area can be expressed as the '''double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>
| |

'''Double Integral-An Example'''

Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.

The limits are from '''0 comma 0''' to '''1 comma 1'''.

This area can be expressed as the double integrals shown here.

Observe the limits and the order of the integrals in terms of the variables.

|-
| |
| | Let us open a new '''GeoGebra''' window.

We will first express '''x''' in terms of '''y''', for both '''functions'''.
|-
| | In the '''input bar''', type '''x=y2''' and press '''Enter'''.
| | In the '''input bar''', type '''x '''equals '''y caret''' 2 and press '''Enter'''.
|-
| | Next, in the '''input bar''', type '''y=x''' and press '''Enter'''.
| | Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.
|-
| | Click on '''View''' tool and select '''CAS'''.
| | Click on '''View''' tool and select '''CAS'''.
|-
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
|-
| | Drag the boundary to make '''CAS''' view bigger.
| | Drag the boundary to make '''CAS''' view bigger.
|-
| | In '''CAS''' view, type '''Int''' in line 1.

Point to the menu that appears.
| | In '''CAS''' view, type '''Int capital I''' in line 1.

A menu with various options appears.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| | Scroll down.

Select the following option.
|-
| | Type '''y''' for the first '''function'''.
| | Type '''y''' for the first '''function'''.
|-
| | Press '''Tab''' and type '''y^2''' for the second '''function'''.
| | Press '''Tab '''and type '''y caret 2''' for the second '''function'''.
|-
| | Press '''Tab''' and type '''y''' as the '''variable'''.
| | Press '''Tab''' and type '''y''' as the '''variable'''.
|-
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
|-
| | Press '''Enter'''.
| | Press '''Enter'''.
|-
| | Point to the value of 1/6 below the entry.

Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.
| | A value 1 divided by 6 appears below the entry.

This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.
|-
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
|-
| | In '''CAS''' view, type '''Int''' and observe the same menu as before.
| | In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| |
|-
| |
| | Now, let us reverse the order of '''functions''' and '''limits'''.
|-
| | Type '''sqrt(x)''' for the first function and '''x''' for the second.
| | Type the following and press '''Enter'''.
|-
| | Point to the '''input bar'''.
| | You can also use the '''input bar''' instead of the '''CAS''' view.
|-
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
|-
| | Drag the boundaries to make '''CAS''' view smaller.
| | Drag the boundaries to make '''CAS''' view smaller.
|-
| |
In the '''input bar''', type '''Int'''.

From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start Value''' and again press '''Tab''' to move to and type 1 as the '''End Value'''.

Press '''Enter'''.

This will also give you an area a of 0.17 or 1 divided by 6.
| |

In the '''input bar''', type '''Int capital I'''.

From menu, select the following option.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start x Value''' and again press '''Tab''' to move to and type 1 as the '''End x Value'''.

Press '''Enter'''.

This will also give you an area '''a''' of 0.17 or 1 divided by 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of:

'''Area Under a Curve''' ('''AUC''')

Area bounded by two '''functions'''
|-
| | '''Slide Number 10'''

'''Assignment'''* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where '''f(x) = 1/(1-x)'''
Calculate <math>{\int }_{x\left(A\right)}^{x\left(B\right)}g\left(x\right)dx</math>and <math>{\int }_{x\left(B\right)}^{x\left(C\right)}g\left(x\right)dx</math>where '''g(x) = 0.5x3+2x2-x-3.75'''

'''A, B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
| | As an '''assignment''':

Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.

Explain the results for '''g of x'''.
|-
| | '''Slide Number 11'''

'''Assignment'''

Calculate the area bounded by the following '''functions''':

[[Image:]]'''y=4x-x2, y=x'''

[[Image:]]'''x2+y2<nowiki>=9, y=3-x</nowiki>'''

[[Image:]]

'''y=1+x2, y=2x2'''
| | As another '''assignment''':

Calculate the shaded areas between these pairs of '''functions'''.
|-
| | '''Slide Number 12'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 13'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials

gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 15'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Radioactive-Dating-Game/English

2019-01-01T06:09:11Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '''interactive PhET simulation.'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Radioactive Dating Game PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Radioactive Dating Game''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school physics and chemistry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''

| |

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Radioactive Dating Game simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.
| | To open the '''jar file''', open the '''terminal'''.
|-
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
|-
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
| | Type '''java space hyphen jar space radioactive-dating-game underscore en dot jar'''.
|-
| | Point to the '''browser''' address.
| | '''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Radioactive Dating Game simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to four '''screens''' in the '''interface'''.
| | The '''interface''' has four '''screens''':

'''Half Life'''

'''Decay Rates'''

'''Measurement'''

'''Dating Game'''
|-
| | Show the '''Half Life''' screen.
| | We are already looking at the '''Half Life''' screen.
|-
| | Point to the graph at the top of the screen.

Point to units of time along the '''x-axis'''.
| | At the top of the screen is an '''Isotope versus Time''' graph.

Pay attention to the units of time.
|-
| | Show '''Choose Isotope''' panel to the right.

Point to the three options in '''Choose Isotope''' panel.
| | On the right side of the screen, you see a '''Choose Isotope''' panel.

It has three options showing unstable nucleus decaying to stable nucleus.
|-
| | Point to the '''Bucket o’ Atoms''' in the '''simulation''' panel.

Point to the '''C-14''' atoms in the bucket and to the default selection of '''C-14'''.

Point to the “'''Add 10'''” button attached to the bottom of the bucket.
| | In the middle is the '''simulation''' panel containing a '''Bucket o’ Atoms'''.

Note that it contains '''C-14''' atoms as the default selection is '''C-14'''.

Attached to the bottom of the bucket is a '''button''' called “'''Add 10'''”.
|-
| | '''Slide Number 7'''

'''Stable and Unstable Nuclei'''

Electrostatic repulsion between protons in nucleus

Strong nuclear force ~ binding energy

Low binding energy; unstable nucleus ~ radioactive

| |

'''Stable and Unstable Nuclei'''

The strong nuclear force overcomes this electrostatic repulsion between protons.

The energy associated with this force is the binding energy.

The lower the binding energy, the more unstable is the nucleus.

Such an unstable nucleus is said to be radioactive.
|-
| | Point to '''Play/Pause button''' and '''Step button''' next to it.
| | Below the '''simulation''' panel is a '''Play/Pause button''' and a '''Step button''' next to it.
|-
| | Point to blue '''Reset All Nuclei button''' in '''simulation''' panel.
| | In the '''simulation''' panel is a blue '''Reset All Nuclei button'''.

It lets you return to the start but with the selected isotope.
|-
| | Point to the white '''Reset All button''' below the right panel.
| | Below the right panel is a white '''Reset All button'''.

It resets the '''simulation''' in this '''screen''' to all the default settings.
|-
| | Point to the '''Isotope versus time''' graph.

Point to vertical red dashed line labeled '''Half Life''' near the 5000 year mark.
| | Observe the '''Isotope versus time''' graph.

There is a vertical red dashed line labeled '''Half Life''' near the 5000 year mark.

The '''half-life''' of '''C-14''' is 5730 years.
|-
| | Point to red '''C-14''' symbol above the blue '''N-14''' symbol along the '''y-axis'''.
| | Along the '''y-axis''', you can see the red '''C-14''' symbol above the blue '''N-14''' symbol.

'''C-14''' atoms will appear in the upper row and '''N-14''' atoms in the lower one.
|-
| | Point to the red circle to the left of the '''Isotope''' label.

Point to '''hash symbols''' on the left of the circle.
| | To the left of the '''Isotope''' label is a red circle.

Numbers of '''C-14''' and '''N-14''' atoms shown by '''hash symbols''' will appear to the left of the circle.
|-
| | '''Slide Number 8'''

'''Radioactive Decay'''
Radioactive Decay

Alpha decay

Beta decay

Gamma decay

'''Radioactive Decay'''

'''Radioactive Decay '''is the spontaneous conversion of an unstable nucleus into a stable nucleus.

It involves the release of subatomic particles and their energy as radiation.

It is of the following types:

'''Alpha decay'''

'''Beta decay'''

'''Gamma decay'''

|-
| | '''Slide Number 9'''

'''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
| | '''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| |
| | Let us get back to the '''simulation'''.
|-
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
|-
| | Point to the 10 '''C-14''' atoms added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
| | Ten '''C-14''' atoms have been added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
|-
| | Show the red '''C-14''' atoms flying across the graph in the upper row.

Show the blue '''N-14''' atoms in the lower row.
| | Observe the atoms moving across the graph in the two rows.

|-
| | Point to the circle changing to blue.
| | Note how the circle changes to blue as more '''N-14''' atoms form.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
| | Keep clicking on '''Step button''' to the right of '''Pause'''.

The circle is half red and half blue.
|-
| | Point to 5 blue '''N-14''' atoms on the left of the dashed half-life line.
| | Observe that there are 5 blue '''N-14''' atoms in the graph.
|-
| | Point to 5 '''C-14''' and '''N-14''' atoms in the '''simulation''' panel.
| | There are 5 '''C-14''' atoms and 5 '''N-14''' atoms in the '''simulation''' panel also.

This is the definition of '''half-life'''.

|-
| | Point to '''Add 10 button''' and to the bucket.
| | If you click again on '''Add 10''', another 10 '''C-14''' atoms will be added to the '''simulation''' panel.
|-
| |
| | Predict the number of '''C-14''' atoms remaining after different periods.
|-
| | Perform the same '''simulation''' for the other nuclei.
| | Perform the same '''simulation''' for the other nuclei.
|-
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

Show the '''interface'''.
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

The '''interface''' has a similar arrangement as the '''Half Life screen'''.

Please explore this '''screen''' in the same way.
|-
| | Click on the '''Measurement tab'''.
| | Now, let us click on the '''Measurement tab''' to go to that '''screen'''.
|-
| | Point to '''Tree''', the default selection under '''Choose an Object''' on the right.
| | In the right panel, under '''Choose an Object''', we will stay with '''Tree''', the default selection.
|-
| | Point to '''Carbon-14''', and '''Objects''' under '''Probe Type''' in top left.
| | In the top left, under '''Probe Type''', we will retain the default selections, '''Carbon-14''' and '''Objects'''.
|-
| | Click on '''Plant Tree button''' in the bottom right corner.

Point to the tree growing right where the '''probe''' is placed.
| | In the bottom right corner, click on '''Plant Tree button'''.

|-
| | Immediately click on the '''Pause button'''.
| | Immediately click on the '''Pause button'''.
|-
| | Point to 100% seen above '''Probe Type''' in the upper left corner.
| | Observe 100% appear above '''Probe Type''' in the upper left corner.
|-
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
| | Keep clicking on the '''Step button''' to the right of '''Pause''' to move the simulation along.
|-
| | Show '''% of C-14''' above the graph.
| | Above the graph, '''% of C-14''' is the default selection.
|-
| | Point to the white box below the graph.
| | The white box below the graph shows the number of years since the tree was planted.
|-
| | Point to the red line at the top of the graph.
| | The red line shows % of '''C-14''' remaining in the tree.
|-
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
|-
| | Point to the red line at the top of the graph.
| | Now the red line shows the '''C-14''' to '''C-12''' ratio in the tree.
|-
| | Click again on the % of '''C-14 radio button''' above the graph.
| | Click again on the % of '''C-14 radio button''' above the graph.
|-
| | Point to % in the top left, the tree and the white box below the graph.
| | Keep track of the % in the top left, the tree and the number of years below the graph.
|-
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
|-
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
|-
| | Point to 50% on '''y-axis''', to red line and to '''x co-ordinate'''.
| | Note the number of years after which you see 50% of '''C-14''' in the tree.
|-
| | Click on '''Rock''' and '''Uranium-238 radio buttons'''.
| | Click on '''Rock''' and '''Uranium-238 radio buttons'''.
|-
| | Click on '''Erupt Volcano''' and '''Cool rock buttons'''.

Measure '''U-238''' levels in the cooled volcanic rock.
| | Click on '''Erupt Volcano''' and '''Cool rock buttons'''.

Measure '''U-238''' levels in the cooled volcanic rock.
|-
| | Click on the '''Air radio button''' to compare isotope levels in objects to air levels.
| | Click on the '''Air radio button''' to compare isotope levels in objects to air levels.
|-
| | Click on the last '''Dating Game tab'''.

Show the '''interface'''.
| | Let us click on the last '''Dating Game tab''' to go to that '''screen'''.

We can measure levels of '''C-14, U-238''' or other '''custom nuclei''' in this '''screen'''.

We see objects on and below the ground on which we can place the probe to measure these levels.
|-
| | '''Slide Number 10'''

'''Radioactive Dating'''

Two isotopes of C: 12C and 14C

Both isotopes → CO2, living organisms

Death of organism, ratio and 14C fall
| |

'''Radioactive Dating'''

Carbon has two isotopes: '''C-12''' and '''C-14'''.

Both are converted to carbon dioxide and are taken in by living organisms.

When an organism dies, it no longer takes in any carbon.

So levels of '''C-14''' and ratio of '''C-14''' to '''C-12''' fall.
|-
| | '''Slide Number 11'''

'''Radioactive Dating-Cont’d'''

Radioactive dating, 14C:12C of samples vs recently dead specimens

Ur-Pb dating for rocks, artefacts etc
| |

'''Radioactive Dating-Continued'''

Radioactive dating compares C-14 C-12 ratio of samples to recently dead specimens.

It estimates how long the organism has been dead.

'''Uranium-lead dating''' is used for rocks, archaeological artefacts etc.
|-
| | Point to the graph on the top.
| | On the top, we see the graph like the ones in the previous screens.
|-
| | Point to these default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
| | Let us keep the following default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
|-
| | Drag the '''probe''' and place it on the animal skull on the ground, to the left.
| | We will drag the '''probe''' and place it on the animal skull on the ground, to the left.
|-
| | Point to the '''pop-up box''' next to the skull.

Show the '''text''', “'''Estimate age of Animal Skull'''”.

Show the empty box and “'''yrs'''” next to it.
| | Observe a '''pop-up box''' that appears next to the skull.

We see the text, “'''Estimate age of Animal Skull'''” and “'''years'''” next to the empty box below.
|-
| | Point to the '''Check Estimate button'''.
| | Below this is a '''Check Estimate button'''.
|-
| | Show 98.2% in the top left side, above '''Probe Type'''.
| | Observe that in the top left side, above '''Probe Type''', we see 98.2%.
|-
| | Drag the double-headed green arrow to the left.

Show 98.2% in the white box above the arrow.
| | Let us drag the double-headed green arrow above the graph.

In the white box above the arrow, % of '''C-14''' should be approximately 98.2%.
|-
| | Show '''t = 123 yrs''' in the white box below % of '''C-14'''.
| | Observe that '''t equals 123 yrs''' appears in the white box below % of '''C-14'''.
|-
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
|-
| | Point to green text-box with 123 years in its place with a green '''smiley''' face next to it.
| | The '''Estimate pop-up box''' disappears.

A green text-box with 123 years appears in its place with a green '''smiley''' face next to it.
|-
| |
| | We have successfully dated the animal skull by measuring the % of '''C-14''' remaining in it.
|-
| | '''Slide Number 12'''

'''Assignment'''

Estimate ages of all objects in '''Dating Game screen'''

Correlate age (years) with percentage of '''unstable''' nucleus

Correlate age (years) with depth at which object found

'''C-14''': animal remains; '''U-238''': rocks, objects
| |

As an '''assignment''',

Estimate ages of all the objects in the '''Dating Game screen'''.

Correlate age in years with percentage of '''unstable''' nucleus.

Correlate age in years with the depth at which the object is found.

Remember to use '''C-14''' for animal remains and '''U-238''' for rocks and other objects.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 13'''

'''Summary'''

We have demonstrated,

'''Radioactive Dating Game PhET simulation'''
| |

In this '''tutorial''', we have demonstrated how to use the '''Radioactive Dating Game PhET simulation'''.
|-
| | '''Slide Number 14'''

'''Summary'''

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
| |

Using this '''simulation''', we looked at:

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Radioactive-Dating-Game/English

2018-12-21T07:44:57Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '''interactive PhET simulation.'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Radioactive Dating Game PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Radioactive Dating Game''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school physics and chemistry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''

| |

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Radioactive Dating Game simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.
| | To open the '''jar file''', open the '''terminal'''.
|-
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
|-
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
|-
| | Point to the '''browser''' address.
| | '''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Radioactive Dating Game simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to four '''screens''' in the '''interface'''.
| | The '''interface''' has four '''screens''':

'''Half Life'''

'''Decay Rates'''

'''Measurement'''

'''Dating Game'''
|-
| | Show the '''Half Life''' screen.
| | We are already looking at the '''Half Life''' screen.
|-
| | Point to the graph at the top of the screen.

Point to units of time along the '''x-axis'''.
| | At the top of the screen is an '''Isotope versus Time''' graph.

Pay attention to the units of time.
|-
| | Show '''Choose Isotope''' panel to the right.

Point to the three options in '''Choose Isotope''' panel.
| | On the right side of the screen, you see a '''Choose Isotope''' panel.

It has three options showing unstable nucleus decaying to stable nucleus.
|-
| | Point to the '''Bucket o’ Atoms''' in the '''simulation''' panel.

Point to the '''C-14''' atoms in the bucket and to the default selection of '''C-14'''.

Point to the “'''Add 10'''” button attached to the bottom of the bucket.
| | In the middle is the '''simulation''' panel containing a '''Bucket o’ Atoms'''.

Note that it contains '''C-14''' atoms as the default selection is '''C-14'''.

Attached to the bottom of the bucket is a '''button''' called “'''Add 10'''”.
|-
| | '''Slide Number 7'''

'''Stable and Unstable Nuclei'''

Electrostatic repulsion between protons in nucleus

Strong nuclear force ~ binding energy

Low binding energy; unstable nucleus ~ radioactive

| |

'''Stable and Unstable Nuclei'''

There is electrostatic repulsion between positively charged protons inside the nucleus.

The strong nuclear force overcomes this electrostatic repulsion between protons.

The energy associated with this force is the binding energy.

The lower the binding energy, the more unstable is the nucleus.

Such an unstable nucleus is said to be radioactive.
|-
| | Point to '''Play/Pause button''' and '''Step button''' next to it.
| | Below this '''simulation''' panel is a '''Play/Pause button''' and a '''Step button'''.
|-
| | Point to blue '''Reset All Nuclei button''' in '''simulation''' panel.
| | In this '''simulation''' panel is a blue '''Reset All Nuclei button'''.

This '''button''' lets you return to the start but with the selected isotope.
|-
| | Point to the white '''Reset All button''' below the right panel.
| | Below the right panel is a white '''Reset All button'''.

This button resets the '''simulation''' in this '''screen''' to all the default settings.
|-
| | Point to the '''Isotope versus time''' graph.

Point to vertical red dashed line labeled '''Half Life''' near the 5000 year mark.
| | Observe the '''Isotope versus time''' graph.

There is a vertical red dashed line labeled '''Half Life''' near the 5000 year mark.

The '''half-life''' of '''C-14''' is 5730 years.
|-
| | Point to red '''C-14''' symbol above the blue '''N-14''' symbol along the '''y-axis'''.
| | Along the '''y-axis''', you can see the red '''C-14''' symbol above the blue '''N-14''' symbol.

'''C-14''' atoms will appear in the upper row and '''N-14''' atoms in the lower one.
|-
| | Point to the red circle to the left of the '''Isotope''' label.

Point to '''hash symbols''' on the left of the circle.
| | To the left of the '''Isotope''' label is a red circle.

Numbers of '''C-14''' and '''N-14''' atoms shown by '''hash symbols''' will appear to the left of the circle.
|-
| | '''Slide Number 8'''

'''Radioactive Decay'''
Radioactive Decay

Alpha decay

Beta decay

Gamma decay

| | '''Radioactive Decay '''is the spontaneous conversion of an unstable nucleus into a stable nucleus.

It involves the release of subatomic particles and their energy as radiation.

It is of the following types:

'''Alpha decay'''

'''Beta decay'''

'''Gamma decay'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| | '''Slide Number 9'''

'''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
| | '''Half-life'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| |
| | Let us get back to the '''simulation'''.
|-
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
|-
| | Point to the 10 '''C-14''' atoms added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
| | Ten '''C-14''' atoms have been added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
|-
| | Show the red '''C-14''' atoms flying across the graph in the upper row.

Show the blue '''N-14''' atoms in the lower row.
| | Observe the atoms moving across the graph in the two rows.

|-
| | Point to the circle changing to blue.
| | Note how the circle changes to blue as more '''N-14''' atoms form.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
|-
| | Point to 5 blue '''N-14''' atoms on the left of the dashed half-life line.
| | Observe that, in the graph, there are 5 blue '''N-14''' atoms on the left of the dashed half-life line.

Out of 10 '''C-14''' atoms, half of them took 5730 years to decay into 5 '''N-14''' atoms.
|-
| | Point to 5 '''C-14''' and '''N-14''' atoms in the '''simulation''' panel.
| | There are 5 '''C-14''' atoms and 5 '''N-14''' atoms in the '''simulation''' panel also.

This is the definition of '''half-life'''.
|-
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
|-
| | Point to the 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
| | Another 5730 years are taken for 5 '''C-14''' atoms to decay to 2.5 '''N-14''' atoms.

We see 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
|-
| |
| | Predict the number of '''C-14''' atoms remaining after different periods.
|-
| | Perform the same '''simulation''' for the other nuclei.
| | Perform the same '''simulation''' for the other nuclei.
|-
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

Show the '''interface'''.
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

The '''interface''' has a similar arrangement as the '''Half Life screen'''.

Please explore this '''screen''' in the same way.
|-
| | Click on the '''Measurement tab'''.
| | Now, let us click on the '''Measurement tab''' to go to that '''screen'''.
|-
| | Point to '''Tree''', the default selection under '''Choose an Object''' on the right.
| | In the right panel, under '''Choose an Object''', we will stay with '''Tree'''.
|-
| | Point to '''Carbon-14''', and '''Objects''' under '''Probe Type''' in top left.
| | In the top left, under '''Probe Type''', we will retain the default selections, '''Carbon-14''', and '''Objects'''.
|-
| | Click on '''Plant Tree button''' in the bottom right corner.

Point to the tree growing right where the '''probe''' is placed.
| | Click on '''Plant Tree button''' in the bottom right corner.

|-
| | Immediately click on the '''Pause button'''.
| | Immediately click on the '''Pause button'''.
|-
| | Point to 100% seen above '''Probe Type''' in the upper left corner.
| | Observe 100% appear above '''Probe Type''' in the upper left corner.
|-
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
|-
| | Point to red line moving at the top of the graph along 100%.
| | Note that a red line moves at the top of the graph along 100%.
|-
| | Show '''% of C-14''' above the graph.
| | Above the graph, '''% of C-14''' is the default selection.
|-
| | Point to the white box below the graph.
| | The white box below the graph shows the number of years since the tree was planted.
|-
| | Point to the red line at the top of the graph.
| | The red line shows % of '''C-14''' remaining in the tree after those years after it was planted.
|-
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
|-
| | Point to the red line at the top of the graph.
| | Now the red line shows the '''C-14''' to '''C-12''' ratio in the tree after those years of planting it
|-
| | Click again on the % of '''C-14 radio button''' above the graph.
| | Click again on the % of '''C-14 radio button''' above the graph.
|-
| | Point to % in the top left, the tree and the white box below the graph.
| | Keep track of the % in the top left, the tree and the number of years below the graph.
|-
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
|-
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
|-
| | Point to 50% on '''y-axis''', to red line and to '''x co-ordinate'''.
| | Note the number of years after which you see 50% of '''C-14''' in the tree.
|-
| | Click on '''Erupt Volcano''' and measure '''U-238''' levels in the cooled volcanic rock.

Click on the '''Air radio button''' to compare isotope levels in objects to air levels.

|-
| | Click on the last '''Dating Game tab'''.

Show the '''interface'''.
| | Let us click on the last '''Dating Game tab''' to go to that '''screen'''.

We can measure levels of '''C-14, U-238''' or other '''custom nuclei''' in this '''screen'''.

We see objects on and below the ground on which we can place the probe to measure these levels.
|-
| | '''Slide Number 10'''

'''Radioactive Dating'''

Two isotopes of C: 12C and 14C

Both isotopes → CO2, living organisms

Death of organism, ratio and 14C fall
| |

'''Radioactive Dating'''

Carbon has two isotopes: '''C-12''' and '''C-14'''.

Both are converted to carbon dioxide and are taken in by living organisms.

When an organism dies, it no longer takes in any carbon.

So levels of '''C-14''' and ratio of '''C-14''' to '''C-12''' fall.
|-
| | '''Slide Number 11'''

'''Radioactive Dating-Cont’d'''

Radioactive dating, 14C:12C of sample vs recently dead specimens

Ur-Pb dating for rocks, artefacts etc
| |

'''Radioactive Dating-Continued'''

Radioactive dating compares C-14 C-12 ratio of sample to recently dead specimens.

It estimates how long the organism has been dead.

'''Uranium-lead dating''' is used for rocks, archaeological artefacts etc
|-
| | Point to the graph on the top.

Point to the two '''radio buttons, % of C-14''' and '''C-14 to C-12''' ratio, for the '''y-axis'''.

Point to the '''x-axis'''.

Point to the vertical red dashed line.
| | On the top, we see the graph like the ones in the previous screens.
|-
| | Point to these default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
| | Let us keep the following default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
|-
| | Drag the '''probe''' and place it on the animal skull on the ground, to the left.
| | We will drag the '''probe''' and place it on the animal skull on the ground, to the left.
|-
| | Point to the '''pop-up box''' next to the skull.

Show the '''text''', “'''Estimate age of Animal Skull'''”.

Show the empty box and “'''yrs'''” next to it.
| | Observe a '''pop-up box''' that appears next to the skull.

We see “'''Estimate age of Animal Skull'''” and “'''yrs'''” next to the empty box below.
|-
| | Point to the '''Check Estimate button'''.
| | Below this is a '''Check Estimate button'''.
|-
| | Show 98.2% in the top left side, above '''Probe Type'''.
| | Observe that in the top left side, above '''Probe Type''', we see 98.2%.
|-
| | Drag the double-headed green arrow to the left.

Show 98.2% in the white box above the arrow.
| | Let us drag the double-headed green arrow above the graph.

In the white box above the arrow, % of '''C-14''' should be approximately 98.2%.
|-
| | Show '''t = 123 yrs''' in the white box below % of '''C-14'''.
| | Observe that '''t equals 123 yrs''' appears in the white box below % of '''C-14'''.
|-
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
|-
| | Point to green text-box with 123 years in its place with a green '''smiley''' face next to it.
| | The '''Estimate pop-up box''' disappears.

A green text-box with 123 years appears in its place with a green '''smiley''' face next to it.
|-
| |
| | We have successfully dated the animal skull by measuring the % of '''C-14''' remaining in it.
|-
| | '''Slide Number 12'''

'''Assignment'''

Estimate ages of all objects in '''Dating Game screen'''

Correlate age (years) with percentage of '''unstable''' nucleus

Correlate age (years) with depth at which object found

'''C-14''': animal remains; '''U-238''': rocks, objects
| |

As an '''assignment''',

Estimate ages of all the objects in the '''Dating Game screen'''.

Correlate age in years with the percentage of '''unstable''' nucleus.

Correlate age in years with the depth at which the object is found.

Remember to use '''C-14''' for animal remains and '''U-238''' for rocks and other objects.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 13'''

'''Summary'''

We have demonstrated,

'''Radioactive Dating Game PhET simulation'''
| |

In this '''tutorial''', we have demonstrated how to use the '''Radioactive Dating Game PhET simulation'''.
|-
| | '''Slide Number 14'''

'''Summary'''

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
| |

Using this '''simulation''', we looked at:

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Radioactive-Dating-Game/English

2018-12-21T07:15:54Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '''interactive PhET simulation.'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Radioactive Dating Game PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Radioactive Dating Game''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school physics and chemistry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''

| |

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Radioactive Dating Game simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.
| | To open the '''jar file''', open the '''terminal'''.
|-
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
|-
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
|-
| | Point to the '''browser''' address.
| | '''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Radioactive Dating Game simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to four '''screens''' in the '''interface'''.
| | The '''interface''' has four '''screens''':

'''Half Life'''

'''Decay Rates'''

'''Measurement'''

'''Dating Game'''
|-
| | Show the '''Half Life''' screen.
| | We are already looking at the '''Half Life''' screen.
|-
| | Point to the graph at the top of the screen.

Point to units of time along the '''x-axis'''.
| | At the top of the screen is an '''Isotope versus Time''' graph.

Pay attention to the units of time.
|-
| | Show '''Choose Isotope''' panel to the right.

Point to the three options in '''Choose Isotope''' panel.
| | On the right side of the screen, you see a '''Choose Isotope''' panel.

It has three options showing unstable nucleus decaying to stable nucleus.
|-
| | Point to the '''Bucket o’ Atoms''' in the '''simulation''' panel.

Point to the '''C-14''' atoms in the bucket and to the default selection of '''C-14'''.

Point to the “'''Add 10'''” button attached to the bottom of the bucket.
| | In the middle is the '''simulation''' panel containing a '''Bucket o’ Atoms'''.

Note that it contains '''C-14''' atoms as the default selection is '''C-14'''.

Attached to the bottom of the bucket is a '''button''' called “'''Add 10'''”.
|-
| | '''Slide Number 7'''

'''Stable and Unstable Nuclei'''

Electrostatic repulsion between protons in nucleus

Strong nuclear force ~ binding energy

High binding energy; stable nucleus

Low binding energy; unstable nucleus ~ radioactive

| |

'''Stable and Unstable Nuclei'''

There is electrostatic repulsion between positively charged protons inside the nucleus.

The strong nuclear force overcomes this electrostatic repulsion between protons.

The energy associated with this force is the binding energy.

The lower the binding energy, the more unstable is the nucleus.

Such an unstable nucleus is said to be radioactive.
|-
| | Point to '''Play/Pause button''' and '''Step button''' next to it.
| | Below this '''simulation''' panel is a '''Play/Pause button''' and a '''Step button'''.
|-
| | Point to blue '''Reset All Nuclei button''' in '''simulation''' panel.
| | In this '''simulation''' panel is a blue '''Reset All Nuclei button'''.

This '''button''' lets you return to the start but with the selected isotope.
|-
| | Point to the white '''Reset All button''' below the right panel.
| | Below the right panel is a white '''Reset All button'''.

This button resets the '''simulation''' in this '''screen''' to all the default settings.
|-
| | Point to the '''Isotope versus time''' graph.

Point to vertical red dashed line labeled '''Half Life''' near the 5000 year mark.
| | Observe the '''Isotope versus time''' graph.

There is a vertical red dashed line labeled '''Half Life''' near the 5000 year mark.

The '''half-life''' of '''C-14''' is 5730 years.
|-
| | Point to red '''C-14''' symbol above the blue '''N-14''' symbol along the '''y-axis'''.
| | Along the '''y-axis''', you can see the red '''C-14''' symbol above the blue '''N-14''' symbol.

'''C-14''' atoms will appear in the upper row and '''N-14''' atoms in the lower one.
|-
| | Point to the red circle to the left of the '''Isotope''' label.

Point to '''hash symbols''' on the left of the circle.
| | To the left of the '''Isotope''' label is a red circle.

Numbers of '''C-14''' and '''N-14''' atoms shown by '''hash symbols''' will appear to the left of the circle.
|-
| | '''Slide Number 8'''

'''Radioactive Decay'''

Alpha decay

Beta decay

Gamma decay

Half-life
| | '''Radioactive Decay '''is the spontaneous conversion of an unstable nucleus into a stable nucleus.

It involves the release of subatomic particles and their energy as radiation.

It is of the following types:

'''Alpha decay'''

'''Beta decay'''

'''Gamma decay'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| |
| | Let us get back to the '''simulation'''.
|-
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
|-
| | Point to the 10 '''C-14''' atoms added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
| | Ten '''C-14''' atoms have been added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
|-
| | Show the red '''C-14''' atoms flying across the graph in the upper row.

Show the blue '''N-14''' atoms in the lower row.
| | Observe the atoms moving across the graph in the two rows.

|-
| | Point to the circle changing to blue.
| | Note how the circle changes to blue as more '''N-14''' atoms form.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red.
|-
| | Point to 5 blue '''N-14''' atoms on the left of the dashed half-life line.
| | Observe that, in the graph, there are 5 blue '''N-14''' atoms on the left of the dashed half-life line.

Out of 10 '''C-14''' atoms, half of them took 5730 years to decay into 5 '''N-14''' atoms.
|-
| | Point to 5 '''C-14''' and '''N-14''' atoms in the '''simulation''' panel.
| | There are 5 '''C-14''' atoms and 5 '''N-14''' atoms in the '''simulation''' panel also.

This is the definition of '''half-life'''.
|-
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
|-
| | Point to the 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
| | Another 5730 years are taken for 5 '''C-14''' atoms to decay to 2.5 '''N-14''' atoms.

We see 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
|-
| |
| | Predict the number of '''C-14''' atoms remaining after different periods.
|-
| | Perform the same '''simulation''' for the other nuclei.
| | Perform the same '''simulation''' for the other nuclei.
|-
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

Show the '''interface'''.
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

The '''interface''' has a similar arrangement as the '''Half Life screen'''.

Please explore this '''screen''' in the same way.
|-
| | Click on the '''Measurement tab'''.
| | Now, let us click on the '''Measurement tab''' to go to that '''screen'''.
|-
| | Point to '''Tree''', the default selection under '''Choose an Object''' on the right.
| | In the right panel, under '''Choose an Object''', we will stay with '''Tree'''.
|-
| | Point to '''Carbon-14''', and '''Objects''' under '''Probe Type''' in top left.
| | In the top left, under '''Probe Type''', we will retain the default selections, '''Carbon-14''', and '''Objects'''.
|-
| | Click on '''Plant Tree button''' in the bottom right corner.

Point to the tree growing right where the '''probe''' is placed.
| | Click on '''Plant Tree button''' in the bottom right corner.

|-
| | Immediately click on the '''Pause button'''.
| | Immediately click on the '''Pause button'''.
|-
| | Point to 100% seen above '''Probe Type''' in the upper left corner.
| | Observe 100% appear above '''Probe Type''' in the upper left corner.
|-
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
| | Keep clicking on the '''Step button''' to the right of '''Pause'''.
|-
| | Point to red line moving at the top of the graph along 100%.
| | Note that a red line moves at the top of the graph along 100%.
|-
| | Show '''% of C-14''' above the graph.
| | Above the graph, '''% of C-14''' is the default selection.
|-
| | Point to the white box below the graph.
| | The white box below the graph shows the number of years since the tree was planted.
|-
| | Point to the red line at the top of the graph.
| | The red line shows % of '''C-14''' remaining in the tree after those years after it was planted.
|-
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
|-
| | Point to the red line at the top of the graph.
| | Now the red line shows the '''C-14''' to '''C-12''' ratio in the tree after those years of planting it
|-
| | Click again on the % of '''C-14 radio button''' above the graph.
| | Click again on the % of '''C-14 radio button''' above the graph.
|-
| | Point to % in the top left, the tree and the white box below the graph.
| | Keep track of the % in the top left, the tree and the number of years below the graph.
|-
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
|-
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
| | Click on the '''Play''' or '''Step buttons''' to get approximately 50% in the top left window.
|-
| | Point to 50% on '''y-axis''', to red line and to '''x co-ordinate'''.
| | Note the number of years after which you see 50% of '''C-14''' in the tree.
|-
| | Click on '''Erupt Volcano''' and measure '''U-238''' levels in the cooled volcanic rock.

Click on the '''Air radio button''' to compare isotope levels in objects to air levels.

|-
| | Click on the last '''Dating Game tab'''.

Show the '''interface'''.
| | Let us click on the last '''Dating Game tab''' to go to that '''screen'''.

We can measure levels of '''C-14, U-238''' or other '''custom nuclei''' in this '''screen'''.

We see objects on and below the ground on which we can place the probe to measure these levels.
|-
| | '''Slide Number 9'''

'''Radioactive Dating'''

Two isotopes of C: 12C and 14C

Both isotopes → CO2, living organisms

Death of organism, ratio and 14C fall
| |

'''Radioactive Dating'''

Carbon has two isotopes: '''C-12''' and '''C-14'''.

Both are converted to carbon dioxide and are taken in by living organisms.

When an organism dies, it no longer takes in any carbon.

So levels of '''C-14''' and ratio of '''C-14''' to '''C-12''' fall.
|-
| | '''Slide Number 10'''

'''Radioactive Dating-Cont’d'''

Radioactive dating, 14C:12C of sample vs recently dead specimens

Ur-Pb dating for rocks, artefacts etc
| |

'''Radioactive Dating-Continued'''

Radioactive dating compares C-14 C-12 ratio of sample to recently dead specimens.

It estimates how long the organism has been dead.

'''Uranium-lead dating''' is used for rocks, archaeological artefacts etc
|-
| | Point to the graph on the top.

Point to the two '''radio buttons, % of C-14''' and '''C-14 to C-12''' ratio, for the '''y-axis'''.

Point to the '''x-axis'''.

Point to the vertical red dashed line.
| | On the top, we see the graph like the ones in the previous screens.
|-
| | Point to these default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
| | Let us keep the following default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
|-
| | Drag the '''probe''' and place it on the animal skull on the ground, to the left.
| | We will drag the '''probe''' and place it on the animal skull on the ground, to the left.
|-
| | Point to the '''pop-up box''' next to the skull.

Show the '''text''', “'''Estimate age of Animal Skull'''”.

Show the empty box and “'''yrs'''” next to it.
| | Observe a '''pop-up box''' that appears next to the skull.

We see “'''Estimate age of Animal Skull'''” and “'''yrs'''” next to the empty box below.
|-
| | Point to the '''Check Estimate button'''.
| | Below this is a '''Check Estimate button'''.
|-
| | Show 98.2% in the top left side, above '''Probe Type'''.
| | Observe that in the top left side, above '''Probe Type''', we see 98.2%.
|-
| | Drag the double-headed green arrow to the left.

Show 98.2% in the white box above the arrow.
| | Let us drag the double-headed green arrow above the graph.

In the white box above the arrow, % of '''C-14''' should be approximately 98.2%.
|-
| | Show '''t = 123 yrs''' in the white box below % of '''C-14'''.
| | Observe that '''t equals 123 yrs''' appears in the white box below % of '''C-14'''.
|-
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
|-
| | Point to green text-box with 123 years in its place with a green '''smiley''' face next to it.
| | The '''Estimate pop-up box''' disappears.

A green text-box with 123 years appears in its place with a green '''smiley''' face next to it.
|-
| |
| | We have successfully dated the animal skull by measuring the % of '''C-14''' remaining in it.
|-
| | '''Slide Number 11'''

'''Assignment'''

Estimate ages of all objects in '''Dating Game screen'''

Correlate age (years) with percentage of '''unstable''' nucleus

Correlate age (years) with depth at which object found

'''C-14''': animal remains; '''U-238''': rocks, objects
| |

As an '''assignment''',

Estimate ages of all the objects in the '''Dating Game screen'''.

Correlate age in years with the percentage of '''unstable''' nucleus.

Correlate age in years with the depth at which the object is found.

Remember to use '''C-14''' for animal remains and '''U-238''' for rocks and other objects.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''

We have demonstrated,

'''Radioactive Dating Game PhET simulation'''
| |

In this '''tutorial''', we have demonstrated how to use the '''Radioactive Dating Game PhET simulation'''.
|-
| | '''Slide Number 13'''

'''Summary'''

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
| |

Using this '''simulation''', we looked at:

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| | '''Slide Number 14'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 15'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 16'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

2018-12-21T05:34:56Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate area:

'''Under a curve (AUC)'''

Bounded by two '''functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Integration

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''[a,b]''' above '''x-axis'''

'''a''' is lower limit, b is upper limit

<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>

Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
| |

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''a b''' above the '''x-axis'''.

'''a''' and '''b''' are called the lower and upper limits of the integral.

Integral of '''f of x''' from '''a''' to '''b''' with respect to '''x''' is the notation for this definite integral.

It is the area bounded by '''y''' equals '''f of x, x''' equals '''a, x''' equals '''b''' and the '''x-axis'''.
|-
| | '''Slide Number 6'''

'''Calculation of a Definite Integral'''

Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
| |

Let us calculate the definite integral of this function with respect to '''x'''.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''g(x)= ‑ 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.
| | Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.
|-
| | Click on '''Slider''' tool and click in '''Graphics''' view.
| | Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.

It should range from 1 to 50 in increments of 1.
|-
| | Leave the '''Number''' radio button checked.
| |
|-
| | Type '''n''' in the '''Name''' field.
|-
| | Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''
|-
| | Point to '''slider n''' in '''Graphics''' view.

|-
| | Drag '''slider n''' to 5.
| | Drag the resulting '''slider n''' to 5.
|-
| | Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.
| | Under '''Point''', click on '''Point on Object''' and click at ‑1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.
|-
| |
| | Let us look at a few ways to approximate '''area under the curve'''.

These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.

We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
|-
| | Type '''uppersum=Upp''' in the '''Input Bar'''.

Show option.

'''UpperSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles> )'''

Click on it.
| | In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.

The following option appears.

Click on it.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Type '''g''' instead of highlighted '''<Function>'''.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
|-
| | Type '''x(A)'''.
| | Type '''x A in parentheses'''.
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
| | Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.

Press '''Enter'''.
|-
| | Point to five rectangles between '''x'''<nowiki= -1 and 2. </nowiki>
| | Note that five rectangles appear between '''x''' equals -1 and 2.
|-
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
|-
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
|-
| | '''Point''' to '''upper sum area under the curve (AUC).'''
| | The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
|-
| | Point to the rectangles extending above the curve.
| | It is an overestimation of the area under the curve.

This is because some portion of each rectangle extends above the curve.
|-
| | Drag the background to move the graph to the left.
| | Drag the background to move the graph to the left.
|-
| |
| | Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.
|-
| | Type '''trapsum=Tra''' in the '''Input bar'''.
| | In the '''input bar''', type '''trapsum''' is equal to '''Tra'''.
|-
| | Point to the menu that appears.
| | A menu with various options appears.
|-
| | Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
| | Select the following option.
|-
| |
| | We will type the same values as before and press '''Enter'''.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| |
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A)'''.
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.
| |
|-
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Point to trapezoids.
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Note the shape of the trapezoids.
|-
| |
| | Let us now look at the integral as the area under the curve.
|-
| | Finally, type '''Int''' in the '''Input Bar'''.
| | Finally, in the '''input bar''', type '''Int'''.
|-
| | '''Point''' to the menu with various options.
| | A menu with various options appears
|-
| | Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.
| | Select the following option.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Again, we will enter the same values as before.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A)'''.
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value'''.

Press '''Enter.'''
| |
|-
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
|-
| | Point to the integrated''' AUC'''.
| | For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals ‑1 to 2.
|-
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
|-
| | Click on '''Text''' tool under '''Slider''' tool.
| | Under '''Slider''', click on '''Text'''.
|-
| | Click in '''Graphics''' view to open a '''text box'''.
| | Click in '''Graphics''' view to open a '''text box'''.
|-
| | In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
| | In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
| | Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

Click '''OK''' in the '''text box'''.
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

In the '''text box''', click '''OK'''.
|-
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
|-
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
|-
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
|-
| | Point to text box and to '''slider n'''.
| | Observe the values in the '''text box''' as you drag '''slider n'''.
|-
| | Point to '''Graphics''' view.
| | '''Trapsum''' is a better approximation of '''AUC''' at high '''n''' values.

'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.

'''F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x2 + C'''
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window
|-
| |
| | We will look at the relationship between '''differentiation''' and '''integration'''.

Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.
|-
| | Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| |
| | Let us call '''integral''' of '''f of x capital F of x'''.
|-
| | Type '''F(x)=Integral(f)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.

Point to equation for '''F(x)=1/3 x3+ x2+x''' appears in '''Algebra''' view.
| | The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.

Its equation for '''capital F of x''' appears in '''Algebra''' view.

Confirm that this is the integral of '''f of x'''.
|-
| | Drag the boundary to see the equations properly.
| | Drag the boundary to see the equations properly.
|-
| | Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following and press '''Enter'''.
|-
| | Point to '''F'(x)''' and '''f(x)'''.
| | Note that this graph coincides with '''f of x'''.

The equations for '''f of x''' and '''h of x''' are the same.

Thus, we can see that '''integration''' is the inverse process of '''differentiation'''.

Taking the derivative of an integral, gives back the original '''function'''.
|-
| | Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
| | Click on '''Point''' tool and create a point at '''1 comma 3'''.
|-
| | Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following and press '''Enter'''.
|-
| | Click on '''Create Sliders''' in the window that pops up.
| | Click on '''Create Sliders''' in the window that pops up.
|-
| | Point to '''slider k'''.
| | A '''slider k''' appears.
|-
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.

Close the '''Preferences''' window.
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5.

Scroll right to set the '''Increment''' to 0.01.

Close the '''Preferences''' box.
|-
| | Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
| | In '''Algebra''' view. double-click on '''i of x''' and on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
|-
| | Drag '''k''' to make '''i(x)''' pass through point '''A'''.

Point to integral function '''(1/3)x3+x2+x+0.7'''.
| | Drag '''k''' to make '''i of x''' pass through point '''A'''.
|-
| | Drag the boundary to see '''i of x''' properly.
| | Drag the boundary to see '''i of x''' properly.
|-
| | Point to '''F(x)+0.7''': the curve and equation.
| | This function is '''capital F of x''' plus 0.7.
|-
| | '''Slide Number 7'''

'''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z=f(x,y)'''
| | '''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z''' which is equal to '''f of x and y'''
|-
| | '''Slide Number 8'''

'''Double Integral-An Example'''

Let us find the area between parabola '''x=y2''' and the line '''y=x'''.

The '''limits''' are from '''(0,0)''' to '''(1,1)'''.

This area can be expressed as the '''double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>
| |

'''Double Integral-An Example'''

Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.

The limits are from '''0 comma 0''' to '''1 comma 1'''.

This area can be expressed as the double integrals shown here.

Observe the limits and the order of the integrals in terms of the variables.

|-
| |
| | Let us open a new '''GeoGebra''' window.

We will first express '''x''' in terms of '''y''', for both '''functions'''.
|-
| | In the '''input bar''', type '''x=y2''' and press '''Enter'''.
| | In the '''input bar''', type '''x '''equals '''y caret''' 2 and press '''Enter'''.
|-
| | Next, in the '''input bar''', type '''y=x''' and press '''Enter'''.
| | Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.
|-
| | Click on '''View''' tool and select '''CAS'''.
| | Click on '''View''' tool and select '''CAS'''.
|-
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
|-
| | Drag the boundary to make '''CAS''' view bigger.
| | Drag the boundary to make '''CAS''' view bigger.
|-
| | In '''CAS''' view, type '''Int''' in line 1.

Point to the menu that appears.
| | In '''CAS''' view, type '''Int capital I''' in line 1.

A menu with various options appears.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| | Scroll down.

Select the following option.
|-
| | Type '''y''' for the first '''function'''.
| | Type '''y''' for the first '''function'''.
|-
| | Press '''Tab''' and type '''y^2''' for the second '''function'''.
| | Press '''Tab '''and type '''y caret 2''' for the second '''function'''.
|-
| | Press '''Tab''' and type '''y''' as the '''variable'''.
| | Press '''Tab''' and type '''y''' as the '''variable'''.
|-
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
|-
| | Press '''Enter'''.
| | Press '''Enter'''.
|-
| | Point to the value of 1/6 below the entry.

Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.
| | A value 1 divided by 6 appears below the entry.

This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.
|-
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
|-
| | In '''CAS''' view, type '''Int''' and observe the same menu as before.
| | In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| |
|-
| |
| | Now, let us reverse the order of '''functions''' and '''limits'''.
|-
| | Type '''sqrt(x)''' for the first function and '''x''' for the second.
| | Type the following and press '''Enter'''.
|-
| | Point to the '''input bar'''.
| | You can also use the '''input bar''' instead of the '''CAS''' view.
|-
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
|-
| | Drag the boundaries to make '''CAS''' view smaller.
| | Drag the boundaries to make '''CAS''' view smaller.
|-
| |
In the '''input bar''', type '''Int'''.

From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start Value''' and again press '''Tab''' to move to and type 1 as the '''End Value'''.

Press '''Enter'''.

This will also give you an area a of 0.17 or 1 divided by 6.
| |

In the '''input bar''', type '''Int capital I'''.

From menu, select the following option.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start x Value''' and again press '''Tab''' to move to and type 1 as the '''End x Value'''.

Press '''Enter'''.

This will also give you an area '''a''' of 0.17 or 1 divided by 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of '''area''':

'''Under a curve''' ('''AUC''')

Bounded by two '''functions'''
|-
| | '''Slide Number 10'''

'''Assignment'''* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where '''f(x) = 1/(1-x)'''
Calculate <math>{\int }_{x\left(A\right)}^{x\left(B\right)}g\left(x\right)dx</math>and <math>{\int }_{x\left(B\right)}^{x\left(C\right)}g\left(x\right)dx</math>where '''g(x) = 0.5x3+2x2-x-3.75'''

'''A, B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
| | As an '''assignment''':

Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.

Explain the results for '''g of x'''.
|-
| | '''Slide Number 11'''

'''Assignment'''

Calculate the area bounded by the following '''functions''':

[[Image:]]'''y=4x-x2, y=x'''

[[Image:]]'''x2+y2<nowiki>=9, y=3-x</nowiki>'''

[[Image:]]

'''y=1+x2, y=2x2'''
| | As another '''assignment''':

Calculate the shaded areas between these pairs of '''functions'''.
|-
| | '''Slide Number 12'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 13'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials

gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 15'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English

2018-12-06T06:36:13Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Limits and Continuity of Functions'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:

Understand '''limits''' of '''functions'''

Look at continuity of '''functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

'''Limits'''

'''Elementary calculus'''

For relevant '''tutorials''', please visit our website.
|-
| | Slide Number 5

Limits

[[Image:]][[Image:]]

| | Let us understand the concept of '''limits'''.

Imagine yourself sliding along the curve or line towards a given value of '''x'''.

The height at which you will be, is the corresponding '''y''' value of the '''function'''.

Any value of '''x''' can be approached from two sides.

The left side gives the '''left hand limit'''.

The right side gives the '''right hand limit'''.
|-
| | '''Slide Number 6'''

'''Limit of a rational polynomial function'''

Let us find '''lim (3x2 – x -10)'''

'''x→2 (x2 – 4)'''
| |

Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.

Let us graph functions and look at their limits.
|-
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Type '''(3 x^2-x-10)/(x^2-4)''' in the '''input bar''' >> '''Enter'''
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Note that spaces denote multiplication.

In the '''input bar''', first type the '''numerator'''.

Now, type the '''denominator'''.

Press '''Enter'''.
|-
| | Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
| | The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool.

Click in and drag '''Graphics''' view to see the graph.
| | Click on '''Move Graphics View'''.

Click in and drag '''Graphics''' view to see the graph.
|-
| | Point to the graph in '''Graphics''' view.
| | As '''x''' approaches 2, the '''function''' approaches some value close to 3.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' and select '''Spreadsheet'''.
|-
| | Point to the spreadsheet on the right side of the '''Graphics''' view.
| | This opens a spreadsheet on the right side of the '''Graphics''' view.
|-
| | Click on '''Options''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
| | Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
|-
| |

Remember to press '''Enter''' to go to the next cell.

Type 1.91, 1.93, 1.96, 1.98 and 2 in '''column A''' from '''cells''' 1 to 5.
| | Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' less than but close to 2.

Remember to press '''Enter''' to go to the next '''cell'''.

In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
|-
| |

Type 2.01, 2.03, 2.05, 2.07 and 2.09 in '''column A''' from '''cells''' 6 to 10.
| | Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' greater than but close to 2.

In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
|-
| | In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
| | In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.

First, the numerator in parentheses

'''3 A1''' in parentheses '''caret''' 2 minus A1 minus 10 followed by division slash'''

Now the denominator in parentheses

'''A1''' in parentheses '''caret''' 2 minus 4 and press '''Enter'''.
|-
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
|-
| | Drag and increase column width.
| | Drag and increase column width.
|-
| | Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.

Point to the spreadsheet.

| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.

This is because the '''function''' is undefined at this value.

The reason for this is that the denominator of the '''function''' becomes 0.

Observe that as '''x''' tends to 2, '''y''' tends to 2.75.

Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
|-
| | '''Slide Number 7'''

'''Limits of discontinuous functions'''

[[Image:]]

'''lim h(x) = ?'''

'''x→c'''

'''lim h(x) = L4; lim h(x) = L3'''

'''x→c- x→c+'''
Thus, '''lim h(x)''' Does Not Exist ('''DNE''')

'''x→c'''
| |

In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.

We want to find the '''limit''' of '''h of x''' as '''x''' approaches '''c'''.

So let us look at the '''left''' and '''right hand limits'''.

For the '''left hand limit''', look at the lower limb where the limit is '''L4'''.

For the '''right hand limit''', look at the upper limb where limit of '''h of x''' is '''L3'''.

But as '''x''' approaches '''c''', the two limbs of '''h of x''' approach different values of '''y'''.

These are '''L3''' and '''L4'''.

The '''left''' and '''right hand limits''' exist.

But the limit of '''h of x''' as '''x''' approaches '''c,''' '''does not exist''' ('''DNE''').

|-
| | '''Slide Number 8'''

'''Limit of a discontinuous function'''

Let us find '''lim f(x) = 2x+3, x ≤ 0'''

'''x→0''' '''3(x+1), x > 0'''

and '''lim f(x) = 2x+3, x ≤ 0'''

'''x→1''' '''3(x+1), x > 0'''
| |

Let us find limits of a '''piecewise''' or '''discontinuous function f of x'''.

'''f of x''' is described by '''2x plus 3''' when '''x''' is 0 or less than 0.

But '''f of x''' is described by '''3 times x plus 1''' when '''x''' is greater than 0.

We want to find the limits when '''x''' tends to 0 and 1.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following line.

'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''

This chooses the '''domain''' of '''x''' from minus 5 (for practical purposes) to 0.

Press '''Enter'''.
|-
| | Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.

Drag the boundary to see it properly.

Point to its graph in '''Graphics''' view.
| | The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.

Drag the boundary to see it properly.

Its graph is seen in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

|-
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
|-
| | Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
| | Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
|-
| | Similarly, click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
| | Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
|-
| | Click in and drag the background to see the graph properly.
| | Click in and drag the background to see the graph properly.
|-
| | Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following command.

Remember the space denotes multiplication.

'''b''' equals '''Function''' with capital F

In square brackets, type 3 space '''x''' plus 1 in parentheses comma 0.01 comma 5'''

This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.

For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0.

Press '''Enter'''.
|-
| | Point to the equation '''b(x)=3(x+1) (0.01 ≤ x ≤ 5)''' in '''Algebra''' view.

Point to its graph in '''Graphics''' view.

| | The equation '''b of x equals 3 times x plus 1''' where '''x''' varies from 0.01 to 5 appears in '''Algebra''' view.

Its graph is seen in '''Graphics''' view.
|-
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''

Point to the blue '''function'''.

Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.

Point to the red '''function'''.

| |

|-
| | Double click on the equation '''a(x)=2x+3''' in '''Algebra''' view.
| | In '''Algebra''' view, double click on the equation '''b of x''' equals 3 times '''x''' plus 1.
|-
| | Click on '''Object Properties'''.
| | Click on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select blue.
| | Click on the '''Color''' tab and select blue.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| |
| | Click in and drag the background to see both '''functions''' in '''Graphics''' view.
|-
| |
| | Under '''Move Graphics View''', click on '''Zoom In'''.

Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
|-
| |
| | Note that there is a break between the blue and red '''functions'''.

This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
|-
| |
| | The blue '''function''' has to be considered for '''x''' less than and equal to 0.

When '''x''' tends to 0, '''f of x''' is 3 as the '''function''' intersects the '''y-axis''' at 0 comma 3.
|-
| |
| | The red '''function''' has to be considered for '''x''' greater than 0.

When '''x''' equals 1, the value of '''f of x''' is 6.
|-
| |
| |
|-
| |
| |
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand limits of '''functions'''

Look at continuity of '''functions'''

|-
| | '''Slide Number 10'''

'''Assignment'''

Find the limit of '''(x3-2x2)/(x2-5x+6)''' as '''x''' tends to 2.

Evaluate '''lim sin4x'''

'''x→0''' '''sin 2x'''
| | '''As an Assignment''':

Find the limit of this '''rational polynomial function''' as '''x''' tends to 2.

Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
|-
| | '''Slide Number 11'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team:

<nowiki>* conducts workshops using spoken tutorials and</nowiki>

<nowiki>* gives certificates on passing online tests.</nowiki>

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 14'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Probability-and-Distributions/English

2018-10-31T06:45:49Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Probability and Distributions in..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Probability and Distributions in GeoGebra'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will:

Learn how to use '''Probability Calculator''' in '''GeoGebra'''

Look at different distributions and parameters.
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | To follow this '''tutorial''', you should be familiar with

'''GeoGebra''' interface

Statistics
|-
| | '''Slide Number 5'''

'''Fish Feed'''

A fishery is testing four types of feed formulation on its fish: '''A, B, C '''and''' D'''

Length (mm)
Weight (lbs)
Girth (mm)
| | '''Fish Feed'''

Let us look at an example.

A fishery is testing four types of feed formulation on its fish: '''A, B, C''' and '''D'''.
Data to be collected after feeding the fish for 6 months are:

Length in '''millimeters'''

Weight in '''pounds'''

Girth in '''millimeters'''

Let’s look at some of these data.
|-
| | '''Slide Number 6'''

'''Fish Feed Data'''

[[Image:|top]]
| | '''Fish Feed Data'''

We will use these data for our '''analyses'''.

Please download the '''code file''', '''Fishery-data''', provided along with this '''tutorial'''.
|-
| | '''Slide Number 7'''

'''Probability'''

Probability of an event P(A), from 0 to 1

P(A) is ratio of frequency of event A to number of trials

Sampling distribution (normal, t etc)

Probabilities compare 2 independent sample proportions or means
| |
'''Probability'''

Probability of an event P(A) is the likelihood that event A will occur.
P(A) lies between 0 and 1.

P(A) is the ratio of frequency of event A to the number of trials.

Statistics are calculated for each sample.
The probability distribution of these statistics is called a sampling distribution.
Examples are normal, t etc.

Probabilities compare 2 independent sample proportions or means.
|-
| | '''Slide Number 8'''

'''Hypothesis testing'''

'''Statistical hypothesis'''

'''Hypothesis testing: H0, Ha'''

'''z-, t-, F-tests''' etc
| |

'''Hypothesis testing'''

A '''statistical hypothesis''' is an assumption about a population parameter

'''Hypothesis testing''' asks: should a '''statistical hypothesis''' be accepted or rejected?

'''H zero''', the '''null hypothesis''', says observations arise from pure chance.

'''Ha''', the''' alternative hypothesis''', says observations arise due to non-random causes.

'''z-, t-''' and '''F-'''tests '''test population parameters''' in different situations.
|-
| |
| | Please refer to '''additional material''' provided along with this '''tutorial'''.
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' tool and select '''Spreadsheet'''.
|-
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.

This will close these views.
|-
| | In the '''code file''', use the '''mouse''' to highlight length data in column '''B'''.
| | In the '''code file''', use the '''mouse''' to highlight length data in column '''B'''.
|-
| | Hold '''Ctrl''' key down and press '''C''' to copy the data.
| | Hold '''Control''' key down and press '''C''' to copy the data.
|-
| | Click on '''Spreadsheet''' view in '''GeoGebra'''.
| | Click in the top of the '''Spreadsheet''' in '''GeoGebra'''.
|-
| | Hold '''Ctrl''' key down and press '''V'''.
| | Hold '''Control''' key down and press '''V'''.
|-
| | Point to the data pasted into '''GeoGebra'''.
| | This will copy and paste the highlighted data from the '''code file''' into '''GeoGebra'''.
|-
| | Drag and adjust the column width.
| | Drag and adjust the column width.
|-
| | Click on '''Text''' and change name to '''Length (mm)-A'''.

Close the '''dialog box'''.
| | As shown earlier in the series, right-click on the heading.

Change the name to '''Length (mm) hyphen A'''.

Close the '''dialog box'''.
|-
| | Adjust the '''column''' width.
| | Adjust the '''column''' width.
|-
| | Highlight data in '''columns E, H''' and '''K'''.

Copy and paste data from '''code file''' into '''GeoGebra'''.

Change names of data from '''columns E, H''' and '''K''' to:

'''Length (mm)-B'''

'''Length (mm)-C'''

'''Length (mm)-D'''
| | Repeat this with data in '''columns E, H''' and '''K'''.
|-
| | Drag '''mouse''' to highlight all labels and data in the four '''columns'''.
| | Select all data in the four '''columns''' by dragging.
|-
| | Under the '''menubar''', under '''One Variable Analysis''', click on '''Multiple Variable Analysis'''.
| | Under the '''menubar''', under '''One Variable Analysis''', click on '''Multiple Variable Analysis'''.
|-
| | Show '''Data Source''' popup window.

Click on '''Analyze''' button.
| | A'''Data Source''' popup window appears.

Click on '''Analyze''' button.
|-
| | Show the '''Data Analysis''' window.
| | A '''Data Analysis''' window appears.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Point to '''Stacked box plots''' appearing in all four '''columns'''.
| | '''Stacked box plots''' appear for data for all four '''columns'''.
|-
| | Click anywhere in the '''GeoGebra''' window and then click on '''Show Statistics''' tool.

Point to '''Statistics''' displayed below the '''box plots'''.
| | Click anywhere in the '''GeoGebra''' window and then click on '''Show Statistics''' tool.

'''Statistics''' are displayed below the '''box plots'''.
|-
| | Above the statistics, click on '''menu button''' next to the display.

Select '''ANOVA'''.
| | Above the statistics, click on '''menu button''' next to the word '''Statistics'''.

Select '''ANOVA'''.
|-
| | Drag boundaries to increase size of statistics tables.
| | Drag the boundaries and resize the window to increase size of statistics tables.
|-
| | Place the '''cursor''' on the boundary below the plots.

And drag to increase the size of the tables.
| | Place the '''cursor''' on the boundary below the plots.

And drag to increase the size of the tables.
|-
| | Point to the results.
| | The '''between groups mean square (MS)''' is much greater than '''within groups MS'''.

'''F value''' is the ratio of '''between groups MS''' to '''within groups MS'''. Hence, '''F value''' is quite large (36.5892).

'''P value''' is 0. This means it is probably less than 0.001. The difference in the means of all groups is '''statistically significant'''.

The feed does make a '''statistically significant''' difference to the length of the fish.

Hence, the '''null hypothesis''' can be rejected in this case. The '''null hypothesis''' here is that none of the length means are different. That is, none of the feeds make any difference to the length of the fish.
|-
| | Click on the '''menu button''' next to the '''ANOVA display'''.
| | Next to the '''ANOVA display''', click on the '''menu button'''.
|-
| | Point to two options appearing for '''T Test''' and '''T Estimate'''.
| | Two options appear for '''T Test''': '''Difference of Means''' and '''Paired Differences'''.
|-
| | Point to '''T Test''' in the '''menu'''.
| | The same two options appear for '''T Estimate'''.

'''Difference of Means''' is for '''unpaired T Test'''.

'''Paired Differences''' is for '''paired T Test'''.
|-
| |
| | The '''T Test''' compares two groups at a time.
|-
| | Select '''T Test: Difference of Means'''.
| | Select '''T Test: Difference of Means'''.
|-
| | Point to '''Sample 1''' and '''Sample 2'''.
| | '''Column A''' data are denoted by default as '''Sample 1'''. '''Column B''' data are denoted by default as '''Sample 2'''
|-
| | Click on '''menu buttons''' next to the displays to reverse the order.
| | Click on the '''menu buttons''' next to the displays to reverse the order.

As mean of '''column B''' is greater than mean of '''column A''', '''T values''' and '''limits''' will now be positive.
|-
| | Point to '''t''' and '''P''' values in '''T tests'''.
| | '''T Tests''' give '''t''' and '''P''' values.

Comparing '''A''' and '''B''' gives '''P''' less than 0.001 and '''T value''' greater than 4.

Thus, feeds '''A''' and '''B''' have a significant effect on lengths of fish.
|-
| | Click on the '''menu button''' and choose '''T estimates, Difference of Means'''.
| | Click on the '''menu button''' and choose '''T estimates, Difference of Means'''.
|-
| | Show the statistics tables.

Point to '''Confidence Level''' 0.95.
| | '''T Estimates''' give '''lower''' and '''upper limits''' for the '''mean difference'''.

The '''confidence level''' is 95%.

We can be 95% sure that the '''mean difference''' is between the '''lower''' and '''upper limits'''.

You can change '''column''' pairs for comparison and look at the '''T Test''' results.
|-
| | Close the '''Data Analysis window'''.
| | Close the '''Data Analysis window'''.
|-
| |
| | Now let us look at the '''Probability Calculator'''.
|-
| | Point to '''Spreadsheet''' view.

Use '''mouse''' to drag and highlight length data for '''feed A'''.
| | We are in the '''Spreadsheet''' view.

Use the '''mouse''' to drag and highlight length data for '''feed A'''.
|-
| | Click on '''One Variable Analysis''' tool.
| | Click on '''One Variable Analysis''' tool.

In the '''Data Source''' popup window that appears, click on '''Analyze''' button.
|-
| | At the top of the '''Data Analysis''' window, click on the 2nd '''Show Statistics''' button.

Note down '''mean mu (µ)''' and '''standard deviation sigma (σ)'''. (745.5, 29.0215)
| | At the top of the '''Data Analysis''' window, click on the 2nd '''Show Statistics''' button.

Note down '''mean mu (µ)''' and '''standard deviation sigma (σ)'''.
|-
| | Close the '''Data Analysis''' window and follow the same steps for '''feed B'''. (801.5, 21.2191)
| | Close the '''Data Analysis''' window and follow the same steps for '''feed B'''.
|-
| | Drag and highlight '''feed A''' length data.
| | Again, drag and highlight '''feed A''' length data.
|-
| | Click on '''View''' and then click on '''Probability Calculator'''.
| | Click on '''View''' and then click on '''Probability Calculator'''.
|-
| | Point to the '''Probability Calculator''' window that pops up.
| | The '''Probability Calculator''' window pops up.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Point to the plot and the '''Distribution tab''' above the plot.

Below the plot, point to the '''Normal display box'''.
| | We are looking at a '''normal distribution''' in the '''Distribution''' window.
|-
| | Place your '''cursor''' on the horizontal boundary below the '''distribution curve'''.

Drag the arrow upwards to see the data entry window below the '''curve''' properly.
| | Place your '''cursor''' on the horizontal boundary below the '''distribution curve'''.

Drag the arrow upwards to see the data entry window below the '''curve''' properly.
|-
| |
| | Let us look at a '''normal distribution''' for fish given feed '''A'''.
|-
| | Type 745.5 in the box next to '''mu''' >> press '''Enter'''.
| | In the box next to '''mu (μ)''', type 745.5 and press '''Enter'''.
|-
| | Type 29.0215 in the box next to '''sigma''' >> press '''Enter'''.
| | In the box next to '''sigma (σ)''', type 29.0215 and press '''Enter'''.
|-
| | Point to '''normal distribution''' plot.
| | A '''normal distribution''' plot appears with mean 745.5 and '''sigma''' 29.0215.
|-
| | Click on the 1st of three buttons below the '''mean''' and '''σ''' boxes.
| | Click on the 1st of three buttons below the '''mean''' and '''σ''' boxes.
|-
| | Point to the right side bracket indicates this is the upper limit.
| | The right side bracket indicates this is the upper limit.
|-
| | Type 770 in the box next to '''P (X ≤''' and press '''Enter'''.
| | In the box next to '''P of X''' less than or equal to, type 770 and press '''Enter'''.
|-
| | Point to the '''probability P''' appearing in the box to the right, 0.8007.
| | Note that the '''probability P''' appears in the box to the right, 0.8007.

Thus, 80.07% fish fed feed '''A''' are 770 mm long or shorter.
|-
| | Type 0.09 in the '''P''' box to the right and press '''Enter'''.
| | Let us do the reverse.

In the '''P''' box to the right of the equal to sign, type 0.09.

Press '''Enter'''.
|-
| |
| | We want to know how long 9% of the fish are, on the lower side of the group.
|-
| | Point to '''X''' 706.5893 appearing in the box.
| | When you press '''Enter''', '''X''' less than or equal to 706.5893 appears in the box.
|-
| | Point to the value.
| | Thus, 9% of the fish are shorter than this length.
|-
| | Click on the '''curve symbol''' next to the '''Normal''' display box.
| | Next to the '''Normal''' display box, click on the '''curve symbol'''.
|-
| | Point to the '''cumulative distribution function curve''' appears.
| | The '''cumulative distribution function curve''' appears.
|-
| | Point to '''Probability''' on '''y-axis''' and length of feed on '''x-axis'''.
| | '''Probability''' is plotted on the '''y-axis''', length of '''feed A''' group is plotted on '''x-axis'''.
|-
| | Click on '''curve symbol''' to return to the '''normal distribution bell curve'''.
| | Click on '''curve symbol''' to return to the '''normal distribution bell curve'''.
|-
| | Click on 2nd of three buttons below '''mu''' and '''sigma''' displays.
| | Below '''mu''' and '''sigma''' displays, click on 2nd of the three buttons.
|-
| | Point to the two brackets indicate that lower and upper limits can be specified.
| | The two brackets indicate that lower and upper limits can be specified.
|-
| | Type 705 in the first box and 758 in the second box and press '''Enter'''.
| | In the first box, type 705 and in the second box, 758, and press '''Enter'''.
|-
| | Point to '''P (705 ≤ X ≤ 758) = 0.5852'''.
| | '''P''' equal to 0.5852 appears.

This means 58.52% of fish fed '''feed A''' are 705 to 758 mm long.
|-
| | Click on the 3rd of the three buttons showing a left bracket.
| | Finally, click on the 3rd button showing a left bracket.
|-
| | Type 760 in the box and press '''Enter'''.
| | In the box, type 760 and press '''Enter'''.
|-
| | Point to 0.3087.
| | 30.87% of fish fed '''feed A''' are longer than 760 mm.
|-
| | Click on '''Statistics''' tab next to '''Distribution''' tab.
| | Next to '''Distribution''' tab, click on '''Statistics''' tab.
|-
| | Close the '''Probability Calculator''' window.
| | Close the '''Probability Calculator''' window.
|-
| | Point to '''Spreadsheet''' view.
| | Let us look at the '''Spreadsheet''' in '''GeoGebra'''.
|-
| | Use '''mouse''' to drag and highlight length data in '''columns A''' and '''B'''.
| | Use '''mouse''' to drag and highlight length data in '''columns A''' and '''B'''.
|-
| | Under '''One Variable Analysis''', select '''Probability Calculator'''.
| | Under '''One Variable Analysis''', select '''Probability Calculator'''.
|-
| | Point to '''Statistics''' window.
| | We are looking, as before, at the '''Statistics''' window.
|-
| | From the dropdown menu at the top, select '''T Test, Difference of Means'''.
| | From the dropdown menu at the top, select '''T Test, Difference of Means'''.
|-
| | Type '''means, standard deviation σ''' and '''N=10''' in respective boxes.

Press '''Enter''' after entering all values.
| | You can type '''mean, standard deviation σ''' and total number of samples '''N''' in the boxes.

We will type 10 for '''N''' as each feed group has 10 fish.

Press '''Enter''' after entering all values.
|-
| |
| | '''Feed A''' mean is lower than '''feed B''' mean.
|-
| | Choose '''feed B''' group as '''Sample 1''' and '''feed A''' as '''Sample 2'''.
| | So we will choose '''feed B''' group as '''Sample 1''' and '''feed A''' as '''Sample 2'''.

This will result in positive values for different statistical parameters.
|-
| |
| | Note '''t''', standard error '''SE''', degrees of freedom '''df''' and '''P''' values.

Compare them to results from '''Multiple Variable Analysis'''.

Select different tests for different pairs of columns in the '''Spreadsheet'''.

Interpret the results and compare with your calculations.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''Probability Calculator''' in '''GeoGebra'''.

We looked at different '''distributions''' and '''parameters'''.
|-
| | '''Slide Number 10'''

'''Assignment'''
| | '''Assignment'''

Perform statistical analyses for weight and girth data given in this '''tutorial'''.

Four oils were used to deep fry chips. Six chips were chosen from each batch fried in a given oil. Amount of absorbed fat was measured for these chips. Is any of the oils absorbed more than the others?

[[Image:|top]]
|-
| | '''Slide Number 11'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 14'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Statistics-using-GeoGebra/English

2018-10-30T06:55:25Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Statistics using GeoGebra''' |-..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Statistics using GeoGebra'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to perform:

'''One Variable Analysis''' to calculate different statistical parameters

'''Two Variable Regression Analysis''' to estimate best fit line

'''Multiple Variable Analysis''' to calculate different statistical parameters
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux '''OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Statistics
|-
| | '''Slide Number 5'''

'''Statistics'''

Data analysis and interpretation

'''Measures of central tendency'''

'''Measures of Dispersion'''

Comparing '''variability '''of data series

'''Additional material '''
| |

'''Statistics'''

Statistics deals with

Data analysis and interpretation

Measures of '''central tendency'''

Measures of '''Dispersion'''

Comparing '''variability''' of data series

Please refer to '''additional material''' provided along with this '''tutorial'''.
|-
| | '''Slide Number 6'''

'''Measures of central tendency'''

'''Arithmetic mean '''

'''Median'''

'''Mode'''
| |

'''Measures of central tendency'''

Measures of '''central tendency''' include '''arithmetic mean, median''' and '''mode'''.

They give an idea about centering of data points.
|-
| | '''Slide Number 7'''

'''Measures of Dispersion'''

'''Range'''

'''Quartiles'''

'''Mean'''

'''Standard Deviation (σ) = sqrt (variance)'''
| |

'''Measures of Dispersion''' include '''Range, Quartiles, Mean''' and '''Standard Deviation'''.

'''Range''' is the difference between the highest and lowest data points.

'''Quartiles''' are the data points that divide the list into quarters

'''Mean''' is the sum of all data points divided by the number of points.

'''Standard deviation, sigma''', is the '''squareroot''' of '''variance'''.
It describes how spread out the data are.
|-
| | '''Slide Number 8'''

'''Fish Feed '''

A fishery is testing four feed formulations on its fish: '''A, B, C''' and '''D'''

Length (mm)

Weight (lbs)

Girth (mm)

| | '''Fish Feed '''

Let us look at an example.

A fishery is testing four types of feed formulations on its fish: '''A, B, C''' and '''D'''.

Data to be collected after feeding the fish for 6 months are:

Length in millimeters

Weight in pounds

Girth in millimeters

Let us look at some of these data.
|-
| | '''Slide Number 9'''

'''Fish Feed Data'''

[[Image:]]
| | We will use these data for our '''analyses'''.

Please download the '''code file''', '''Fishery-data''', provided along with this '''tutorial'''.
|-
| | '''Slide Number 10'''

'''Box Plot'''

[[Image:]]
| | '''Box Plot'''

'''Box plot''' is a standardized way of showing data, based on the '''five number summary'''.

'''Minimum''' and '''maximum''' values are the '''whiskers''' of the '''box plot'''.

The '''box''' spans first to third '''quartiles'''.

The segment inside the box shows the '''median'''.
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' tool and select '''Spreadsheet'''.
|-
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.
| | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views.

This will close these views.
|-
| | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''.

Show data in '''columns H''' and '''I'''.

Press '''Ctrl''' key down and press '''C'''.
| | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''.

These are data for fish that have been fed formulation '''C'''.

Hold '''Control''' key down and press '''C'''.
|-
| | Click on '''Spreadsheet''' view in '''GeoGebra'''.
| | Click in the top of the '''Spreadsheet''' in '''GeoGebra'''.
|-
| | Press '''Ctrl''' key down and press '''V'''.
| | Hold '''Control''' key down and press '''V'''.

This will copy and paste the highlighted data from the '''code file''' into '''GeoGebra'''.
|-
| | Place the '''cursor''' on the first column header in '''Spreadsheet''' view.

Drag and adjust '''column A''''s width.
| | Place the '''cursor''' on the first column header in '''Spreadsheet''' view.

Drag and adjust '''column A''''s width.
|-
| | Right-click on '''column A''' heading of '''Length (mm)'''.

Select '''Object Properties'''.

Point to '''dialog box'''.
| | Right-click on '''column A''' heading of '''Length millimetres'''.

Select '''Object Properties'''.

A '''dialog box''' opens.
|-
| | Click on '''Text tab''' and change the name to '''Length (mm)-C'''.

Close the '''dialog box'''.

Similarly, add '''–C''' to '''Weight (lbs)'''.
| | Click on '''Text tab''' and change the name to '''Length millimetres''' hyphen '''C'''.

Close the '''dialog box'''.

Similarly, add '''hyphen C''' to '''Weight pounds'''.
|-
| | Adjust '''column B''' width.
| | Adjust '''column B''' width.
|-
| | Use '''mouse''' to drag and highlight first '''column A'''’s length data and label in '''GeoGebra'''.
| | Click on '''column A''' heading of '''Length millimetres C'''.

Drag to highlight length data in '''Spreadsheet''' view.
|-
| | Below the '''menubar''', click on '''One Variable Analysis''' tool.

Point to '''Data Source''' popup window.

Click on '''Analyze button'''.
| | Below the '''menubar''', click on '''One Variable Analysis''' tool.

A '''Data Source''' popup window appears.

Click on '''Analyze button'''.
|-
| | Point to '''Data Analysis''' window and '''histogram'''.
| | A '''Data Analysis''' window appears.

By default, a '''histogram''' is plotted.
|-
| | Drag the boundary to see the graph properly.
| | Drag the boundary to see the graph properly.
|-
| | Point to length on the '''x-axis''' and frequency on the '''y-axis'''.
| | The length is plotted on the '''x-axis'''.

The number of fish that are of a particular length, the '''frequency''', is plotted on the '''y-axis'''.
|-
| | Point to the '''display box''' above the graph containing the word '''Histogram'''.
| | Note the '''display box''' above the graph containing the word '''Histogram'''.
|-
| | In the '''display box''', click on the '''dropdown menu button'''.
| | In the '''display box''', click on the '''dropdown menu button''' to display the list of plots.
|-
| | Select different options ('''Bar chart, box plot''' etc) and see how they look.
| | Select different plot options ('''Bar chart, box plot''' etc) and observe the plots.
|-
| | Select '''Histogram'''.

Point to '''slider''' to the right of the display.
| | We will stay with the '''histogram''' option.

To the right of the '''dropdown menu''' is a '''slider'''.
|-
| | Drag the '''slider''' from left to right to go to 20.
| | Drag the '''slider''' from left to right to go to 20.
|-
| | Point to rectangles between minimum and maximum values of data.
| | The '''slider''' changes the number of rectangles between the minimum and maximum values of data.
|-
| | Click on '''Options''' button to the right of the '''slider'''.
| | Click on '''Options''' button to the right of the '''slider'''.
|-
| | Under '''Classes''', check '''Set Classes Manually'''.
| | Under '''Classes''', check '''Set Classes Manually check box'''.

This displays '''Start''' and '''Width text-boxes''' to the left of the '''Options button'''.
|-
| | Type 800 in the '''Start text-box''' and press '''Enter'''.

Show the value of 5 in the '''Width text-box'''.
| | As all the fish are over 800 mm long, type 800 in the '''Start text-box''' and press '''Enter'''.

We will stay with the default value of 5 for rectangle '''width'''.
|-
| | Uncheck '''Set Classes Manually'''.
| | Uncheck '''Set Classes Manually''' check box.
|-
| | Click on '''Options button''' again to hide the window.
| | Click on '''Options button''' again to hide the window.
|-
| | Click on '''Show Data''' tool >> point to data highlighted in the '''Spreadsheet'''.
| | Above the '''Histogram text-box''', click on the third '''Show Data tool button'''.

This displays all the data highlighted in the '''Spreadsheet'''.
|-
| | Drag the boundary to see the data properly.
| | Drag the boundary to see the data properly.
|-
| | Click on '''Show Data''' tool again to hide the list.
| | Click on the '''Show Data''' tool again to hide the list.
|-
| | Click on '''Show 2nd Plot''' tool.
| | Above the '''Histogram text-box''', click on the last '''Show 2nd Plot tool button'''.
|-
| | Select '''histogram''' for top plot and '''box plot''' for bottom plot.
| | The same data are graphed in two vertically placed plots.

You can select plot types from the '''dropdown menu''' above each plot.
|-
| | Click on '''Show Statistics''' tool.

Point to '''Statistics''' for both plots.
| | Above the '''Histogram text-box''', click on the second '''Show Statistics tool button'''.

'''Statistics''' for the plot appears as a '''panel''' in the middle.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Click and point to '''Median''', '''Min''', '''Max''', '''Q1''' and '''Q3''' values in the '''box plot'''.
| | Let us compare '''histogram''' and '''box plot'''.

In the '''box plot''', locate the '''Median''', '''Min''', '''Max''', '''Q1''' and '''Q3''' values.
|-
| | Click on the button next to '''Options button''' above the plot.
| | Above each plot, in the upper right corner, click on the '''button''' next to '''Options'''.

A '''dropdown menu''' appears with which you can copy each plot to '''Clipboard''' or '''export''' it as an '''image'''.
|-
| | Click on '''Show Statistics tool button''' to hide the data.
| | Click on '''Show Statistics tool button''' to hide the data.
|-
| | Close the '''Data Analysis''' window.
| | Close the '''Data Analysis''' window.
|-
| | '''Slide Number 11'''

'''Least Squares Linear Regression (LSLR)'''

Changing an independent variable '''x''' changes the dependent variable '''y'''.

'''LSLR''' predicts '''y''' based on '''x''' value.

'''LSRL (best fit line) y = b0 + b1x'''

'''Coefficient of determination R2'''
| | '''Least Squares Linear Regression (LSLR)'''

Changing an independent variable '''x''' changes the dependent variable '''y'''.

'''LSLR''' predicts '''y''' based on '''x''' value.

'''Least Squares Regression Line (LSRL)''' is also called the '''best fit line'''.

It is given by '''y = b0 + b1x'''.

'''b0''', the '''y intercept''', is a '''constant'''.

'''b1''', the '''slope''', is the '''regression coefficient'''.

'''Coefficient of determination R squared'''

'''R squared''' ranges from 0 to 1.
|-
| | Show length and weight data in the '''Spreadsheet''' in the '''GeoGebra'''.
| | Let’s go back to the length and weight data in the '''Spreadsheet''' view in '''GeoGebra'''.
|-
| | Drag '''mouse''' to highlight all labels and data in the two '''columns'''.
| | Drag and select all the data in both '''columns'''.
|-
| | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool.
| | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool.
|-
| | Click '''Analyze button''' in the '''Data Source window''' that pops up.
| | In the '''Data Source window''' that pops up, click '''Analyze button'''.
|-
| | '''Data Analysis''' window appears.
| | A '''Data Analysis''' window appears with two plots.
|-
| | Show both plots.
| | By default, the upper plot is a '''Scatterplot''' and the lower a '''Residual plot'''.
|-
| | Click on '''Show Statistics''' tool to see '''Statistics'''.
| | Click on '''Show Statistics''' tool to see the '''Statistics'''.
|-
| | Drag the boundary to see them properly.
| | Drag the boundary to see them properly.
|-
| | Below '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''.
| | Below the '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''.
|-
| | Point to the red line that is drawn through some points.
| | Note the red line that is drawn through some points in the '''Scatterplot'''.
|-
| | Point to length plotted on '''x-axis''', weight on '''y-axis'''.
| | Length is plotted along '''x-axis''' and weight along '''y-axis'''.
|-
| | Point to equation is given in red, '''y= 0.08x-48.39'''.
| | This is the '''best fit line''' that passes through as many points as possible.

Its equation is given in red, '''y= 0.08x-48.39''', at the bottom.
|-
| | Point to '''R2''' value of 0.7722.
| | Note that '''R squared''' value for this line is 0.7722 in the '''Statistics window.'''

The closer '''R squared '''value is to 1, the better is the prediction of '''y''' from '''x''' by the equation.

This ''' R squared''' value indicates good fit between the model and the actual data.
|-
| | Select other '''regression models''' to see effects on '''R2'''.
| | Select other '''regression models''' to see effects on the '''R squared''' value.
|-
| | Point to the lower '''Residual Plot'''.
| | The lower plot is the '''Residual Plot'''.

It shows '''residuals''' along the '''y-axis''', '''length''' along the '''x-axis'''.

'''Residuals '''are the differences between observed and predicted values of all points.

Absence of '''U''' shape in the '''scatterplot''' means that '''linear regression''' is appropriate for these data.
|-
| | Click on '''Switch Axes button'''.
| | Above the '''Statistics window''', click on the last '''Switch Axes button'''.
|-
| | Point to length now plotted on '''y-axis''' and weight on '''x-axis'''.
| | For the '''scatterplot''', length is now plotted along '''y-axis''' and weight along '''x-axis'''.
|-
| | Point to the '''best fit line''' and '''statistics'''.

Point to equation '''y= 9.91x + 684.3'''.
| | Observe that the '''best fit line''' and many '''statistics''' change.
|-
| | Point to '''r''', '''R2 '''and '''rho (ρ)'''.
| | The only statistics that remain the same are '''r''', '''R squared '''and '''rho (ρ)'''.

Note that '''r''' and '''rho''' are greater than 0.8, indicating positive '''correlation'''.

Weight increases as length increases for fish given '''feed C'''.

The relationship is strong and well predicted by the '''best fit lines'''.
|-
| | Again, click on '''Switch Axes button'''.
| | Again, click on '''Switch Axes button'''.
|-
| | Point to '''Symbolic Evaluation''' at the bottom.
| | At the bottom, in '''Symbolic Evaluation''', you can enter a value for '''x''' to get a prediction for '''y'''.
|-
| | Point at the line in the '''Scatterplot'''.
| | If you choose '''x''' values below the '''x-intercept''' of the line, you will get negative values of '''y'''.

To get logical predictions, we will enter '''x''' values above the '''x-intercept'''.
|-
| | In '''Symbolic Evaluation''', type in a value for '''x''' and press '''Enter'''.
| | In '''Symbolic Evaluation''', in the text-box for '''x''', type 800 and press '''Enter'''.
|-
| | Point to '''y''' value appearing next to the '''display box'''.
| | Note that a '''y''' value appears next to the '''display box'''.

The '''x''' value was substituted in the '''best fit line''' equation to get the '''y''' value.
|-
| | Again, click on '''Show Statistics tool button'''.
| | Again, click on '''Show Statistics tool button'''.
|-
| | Close the '''Data Analysis window'''.
| | Close the '''Data Analysis window'''.
|-
| | Point to length and weight data in the '''Spreadsheet'''.
| | Let’s go back to the length and weight data in the '''Spreadsheet'''.
|-
| | Drag '''mouse''' to highlight all labels and data in the two '''columns'''.
| | In the '''Spreadsheet''', select all the data in both '''columns'''.
|-
| | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool.
| | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool.
|-
| | Click '''Analyze button''' in the '''Data Source window''' that pops up.
| | In the '''Data Source window''' that pops up, click '''Analyze button'''.
|-
| | Point to''' Stacked Box Plots''' in the window and to the cell numbers in each row.
| | '''Stacked Box Plots''' appear in the window.

They are for length and weight data.
|-
| | Click on '''Show Statistics''' tool.

Point to '''Statistics''' for both plots.
| | Above the plot, click on the second '''Show Statistics''' tool.

'''Statistics''' for both plots appear below.
|-
| | Place the '''cursor''' on the boundary between the plot and statistics.

When the arrow appears, drag the boundary to resize the windows.
| | Place the '''cursor''' on the boundary between the plot and statistics.

When the '''arrow''' appears, drag the boundary to resize the windows.
|-
| | Close '''Data Analysis window'''.
| | Close '''Data Analysis window'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to perform:

'''One Variable Analysis''' to calculate different statistical parameters

'''Two Variable Regression Analysis''' to estimate best fit line

'''Multiple Variable Analysis''' to calculate different statistical parameters
|-
| | '''Slide Number 13'''

'''Assignment'''
| | '''Assignment'''

Perform statistical analyses for weight and girth data given in this '''tutorial'''

Four oils were used to deep fry chips. Six chips were chosen from each batch fried in a given oil. Amount of absorbed fat was measured for these chips Is any of the oils absorbed more than the others?

[[Image:]]
|-
| | '''Slide Number 14'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 15'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 16'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Integration-using-GeoGebra/English

2018-10-26T09:09:14Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Integration using GeoGebra''' |-..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Integration using GeoGebra'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will use '''GeoGebra''' to look at integration to estimate area:

'''Under a curve (AUC)'''

Bounded by two''' functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux '''OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

[http://www.spoken-tutorial.org/ www.spoken-tutorial.org]
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Integration

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''[a,b]''' above '''x-axis'''

'''a''' is lower limit, b is upper limit

<math>\underset{a}{\overset{b}{\int }}f\left(x\right)dx</math>

Area bounded by '''y=f(x), x=a, x=b''' and '''x-axis'''
| |

'''Definite Integral'''

Consider '''f''' is a continuous '''function''' over interval '''a b''' above the '''x-axis'''.

'''a''' and '''b''' are called the lower and upper limits of the integral.

Integral of''' f of x '''from''' a '''to''' b '''with respect to''' x '''is the notation for this definite integral.

It is the area bounded by '''y '''equals''' f of x, x '''equals''' a, x '''equals''' b''' and the '''x-axis'''.
|-
| | '''Slide Number 6'''

'''Calculation of a Definite Integral'''

Let us calculate the definite integral<math>{\int }_{-1}^{2}(-0.5x\hat{3}+2x\hat{2}-x+1)dx</math>
| |

Let us calculate the definite integral of this function with respect to '''x'''.

Lower and upper limits are minus 1 and 2.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''g(x)= ‑ 0.5 x^3+ 2 x^2-x+1''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.
|-
| | Point to the graph in '''Graphics''' view and its equation in '''Algebra''' view.
| | Note the graph in '''Graphics''' view and its equation in '''Algebra''' view.
|-
| | Click on '''Slider''' tool and click in '''Graphics''' view.
| | Using the '''Slider''' tool, create a number '''slider n''' in '''Graphics''' view.

It should range from 1 to 50 in increments of 1.
|-
| | Leave the '''Number''' radio button checked.
| |

|-
| | Type '''n''' in the '''Name''' field.
|-
| | Set 1 as '''Min''', 50 as the '''Max''' and 1 as '''Increment''' >> '''OK'''
|-
| | Point to '''slider n''' in '''Graphics''' view.

|-
| | Drag '''slider n''' to 5.
| | Drag the resulting '''slider n''' to 5.
|-
| | Click on '''Point on Object''' tool and click at ('''-1,0) '''and '''(2,0) '''to create '''A''' and '''B'''.
| | Under '''Point''', click on '''Point on Object''' and click at ‑1 comma 0 and 2 comma 0 to create '''A''' and '''B'''.
|-
| |
| | Let us look at a few ways to approximate '''area under the curve'''.

These will include '''upper Riemann''' and '''trapezoidal sums''' as well as '''integration'''.

We will first assign the variable label '''uppersum''' to the '''Upper Riemann Sum''' in '''GeoGebra'''.
|-
| | Type '''uppersum=Upp''' in the '''Input Bar'''.

Show option.

'''UpperSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles> )'''

Click on it.
| | In the '''input bar''', type '''uppersum '''is equal to''' capital U p p'''.

The following option appears.

Click on it.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Type '''g''' instead of highlighted '''<Function>'''.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
|-
| | Type '''x(A) '''.
| | Type '''x A in parentheses'''.
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles''' >> '''Enter'''
| | Similarly, type '''x B in parentheses''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.

Press '''Enter'''.
|-
| | Point to five rectangles between '''x'''<nowiki>= -1 and 2. </nowiki>
| | Note that five rectangles appear between '''x''' equals -1 and 2.
|-
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
| | Under '''Move Graphics View,''' click on '''Zoom In '''and click in '''Graphics''' view.
|-
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
| | Again click on '''Move Graphics View''' and drag the background to see all the rectangles properly.
|-
| | '''Point''' to '''upper sum area under the curve (AUC).'''
| | The '''upper sum area under the curve (AUC)''' adds the area of all these rectangles.
|-
| | Point to the rectangles extending above the curve.
| | It is an overestimation of the area under the curve.

This is because some portion of each rectangle extends above the curve.
|-
| | Drag the background to move the graph to the left.
| | Drag the background to move the graph to the left.
|-
| |
| | Let us now assign the variable label '''trapsum''' to the '''Trapezoidal Sum'''.
|-
| | Type '''trapsum=Tra''' in the '''Input bar'''.
| | In the '''input bar''', type '''trapsum '''is equal to'''Tra'''.
|-
| | Point to the menu that appears.
| | A menu with various options appears.
|-
| | Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
| | Select '''TrapezoidalSum( <Function>, <Start x-Value>, <End x-Value>, <Number of Trapezoids> ).'''
|-
| |
| | We will type the same values as before and press '''Enter'''.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| |
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A)'''.
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value''' and '''n''' as '''Number of Rectangles'''.
| |
|-
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Point to trapezoids.
| | In '''Algebra''' view, uncheck '''uppersum''' to hide it in '''Graphics''' view.

Note the shape of the trapezoids.
|-
| |
| | Let us now look at the integral as the area under the curve.
|-
| | Finally, type '''Int''' in the '''Input Bar'''.
| | Finally, in the '''input bar''', type '''Int'''.
|-
| | '''Point''' to the menu with various options.
| | A menu with various options appears
|-
| | Select '''Integral( <Function>, <Start x-Value>, <End x-Value>)'''.
| | '''Select Integral( <Function>, <Start x-Value>, <End x-Value>)'''.
|-
| | Type '''g''' instead of highlighted '''<Function>'''.
| | Again, we will enter the same values as before.
|-
| | Press '''Tab''' to highlight '''<Start x-Value>'''.
| |
|-
| | Type '''x(A). '''
| |
|-
| | Similarly, type '''x(B)''' for '''End x-Value'''.

Press '''Enter.'''
| |
|-
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''trapsum''' to hide it in '''Graphics''' view.
|-
| | Point to the integrated''' AUC'''.
| | For the integral, the curve is the upper bound of the '''AUC''' from '''x''' equals ‑1 to 2.
|-
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
| | In '''Algebra''' view, uncheck '''integral a''' to hide it in '''Graphics''' view.
|-
| | Click on '''Text''' tool under '''Slider''' tool.
| | Under '''Slider''', click on '''Text'''.
|-
| | Click in '''Graphics''' view to open a '''text box'''.
| | Click in '''Graphics''' view to open a '''text box'''.
|-
| | In the '''Edit''' field, type '''Upper Sum = ''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
| | In the '''Edit''' field, type '''Upper space Sum equals''' and in '''Algebra''' view, click on '''uppersum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Trapezoidal Sum =''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
| | Type '''Trapezoidal space Sum equals''' and in '''Algebra''' view, click on '''trapsum'''.

Click again in the '''text box''' and press '''Enter'''.
|-
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

Click '''OK''' in the '''text box'''.
| | Type '''Integral a equals''' and in '''Algebra''' view, click on '''a'''.

In the '''text box''', click '''OK'''.
|-
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
| | Click on '''Move''' and drag the '''text box''' in case you need to see it better.
|-
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
| | Now, click on the '''text box''' and click on the '''Graphics''' panel and select '''bold''' to make the text bold.
|-
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
| | In '''Algebra''' view, check '''a, trapsum''' and '''uppersum''' to show all of them.
|-
| | In '''Graphics''' view, double click on an '''uppersum''' rectangle.
| | In '''Graphics''' view, double click on an '''uppersum''' rectangle.
|-
| | In the '''Redefine text box''' that opens, click on '''Object Properties'''.
| | In the '''Redefine text box''' that opens, click on '''Object Properties'''.
|-
| | Under '''Color''' tab, choose yellow.

Under '''Basic''' tab, uncheck '''Show Label'''.
| | Under '''Color''' tab, choose yellow.

Under '''Basic''' tab, uncheck '''Show Label'''.
|-
| | In the left panel, now click on '''trapsum'''.

Again, under '''Basic''' tab, uncheck '''Show Label'''.

Click on '''Color''' tab.

Let us leave the color as the default brown.
| | In the left panel, now click on '''trapsum'''.

Again, under '''Basic''' tab, uncheck '''Show Label'''.

Click on '''Color''' tab.

Let us leave the color as the default brown.
|-
| | Finally, in the left panel, click on '''a'''.

Under '''Basic''' tab, uncheck '''Show Label'''.

Click on '''Color''' tab and choose blue.
| | Finally, in the left panel, click on '''a'''.

Under '''Color''' tab, choose blue.

Under '''Basic''' tab, uncheck '''Show Label'''.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Click on '''Options''' tool and select '''Rounding''' , '''5 decimal places'''.
| | Click on '''Options''' tool and select '''Rounding''', '''5 decimal places'''.
|-
| | Point to text box and to '''slider n'''.

Drag '''slider n''' to 10, then 20, 30, 40 and 50.
| | Observe the values in the '''text box''' as you drag '''slider n'''.

Drag '''slider n''' to 10, then 20, 30, 40 and 50.
|-
| | Point to all values in '''Graphics''' view.
| | Observe that the '''upper Reimann''' and '''trapezoidal sums''' remain higher than the '''integral'''.
|-
| | Point to '''Graphics''' view.
| | As '''n''' increases, the '''upper sum''' decreases.

'''Integral''' does not change as it is not broken into '''n''' shapes.

Thus, '''trapsum''' is a better approximation of '''AUC''' at high '''n''' values.

'''Integrating''' such '''sums''' from '''A''' to '''B''' at high values of '''n''' will give us the '''AUC'''.
|-
| |
| | Let us look at the geometrical representation of an indefinite integral.
|-
| | '''Slide Number 6'''

'''Indefinite Integral: Geometrical Interpretation'''

[[Image:]]

Family of '''integrals''' of '''f(x) = 2x'''.

'''F(x)''' where '''y=x2+C''', '''C''' is any constant.

'''F(x) =<math>\underset{❑}{\overset{❑}{\int }}f\left(x\right)dx</math> = <math>\underset{❑}{\overset{❑}{\int }}2xdx</math> = x2 + C'''

| | '''Indefinite integral: Geometrical interpretation'''

Parabolas in this figure are members of a family of '''integrals''' of '''f of x,''' which equals '''2x'''.

The family is represented by '''capital F of x'''.

'''y''' is equal to '''x squared plus capital C''', where '''capital C''' is any constant.

'''capital F of x''' is equal to '''integral of f of x''' '''with respect''' '''to x'''.

This is equal to '''integral of 2x with respect to x'''.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window
|-
| |
| | We will look at the relationship between '''differentiation''' and '''integration'''.

Also we will look at finding the '''integral function''' through a point '''A 1 comma 3'''.
|-
| | Type '''f(x)=x^2+2 x+1''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.

'''f x in parentheses equals x caret 2 plus 2 space x plus 1'''
|-
| | Drag the boundary to see the equation in '''Algebra''' view.
| | Drag the boundary to see the equation in '''Algebra''' view.
|-
| |
| | Let us call '''integral''' of '''f of x capital F of x'''.
|-
| | Type '''F(x)=Integral(f)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.

'''capital F x in parentheses equals capital I integral f in parentheses'''
|-
| | Point to the red '''integral''' curve of '''f(x)''' in '''Graphics''' view.

Point to equation for '''F(x)=1/3 x3+ x2+x''' appears in '''Algebra''' view.
| | The '''integral''' curve of '''f of x''' is red in '''Graphics''' view.

Its equation for '''capital F of x''' appears in '''Algebra''' view.

Confirm that this is the integral of '''f of x'''.
|-
| | Drag the boundary to see the equations properly.
| | Drag the boundary to see the equations properly.
|-
| | Type '''h(x)=F'(x)''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type the following and press '''Enter'''.

'''h x in parentheses equals capital F prime x in parentheses'''
|-
| | Point to '''F'(x)''' and '''f(x)'''.
| | Note that this graph coincides with '''f of x'''.

The equations for '''f of x''' and '''h of x''' are the same.

Thus, we can see that '''integration''' is the inverse process of '''differentiation'''.

Taking the derivative of an integral, gives back the original '''function'''.
|-
| | Click on '''Point''' tool and create point '''A''' at '''(1,3)'''.
| | Click on '''Point''' tool and create a point at '''1 comma 3'''.
|-
| | Type '''i(x)=F(x)+k''' in the '''Input Bar''' >> '''Enter'''.
| | In the '''input bar''', type '''i x in parentheses equals capital F x in parentheses plus k''' and press '''Enter'''.
|-
| | Click on '''Create Sliders''' in the window that pops up.
| | Click on '''Create Sliders''' in the window that pops up.
|-
| | Point to '''slider k'''.
| | A '''slider k''' appears.
|-
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5 and '''Increment''' to 0.01.

Close the '''Preferences''' window.
| | Double click on '''slider k'''.

Set '''Min''' at 0, '''Max''' at 5.

Scroll right to set the '''Increment''' to 0.01.

Close the '''Preferences''' box.
|-
| | Double click on '''i(x)''' in '''Algebra''' view and on '''Object Properties'''.
| | In '''Algebra''' view. double-click on '''i of x''' and on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
| | Click on '''Color''' tab and select green.

Close the '''Preferences''' box.
|-
| | Drag '''k''' to make '''i(x)''' pass through point '''A'''.

Point to integral function '''(1/3)x3+x2+x+0.7'''.
| | Drag '''k''' to make '''i of x''' pass through point '''A'''.
|-
| | Drag the boundary to see '''i of x''' properly.
| | Drag the boundary to see '''i of x''' properly.
|-
| | The '''integral function x cubed divided by 3 plus x squared plus x plus 0.7''' passes through '''A'''.
| | The '''integral function x cubed divided by 3 plus x squared plus x plus 0.7''' passes through '''A'''.
|-
| | Point to '''F(x)+0.7''': the curve and equation.
| | This function is '''capital F of x''' plus 0.7.
|-
| | '''Slide Number 7'''

'''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z=f(x,y)'''
| | '''Double Integrals'''

'''Double integrals''' can be used to find:

The '''area under a curve''' along '''x''' and '''y''' '''axes'''’ directions

The volume under a surface '''z''' which is equal to '''f of x and y'''
|-
| | '''Slide Number 8'''

'''Double Integral-An Example'''

Let us find the area between parabola '''x=y2''' and the line '''y=x'''.

The '''limits''' are from '''(0,0)''' to '''(1,1)'''.

This area can be expressed as the '''double integral =<math>{\left({\int }_{0}^{1}{\int }_{y\hat{2}}^{y}dxdy\right)}^{}</math><nowiki>= </nowiki>'''<math>\left({\int }_{0}^{1}{\int }_{x}^{x\hat{0.5}}dydx\right)</math>
| |

'''Double Integral-An Example'''

Let us find the area between a parabola '''x equals y squared''' and the line '''y equals x'''.

The limits are from '''0 comma 0''' to '''1 comma 1'''.

This area can be expressed as the double integrals shown here.

Observe the limits and the order of the integrals in terms of the variables.

'''Double integral''' from 0 to 1 and from '''y2''' to '''y''' with respect to '''x''' then '''y'''.

This is equal to the '''double integral''' from 0 to 1 and from '''x''' to '''squareroot of x'''.

But in the reverse order so that it is first with respect to '''y''' then '''x'''.
|-
| |
| | Let us open a new '''GeoGebra''' window.

We will first express '''x''' in terms of '''y''', for both '''functions'''.
|-
| | In the '''input bar''', type '''x=y2''' and press '''Enter'''.
| | In the '''input bar''', type '''x '''equals '''y caret''' 2 and press '''Enter'''.
|-
| | Next, in the '''input bar''', type '''y=x''' and press '''Enter'''.
| | Next, in the '''input bar''', type '''y equals x''' and press '''Enter'''.
|-
| | Point to the area between the parabola and the line, from '''(0,0) '''to '''(1,1)'''.
| | We want to find the area between the parabola and the line from 0 comma 0 to 1 comma 1.
|-
| | Click on '''View''' tool and select '''CAS'''.
| | Click on '''View''' tool and select '''CAS'''.
|-
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
| | In '''Algebra''' view, click top right button to close '''Algebra''' view.
|-
| | Drag the boundary to make '''CAS''' view bigger.
| | Drag the boundary to make '''CAS''' view bigger.
|-
| | In '''CAS''' view, type '''Int''' in line 1.

Point to the menu that appears.
| | In '''CAS''' view, type '''Int capital I''' in line 1.

A menu with various options appears.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| | Scroll down.

Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
|-
| | Type '''y''' for the first '''function'''.
| | Type '''y''' for the first '''function'''.
|-
| | Press '''Tab''' and type '''y^2''' for the second '''function'''.
| | Press '''Tab '''and type '''y caret 2''' for the second '''function'''.
|-
| | Press '''Tab''' and type '''y''' as the '''variable'''.
| | Press '''Tab''' and type '''y''' as the '''variable'''.
|-
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
| | Press '''Tab''' and type 0 and 1 as '''start''' and '''end values''' of '''y'''.
|-
| | Press '''Enter'''.
| | Press '''Enter'''.
|-
| | Point to the value of 1/6 below the entry.

Point to the area between the parabola and the line from '''(0,0)''' to '''(1,1)'''.
| | A value 1 divided by 6 appears below the entry.

This is the area between the parabola and the line from '''0 comma 0''' to '''1 comma 1'''.
|-
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
| | Let us now express '''y''' in terms of '''x''' for both '''functions'''.
|-
| | In '''CAS''' view, type '''Int''' and observe the same menu as before.
| | In '''CAS''' view, type '''Int capital I''' and choose the same option from the menu as before.
|-
| | Select '''IntegralBetween( <Function>, <Function>, <Variable>, <Start Value>, <End Value> )'''.
| |
|-
| |
| | Now, let us reverse the order of '''functions''' and '''limits'''.
|-
| | Type '''sqrt(x)''' for the first function and '''x''' for the second.
| | Type '''sqrt x in parentheses''' for the first '''function''' and '''x''' for the second.
|-
| | Type '''x''' as the variable and enter 0 and 1 as '''start''' and '''end values''' of '''x'''.
| | Type '''x''' as the variable and enter 0 and 1 as '''start''' and '''end values''' of '''x'''.
|-
| | When you press '''Enter''', point to the same output of 1/6.
| | When you press '''Enter''', you see the same output of 1 divided by 6 as the area.
|-
| | Point to the '''input bar'''.
| | You can also use the '''input bar''' instead of the '''CAS''' view.
|-
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
| | Under '''View,''' click on '''Algebra''' to see '''Algebra''' view again.
|-
| | Drag the boundaries to make '''CAS''' view smaller.
| | Drag the boundaries to make '''CAS''' view smaller.
|-
| |

In the '''input bar''', type '''Int'''.

From the menu, select '''IntegralBetween( <Function>, <Function>, <Start Value>, <End Value> )'''.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start Value''' and again press '''Tab''' to move to and type 1 as the '''End Value'''.

Press '''Enter'''.

This will also give you an area a of 0.17 or 1 divided by 6.
| |

In the '''input bar''', type '''Int capital I'''.

From menu, select '''IntegralBetween Function, Function, Start x Value, End x Value'''.

Type '''y''' for the first '''function'''.

Press '''Tab''', type '''y caret 2''' for the second '''function'''.

Press '''Tab''', type 0 as the '''Start x Value''' and again press '''Tab''' to move to and type 1 as the '''End x Value'''.

Press '''Enter'''.

This will also give you an area '''a''' of 0.17 or 1 divided by 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have used '''GeoGebra''' to understand '''integration''' as estimation of '''area''':

'''Under a curve''' ('''AUC''')

Bounded by two '''functions'''
|-
| | '''Slide Number 10'''

'''Assignment'''* Calculate <math>{\int }_{0}^{0.5}f\left(x\right)dx</math>where '''f(x) = 1/(1-x)'''
Calculate <math>{\int }_{x\left(A\right)}^{x\left(B\right)}g\left(x\right)dx</math>and <math>{\int }_{x\left(B\right)}^{x\left(C\right)}g\left(x\right)dx</math>where '''g(x) = 0.5x3+2x2-x-3.75'''

'''A, B''' and '''C''' are points where the curve intersects '''x-axis''' (left to right); explain the results
| | As an '''assignment''':

Calculate the integrals of '''f of x''' and '''g of x''' between the limits shown with respect to '''x'''.

Explain the results for '''g of x'''.
|-
| | '''Slide Number 11'''

'''Assignment'''

Calculate the area bounded by the following '''functions''':

[[Image:]]'''y=4x-x2, y=x'''

[[Image:]]'''x2+y2<nowiki>=9, y=3-x</nowiki>'''

[[Image:]]

'''y=1+x2, y=2x2'''
| | As another '''assignment''':

Calculate the shaded areas between these pairs of '''functions'''.
|-
| | '''Slide Number 12'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 13'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

conducts workshops using spoken tutorials

gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 14'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 15'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English

2018-10-17T06:45:36Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Limits and Continuity of Functions'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:

Understand '''limits''' of '''functions'''

Look at continuity of '''functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

'''Limits'''

'''Elementary calculus'''

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Limits'''

[[Image:]][[Image:]]

[[Image:]]
| | Let us understand the concept of '''limits''' by looking at three graphs '''A, B''' and '''C'''.

Imagine yourself sliding along the curve or line towards a given value of '''x'''.

The height at which you will be, is the corresponding '''y''' value of the '''function'''.

Any value of '''x''' can be approached from two sides.

The left side gives the '''left hand limit'''.

The right side gives the '''right hand limit'''.
|-
| | '''Slide Number 6'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b) f(x) = ?'''

'''lim_(x→ b-) f(x) = L1; lim_(x→b+) f(x) = L1 = f(b)'''

| |

In graph '''A''', let us find the '''limit''' of '''f of x''' as '''x''' approaches or tends to '''b'''.

'''f of x''' is a continuous line.

The '''left hand limit''' of '''f of x''' as '''x''' tends to '''b''' is '''L1'''.

And the '''right hand limit''' of '''f of x''' as '''x''' tends to '''b''' is also '''L1'''.

Thus, the '''limit''' of '''f of x''' as '''x''' approaches '''b''' is '''L1'''.

It is the same as evaluating '''f of x''' at '''x equals b''', that is, '''f of b.'''
|-
| | '''Slide Number 7'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b1) g(x) =?'''

'''lim_(x→b1-) g(x) = lim_(x→b1+) g(x) = L2''''

But '''g(b1)''' does not exist ('''DNE''')

'''lim_(x→b) g(x) = g(b) = L2; lim_(x→a) g(x) = g(a) = L1'''

| |

What is the '''limit''' of '''g of x''' as '''x''' tends to '''b1'''?

In graph '''A''', note that '''g of x''' has an open circle at '''b1 comma L2 prime'''.

This means that '''g of x''' does not exist at this point.

Let us find the '''limit''' of '''g of x''' as '''x''' approaches '''b1'''.

The '''left hand''' and '''right hand limits''' are '''L2'''' as '''x''' approaches '''b1'''.

But '''g of x''' itself does not exist at '''x equals b1'''.

However, '''g of x''' can be evaluated at '''x equals b''' and '''x equals a'''.

And these values are the same as the '''limits''' of '''g of x''' as '''x''' approaches '''b''' and '''a'''.
|-
| | '''Slide Number 8'''

'''Limits of discontinuous functions'''

[[Image:]]

'''lim_(x→c) h(x) = ?'''

'''lim_(x→c-) h(x) = L4; lim_(x→c+) h(x) = L3'''

Thus, '''lim_(x→c) h(x)''' does not exist ('''DNE''')

| |

In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.

We want to find the '''limit''' of '''h of x''' as '''x''' approaches '''c'''.

So let us look at the '''left''' and '''right hand limits'''.

For the '''left hand limit''', look at the lower limb where the limit is '''L4'''.

For the '''right hand limit''', look at the upper limb where limit of '''h of x''' is '''L3'''.

But as '''x''' approaches '''c''', the two limbs of '''h of x''' approach different values of '''y'''.

These are '''L3''' and '''L4'''.

The '''left''' and '''right hand limits''' exist.

But the limit of '''h of x''' as '''x''' approaches '''c,''' '''does not exist''' ('''DNE''').
|-
| | '''Slide Number 10'''

'''Limits at infinity'''

[[Image:]]

'''lim_(x→∞) i(x) = ? lim_(x→-∞) i(x) = ?'''

'''lim_(x→∞) i(x) = 2; lim_(x→-∞) i(x) = 1'''

| |

In graph '''C''', '''i of x''' has two parts.

The first part is the upper right one.

Both arms extend towards '''infinity''' ('''∞''').

The second part is the lower left one.

Both arms extend towards '''negative infinity''' ('''-∞''').

What are the limits of '''i of x''' as '''x''' tends to '''infinity''' and '''minus infinity'''?

The limit of '''i of x''' as '''x''' approaches '''infinity''' is 2.

And the limit of '''i of x''' as '''x''' approaches '''negative infinity''' is 1.
|-
| | '''Slide Number 11'''

'''Limit of a rational polynomial function'''

Let us find '''lim_(x→2) (3x2 – x -10)/(x2 – 4)'''

| |

Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.

|-
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Type '''(3 x2-x-10)/(x2-4)''' in the '''input bar''' >> '''Enter'''
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Note that spaces denote multiplication.

In the '''input bar''', first type the '''numerator'''.

Now, type the '''denominator'''.

Press '''Enter'''.
|-
| | Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
| | The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool.

Click in and drag '''Graphics''' view to see the graph.
| | Click on '''Move Graphics View'''.

Click in and drag '''Graphics''' view to see the graph.
|-
| | Point to the graph in '''Graphics''' view.
| | As '''x''' approaches 2, the '''function''' approaches some value close to 3.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' and select '''Spreadsheet'''.
|-
| | Point to the spreadsheet on the right side of the '''Graphics''' view.
| | This opens a spreadsheet on the right side of the '''Graphics''' view.
|-
| | Click on '''Options''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
| | Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
|-
| |

Remember to press '''Enter''' to go to the next cell.

Type 1.91, 1.93, 1.96, 1.98 and 2 in '''column A''' from '''cells''' 1 to 5.
| | Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' less than but close to 2.

Remember to press '''Enter''' to go to the next '''cell'''.

In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
|-
| |

Type 2.01, 2.03, 2.05, 2.07 and 2.09 in '''column A''' from '''cells''' 6 to 10.
| | Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' greater than but close to 2.

In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
|-
| | In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
| | In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.

First, the numerator in parentheses

'''3 A1''' in parentheses '''caret''' 2 minus A1 minus 10 followed by division slash'''

Now the denominator in parentheses

'''A1''' in parentheses '''caret''' 2 minus 4 and press '''Enter'''.
|-
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
|-
| | Drag and increase column width.
| | Drag and increase column width.
|-
| | Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.

Point to the spreadsheet.

| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.

This is because the '''function''' is undefined at this value.

Observe that as '''x''' tends to 2, '''y''' tends to 2.75.

Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
|-
| | '''Slide Number 12'''

'''Limit of a rational polynomial function'''

'''lim_(x→2) (3x2 – x -10)/(x2 – 4) = 2.75'''

| |

Thus, the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2 is 2.75.
|-
| | '''Slide Number 13'''

'''Limit of a discontinuous function'''

Let us find '''lim_(x→0) f(x) = 2x+3, x ≤ 0'''

................................ ='''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0'''

...........................= '''3(x+1), x > 0'''
| |

Let us find limits of a '''piecewise''' or '''discontinuous function f of x'''.

'''f of x''' is described by '''2x plus 3''' when '''x''' is 0 or less than 0.

But '''f of x''' is described by '''3 times x plus 1''' when '''x''' is greater than 0.

We want to find the limits when '''x''' tends to 0 and 1.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following line.

This chooses the '''domain''' of '''x''' from minus 5 (for practical purposes) to 0.

Press '''Enter'''.
|-
| | Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.

Drag the boundary to see it properly.

Point to its graph in '''Graphics''' view.
| | The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.

Drag the boundary to see it properly.

Its graph is seen in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

|-
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
|-
| | Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
| | Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
|-
| | Similarly, click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
| | Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
|-
| | Click in and drag the background to see the graph properly.
| | Click in and drag the background to see the graph properly.
|-
| | Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following command and press '''Enter'''.

Remember the space denotes multiplication.

This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.

For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0.
|-
| | Point to the equation '''b(x)=3(x+1) (0.01 ≤ x ≤ 5)''' in '''Algebra''' view.

Point to its graph in '''Graphics''' view.

| | The equation '''b of x equals 3 times x plus 1''' where '''x''' varies from 0.01 to 5 appears in '''Algebra''' view.

Its graph is seen in '''Graphics''' view.
|-
| | Double click on the equation '''a(x)=2x+3''' in '''Algebra''' view.
| | In '''Algebra''' view, double click on the equation '''b of x''' equals 3 times '''x''' plus 1.
|-
| | Click on '''Object Properties'''.
| | Click on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select blue.
| | Click on the '''Color''' tab and select blue.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| |
| | Click in and drag the background to see both '''functions''' in '''Graphics''' view.
|-
| |
| | Under '''Move Graphics View''', click on '''Zoom In'''.

Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
|-
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''
| | Note that there is a break between the blue and red '''functions'''.

This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
|-
| | Point to the blue '''function'''.

Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.

| | The blue '''function''' has to be considered for '''x''' less than and equal to 0.

When '''x''' tends to 0, '''f of x''' is 3 as the '''function''' intersects the '''y-axis''' at 0 comma 3.
|-
| | Point to the red '''function'''.
| | The red '''function''' has to be considered for '''x''' greater than 0.

When '''x''' equals 1, the value of '''f of x''' is 6.
|-
| | '''Slide Number 14'''

'''Limit of a discontinuous function'''

'''lim_(x→0) f(x) = 2x+3, x ≤ 0 }=3'''

..........................= '''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0 }=6'''

.........................= '''3(x+1), x > 0'''
| |

Thus, for this '''discontinuous function''', '''f of x''' is 3 when '''x''' is 0.

When '''x''' is 1, '''f of x''' is 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 15'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand limits of '''functions'''

Look at continuity of '''functions'''

|-
| | '''Slide Number 16'''

'''Assignment'''

Find the limit of '''(x3-2x2)/(x2-5x+6)''' as '''x''' tends to 2.

Evaluate '''lim_(x→0) sin 4x/sin 2x'''

| | '''As an Assignment''':

Find the limit of this '''rational polynomial function''' as '''x''' tends to 2.

Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
|-
| | '''Slide Number 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team:

<nowiki>* conducts workshops using spoken tutorials and</nowiki>

<nowiki>* gives certificates on passing online tests.</nowiki>

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 20'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

PhET/C3/Radioactive-Dating-Game/English

2018-10-11T06:59:36Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on''' Radioactive Dating Game, '''an '''interactive PhET simulation.'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Radioactive Dating Game PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Radioactive Dating Game''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school physics and chemistry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''

| |

Using this '''simulation''', we will look at

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Radioactive Dating Game simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.
| | To open the '''jar file''', open the '''terminal'''.
|-
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
| | At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.
|-
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
| | Type '''java space hyphen jar space radioactive-dating-game_en.jar'''.
|-
| | Point to the '''browser''' address.
| | '''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Radioactive Dating Game simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to four '''screens''' in the '''interface'''.
| | The '''interface''' has four '''screens''':

'''Half Life'''

'''Decay Rates'''

'''Measurement'''

'''Dating Game'''
|-
| | Show the '''Half Life''' screen.
| | We are already looking at the '''Half Life''' screen.
|-
| | Point to the graph at the top of the screen.

Point to units of time along the '''x-axis'''.
| | At the top of the screen is an '''Isotope versus Time''' graph.

Pay attention to the units of time.
|-
| | Show '''Choose Isotope''' panel to the right.

Point to the three options in '''Choose Isotope''' panel.
| | On the right side of the screen, you see a '''Choose Isotope''' panel.

It has three options showing unstable nucleus decaying to stable nucleus.
|-
| | Point to the '''Bucket o’ Atoms''' in the '''simulation''' panel.

Point to the '''C-14''' atoms in the bucket and to the default selection of '''C-14'''.

Point to the “'''Add 10'''” button attached to the bottom of the bucket.
| | In the middle is the '''simulation''' panel containing a '''Bucket o’ Atoms'''.

Note that it contains '''C-14''' atoms as the default selection is '''C-14'''.

Attached to the bottom of the bucket is a '''button''' called “'''Add 10'''”.
|-
| | '''Slide Number 7'''

'''Stable and Unstable Nuclei'''

Electrostatic repulsion between protons in nucleus

Strong nuclear force ~ binding energy

High binding energy; stable nucleus

Low binding energy; unstable nucleus ~ radioactive

| |

'''Stable and Unstable Nuclei'''

There is electrostatic repulsion between positively charged protons inside the nucleus.

The strong nuclear force overcomes this electrostatic repulsion between protons.

The energy associated with this force is the binding energy.

The lower the binding energy, the more unstable is the nucleus.

Such an unstable nucleus is said to be radioactive.
|-
| | Point to '''Play/Pause button''' and '''Step button''' next to it.
| | Below this '''simulation''' panel is a '''Play/Pause button''' and a '''Step button''' next to it.
|-
| | Point to blue '''Reset All Nuclei button''' in '''simulation''' panel.
| | In this '''simulation''' panel is a blue '''Reset All Nuclei button'''.

This '''button''' lets you return to the start but with the selected isotope.
|-
| | Point to the white '''Reset All button''' below the right panel.
| | Below the right panel is a white '''Reset All button'''.

This button resets the '''simulation''' in this '''screen''' to all the default settings.
|-
| | Point to the default '''Isotope''' selection of '''C-14 to N-14'''.
| | The default '''Isotope''' selection is '''C-14 to N-14'''.
|-
| | Point to the '''Isotope versus time''' graph.

Point to vertical red dashed line labeled '''Half Life''' near the 5000 year mark.
| | Observe the '''Isotope versus time''' graph.

There is a vertical red dashed line labeled '''Half Life''' near the 5000 year mark.

The '''half-life''' of '''C-14''' is 5730 years.
|-
| | Point to red '''C-14''' symbol above the blue '''N-14''' symbol along the '''y-axis'''.
| | Along the '''y-axis''', you can see the red '''C-14''' symbol above the blue '''N-14''' symbol.

'''C-14''' atoms will appear in the upper row and '''N-14''' atoms in the lower one.
|-
| | Point to the red circle to the left of the '''Isotope''' label.

Point to '''hash symbols''' on the left of the circle.
| | To the left of the '''Isotope''' label is a red circle.

Numbers of '''C-14''' and '''N-14''' atoms shown by '''hash symbols''' will appear to the left of the circle.
|-
| | '''Slide Number 8'''

'''Radioactive Decay'''

Alpha decay

Beta decay

Gamma decay

Half-life
| | '''Radioactive Decay '''is the spontaneous conversion of an unstable nucleus into a stable nucleus.

It involves the release of subatomic particles and their energy as radiation.

It is of the following types:

'''Alpha decay'''

'''Beta decay'''

'''Gamma decay'''

'''Half-life''' is the time taken for half of the nuclei in a radioactive material to decay.
|-
| |
| | Let us get back to the '''simulation'''.
|-
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
| | Click on '''Add 10''' and immediately click on the '''Pause button'''.
|-
| | Point to the 10 '''C-14''' atoms added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
| | Ten '''C-14''' atoms have been added to the '''simulation''' panel.

Almost immediately, red '''C-14''' has started to decay to give blue '''N-14'''.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
| | Keep clicking on '''Step button''' to the right of '''Pause'''.
|-
| | Show the red '''C-14''' atoms flying across the graph in the upper row.

Show the blue '''N-14''' atoms in the lower row.
| | Observe the red '''C-14''' atoms moving across the graph in the upper row.

In the lower row, blue '''N-14''' atoms appear as the '''C-14''' atoms decay.
|-
| | Point to the circle changing to blue.
| | Note how the circle changes to blue as more '''N-14''' atoms form.
|-
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red and half blue.
| | Keep clicking on '''Step button''' to the right of '''Pause''' until the circle is half red and half blue.
|-
| | Point to 5 blue '''N-14''' atoms on the left of the dashed half-life line.
| | Observe that, in the graph, there are 5 blue '''N-14''' atoms on the left of the dashed half-life line.

Out of 10 '''C-14''' atoms, half of them took 5730 years to decay into 5 '''N-14''' atoms.
|-
| | Point to 5 '''C-14''' and '''N-14''' atoms in the '''simulation''' panel.
| | Observe that there are 5 '''C-14''' atoms and 5 '''N-14''' atoms in the '''simulation''' panel also.

This is the definition of '''half-life'''.
|-
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
| | Click on '''Step button''' until you have 7 '''N-14 '''and 3 '''C-14''' atoms.
|-
| | Point to the 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
| | Another 5730 years are taken for the remaining 5 '''C-14''' atoms to decay to 2.5 '''N-14''' atoms.

We see 2 '''N-14''' atoms between the red dashed line and the 10000 mark.
|-
| | Point to '''Add 10''' '''button''' and to the bucket.
| | If you click again on '''Add 10''', another 10 '''C-14''' atoms will be added to the '''simulation''' panel.

In this way, you can keep clicking '''Add 10''' until the bucket is empty.
|-
| |
| | Predict the number of '''C-14''' atoms remaining after different periods.
|-
| | Perform the same '''simulation''' for the other nuclei.
| | Perform the same '''simulation''' for the other nuclei.
|-
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

Show the '''interface'''.
| | Click on the '''Decay Rates tab''' to go to that '''screen'''.

The '''interface''' has a similar arrangement as the '''Half Life screen'''.

Please explore this '''screen''' in the same way.
|-
| | Click on the '''Measurement tab'''.
| | Now, let us click on the '''Measurement tab''' to go to that '''screen'''.
|-
| | Point to '''Tree''', the default selection under '''Choose an Object''' on the right.
| | In the right panel, under '''Choose an Object''', we will stay with '''Tree''', the default selection.
|-
| | Point to '''Carbon-14''', and '''Objects''' under '''Probe Type''' in top left.
| | In the top left, under '''Probe Type''', we will retain the default selections, '''Carbon-14''', and '''Objects'''.
|-
| | Click on '''Plant Tree button''' in the bottom right corner.

Point to the tree growing right where the '''probe''' is placed.
| | Click on '''Plant Tree button''' in the bottom right corner.

Observe that a tree grows right where the '''probe''' is placed.
|-
| | Immediately click on the '''Pause button'''.
| | Immediately click on the '''Pause button'''.
|-
| | Point to 100% seen above '''Probe Type''' in the upper left corner.
| | Observe 100% appear above '''Probe Type''' in the upper left corner.
|-
| | Keep clicking on the '''Step button''' to the right of '''Pause''' to move the '''simulation''' along.
| | Keep clicking on the '''Step button''' to the right of '''Pause''' to move the '''simulation''' along.
|-
| | Point to red line moving at the top of the graph along 100%.
| | Note that a red line moves at the top of the graph along 100%.
|-
| | Show '''% of C-14''' above the graph.
| | Above the graph, '''% of C-14''' is the default selection.
|-
| | Point to the white box below the graph.
| | The white box below the graph shows the number of years since the tree was planted.
|-
| | Point to the red line at the top of the graph.
| | The red line shows % of '''C-14''' remaining in the tree after those years after it was planted.
|-
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the second '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
|-
| | Point to the red line at the top of the graph.
| | Now the red line shows the '''C-14''' to '''C-12''' ratio in the tree after those years of planting it
|-
| | Click again on the % of '''C-14 radio button''' above the graph.
| | Click again on the % of '''C-14 radio button''' above the graph.
|-
| | Point to % in the top left, the tree and the white box below the graph.
| | Keep track of the % in the top left, the tree and the number of years below the graph.
|-
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
| | Note down the number of years and % of '''C-14''' when the tree

Loses its green color

Loses all its leaves

Falls over
|-
| | Click on the '''Play button''' and let the '''simulation''' run until you see 55% above '''Probe Type'''.
| | Click on the '''Play button''' and let the '''simulation''' run until you see 55% above '''Probe Type'''.
|-
| | Keep clicking on the '''Step button''' until you see 50% in the top left window.
| | Keep clicking on the '''Step button''' until you see 50% in the top left window.
|-
| | Point to the tree and the number of years in the white box.
| | Observe the tree and the number of years seen below the graph.
|-
| | Point to 50% on '''y-axis''', to red line and to '''x co-ordinate'''.
| | Note the number of years after which you see 50% of '''C-14''' in the tree.
|-
| | Click on the '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.
| | Click on the '''C-14''' to '''C-12''' ratio '''radio button''' above the graph.

Observe that this ratio has also dropped to half of what it was when the tree was planted.
|-
| | '''Slide Number 9'''

'''Other Measurements'''

Select '''Rock''' and '''Uranium-238'''

'''Objects''' selection

'''Probe''' → volcano → '''Erupt Volcano'''

When do you see 50% '''U-238''' in cooled volcanic rock?

'''Half-life''' of '''U-238''' is 4.5 billion years

Air '''radio button''' ('''probe''' in air)
| |

Choose the options shown here to carry out other measurements.
|-
| | Click on the last '''Dating Game tab'''.

Show the '''interface'''.
| | Let us click on the last '''Dating Game tab''' to go to that '''screen'''.

We can measure the levels of '''C-14, U-238''' or other '''custom nuclei''' in this '''screen'''.

We see objects on and below the ground on which we can place the probe to measure these levels.
|-
| | '''Slide Number 10'''

'''Radioactive Dating'''

Two isotopes of C: 12C and 14C

Both isotopes → CO2, living organisms

14C → 14N, 14C:12C organisms ~ atmosphere
| |

'''Radioactive Dating'''

Carbon has two isotopes: '''C-12''' and '''C-14'''.

Both are converted to carbon dioxide and are taken in by living organisms.

'''C-14''' decays to '''N-14''' but the ratio of '''C-14''' to '''C-12''' remains same in organisms and the atmosphere.
|-
| | '''Slide Number 11'''

'''Radioactive Dating-Cont’d'''

Death of organism, ratio and 14C fall

Radioactive dating, 14C:12C of sample vs recently dead specimens

Ur-Pb dating for rocks, artefacts etc
| |

'''Radioactive Dating-Continued'''

When an organism dies, it no longer takes in any carbon.

So levels of '''C-14''' and ratio of '''C-14''' to '''C-12''' fall.

Radioactive dating compares C-14 C-12 ratio of sample to recently dead specimens.

It estimates how long the organism has been dead.

'''Uranium-lead dating''' is used for rocks, archaeological artefacts etc
|-
| | Point to the graph on the top.

Point to the two '''radio buttons, % of C-14''' and '''C-14 to C-12''' ratio, for the '''y-axis'''.

Point to the '''x-axis'''.

Point to the vertical red dashed line.
| | On the top, we see the graph.

As before, we have two '''radio buttons, % of C-14''' and '''C-14 to C-12''' ratio, for the '''y-axis'''.

'''x-axis''' gives time in years.

The vertical red dashed line is the '''half-life'''.
|-
| | Point to these default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
| | Let us keep the following default selections:

Under '''Probe Type''', '''Carbon-14'''

'''Objects'''

'''% of C-14'''
|-
| | Drag the '''probe''' and place it on the animal skull on the ground, to the left.
| | We will drag the '''probe''' and place it on the animal skull on the ground, to the left.
|-
| | Point to the '''pop-up box''' next to the skull.

Show the '''text''', “'''Estimate age of Animal Skull'''”.

Show the empty box and “'''yrs'''” next to it.
| | Observe a '''pop-up box''' that appears next to the skull.

We see the '''text''', “'''Estimate age of Animal Skull'''” and below that an empty box and “'''yrs'''” next to it.
|-
| | Point to the '''Check Estimate button'''.
| | Below this is a '''Check Estimate button'''.
|-
| | Show 98.2% in the top left side, above '''Probe Type'''.
| | Observe that in the top left side, above '''Probe Type''', we see 98.2%.
|-
| | Drag the double-headed green arrow to the left.

Show 98.2% in the white box above the arrow.
| | Let us drag the double-headed green arrow above the graph.

In the white box above the arrow, % of '''C-14''' should be approximately 98.2%.
|-
| | Show '''t = 123 yrs''' in the white box below % of '''C-14'''.
| | Observe that it '''equals 123 yrs''' appears in the white box below % of '''C-14'''.

So after 123 years, approximately 98.2% '''C-14''' is left in the animal skull.
|-
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
| | Type 123 in the empty box below '''Estimate age of Animal Skull'''.

Click '''Check Estimate button'''.
|-
| | Point to green text-box with 123 years in its place with a green '''smiley''' face next to it.
| | The '''Estimate pop-up box''' disappears.

A green text-box with 123 years appears in its place with a green '''smiley''' face next to it.
|-
| |
| | We have successfully dated the animal skull by measuring the % of '''C-14''' remaining in it.
|-
| | '''Slide Number 12'''

'''Assignment'''

Estimate ages of all objects in '''Dating Game screen'''

Correlate age (years) with percentage of '''unstable''' nucleus

Correlate age (years) with depth at which object found

'''C-14''': animal remains; '''U-238''': rocks, objects
| |

As an '''assignment''',

Estimate ages of all the objects in the '''Dating Game screen'''.

Correlate age in years with the percentage of '''unstable''' nucleus.

Correlate age in years with the depth at which the object is found.

Remember to use '''C-14''' for animal remains and '''U-238''' for rocks and other objects.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 13'''

'''Summary'''

We have demonstrated,

'''Radioactive Dating Game PhET simulation'''
| |

In this '''tutorial''', we have demonstrated how to use the '''Radioactive Dating Game PhET simulation'''.
|-
| | '''Slide Number 14'''

'''Summary'''

Radioactive decay and half life

Decay rates

Measurement of radioactivity

Radioactive dating
| |

Using this '''simulation''', we looked at:

'''Radioactive decay''' and '''half life'''

Decay rates

Measurement of radioactivity

'''Radioactive dating'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Differentiation-using-GeoGebra/English

2018-10-08T09:37:16Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Differentiation using GeoGebra'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn how to use '''GeoGebra''' to:

Understand Differentiation

Draw graphs of derivative of functions

|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d

|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Differentiation

For relevant '''tutorials''', please visit our website.

|-
| | '''Slide Number 5'''

'''Differentiation: First Principles'''

[[Image:]]

'''f(x) = x2-x'''

'''f'(x)''' is derivative of '''f(x)'''

'''A (x, f(x)), B (x+j, f(x+j))'''
| |

Let us understand differentiation using '''first principles''' for the '''function f of x'''.

'''f of x''' is equal to '''x squared''' minus '''x'''

'''f prime x''' is the derivative of '''f of x'''.

Consider 2 points, '''A''' and '''B'''.

'''A''' is '''x''' comma '''f of x''' and '''B''' is '''x''' plus '''j''' comma '''f of x''' plus '''j'''
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Type '''f(x)=x^2-x''' in the '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line and press '''Enter'''.

For the '''caret symbol''', hold the '''Shift''' key down and press 6.

'''f x''' in parentheses equals '''x caret 2''' minus '''x'''
|-
| | Point to the equation in '''Algebra''' view.

Point to '''parabola''' in '''Graphics''' view.
| | The equation appears in the '''Algebra''' view and the '''function f''' is graphed as a '''parabola'''.
|-
| | Point to '''parabola''' in '''Graphics''' view.
| | It opens upwards and intersects the '''x-axis''' at two points.
|-
| | Click on '''Point on Object''' tool >> click on the parabola at '''(2,2)'''.

Point to '''A''' at '''(2,2)'''.
| | Under '''Point''', click on '''Point on Object''' and click on the parabola at 2 comma 2.

This creates point '''A'''.
|-
| | Click on '''Point''' tool and click on '''(3,6)'''.
| | Create a point '''B''' at 3 comma 6.
|-
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
|-
| | Click on the '''Move''' tool.

Double click on the resulting '''line g''' and click on '''Object Properties'''.

Click on '''Color''' tab and select blue.

Click on '''Style''' tab and select '''dashed style'''.
| | Let us make this line '''g '''blue and dashed.
|-
| | Click on '''Tangents''' tool under '''Perpendicular Line''' tool.
| | Under '''Perpendicular Line''', click on '''Tangents'''.
|-
| | Click on '''A''' and then on the '''parabola'''.
| | Then click on '''A''' and then on the '''parabola'''.
|-
| | Point to '''tangent h''' at point '''A''' to the '''parabola'''.
| | This draws a '''tangent h''' at point '''A''' to the '''parabola'''.
|-
| | Double click on '''tangent h''' and click on '''Object Properties'''.

Under '''Color''' tab, select red.

Close the '''Preferences''' box.
| | Let us make '''tangent h''' a red line.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.
| | Click on the '''Point''' tool and click anywhere in '''Graphics''' view.
|-
| | Point to point '''C'''.
| | This creates point '''C'''.
|-
| | Double click on point '''C''' in '''Algebra''' view and change its '''coordinates''' to '''(x(B),y(A))'''.
| | In '''Algebra''' view, double-click on '''C''' and change its '''coordinates''' to the following ones.

In parentheses, '''x B''' in parentheses comma '''y A''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''C'''.
| | Now C has the same '''x coordinate''' as point '''B''' and the same '''y coordinate''' as point '''A'''.
|-
| |
| | Let us draw segments '''BC''' and '''AC.'''
|-
| | Under '''Line''', click on '''Segment''' and click on '''B '''and '''C''', and then on '''A''' and '''C'''.
| | Under '''Line''', click on '''Segment''' and click on '''B''' and '''C'''.

Then, click on '''A''' and '''C''' to draw '''AC'''.

Note that '''BC''' and '''AC''' are called '''i''' and '''j''' in the order of their creation.
|-
| | Right-click on '''AC''' >> Select '''Object Properties''' >> '''Color''' tab >> Purple

Click on '''Style''' tab >> select dashed line
| | We will make '''AC''' and '''BC''' purple and dashed segments.
|-
| | Under '''Basic''' tab >> choose '''Name and Value''' >> '''Show Label''' check box.
| | Under '''Basic''' tab, choose '''Name and Value''' from the dropdown menu next to the '''Show Label''' check box.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| | Click on '''Move''' and drag '''B''' towards '''A''' on the '''parabola'''.
| | Click on '''Move''' and drag '''B''' towards '''A''' on the '''parabola'''.
|-
| | Point to the value of '''j''' (length of '''AC''') and lines '''g''' and '''h'''.
| | Observe lines '''g''' and '''h''' and the value of '''j''' (length of '''AC''').

As '''j''' approaches 0, points '''B'''and '''A''' begin to overlap.

Lines '''g''' and '''h''' also begin to overlap.
|-
| | Point to line '''g''', '''BC''' and '''AC'''.
| | Slope of line '''g''' is the ratio of length of '''BC''' to length of '''AC'''.
|-
| | Point to all the points on the parabola.
| | Derivative of the parabola is the slopes of tangents at all points on curve.
|-
| | Point to the lines '''g''' and '''h'''.
| | When '''j''' equals 0, line '''g''' through '''A''' and '''B''' coincides with the tangent line '''h''' at '''A'''.

But slope of '''g''' is undefined as the denominator of the ratio of '''i''' to '''j''' is 0.
|-
| | Point to text-box that appears in '''GeoGebra''' window.

As '''B''' approaches '''A''', slope '''AB''' approaches slope of tangent at '''A'''.
| | As '''B''' approaches '''A''' on '''f of x''', slope of '''AB''' approaches the slope of tangent at '''A'''.
|-
| |
| | Now let us look at the '''Algebra''' behind these concepts.
|-
| | '''Slide Number 7'''

'''Differentiation: First Principles, the Algebra'''

'''f'(x) = lim_(j→0) length of Segment BC/length of Segment AC'''

.......''' = lim_(j→0) [(f(x+j) – f(x)]/[(x+j) – x]'''

Remember '''f(x) = x2-x, (x+j)2 = x2+2xj+j2'''

'''f'(x) = lim_(j→0) [(x+j)2-(x+j)-(x2-x)]/[(x+j-x]'''

| | Slope of line '''AB''' equals the ratio of the lengths of '''BC''' to '''AC'''.

Line '''AB''' becomes the tangent at point '''A''' as distance '''j''' between '''A''' and '''B''' approaches 0.

'''BC''' is the difference between '''y' coordinates''', '''f of x''' plus '''j''' and '''f of x''', for '''A''' and '''B'''.

'''AC''' is the difference between the '''x-coordinates''', '''x''' plus '''j''' and '''x'''.

Let us rewrite '''f of x''' plus '''j''' and '''f of x''' in terms of '''x squared''' minus '''x'''.

We will expand the terms in the numerator.
|-
| | '''Slide 8 Differentiation: First Principles—the Algebra-Cont’d'''

'''f'(x) = lim_(j→0) [x2+2xj+j2-x-j-x2+x]/j'''

..............'''= lim_(j→0) [2xj+j2-j]/j = lim_(j→0) [j(2x+j-1)]/j'''

..................'''= lim_(j→0) [2x+j-1] = 2x-1'''

'''f'(x2-x) = 2x - 1'''
| | After expanding the terms in the numerator, we will cancel out similar terms with opposite signs.

As '''j''' is a common factor in the numerator, we will pull it out.

Now, we can cancel '''j''' from both the numerator and denominator.

Note that as '''j''' approaches 0, '''j''' can be ignored so that '''2x''' plus '''j''' minus 1 approaches '''2x''' minus 1.

As we know, derivative of '''x squared''' minus '''x''' is '''2x''' minus 1.

Thus, the derivative of a '''function''' is the slope of the tangent at a point on the function.
|-
| |
| | Let us look at derivative graphs for some '''functions'''.
|-
| | '''Slide Number 10'''

'''Differentiation of a Polynomial Function'''

Consider '''g(x)=5+12x-x3'''

'''d(5+12x-x3)/dx = d(5)/dx + d(12x)/dx - d(x3)/dx = 0 + 12 - 3x 2 = -3x 2 +12'''

For '''g(x)=5+12x-x3, g'(x) = -3x2 +12'''
| |
Consider '''g of x''' equals 5 plus '''12 x '''minus '''x cubed'''.

Derivative '''g prime x''' is the sum and difference of derivatives of the individual components.

'''g prime x''' equals 5 plus '''12 x''' minus '''x cubed''' is calculated by applying these rules.
|-
| |
| | Let us differentiate '''g of x''' in '''GeoGebra'''.
|-
| | Open a new '''GeoGebra''' window.
| | Open a new '''GeoGebra''' window.
|-
| | Type '''g(x)=5+12x-x^3''' in '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line and press '''Enter'''.

'''g x''' in parentheses equals 5 plus '''12 x''' minus '''x caret''' 3
|-
| | Right-click in '''Graphics''' view and select '''xAxis : yAxis''' option.

Select '''1:5'''.
| | Right-click in '''Graphics''' view and select '''xAxis''' is to '''yAxis''' option.

Select 1 is to 5.
|-
| | Click on '''Point on Object''' tool and click on the curve to create point '''A'''.
| | Click on '''Point on Object''' tool and click on curve '''g of x''' to create point '''A'''.
|-
| | Click on '''Tangent''' under '''Perpendicular Line'''.

Click on point '''A''' and the curve.
| | Under '''Perpendicular Line''', click on '''Tangent'''.

Now click on point '''A''' and the curve '''g of x'''.
|-
| | Point to tangent line '''f''' to the curve at point '''A'''.
| | This draws a tangent line '''f''' to the curve at point '''A'''.
|-
| | Click on '''Slope''' tool under '''Angle''' tool and on tangent line '''f'''.
| | Under '''Angle''', click on '''Slope''' and on tangent line '''f'''.
|-
| | Point to slope of '''line f''' at '''A''' appearing as '''m''' value in '''Graphics''' view.
| | Slope of line '''f'''at '''A''' appears as '''m''' value in '''Graphics''' view.
|-
| | Click on '''Point''' tool and in '''Graphics''' view to create point '''B'''.
| | Click on '''Point''' tool and click in '''Graphics''' view to create point '''B'''.
|-
| | Double click on point '''B''' in '''Algebra''' view and change '''coordinates''' to ('''x(A), m)'''.
| | In '''Algebra''' view, double click on '''B''' and change its '''coordinates''' to '''x A''' in parentheses comma '''m'''.

Press '''Enter'''.
|-
| | Point to points '''A''' and '''B''' and slope ''' m''' of tangent line '''g'''.
| | This makes '''x coordinate''' of '''B''' the same as that of '''A'''.

And slope '''m''' of tangent line '''f''' is its '''y coordinate'''.
|-
| | Right-click on point '''B''' and select '''Trace On''' option.
| | Right-click on point '''B''' and select '''Trace On''' option
|-
| | Click on '''Move''' tool and move point '''A''' on curve.

Observe the curve traced by point '''B'''.
| | Click on '''Move''' tool and move point '''A''' on the curve.

Observe the curve traced by point '''B'''.
|-
| | Type '''Deri''' in '''input bar''' >> select '''Derivative( <Function> )''' >> Type '''g''' instead of highlighted '''<Function>''' >> press '''Enter'''
| | In the '''input bar''', type '''capital D e r i'''.

From the menu that appears, select '''Derivative Function''' option.

Type '''g''' to replace the highlighted word '''<Function>'''.

Press '''Enter'''.
|-
| | Point to the derivative graph.
| | This will confirm that you have the correct derivative graph.
|-
| | Point to the equation of '''g prime x''' in '''Algebra''' view.
| | Note the equation of '''g prime x''' in '''Algebra''' view.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Compare slide’s calculations with equation of '''g'(x)''' in '''Algebra''' view.
| | Compare the calculations in the previous slide with the equation of '''g prime x'''
|-
| |
| | Let us use differentiation to find the maxima and minima of a '''function'''.

Let us look at '''g of x'''.
|-
| | Point to derivative curve '''g'(x)''' above the '''x-axis''' and to '''g(x)'''.
| | Derivative curve '''g prime x''' remains above the '''x-axis''' (is positive) as long as '''g of x''' is increasing.
|-
| | Point to derivative curve '''g'(x)''' below the '''x-axis''' and to '''g(x)'''.
| | '''g prime x''' remains below the '''x-axis''' (is negative) as long as '''g of x''' is decreasing.
|-
| | Point to derivative curve '''g'(x)''' intersecting '''x-axis''' at '''x = -2 '''and''' x = 2'''.
| | 2 and -2 are the values of '''x''' when '''g prime x''' equals 0.
|-
| | Point to the "peak" and "valley" on '''g of x'''.
| | Slope of the tangents at the corresponding points on '''g of x''' is 0.

These points on '''g of x''' are maxima or a minima.
|-
| |

Point to '''(-2,-11)''' and '''(2,21)'''.
| | Hence, for '''g of x,''' -2 comma -11 is the minimum and 2 comma 21 is the maximum.
|-
| | Point to minimum of '''g(x)''' and '''x=-3''' and '''x = -1'''.
| | In '''GeoGebra''', we can see that the minimum value of '''g of x''' lies between '''x''' equals -3 and '''x''' equals -1.
|-
| | In the '''input bar''', type '''Min''' with capital M.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type -3.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type -1.

Press '''Enter'''.
| | In the '''input bar''', type '''Min''' with capital M.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type -3.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type -1.

Press '''Enter'''.
|-
| | Point to minimum '''C''' in '''Graphics '''view and its '''co-ordinates''' in '''Algebra''' view.
| | In '''Graphics''' view, we see the minimum on '''g of x'''.

Its '''co-ordinates''' are -2 comma -11 in '''Algebra''' view.
|-
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type 1.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type 4.

Press '''Enter.'''
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type 1.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type 4.

Press '''Enter'''.
|-
| | Point to maximum '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
| | In '''Graphics''' view, we see the maximum on '''g of x'''.

Its '''co-ordinates''' are 2 comma 21 in '''Algebra''' view.
|-
| |
| | Finally, let us take a look at a practical application of differentiation.
|-
| | '''Slide Number 16'''

'''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard

We have to convert it into a box

Squares have to be cut from the four corners

What size squares should we cut out to get the maximum volume of the box?
| | '''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard.

We have to convert it into a box.

Squares have to be cut from the four corners.

What size squares should we cut out to get the maximum volume of the box?
|-
| | '''Slide Number 17'''

'''A Sketch of the Cardboard'''

Let’s draw the cardboard:

[[Image:]]

The volume function here is '''(24-2x)*(15-2x)*x''' cubic inches.
| | '''A Sketch of the Cardboard'''

Let us draw the cardboard:

This is the volume '''function''' here.

You could expand it into a '''cubic polynomial'''<nowiki>; but we will leave it as it is. </nowiki>
|-
| | Open a new '''GeoGebra''' window.
| | Open a new '''GeoGebra''' window.
|-
| | Type '''(24-2 x) (15-2 x) x''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.

In parentheses, 24 minus 2 space '''x''' space in parentheses 15 minus 2 space '''x''' space '''x'''
|-
| | Drag the boundary to see the equation properly in '''Algebra''' view.
| | Drag the boundary to see the equation properly in '''Algebra''' view.
|-
| | Right-click in '''Graphics''' view and set '''xAxis : yAxis''' to '''1:50'''.
| | Right-click in '''Graphics''' view and set '''xAxis''' is to '''yAxis''' to 1 is to 50.
|-
| | Point to the graph for this volume '''function''' in '''Graphics''' view.

Click in and drag the background to move '''Graphics''' view to see the maximum.
| | Observe the graph that is plotted for this volume '''function''' in '''Graphics''' view.

Click in and drag the background to move '''Graphics''' view to see the maximum.
|-
| | Point to the maximum on top of the broad peak and to '''x''' = 0 and '''x''' = 7.
| | Note that the maximum is on the top of a broad peak from '''x''' equals 0 to '''x''' equals 7.
|-
| | Point to both axes.
| | The length of the square side is plotted along the '''x-axis'''.

Volume of the box is plotted along the '''y-axis'''.
|-
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-value'''.

Instead of highlighted '''Function''', type '''f'''.

Press '''Tab''' to move and highlight '''Start x-Value''' and type 0.

Again, press '''Tab''' to move and highlight '''End x-Value''' and type 10.

Press '''Enter'''.
| | As before, let us find the maximum of this '''function'''.
|-
| | Point to the maximum,''' A''', in '''Graphics''' view and its '''coordinates''' in '''Algebra''' view.
| | This maps the maximum, point '''A''', on the curve.

Click in and drag the background to see

Its '''coordinates''' 3 comma 486 appear in '''Algebra''' view.

Thus, we have to cut out 3 inches squares from all corners.

This will give the maximum possible volume of 486 cubic inches for the cardboard box.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 19'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand differentiation

Draw graphs of derivatives of '''functions'''
|-
| | '''Slide Number 16'''

'''Assignment'''

Draw graphs of derivatives of the following functions in '''GeoGebra''':

'''h(x)=ex'''

'''i(x)=ln(x)'''

'''j(x)=(5x3+3x-1)/(x-1)'''

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
| | As an assignment:

Draw graphs of derivatives of the following functions in '''GeoGebra'''.

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
|-
| | '''Slide Number 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

Conducts workshops using spoken tutorials and
Gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 20'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from''' IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Differentiation-using-GeoGebra/English

2018-10-08T09:32:55Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Differentiation using GeoGebra'''. |- |..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Differentiation using GeoGebra'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn how to use '''GeoGebra''' to:

Understand Differentiation

Draw graphs of derivative of functions

|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d

|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Differentiation

For relevant '''tutorials''', please visit our website.

|-
| | '''Slide Number 5'''

'''Differentiation: First Principles'''

[[Image:]]

'''f(x) = x2-x'''

'''f'(x)''' is derivative of '''f(x)'''

'''A (x, f(x)), B (x+j, f(x+j))'''
| |

Let us understand differentiation using '''first principles''' for the '''function f of x'''.

'''f of x''' is equal to '''x squared''' minus '''x'''

'''f prime x''' is the derivative of '''f of x'''.

Consider 2 points, '''A''' and '''B'''.

'''A''' is '''x''' comma '''f of x''' and '''B''' is '''x''' plus '''j''' comma '''f of x''' plus '''j'''
|-
| | Show the '''GeoGebra''' window.
| | I have opened the '''GeoGebra''' interface.
|-
| | Type '''f(x)=x^2-x''' in the '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line and press '''Enter'''.

For the '''caret symbol''', hold the '''Shift''' key down and press 6.

'''f x''' in parentheses equals '''x caret 2''' minus '''x'''
|-
| | Point to the equation in '''Algebra''' view.

Point to '''parabola''' in '''Graphics''' view.
| | The equation appears in the '''Algebra''' view and the '''function f''' is graphed as a '''parabola'''.
|-
| | Point to '''parabola''' in '''Graphics''' view.
| | It opens upwards and intersects the '''x-axis''' at two points.
|-
| | Click on '''Point on Object''' tool >> click on the parabola at '''(2,2)'''.

Point to '''A''' at '''(2,2)'''.
| | Under '''Point''', click on '''Point on Object''' and click on the parabola at 2 comma 2.

This creates point '''A'''.
|-
| | Click on '''Point''' tool and click on '''(3,6)'''.
| | Create a point '''B''' at 3 comma 6.
|-
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
| | Click on '''Line''' tool and click on points '''B''' and '''A'''.
|-
| | Click on the '''Move''' tool.

Double click on the resulting '''line g''' and click on '''Object Properties'''.

Click on '''Color''' tab and select blue.

Click on '''Style''' tab and select '''dashed style'''.
| | Let us make this line '''g '''blue and dashed.
|-
| | Click on '''Tangents''' tool under '''Perpendicular Line''' tool.
| | Under '''Perpendicular Line''', click on '''Tangents'''.
|-
| | Click on '''A''' and then on the '''parabola'''.
| | Then click on '''A''' and then on the '''parabola'''.
|-
| | Point to '''tangent h''' at point '''A''' to the '''parabola'''.
| | This draws a '''tangent h''' at point '''A''' to the '''parabola'''.
|-
| | Double click on '''tangent h''' and click on '''Object Properties'''.

Under '''Color''' tab, select red.

Close the '''Preferences''' box.
| | Let us make '''tangent h''' a red line.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.
| | Click on the '''Point''' tool and click anywhere in '''Graphics''' view.
|-
| | Point to point '''C'''.
| | This creates point '''C'''.
|-
| | Double click on point '''C''' in '''Algebra''' view and change its '''coordinates''' to '''(x(B),y(A))'''.
| | In '''Algebra''' view, double-click on '''C''' and change its '''coordinates''' to the following ones.

In parentheses, '''x B''' in parentheses comma '''y A''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''C'''.
| | Now C has the same '''x coordinate''' as point '''B''' and the same '''y coordinate''' as point '''A'''.
|-
| |
| | Let us draw segments '''BC''' and '''AC.'''
|-
| | Under '''Line''', click on '''Segment''' and click on '''B '''and '''C''', and then on '''A''' and '''C'''.
| | Under '''Line''', click on '''Segment''' and click on '''B''' and '''C'''.

Then, click on '''A''' and '''C''' to draw '''AC'''.

Note that '''BC''' and '''AC''' are called '''i''' and '''j''' in the order of their creation.
|-
| | Right-click on '''AC''' >> Select '''Object Properties''' >> '''Color''' tab >> Purple

Click on '''Style''' tab >> select dashed line
| | We will make '''AC''' and '''BC''' purple and dashed segments.
|-
| | Under '''Basic''' tab >> choose '''Name and Value''' >> '''Show Label''' check box.
| | Under '''Basic''' tab, choose '''Name and Value''' from the dropdown menu next to the '''Show Label''' check box.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| | Click on '''Move''' and drag '''B''' towards '''A''' on the '''parabola'''.
| | Click on '''Move''' and drag '''B''' towards '''A''' on the '''parabola'''.
|-
| | Point to the value of '''j''' (length of '''AC''') and lines '''g''' and '''h'''.
| | Observe lines '''g''' and '''h''' and the value of '''j''' (length of '''AC''').

As '''j''' approaches 0, points '''B'''and '''A''' begin to overlap.

Lines '''g''' and '''h''' also begin to overlap.
|-
| | Point to line '''g''', '''BC''' and '''AC'''.
| | Slope of line '''g''' is the ratio of length of '''BC''' to length of '''AC'''.
|-
| | Point to all the points on the parabola.
| | Derivative of the parabola is the slopes of tangents at all points on curve.
|-
| | Point to the lines '''g''' and '''h'''.
| | When '''j''' equals 0, line '''g''' through '''A''' and '''B''' coincides with the tangent line '''h''' at '''A'''.

But slope of '''g''' is undefined as the denominator of the ratio of '''i''' to '''j''' is 0.
|-
| | Point to text-box that appears in '''GeoGebra''' window.

As '''B''' approaches '''A''', slope '''AB''' approaches slope of tangent at '''A'''.
| | As '''B''' approaches '''A''' on '''f of x''', slope of '''AB''' approaches the slope of tangent at '''A'''.
|-
| |
| | Now let us look at the '''Algebra''' behind these concepts.
|-
| | '''Slide Number 7'''

'''Differentiation: First Principles, the Algebra'''

'''f'(x) = lim_(j→0) length of Segment BC/length of Segment AC'''

.......''' = lim_(j→0) [(f(x+j) – f(x)]/[(x+j) – x]'''

Remember '''f(x) = x2-x, (x+j)2 = x2+2xj+j2'''

'''f'(x) = lim_(j→0) [(x+j)2-(x+j)-(x2-x)]/[(x+j-x]'''

| | Slope of line '''AB''' equals the ratio of the lengths of '''BC''' to '''AC'''.

Line '''AB''' becomes the tangent at point '''A''' as distance '''j''' between '''A''' and '''B''' approaches 0.

'''BC''' is the difference between '''y' coordinates''', '''f of x''' plus '''j''' and '''f of x''', for '''A''' and '''B'''.

'''AC''' is the difference between the '''x-coordinates''', '''x''' plus '''j''' and '''x'''.

Let us rewrite '''f of x''' plus '''j''' and '''f of x''' in terms of '''x squared''' minus '''x'''.

We will expand the terms in the numerator.
|-
| | '''Slide 8 Differentiation: First Principles—the Algebra-Cont’d'''

'''f'(x) = lim_(j→0) [x2+2xj+j2-x-j-x2+x]/j'''

..............'''= lim_(j→0) [2xj+j2-j]/j = lim_(j→0) [j(2x+j-1)]/j'''

..................'''= lim_(j→0) [2x+j-1] = 2x-1'''

'''f'(x2-x) = 2x - 1'''
| | After expanding the terms in the numerator, we will cancel out similar terms with opposite signs.

As '''j''' is a common factor in the numerator, we will pull it out.

Now, we can cancel '''j''' from both the numerator and denominator.

Note that as '''j''' approaches 0, '''j''' can be ignored so that '''2x''' plus '''j''' minus 1 approaches '''2x''' minus 1.

As we know, derivative of '''x squared''' minus '''x''' is '''2x''' minus 1.

Thus, the derivative of a '''function''' is the slope of the tangent at a point on the function.
|-
| |
| | Let us look at derivative graphs for some '''functions'''.
|-
| | '''Slide Number 10'''

'''Differentiation of a Polynomial Function'''

Consider '''g(x)=5+12x-x3'''

'''d(5+12x-x3)/dx = d(5)/dx + d(12x)/dx - d(x3)/dx = 0 + 12 - 3x 2 = -3x 2 +12'''

For '''g(x)=5+12x-x3, g'(x) = -3x2 +12'''
| |
Consider '''g of x''' equals 5 plus '''12 x '''minus '''x cubed'''.

Derivative '''g prime x''' is the sum and difference of derivatives of the individual components.

'''g prime x''' equals 5 plus '''12 x''' minus '''x cubed''' is calculated by applying these rules.
|-
| |
| | Let us differentiate '''g of x''' in '''GeoGebra'''.
|-
| | Open a new '''GeoGebra''' window.
| | Open a new '''GeoGebra''' window.
|-
| | Type '''g(x)=5+12x-x^3''' in '''input bar''' >> '''Enter'''
| | In the '''input bar''', type the following line and press '''Enter'''.

'''g x''' in parentheses equals 5 plus '''12 x''' minus '''x caret''' 3
|-
| | Right-click in '''Graphics''' view and select '''xAxis : yAxis''' option.

Select '''1:5'''.
| | Right-click in '''Graphics''' view and select '''xAxis''' is to '''yAxis''' option.

Select 1 is to 5.
|-
| | Click on '''Point on Object''' tool and click on the curve to create point '''A'''.
| | Click on '''Point on Object''' tool and click on curve '''g of x''' to create point '''A'''.
|-
| | Click on '''Tangent''' under '''Perpendicular Line'''.

Click on point '''A''' and the curve.
| | Under '''Perpendicular Line''', click on '''Tangent'''.

Now click on point '''A''' and the curve '''g of x'''.
|-
| | Point to tangent line '''f''' to the curve at point '''A'''.
| | This draws a tangent line '''f''' to the curve at point '''A'''.
|-
| | Click on '''Slope''' tool under '''Angle''' tool and on tangent line '''f'''.
| | Under '''Angle''', click on '''Slope''' and on tangent line '''f'''.
|-
| | Point to slope of '''line f''' at '''A''' appearing as '''m''' value in '''Graphics''' view.
| | Slope of line '''f'''at '''A''' appears as '''m''' value in '''Graphics''' view.
|-
| | Click on '''Point''' tool and in '''Graphics''' view to create point '''B'''.
| | Click on '''Point''' tool and click in '''Graphics''' view to create point '''B'''.
|-
| | Double click on point '''B''' in '''Algebra''' view and change '''coordinates''' to ('''x(A), m)'''.
| | In '''Algebra''' view, double click on '''B''' and change its '''coordinates''' to '''x A''' in parentheses comma '''m'''.

Press '''Enter'''.
|-
| | Point to points '''A''' and '''B''' and slope ''' m''' of tangent line '''g'''.
| | This makes '''x coordinate''' of '''B''' the same as that of '''A'''.

And slope '''m''' of tangent line '''f''' is its '''y coordinate'''.
|-
| | Right-click on point '''B''' and select '''Trace On''' option.
| | Right-click on point '''B''' and select '''Trace On''' option
|-
| | Click on '''Move''' tool and move point '''A''' on curve.

Observe the curve traced by point '''B'''.
| | Click on '''Move''' tool and move point '''A''' on the curve.

Observe the curve traced by point '''B'''.
|-
| | Type '''Deri''' in '''input bar''' >> select '''Derivative( <Function> )''' >> Type '''g''' instead of highlighted '''<Function>''' >> press '''Enter'''
| | In the '''input bar''', type '''capital D e r i'''.

From the menu that appears, select '''Derivative Function''' option.

Type '''g''' to replace the highlighted word '''<Function>'''.

Press '''Enter'''.
|-
| | Point to the derivative graph.
| | This will confirm that you have the correct derivative graph.
|-
| | Point to the equation of '''g prime x''' in '''Algebra''' view.
| | Note the equation of '''g prime x''' in '''Algebra''' view.
|-
| | Drag the boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Compare slide’s calculations with equation of '''g'(x)''' in '''Algebra''' view.
| | Compare the calculations in the previous slide with the equation of '''g prime x'''
|-
| |
| | Let us use differentiation to find the maxima and minima of a '''function'''.

Let us look at '''g of x'''.
|-
| | Point to derivative curve '''g'(x)''' above the '''x-axis''' and to '''g(x)'''.
| | Derivative curve '''g prime x''' remains above the '''x-axis''' (is positive) as long as '''g of x''' is increasing.
|-
| | Point to derivative curve '''g'(x)''' below the '''x-axis''' and to '''g(x)'''.
| | '''g prime x''' remains below the '''x-axis''' (is negative) as long as '''g of x''' is decreasing.
|-
| | Point to derivative curve '''g'(x)''' intersecting '''x-axis''' at '''x = -2 '''and''' x = 2'''.
| | 2 and -2 are the values of '''x''' when '''g prime x''' equals 0.
|-
| |
| | Slope of the tangents at the corresponding points on '''g of x''' is 0.

These points on '''g of x''' are maxima or a minima.
|-
| |

Point to '''(-2,-11)''' and '''(2,21)'''.
| | Hence, for '''g of x,''' -2 comma -11 is the minimum and 2 comma 21 is the maximum.
|-
| | Point to minimum of '''g(x)''' and '''x=-3''' and '''x = -1'''.
| | In '''GeoGebra''', we can see that the minimum value of '''g of x''' lies between '''x''' equals -3 and '''x''' equals -1.
|-
| | In the '''input bar''', type '''Min''' with capital M.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type -3.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type -1.

Press '''Enter'''.
| | In the '''input bar''', type '''Min''' with capital M.

From the menu that appears, select '''Min Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type -3.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type -1.

Press '''Enter'''.
|-
| | Point to minimum '''C''' in '''Graphics '''view and its '''co-ordinates''' in '''Algebra''' view.
| | In '''Graphics''' view, we see the minimum on '''g of x'''.

Its '''co-ordinates''' are -2 comma -11 in '''Algebra''' view.
|-
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type 1.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type 4.

Press '''Enter.'''
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-Value''' option.

Type '''g''' instead of the highlighted word '''Function'''.

Press '''Tab''' to move to and highlight '''Start x-Value''' and type 1.

Again, press '''Tab''' to move to and highlight '''End x-Value''' and type 4.

Press '''Enter'''.
|-
| | Point to maximum '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
| | In '''Graphics''' view, we see the maximum on '''g of x'''.

Its '''co-ordinates''' are 2 comma 21 in '''Algebra''' view.
|-
| |
| | Finally, let us take a look at a practical application of differentiation.
|-
| | '''Slide Number 16'''

'''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard

We have to convert it into a box

Squares have to be cut from the four corners

What size squares should we cut out to get the maximum volume of the box?
| | '''A Practical Application of Differentiation'''

We have a 24 inches by 15 inches piece of cardboard.

We have to convert it into a box.

Squares have to be cut from the four corners.

What size squares should we cut out to get the maximum volume of the box?
|-
| | '''Slide Number 17'''

'''A Sketch of the Cardboard'''

Let’s draw the cardboard:

[[Image:]]

The volume function here is '''(24-2x)*(15-2x)*x''' cubic inches.
| | '''A Sketch of the Cardboard'''

Let us draw the cardboard:

This is the volume '''function''' here.

You could expand it into a '''cubic polynomial'''<nowiki>; but we will leave it as it is. </nowiki>
|-
| | Open a new '''GeoGebra''' window.
| | Open a new '''GeoGebra''' window.
|-
| | Type '''(24-2 x) (15-2 x) x''' in the '''input bar''' >> '''Enter'''.
| | In the '''input bar''', type the following line and press '''Enter'''.

In parentheses, 24 minus 2 space '''x''' space in parentheses 15 minus 2 space '''x''' space '''x'''
|-
| | Drag the boundary to see the equation properly in '''Algebra''' view.
| | Drag the boundary to see the equation properly in '''Algebra''' view.
|-
| | Right-click in '''Graphics''' view and set '''xAxis : yAxis''' to '''1:50'''.
| | Right-click in '''Graphics''' view and set '''xAxis''' is to '''yAxis''' to 1 is to 50.
|-
| | Point to the graph for this volume '''function''' in '''Graphics''' view.

Click in and drag the background to move '''Graphics''' view to see the maximum.
| | Observe the graph that is plotted for this volume '''function''' in '''Graphics''' view.

Click in and drag the background to move '''Graphics''' view to see the maximum.
|-
| | Point to the maximum on top of the broad peak and to '''x''' = 0 and '''x''' = 7.
| | Note that the maximum is on the top of a broad peak from '''x''' equals 0 to '''x''' equals 7.
|-
| | Point to both axes.
| | The length of the square side is plotted along the '''x-axis'''.

Volume of the box is plotted along the '''y-axis'''.
|-
| | In the '''input bar''', type '''Max''' with capital M.

From the menu that appears, select '''Max Function Start x-Value End x-value'''.

Instead of highlighted '''Function''', type '''f'''.

Press '''Tab''' to move and highlight '''Start x-Value''' and type 0.

Again, press '''Tab''' to move and highlight '''End x-Value''' and type 10.

Press '''Enter'''.
| | As before, let us find the maximum of this '''function'''.
|-
| | Point to the maximum,''' A''', in '''Graphics''' view and its '''coordinates''' in '''Algebra''' view.
| | This maps the maximum, point '''A''', on the curve.

Click in and drag the background to see

Its '''coordinates''' 3 comma 486 appear in '''Algebra''' view.

Thus, we have to cut out 3 inches squares from all corners.

This will give the maximum possible volume of 486 cubic inches for the cardboard box.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 19'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand differentiation

Draw graphs of derivatives of '''functions'''
|-
| | '''Slide Number 16'''

'''Assignment'''

Draw graphs of derivatives of the following functions in '''GeoGebra''':

'''h(x)=ex'''

'''i(x)=ln(x)'''

'''j(x)=(5x3+3x-1)/(x-1)'''

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
| | As an assignment:

Draw graphs of derivatives of the following functions in '''GeoGebra'''.

Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
|-
| | '''Slide Number 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team:

Conducts workshops using spoken tutorials and
Gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 20'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from''' IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English

2018-10-08T08:44:07Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Limits and Continuity of Function..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Limits and Continuity of Functions'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:

Understand '''limits''' of '''functions'''

Look at continuity of '''functions'''
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

'''Limits'''

'''Elementary calculus'''

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Limits'''

[[Image:]][[Image:]]

[[Image:]]
| | Let us understand the concept of '''limits''' by looking at three graphs '''A, B''' and '''C'''.

Imagine yourself sliding along the curve or line towards a given value of '''x'''.

The height at which you will be, is the corresponding '''y''' value of the '''function'''.

Any value of '''x''' can be approached from two sides.

The left side gives the '''left hand limit'''.

The right side gives the '''right hand limit'''.
|-
| | '''Slide Number 6'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b) f(x) = ?'''

'''lim_(x→ b-) f(x) = L1; lim_(x→b+) f(x) = L1 = f(b)'''

| |

In graph '''A''', let us find the '''limit''' of '''f of x''' as '''x''' approaches or tends to '''b'''.

'''f of x''' is a continuous line.

The '''left hand limit''' of '''f of x''' as '''x''' tends to '''b''' is '''L1'''.

And the '''right hand limit''' of '''f of x''' as '''x''' tends to '''b''' is also '''L1'''.

Thus, the '''limit''' of '''f of x''' as '''x''' approaches '''b''' is '''L1'''.

It is the same as evaluating '''f of x''' at '''x equals b''', that is, '''f of b.'''
|-
| | '''Slide Number 7'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b1) g(x) =?'''

'''lim_(x→b1-) g(x) = lim_(x→b1+) g(x) = L2''''

But '''g(b1)''' does not exist ('''DNE''')

'''lim_(x→b) g(x) = g(b) = L2; lim_(x→a) g(x) = g(a) = L1'''

| |

What is the '''limit''' of '''g of x''' as '''x''' tends to '''b1'''?

In graph '''A''', note that '''g of x''' has an open circle at '''b1 comma L2 prime'''.

This means that '''g of x''' does not exist at this point.

Let us find the '''limit''' of '''g of x''' as '''x''' approaches '''b1'''.

The '''left hand''' and '''right hand limits''' are '''L2'''' as '''x''' approaches '''b1'''.

But '''g of x''' itself does not exist at '''x equals b1'''.

However, '''g of x''' can be evaluated at '''x equals b''' and '''x equals a'''.

And these values are the same as the '''limits''' of '''g of x''' as '''x''' approaches '''b''' and '''a'''.
|-
| | '''Slide Number 8'''

'''Limits of discontinuous functions'''

[[Image:]]

'''lim_(x→c) h(x) = ?'''

'''lim_(x→c-) h(x) = L4; lim_(x→c+) h(x) = L3'''

Thus, '''lim_(x→c) h(x)''' does not exist ('''DNE''')

| |

In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.

We want to find the '''limit''' of '''h of x''' as '''x''' approaches '''c'''.

So let us look at the '''left''' and '''right hand limits'''.

For the '''left hand limit''', look at the lower limb where the limit is '''L4'''.

For the '''right hand limit''', look at the upper limb where limit of '''h of x''' is '''L3'''.

But as '''x''' approaches '''c''', the two limbs of '''h of x''' approach different values of '''y'''.

These are '''L3''' and '''L4'''.

The '''left''' and '''right hand limits''' exist.

But the limit of '''h of x''' as '''x''' approaches '''c,''' '''does not exist''' ('''DNE''').
|-
| | '''Slide Number 9'''

'''Criteria for continuous functions'''

Note that for a '''function f(x)''' to be called continuous,

'''f(a)''' should be defined

'''lim_(x→ a) f(x)''' exists

'''lim_(x→ a) f(x) = f(a)'''

| |

Note that for a '''function f of x''' to be called continuous,

'''f of a''' should be defined

limit of '''f of x''' as '''x''' tends to '''a''' exists

limit of '''f of x''' as '''x''' tends to '''a''' is '''f of a'''
|-
| | '''Slide Number 10'''

'''Limits at infinity'''

[[Image:]]

'''lim_(x→∞) i(x) = ? lim_(x→-∞) i(x) = ?'''

'''lim_(x→∞) i(x) = 2; lim_(x→-∞) i(x) = 1'''

| |

In graph '''C''', '''i of x''' has two parts.

The first part is the upper right one.

Both arms extend towards '''infinity''' ('''∞''').

The second part is the lower left one.

Both arms extend towards '''negative infinity''' ('''-∞''').

What are the limits of '''i of x''' as '''x''' tends to '''infinity''' and '''minus infinity'''?

The limit of '''i of x''' as '''x''' approaches '''infinity''' is 2.

And the limit of '''i of x''' as '''x''' approaches '''negative infinity''' is 1.
|-
| | '''Slide Number 11'''

'''Limit of a rational polynomial function'''

Let us find '''lim_(x→2) (3x2 – x -10)/(x2 – 4)'''

| |

Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.

Let us graph functions and look at their limits.
|-
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Type '''(3 x2-x-10)/(x2-4)''' in the '''input bar''' >> '''Enter'''
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

Note that spaces denote multiplication.

In the '''input bar''', first type the '''numerator'''.

In parentheses, 3 space '''x caret''' 2 minus '''x''' minus 10 followed by division slash'''

Now, type the '''denominator'''.

In parentheses, '''x caret''' 2 minus 4

Press '''Enter'''.
|-
| | Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
| | The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool.

Click in and drag '''Graphics''' view to see the graph.
| | Click on '''Move Graphics View'''.

Click in and drag '''Graphics''' view to see the graph.
|-
| | Point to the graph in '''Graphics''' view.
| | As '''x''' approaches 2, the '''function''' approaches some value close to 3.
|-
| | Click on '''View''' tool and select '''Spreadsheet'''.
| | Click on '''View''' and select '''Spreadsheet'''.
|-
| | Point to the spreadsheet on the right side of the '''Graphics''' view.
| | This opens a spreadsheet on the right side of the '''Graphics''' view.
|-
| | Click on '''Options''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
| | Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
|-
| |

Remember to press '''Enter''' to go to the next cell.

Type 1.91, 1.93, 1.96, 1.98 and 2 in '''column A''' from '''cells''' 1 to 5.
| | Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' less than but close to 2.

Remember to press '''Enter''' to go to the next '''cell'''.

In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
|-
| |

Type 2.01, 2.03, 2.05, 2.07 and 2.09 in '''column A''' from '''cells''' 6 to 10.
| | Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.

We will choose values of '''x''' greater than but close to 2.

In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
|-
| | In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
| | In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.

First, the numerator in parentheses

'''3 A1''' in parentheses '''caret''' 2 minus A1 minus 10 followed by division slash'''

Now the denominator in parentheses

'''A1''' in parentheses '''caret''' 2 minus 4 and press '''Enter'''.
|-
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
| | Click on '''cell B1''' to highlight it.

Place the '''cursor''' at the bottom right corner of the '''cell'''.

Drag the '''cursor''' to highlight cells until '''B10'''.

This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
|-
| | Drag and increase column width.
| | Drag and increase column width.
|-
| | Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.

Point to the spreadsheet.

| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.

This is because the '''function''' is undefined at this value.

The reason for this is that the denominator of the '''function''' becomes 0.

Observe that as '''x''' tends to 2, '''y''' tends to 2.75.

Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
|-
| | '''Slide Number 12'''

'''Limit of a rational polynomial function'''

'''lim_(x→2) (3x2 – x -10)/(x2 – 4) = 2.75'''

| |

Thus, the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2 is 2.75.
|-
| | '''Slide Number 13'''

'''Limit of a discontinuous function'''

Let us find '''lim_(x→0) f(x) = 2x+3, x ≤ 0'''

................................ ='''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0'''

...........................= '''3(x+1), x > 0'''
| |

Let us find limits of a '''piecewise''' or '''discontinuous function f of x'''.

'''f of x''' is described by '''2x plus 3''' when '''x''' is 0 or less than 0.

But '''f of x''' is described by '''3 times x plus 1''' when '''x''' is greater than 0.

We want to find the limits when '''x''' tends to 0 and 1.
|-
| | Open a new '''GeoGebra''' window.
| | Let us open a new '''GeoGebra''' window.
|-
| | Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following line.

'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''

This chooses the '''domain''' of '''x''' from minus 5 (for practical purposes) to 0.

Press '''Enter'''.
|-
| | Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.

Drag the boundary to see it properly.

Point to its graph in '''Graphics''' view.
| | The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.

Drag the boundary to see it properly.

Its graph is seen in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

|-
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
|-
| | Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
| | Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.

When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
|-
| | Similarly, click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
| | Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.

When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
|-
| | Click in and drag the background to see the graph properly.
| | Click in and drag the background to see the graph properly.
|-
| | Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''

| | In the '''input bar''', type the following command and press '''Enter'''.

Remember the space denotes multiplication.

'''b''' equals '''Function''' with capital F

In square brackets, type 3 space '''x''' plus 1 in parentheses comma 0.01 comma 5'''

This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.

For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0.
|-
| | Point to the equation '''b(x)=3(x+1) (0.01 ≤ x ≤ 5)''' in '''Algebra''' view.

Point to its graph in '''Graphics''' view.

| | The equation '''b of x equals 3 times x plus 1''' where '''x''' varies from 0.01 to 5 appears in '''Algebra''' view.

Its graph is seen in '''Graphics''' view.
|-
| | Double click on the equation '''a(x)=2x+3''' in '''Algebra''' view.
| | In '''Algebra''' view, double click on the equation '''b of x''' equals 3 times '''x''' plus 1.
|-
| | Click on '''Object Properties'''.
| | Click on '''Object Properties'''.
|-
| | Click on '''Color''' tab and select blue.
| | Click on the '''Color''' tab and select blue.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| |
| | Click in and drag the background to see both '''functions''' in '''Graphics''' view.
|-
| |
| | Under '''Move Graphics View''', click on '''Zoom In'''.

Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
|-
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''
| | Note that there is a break between the blue and red '''functions'''.

This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
|-
| | Point to the blue '''function'''.

Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.

| | The blue '''function''' has to be considered for '''x''' less than and equal to 0.

When '''x''' tends to 0, '''f of x''' is 3 as the '''function''' intersects the '''y-axis''' at 0 comma 3.
|-
| | Point to the red '''function'''.
| | The red '''function''' has to be considered for '''x''' greater than 0.

When '''x''' equals 1, the value of '''f of x''' is 6.
|-
| | '''Slide Number 14'''

'''Limit of a discontinuous function'''

'''lim_(x→0) f(x) = 2x+3, x ≤ 0 }=3'''

..........................= '''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0 }=6'''

.........................= '''3(x+1), x > 0'''
| |

Thus, for this '''discontinuous function''', '''f of x''' is 3 when '''x''' is 0.

When '''x''' is 1, '''f of x''' is 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 15'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand limits of '''functions'''

Look at continuity of '''functions'''

|-
| | '''Slide Number 16'''

'''Assignment'''

Find the limit of '''(x3-2x2)/(x2-5x+6)''' as '''x''' tends to 2.

Evaluate '''lim_(x→0) sin 4x/sin 2x'''

| | '''As an Assignment''':

Find the limit of this '''rational polynomial function''' as '''x''' tends to 2.

Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
|-
| | '''Slide Number 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team:

<nowiki>* conducts workshops using spoken tutorials and</nowiki>

<nowiki>* gives certificates on passing online tests.</nowiki>

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 20'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/3D-Geometry/English

2018-10-08T06:23:18Z

Vidhya: Created page with "{|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''3D Geometry'''. |- | | '''Slide Number 2..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''3D Geometry'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to view:

And construct different structures in '''3D space'''

Solids of rotation of polynomial functions

Trigonometric functions in 3D space
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0-d
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:

'''GeoGebra''' interface

Geometry

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Rectangular Co-ordinate System'''

[[Image:]]
| | This image shows the '''rectangular coordinate system'''.

It is made up of mutually perpendicular axes and planes formed by them.

The axes are '''x''' (in red), '''y''' (in green) and '''z''' (in blue).

All points in '''3D''' space are denoted by their '''x y z coordinates'''.

The point of intersection of the three axes is the '''origin O 0 comma 0 comma 0'''.

The gray rectangle in the image depicts the '''XY''' plane.

The planes divide space into 8 octants.

Point '''A''' is in the '''XOYZ''' octant and has the '''coordinates 4 comma 4 comma 2'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us draw a '''3D''' pyramid in '''GeoGebra'''.

I have already opened a new window in '''GeoGebra'''.

This time, we work with '''Algebra, 2D Graphics''' and '''3D Graphics''' views.
|-
| | Under '''View''', select '''3D Graphics'''.
| | Under '''View''', select '''3D Graphics'''.
|-
| | Click in '''2D Graphics View''' to draw in '''2D'''.
| | Click in '''2D Graphics View''' to draw in '''2D'''.
|-
| | Drag the boundary to see '''2D Graphics''' properly.
| | Drag the boundary to see '''2D Graphics''' properly.
|-
| | Click in '''2D Graphics'''.
| | Click in '''2D Graphics'''.
|-
| | In '''2D Graphics''' view, click on '''Polygon''' tool and click on '''origin (0,0)'''.

Point to '''A'''.
| | In '''2D Graphics''' view, click on the '''Polygon''' tool and click on origin 0 comma 0.

This creates point '''A''' at the origin.
|-
| | Then click on '''(2,0)''' to create point '''B'''.
| | Then click on 2 comma 0 to create point '''B'''.
|-
| | Click on '''(2,2)''' for point '''C''', and on '''(0,2)''' to draw point '''D'''.
| | Click on 2 comma 2 for '''C''' and on 0 comma 2 to draw '''D'''.
|-
| | Click again on point '''A'''.
| | Finally, click again on '''A'''.
|-
| | Point to quadrilateral '''q1''' in '''2D''' and '''3D Graphics''' views.
| | Note that a quadrilateral '''q1''' is seen in '''2D''' and '''3D Graphics''' views.

The length of each side is 2 units.
|-
| | Click on '''Move''' tool.
| | Click on the '''Move''' tool.
|-
| | Click in '''2D Graphics''' and drag the background.
| | Click in '''2D Graphics''' and drag the background.
|-
| | Drag the boundary to see '''3D Graphics''' properly.
| | Drag the boundary to see '''3D Graphics''' properly.
|-
| | Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.

Click on the square in '''3D Graphics''' view.
| | Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.

In '''3D Graphics''' view, click on the square.
|-
| | An '''Altitude''' text box opens up, type 3 and click '''OK'''.
| | In the '''Altitude''' text-box that opens, type 3 and click '''OK'''.
|-
| | Point to pyramid '''e''' in '''3D Graphics''' view.

Point to base.

Point to '''E (1,1,3)'''.

Show altitude.
| | A pyramid '''e''' appears in '''3D Graphics''' view.

Its base is the quadrilateral '''q1'''.

Its apex is '''E''' 1 comma 1 comma 3.

Its altitude or height is 3 units.
|-
| | '''Slide Number 6'''

'''Rotation of a Polynomial'''

Let us rotate '''f(x)= ¬2x4-x3+3x2'''

Part in second '''quadrant''' ('''XY''' plane) about '''x-axis'''
| | '''Rotation of a Polynomial'''

Let us rotate '''f of x''' equals minus '''2 x raised to 4''' minus '''x cubed''' plus '''3 x squared'''.

We will rotate the part that lies in the second '''quadrant''', in '''XY''' plane, about the '''x-axis'''.
|-
| | Show the '''GeoGebra''' window.

| | I have already opened a new window in '''GeoGebra'''.

We will initially work with '''Algebra''' and '''2D Graphics''' views and open '''3D Graphics''' view later.
|-
| | In '''input bar''', type the following line and press '''Enter'''.

'''f(x) = -2 x^4 -x^3+3 x^2'''
| | In the '''input bar''', type the following line.

To type the '''caret symbol''', hold '''Shift''' key down and press 6.

'''f x''' in parentheses equals minus 2 space '''x caret''' 4 minus '''x caret''' 3 plus 3 space '''x caret''' 2

Spaces here denote multiplication.

Press '''Enter'''.
|-
| | Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
| | Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
|-
| | Keeping the '''cursor''' on the '''x-axis'''.
| | Keep the '''cursor''' on the '''x-axis'''.
|-
| | Drag it along until you reach the intersection of '''f''' and '''x-axis'''.

Point to the label '''function f, x-axis''' that appears.
| | Drag it along until you see '''function f, x-axis''' at the intersection of '''f''' and '''x-axis'''.
|-
| | Click on this intersection point.
| | Click on this intersection point.
|-
| | Point to '''A'''.
| | Point '''A''' appears.
|-
| | Click on '''Slider''' and in '''Graphics''' view.
| | Click on '''Slider''' and in '''Graphics''' view.
|-
| | A '''Slider''' dialog box opens.
| | A '''Slider''' dialog-box opens.
|-
| | Leave '''a''' as the '''Name'''.
| | Leave '''a''' as the '''Name'''.
|-
| | Change '''Min''' value to -1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
| | Change '''Min''' value to '''minus''' 1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
|-
| | Click '''OK'''.
| | Click '''OK'''.
|-
| | Point to '''slider a'''.

Point to the part of '''function''' and '''x-axis''' in the second '''quadrant'''.
| | This creates '''slider a''', which changes the value of '''a''' from -1.5 to 0.

It will focus on the part of the graph in the second '''quadrant'''.
|-
| | In the '''input bar''', type '''(a,f(a))'''.

Press '''Enter'''.
| | In the '''input bar''', type the following in parentheses.

'''a''' comma '''f a''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''B'''.
| | This creates point '''B''' whose '''x coordinate''' is the value of '''a'''.

Its '''y-coordinate''' lies along the curve described by the '''function f''' between '''x''' equals 1.5 and 0.
|-
| | Right-click on '''slider a''' and check '''Animation On'''.
| | Right-click on '''slider a''' and check '''Animation On'''.
|-
| | Point to '''B''' and '''slider a'''.
| | Point '''B''' travels along '''function f''' as '''a''' changes.
|-
| | Right-click on '''slider a''' and uncheck '''Animation On'''.
| | Right-click on '''slider a''' and uncheck '''Animation On'''.
|-
| | In the '''input bar''', type '''(a,0)''' and press '''Enter'''.
| | In the '''input bar''', type '''a''' comma 0 in parentheses and press '''Enter'''.
|-
| | Point to '''C''' and '''slider a'''.
| | This creates point '''C'''.

As its '''x co-ordinate a''' changes, '''C''' moves below point '''B''' along the '''x-axis'''.
|-
| | Under '''Line''', click on '''Segment''' and click on '''B''' and '''C''' to join them.
| | Under '''Line''', click on '''Segment''' and click on '''B''' and '''C''' to join them.
|-
| | Click on '''Move Graphics View''' and drag background to the left.
| | Click on '''Move Graphics View''' and drag the background to the left.
|-
| | Click on '''View''' and check '''3D Graphics''' to see the '''3D Graphics''' view.
| | Click on '''View''' and check '''3D Graphics''' to see the '''3D Graphics''' view.
|-
| | Point to '''2D Graphics''' and '''3D Graphics''' views.
| | Note that what is drawn in '''2D Graphics''' appears in the '''XY''' plane, in '''3D Graphics'''.
|-
| | Click in '''3D Graphics''' view and on '''Rotate 3D Graphics View'''.

Rotate the '''3D Graphics''' view to see the curve properly.
| | Click in '''3D Graphics''' view and on '''Rotate 3D Graphics View'''.

Rotate '''3D Graphics''' to see the curve properly.
|-
| | Place '''cursor''' on '''y-axis''' (in green).

Click to see an arrow aligned with the '''y-axis'''.
| | Place the '''cursor''' on the '''y-axis''' (in green).

Click to see an arrow aligned with the '''y-axis'''.
|-
| | Then, drag along the '''y axis''' to see the curve properly.
| | Drag to pull the '''y-axis''' in or outwards to see the curve.
|-
| | Type '''Circle[C,f(a),xAxis]''' in the '''input bar''' and press '''Enter'''.
| | In the '''input bar''', type the following line

'''Circle''' open square brackets '''capital C '''comma '''f a''' in parentheses comma '''x Axis''' with '''capital A'''.

Close square brackets.
|-
| | Point to circle '''c''' with center at point '''C'''.
| | This creates circle '''c''' with center at point '''C'''.

Its radius is equal to '''f of a''' corresponding to the value of '''a''' on '''slider a'''.

Its rotation is around the '''x-axis'''.

Press '''Enter'''.
|-
| | Right-click on circle '''c''' in '''Algebra''' view and check '''Trace On''' option.
| | In '''Algebra''' view, right-click on circle '''c''' and check '''Trace On''' option.
|-
| | Right-click on '''slider a''' and select '''Animation On''' option.
| | Right-click on '''slider a''' and select '''Animation On''' option.
|-
| | Point to the solid traced as '''a''' changes.
| | Observe the solid traced as '''a''' changes.
|-
| | Point to '''2D''' and '''3D Graphics''' views.
| | Watch both '''2D''' and '''3D Graphics''' views.
|-
| | Point to Segment '''BC''', '''x-axis''' and '''function f'''.
| | Segment '''BC''' moves between the '''x-axis''' and '''function f'''.
|-
| | Point to the part of '''f''' in the second '''quadrant'''.
| | The part of '''function f''' that is in the second '''quadrant''' in '''2D,''' rotates around the '''x-axis'''.
|-
| | Drag '''3D Graphics''' to see it from another angle.
| | Drag '''3D Graphics''' to see it from another angle.
|-
| | Show the '''GeoGebra''' window.
| | Finally, let us look at '''trigonometric functions''' in '''3D.'''

I have already opened a new window in '''GeoGebra'''.
|-
| | Click on '''3D Graphics''' tool under '''View'''.
| | Under '''View''', click on '''3D Graphics'''.
|-
| | Drag the boundary to see '''2D Graphics''' properly.
| | Drag the boundary to see '''2D Graphics''' properly.
|-
| | Click in '''2D Graphics''', then on the '''Slider''' tool and in '''Graphics''' view.
| | Click in '''2D Graphics''', then on the '''Slider''' tool and in '''Graphics''' view.
|-
| | Point to the '''slider''' dialog box and '''Number''' radio button selection.

Type '''t''' in the '''Name''' field.
| | A '''slider''' dialog-box opens.

By default, the '''Number''' radio-button is selected.

In the '''Name''' field, type '''t'''.
|-
| | Set '''Min''' to -6, '''Max''' to 16 and '''increment''' of 0.1.

Click '''OK'''.
| | Set '''Min''' to minus 6, '''Max''' to 16 and '''increment''' of 0.1.

Click '''OK'''.
|-
| | Point to '''slider t'''.
| | This creates a '''slider t''' which will change '''t''' from minus 6 to 16.
|-
| | In the '''input bar''', type '''f(t)=cos(t)''' and press '''Enter'''.
| | In the '''input bar''', type '''f t''' in parentheses equals '''cos t''' in parentheses and press '''Enter'''.
|-
| | Click in '''2D Graphics'''.
| | Click in '''2D Graphics'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
|-
| | Click on '''Move Graphics View''' and drag the background.
| | Click on '''Move Graphics View''' and drag the background.
|-
| | Point to the '''cosine function''' of '''f(t)''' in '''2D''' and '''3D Graphics''' views.
| | You can see the graph of the '''cosine function''' of '''f of t''', in '''2D''' and '''3D Graphics''' views.
|-
| | Similarly, type '''g(t)=sin(t)''' in the '''input bar''' and press '''Enter'''.
| | Similarly, in the '''input bar''', type '''g t''' in parentheses equals '''sin t''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''sine function''' graph ('''g(t)''').
| | '''Sine function''' graph of '''g of t''' appears.
|-
| | Now, type '''h(t)=t/4''' in the '''input bar''' and press '''Enter'''.
| | In the '''input bar''', type '''h t''' in parentheses equals '''t''' divided by 4 and press '''Enter'''.
|-
| | Point to line ('''h(t)''').
| | Line '''h of t''' is of the form '''y''' equals '''mx''' where slope '''m''' is 1 divided by 4.
|-
| | Click in '''3D Graphics''' view.
| | Click in '''3D Graphics''' view.
|-
| | Click on '''Point''' tool and click in the gray area in '''3D Graphics''' view.

Point to point '''A'''.
| | Click on the '''Point''' tool and click in the gray area in '''3D Graphics''' view.

This creates point '''A'''.
|-
| | Drag the boundary to see its '''co-ordinates''' properly.
| | Drag the boundary to see its '''co-ordinates''' properly.
|-
| | Double click on point '''A''' in '''Algebra''' view.

Change the '''co-ordinates''' to '''(f(t),g(t),h(t))'''.

Press '''Enter'''.
| | In '''Algebra''' view, double-click on '''A'''.

Change the '''coordinates''' to the following.
|-
| | Point to '''A'''.
| | The '''x- coordinate''' of '''A''' is '''cos t'''.

The '''y-coordinate''' is '''sin t''' and '''t''' divided by '''4''' is its '''z coordinate'''.
|-
| | Right-click on '''slider t''' and click on '''Object Properties'''.
| | Right-click on '''slider t''' and click on '''Object Properties'''.
|-
| | A '''Preferences''' dialog-box opens.

Click on '''Slider''' tab.

Under '''Animation''', for '''Repeat''', choose option “'''Increasing'''” from dropdown menu.
| | A '''Preferences''' dialog-box opens.

Click on '''Slider''' tab.

Under '''Animation''', for '''Repeat''', choose option “'''Increasing'''” from the dropdown menu.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| | Right-click on point '''A''' in '''Algebra''' view and select '''Trace On'''.
| | In '''Algebra''' view, right-click on '''A''' and select '''Trace On'''.
|-
| | Right-click on '''slider t''' and check '''Animation On'''.
| | Right-click on '''slider t''' and check '''Animation On'''.
|-
| | Point to point '''A''' and the '''helix''' in '''3D Graphics''' view.

Point to point '''A’s co-ordinates''' in '''Algebra''' view.
| | Point '''A''' traces a '''helix''' in '''3D''' space with '''coordinates''' mentioned earlier.
|-
| | Click in '''Rotate 3D Graphic View''' and rotate the background.

Rotate '''3D Graphics''' view.
| | Click in '''Rotate 3D Graphic View''' and rotate the background.

Rotate '''3D Graphics''' view so you are looking down the '''z-axis''' at the '''XY''' plane.
|-
| | Point to the traces of point '''A (cos(t), sin(t))'''.
| | Note that the traces of '''A''' are the circumference of a '''unit circle'''.

Point '''A''' moves along the circle as angle '''t''' changes.

In '''2D''', its '''coordinates''' are '''cos t''' comma '''sin t'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to view:

And construct different structures in '''3D''' space

Solids of rotation of '''polynomial functions'''

'''Trigonometric functions''' in '''3D''' space
|-
| | '''Slide Number 7'''

'''Assignment'''

Construct a prism and a cylinder.

Draw lines to pierce the structures and find their intersection points.

Graph the '''polynomial, f(x)=x5-7x4+9x3+23x2-50x+24'''.

Show the solid formed due to rotation of peak in first '''quadrant''' in '''XY''' plane.
| | As an assignment:

Construct a prism and a cylinder anywhere in '''3D''' space.

Draw lines to pierce the structures and find their intersection points.

Graph the given '''polynomial'''.

Show the solid formed due to rotation of the peak, in the first '''quadrant''', in the '''XY''' plane.
|-
| | '''Slide Number 8'''

'''Assignment'''

You tried to fly a kite off a cliff. The kite got dumped into the lake below

You gave out 325 feet of string

The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees

How high is the cliff?
| | As another assignment,

You tried to fly a kite off a cliff. The kite got dumped into the lake below.

You gave out 325 feet of string.

The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees.

How high is the cliff?
|-
| | '''Slide Number 8'''

'''About Spoken Tutorial Project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 9'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial '''project''' '''team:

<nowiki* conducts workshops using spoken tutorials and</nowiki>

<nowiki>* gives certificates on passing online tests.</nowiki>

For more details, please write to us.
|-
| | '''Slide Number 10'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 11'''

'''Acknowledgement'''
| | '''Spoken Tutorial''' project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

PhET/C3/Natural-Selection/English

2018-09-24T10:16:38Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Natural Selection''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Natural Selection PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Natural Selection, an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux 'OS ''' version 16.04

'''Java ''' version 1.8.0

'''Firefox Web Browser ''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java ''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with biology and ecology.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

We will look at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
| |
Using this '''simulation''', we will look at a population of rabbits for effects,

Of mutations and selection factors

Of environment

On pedigree
|-
| |
| | Let us begin.
|-
| | '''Slide Number 6'''

'''Mutations'''

Mutations

Changes in phenotype may be visible

Dominant

Recessive
| |

'''Mutations'''

Mutations are alterations in the '''nucleotide''' sequence of any genetic element.

They are passed onto offspring.

Mutations may or may not change the observable traits or phenotype of an organism.

Inheritance of mutations can be dominant or recessive.
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Natural Selection simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space Natural-Selection_en.jar'''.

Point to the '''browser''' address.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space natural-selection_en.jar'''.

'''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Natural Selection simulation'''.
|-
| | Point to the rabbit hopping in the '''simulation''' panel.

Click '''Pause button''' at the bottom of the '''interface'''.
| | Observe the rabbit hopping in the '''simulation''' panel.

Click the '''Pause button''' at the bottom of the '''interface'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to each section in the '''interface'''.

Point to the '''Reset All button'''.
| | The '''interface''' has :

'''Simulation''' panel

On the left side,

'''Add Mutation'''

'''Edit Genes'''

In the middle,

'''Graph'''

'''Time until next generation progress bar'''

'''Play/Pause button, Step button'''

On the right side,

'''Selection Factor'''

'''Environment'''

'''Chart'''

Clicking on the '''Reset All button''' takes you back to the starting point.
|-
| | Under '''Selection Factor''', point to the default selection '''None'''.

Under '''Environment''', point to the default selection of '''Equator'''.

Under '''Chart''', point to the default selection of '''Population'''.
| | Let us keep the default settings:

'''None''' for '''Selection Factor'''

'''Equator''' for '''Environment'''

'''Population''' for '''Chart'''
|-
| | Point to the '''Population vs Time''' graph.

| | Observe the '''Population versus Time''' graph.

It shows the number of rabbits plotted on the '''y axis''' and time on the '''x axis'''.
|-
| | Click '''Play button''' at the bottom of the '''interface'''.
| | Click '''Play button''' at the bottom of the '''interface'''.
|-
| | Point to the black line in the '''Population''' graph.
| | In the graph, observe how the black line moves to the right.
|-
| | Keep clicking on '''Step button'''.
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
|-
| | Point to the '''simulation''' panel and the '''Game Over pop-up box'''.

Point to the text “'''All of the bunnies died!'''
| | If there are no rabbits left, you will see a '''Game Over pop-up box''' like this.

The text will read “'''All of the bunnies died!”'''
|-
| | Click on '''Play Again button'''.
| | Click on the '''Play Again button''' to resume the '''simulation'''.
|-
| | Click on '''Add a Friend''' to add another rabbit to the '''simulation'''.
| | Click on '''Add a Friend''' to add another rabbit to the '''simulation'''.
|-
| | Click on '''Pause''' at the bottom of the interface.
| | Click on '''Pause''' at the bottom of the interface.
|-
| | '''Slide Number 8'''

'''Generations of Progeny'''

[[Image:]]
| | '''Generations of Progeny'''

Observe the labels on the right and roman numerals on the left of each row.

Generation '''P''' is shown in row I (one).

The '''F1''' generation in row II (two) is made up of the progeny or children of generation P.

And so on until row V (five).
|-
| | Point to the graph.
| | We will allow the population to grow until F3.

This would be three steps after the mating pair.
|-
| | Click '''Play''' at the bottom of the interface.
| | Click '''Play''' at the bottom of the interface.
|-
| | Point to the '''progress bar'''.
| | Note how the '''progress bar''' is full when a new generation begins.

The '''progress bar''' starts moving to the left as a generation starts growing.
|-
| | Click '''Pause button'''.
| | Click '''Pause button'''.
|-
| | Click the second button at the top left corner of the graph to '''zoom out'''.
| | Click the second button at the top left corner of the graph to '''zoom out'''.

You can now see the height of the next step.
|-
| | Point to the black line showing the total number of rabbits.
| | The black line shows the total number of rabbits in the graph.
|-
| | Under '''Selection Factor''', click the '''Food radio button'''.
| | Under '''Selection Factor''', click the '''Food radio button'''.
|-
| | Under '''Add Mutation''', click the '''Long Teeth button'''.
| | Under '''Add Mutation''', click the '''Long Teeth button'''.
|-
| | Point to the '''text-box''', “'''Mutation coming'''” at the bottom of the '''simulation''' panel.

Point to the yellow triangle with the lightning to the left of the text.

Point to the picture of long teeth in the text-box.
| | A '''text-box''', “'''Mutation coming'''” appears at the bottom of the '''simulation''' panel.

Note the yellow triangle with the lightning inside.

This indicates a mutation.

You can see a picture of long teeth in the text-box.
|-
| | Under '''Edit Genes''', point to the rows next to the '''Teeth''' label.
| | Under '''Edit Genes''', the rows next to the '''Teeth''' label have become active.

The '''radio buttons''' can now be clicked.
|-
| | Point to each row with two options appearing under the '''Dominant''' and '''Recessive''' columns.

Point to the upper row with a picture of short teeth to the left.

Point to the lower row with a picture of long teeth to the left.
| | For each row, two options appear under the '''Dominant''' and '''Recessive columns'''.

To the left of each row are the pictures of long and short teeth.
|-
| | Point to default selections of dominant mutation for long teeth and recessive for short teeth.
| | Default selections are dominant mutation for long teeth and recessive for short teeth.
|-
| | Point to the '''progress bar''' and the graph.
| | Observe the '''progress bar''' and the graph after the mutation has been added.
|-
| | Point to the graph.

Keep clicking on the '''Step button'''.

Point to the interval between two narrow steps after the mutation in the graph.
| | Let us allow the population to grow for another three generations after the mutation.

Keep clicking on the '''Step button'''.

The interval between two narrow steps in the graph corresponds to a generation.
|-
| | Click on the second '''Zoom Out button''' until you see the steps.
| | Click on the second '''Zoom Out button''' in the graph until you see the steps.
|-
| | Point to differently colored lines appearing in the graph after mutation and food selection.
| | Observe how differently colored lines appear in the graph after mutation and food selection.
|-
| |
Point to the legend below the graph.
| | Note that the timing of mutations and selection factors will affect population growth.

The legend below the graph gives the different colors and what they mean.
|-
| | Point to dominant '''radio button''' selection next to long teeth.

Point to the magenta line for long teeth, the olive line for short teeth.
| | We introduced a dominant mutation for long teeth.

Let us look at the magenta line for long teeth, the olive line for short teeth.
|-
| | Point to upper olive line and a lower magenta line.
| | Initially, the olive line is above the magenta line.

The number of short-toothed rabbits is higher than that of the long-toothed rabbits.
|-
| | Point to upper magenta line and a lower olive line.
| | Later, the magenta line is above the olive line.

The number of long-toothed rabbits has increased relative to the short-toothed ones.
|-
| |
| | This means that long teeth help rabbits survive by eating the available food.
|-
| | '''Slide Number 9'''

'''Fur Mutation'''

Set up conditions to study effects of a fur mutation on survival

Brown fur dominant, white fur recessive

Selection factor = Wolves

Three generations after mutation
| | '''Fur Mutation'''

Set up conditions to study effects of a fur mutation on survival of rabbits.

Keep brown fur as the dominant mutation and white fur as the recessive mutation.

Choose wolves as the selection factor.

Allow the population to grow for another 3 generations after the mutation.
|-
| | Show the wolves move in and out killing the rabbits.

Show the brown rabbits that begin to appear after the mutation.
| | Observe how the wolves move in and out, killing the rabbits.

Brown rabbits begin to appear after the mutation was introduced.
|-
| | Show the '''radio button''' selections for dominant mutations for long teeth and brown fur.

Show the graph.
| | Note that we are still looking at effects of the dominant long teeth mutation besides brown fur.
|-
| | Point to the graph and '''simulation''' panel.
| | Compare numbers of rabbits having white and brown fur in the graph and '''simulation''' panel.
|-
| | Point to the red line above the cyan one.
| | Initially, there are more white rabbits than brown rabbits.
|-
| | Point to later steps showing the cyan line above the red one.
| | Later, the number of brown rabbits has increased relative to white ones.
|-
| |
| | At the equator, with wolves killing the rabbits, brown fur is an advantage for survival.

This strategy to blend with the environment is called '''camouflage'''.
|-
| |
| | What can you say about the numbers of long- and short-toothed rabbits?

Sometimes a mutation changes the phenotype of all rabbits.

If so, the graph will not compare the mutation versus the wild type (unmutated) phenotypes.

Here, there are more long-toothed rabbits than short-toothed ones.
|-
| | '''Slide Number 10'''

'''Tail Mutation'''

Set up conditions to study effects of a tail mutation on survival

Long tail dominant, short tail recessive

Selection factor = Wolves

Three generations after mutation
| | '''Tail Mutation'''

Set up conditions to study effects of a tail mutation on survival of rabbits.

Keep long tail as the dominant mutation and short tail as the recessive mutation.

Choose wolves as the selection factor.

Allow the population to grow for another 3 generations after the mutation.
|-
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
|-
| | Click on the 2nd '''Zoom Out button'''.
| | Click on the 2nd '''Zoom Out button'''.
|-
| | Point to the differently colored lines in the graph.
| | Note the number of rabbits with brown fur, long teeth and short tails.

It is higher than that of rabbits with white fur, short teeth and long tails.
|-
| |
| | Brown fur and short tails help escape from wolves.

Long teeth help survival by making it easier to eat vegetation.
|-
| | Under '''Chart''', select '''Pedigree'''.
| | Under '''Chart''', click '''Pedigree'''.
|-
| | Point to the text, “'''Click a Bunny'''” at the top.

Click on the rabbit at the left bottom corner of the '''simulation''' panel.

Point to the rabbit framed inside a blue rectangle.
| | Note the text, “'''Click a Bunny'''” at the top.

Let us click on the rabbit at the left bottom corner of the '''simulation''' panel.

Observe how the selected rabbit is framed inside a blue rectangle.
|-
| | Point to the '''pedigree chart''' for the rabbit framed in the blue rectangle.
| | The '''pedigree chart''' appears for the rabbit framed in the blue rectangle.
|-
| | Click on the top right '''button''' in the '''Pedigree''' window.
| | Click on the top right '''button''' in the '''Pedigree''' window.
|-
| | Point to the '''Pedigree''' window that has separated.

Point to the '''Population''' chart behind it.

Resize the '''Pedigree''' window.
| | The '''Pedigree''' window is separated and the '''Population''' chart appears behind it.

We can now resize the '''Pedigree''' window.
|-
| | Point to the color of the previous generations of rabbits.

Point to the red crosses on the rabbits.

Point to the yellow triangle with the lightning symbol inside.
| | Note the color of the previous four generations of rabbits above the selected rabbit.

Red crosses on the rabbits indicate that they are dead.

The yellow triangle with the lightning symbol inside indicates a mutation.

It shows that that rabbit underwent a mutation so its genotype and phenotype changed.
|-
| |
| | '''Pedigree''' analysis allows study of inheritance of genes based on data about phenotypes.
|-
| | Click repeatedly on the '''Step button'''.

Show the window where rabbits can be seen all over the planet Earth.
| | Click repeatedly on the '''Step button'''.

Rabbits can be seen all over the continents on the planet Earth.
|-
| | Point to the caption, “'''Bunnies have taken over the world!'''”
| | Observe the caption, “'''Bunnies have taken over the world!'''”

These are the long-term effects of the '''simulation''' under these conditions.
|-
| |
| | Do refer to '''Additional material''' provided with this '''tutorial'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 11'''

'''Summary'''

We have demonstrated,

'''Natural Selection PhET simulation'''
| | In this '''tutorial''', we have demonstrated how to use the '''Natural Selection PhET simulation'''.
|-
| | '''Slide Number 12'''

'''Summary'''

We looked at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
| | Using this '''simulation''', we looked at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
|-
| | '''Slide Number 13'''

'''Assignment'''

Observe the rabbits:

For effects of mutations and selection factors in '''Arctic''' environment

After switching mutations from dominant to recessive and ''vice versa''

For changes in '''pedigree''' under different conditions
| | As an '''assignment''', observe the rabbits:

For effects of mutations and selection factors in '''Arctic''' environment

After switching mutations from dominant to recessive and ''vice versa''.

For changes in '''pedigree''' under different conditions
|-
| | '''Slide Number 14'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 15'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 16'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C2/Equation-Grapher/English

2018-09-17T08:43:09Z

Vidhya:

{|border=1
|| '''Visual Cue'''
|| '''Narration'''

|-
|| '''Slide Number 1'''

'''Title Slide'''
|| Welcome to this tutorial on '''Equation Grapher'''.
|-
|| '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate '''PhET simulation''',

'''Equation Grapher'''
|| In this tutorial, we will demonstrate '''Equation Grapher PhET simulation'''.
|-
|| '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
|| Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
|| '''Slide Number 4'''

'''Pre-requisites'''
|| Learner should be familiar with topics in high school mathematics.
|-
|| '''Slide Number 5'''

'''Learning Goals'''

Lines '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y = ax2 + bx + c'''.

|| Using this '''simulation''' we will look at,

Lines of the form '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y equals ax squared plus bx plus c'''
|-
|| '''Slide Number 6'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' ∈ ℝ, index ''n'' is a positive integer, ''0 ≤ r ≤n, then,''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

'''Reminder:''''' nC1 = n!/[1! (n-1)!]''
|| '''Binomial Theorem'''

'''a''' and '''b''' are real numbers, '''index''' '''n''' is a positive integer.

'''r''' lies between 0 and '''n'''. Then,

'''Binomial theorem''' states that '''a''' plus '''b''' raised to '''n''' can be expanded as shown.
|-
|| '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
|| Use the given link to download the simulation.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
||
|| I have already downloaded '''Equation Grapher''' simulation to my '''Downloads''' folder.
|-
|| Press Ctrl+Alt+T to open the terminal.

Type '''cd Downloads''' >> press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Point to the opened '''file format'''.
|| To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Press '''Enter'''.

'''File''' opens in the '''browser''' in '''html format'''.
|-
|| Cursor on the '''interface'''.
|| This is the '''interface''' for the '''Equation Grapher simulation'''.
|-
|| Point to the '''interface'''.

|| The interface shows '''Cartesian co-ordinate system''' of '''x''' and '''y axes'''.
|-
|| Point to the first '''quadrant'''.

Point to the '''quadratic function''', '''y = ax2 + bx + c'''.

Point to the '''sliders''' and '''display boxes'''.

Point to the red '''Zero''' button.

Point to the green '''Save''' button.

Point to the equation.
|| The first quadrant contains:

The red-colored '''quadratic equation''', '''y equals ax squared plus bx plus c'''

Three '''sliders''' and display boxes under '''ax2, bx''' and '''c'''

The '''sliders''' allow you to change the values of the coefficients, '''a''', '''b''' and '''c'''.

The display boxes show these values and can be used to enter values.

A red '''Zero''' button to set all '''sliders''' at '''0'''

A green '''Save''' button to save the '''equation'''

The updated equation in red is shown below the '''sliders'''.
|-
|| Point to the fourth quadrant.

Point to the '''quadratic equation'''.

Point to the check boxes.

Point to the violet, green and blue terms.

|| The fourth quadrant contains the

'''Quadratic''' equation '''y = ax2+bx+c'''

Three check boxes under '''ax2, bx''' and '''c'''

Note that the '''ax squared''' term is violet, '''bx''' is green and '''c''' is blue.
|-
|| In the first quadrant, in the display box below '''ax2''', type 1.

Point to the '''slider''' under '''ax2'''.
|| In the first quadrant, in the display box below '''ax squared''', type 1.

Observe how the '''slider''' under '''ax squared''' also moves to 1.
|-
|| Point to the red parabola and origin '''(0,0)'''.
|| A red parabola with vertex at origin '''0 comma 0''' appears in the window.

It opens upwards.
|-
|| In the first quadrant, in the display box below '''bx''', type 1
|| In the first quadrant, in the display box below '''bx''', type 1.
|-
|| Point to the parabola.
|| Observe how the parabola shifts downwards and to the left.
|-
|| In the first quadrant, in the display box below '''c''', type 1.
|| In the first quadrant, in the display box below '''c''', type 1.
|-
|| Point to the parabola.
|| Observe how the parabola moves upwards.
|-
|| In the fourth quadrant, check the box below the violet coloured '''ax2''' term.
|| In the fourth quadrant, check the box below the violet coloured '''ax squared''' term.
|-
|| Point to the violet and red parabolas.

Point to the equation.
|| A violet parabola appears next to the red parabola.

This violet parabola corresponds to the '''y equals ax squared''' part of the red equation.
|-
|| Point to the equation, '''y = x2''', in the first quadrant.
|| The equation for the violet parabola is '''y equals x squared'''.
|-
|| Check the box below the green '''bx''' term in the fourth quadrant.
|| Now, in the fourth quadrant, check the box below the green '''bx''' term.
|-
|| Point to the green line.

Point to the origin '''(0,0)'''.

Point to the equation, '''y = x''', in the first quadrant.
|| Observe how a green line appears in the '''Cartesian''' plane.

It passes through the origin '''0 comma 0'''.

It corresponds to the '''x''' term and its equation is '''y equals x'''.
|-
|| Check the box below the blue '''c''' term in the fourth quadrant.
|| Now, in the fourth quadrant,check the box below the blue '''c''' term .
|-
|| Point to the blue line.

Point to the equation, '''y=c'''.
|| Observe how a blue line appears in the '''Cartesian''' plane.

Its equation is '''y equals c''' and it corresponds to the constant term of the equation.
|-
|| Click on the green '''Save''' button.
|| Click on the green '''Save''' button.
|-
|| Point to the blue saved parabola, '''y = x2+ x + 1'''.
|| This saves the equation '''y equals x squared plus x plus 1'''.
|-
|| Change values for '''a''', '''b'''and '''c'''.

Point to the '''sliders''' and the display boxes below the terms.

Point to the graphs.
|| Change the values for '''a''', '''b''' and '''c'''.

You can either use the '''sliders''' or type in the display boxes below the terms.

Observe the effects of these changes on the graphs.
|-
|| Point to the blue saved parabola, '''y = x2 + x + 1'''.
|| Note that as you change '''a''', '''b''' and '''c''', you can still see the parabola '''y equals x squared plus x plus 1'''.

This is because we saved this equation.
|-
||
|| Save other graphs that you want to compare to see the effects of '''a''', '''b ''' and '''c'''.

You can only save one equation at a time.
|-
|| Point to the blue '''Erase''' button.
|| Note that after you have saved an equation, a blue '''Erase''' button appears.

This will erase the saved equation.
|-
|| Click on the red '''Zero''' button.
|| Click on the red '''Zero''' button.

This resets all coefficients '''a''', '''b''' and '''c''' to 0.
|-
|| '''Slide Number 8'''

'''Assignment'''
|| As an '''assignment''', compare the parabolas graphed for different combinations of:

'''a''' <0 and '''a''' >0

'''b''' <0 and '''b''' >0

'''c''' <0 and '''c''' >0
|-
|| '''Slide Number 9'''

'''Summary'''
|| In this '''tutorial''', we have demonstrated the

'''Equation Grapher PhET simulation'''
|-
|| '''Slide Number 10'''

'''Summary'''
|| Using this '''simulation''', we have looked at:

Lines of the form '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y = ax2 + bx + c'''
|-
|| '''Slide Number 11'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
|| The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
|| '''Slide Number 12'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project''' team conducts workshops using spoken tutorials and gives certificates on passing online tests.

For more details, please write to us.
|-
|| '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
|| Please post your timed queries in this forum.
|-
|| '''Slide Number 14'''

'''Acknowledgement'''
|| This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''
|-
|| '''Slide Number 15'''

'''Acknowledgement'''
|| '''Spoken Tutorial Project''' is funded by '''NMEICT''', MHRD, Government of India.

More information on this mission is available at this link.
|-
||
|| This is '''Vidhya Iyer''' from '''IIT Bombay''' signing off.

Thank you for joining.
|-
|}

PhET/C2/Color-Vision/English

2018-09-17T08:15:20Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on''' Color Vision, '''an '''interactive PhET..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on''' Color Vision, '''an '''interactive PhET simulation'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Color Vision PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Color Vision, '''an interactive '''PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

Ubuntu Linux OS version 16.04

Java version 1.8.0

Firefox Web Browser version 60.0.2
| | Here I am using,

Ubuntu Linux''' '''OS version 16.04

Java version 1.8.0

Firefox Web Browser version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with high school biology and physics.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

White light

Colours from visible '''spectrum'''

Light and filters of different colours of visible '''spectrum'''

Red, green and blue light, separately or in combination
| | Using this simulation, we will look at colour vision when the human eye sees:

White light

Light of different colours from the visible '''spectrum'''

Light and filters of different colours of the visible '''spectrum'''

Red, green and blue light, separately or in combination
|-
| |
| | Please refer to the '''additional material''' provided with this '''tutorial'''.

Let us begin.
|-
| | '''Slide Number 6'''

'''Visible Light'''

[[Image:]]

Electromagnetic '''spectrum''', 380 -760 nm, visible light.

'''VIBGYOR'''

Lowest wavelength (highest frequency) appears violet

Highest wavelength (lowest frequency) appears red

All colours --> white light
| |'''Visible Light'''

A portion of the electromagnetic '''spectrum''' is detected by human eye as visible light.

This portion extends from 380 to 760 '''nanometers'''.

The colours of the visible '''spectrum''' can be remembered as '''VIBGYOR'''.

Violet-indigo-blue-green-yellow-orange-red

The lowest wavelength (highest frequency) appears violet.

The highest wavelength (lowest frequency) appears red.

Combining all the colours gives white light.
|-
| |
| |
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colourado.edu]
|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Color Vision simulation''' to my '''Downloads folder'''.
|-
| | Right click on '''color-vision_en.html''' file.

Select Open With Firefox Web Browser option.

Point to the browser address.
| | To open the simulation, right click on the '''color-vision_en.html''' file.

Select the '''Open With Firefox Web Browser''' option.

The file opens in the browser.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Color Vision simulation'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to two '''screens''' in the '''interface'''.
| | The '''interface''' has two '''screens'''

'''Single Bulb'''

'''RGB Bulbs'''
|-
| | Click on '''Single Bulb screen'''.
| | Click on '''Single Bulb screen'''.
|-
| | In the '''Single Bulb screen''', point to the person facing the right.

Point to two small images below the neck.
| | In the '''Single Bulb screen''', you can see a person facing the right.

Just below the neck are two small images.
|-
| | Point to the first highlighted image.

Point to the person’s face.
| | The first one is highlighted.

It allows you to see the person’s face.
|-
| | Click on the second image.

Point to the highlighted image and the cross-section of the person’s brain.

Show the '''optic''' nerves extending from the eyes into different parts of the brain.
| | Click on the second image.

On being highlighted, it shows the cross-section of the person’s brain.

Observe the '''optic''' nerves extending from the eyes into different parts of the brain.
|-
| | '''Slide Number 8'''

'''Color Vision'''

[[Image:]]

'''Cone '''receptors of '''retina'''

'''Trichromatic''' colour vision ('''S ~B, M~G, L~R''')
| |
'''Cone''' cells are photoreceptors in the '''retina''' that are sensitive to a range of wavelengths.

The leaf absorbs all wavelengths of visible light except green light.

Green light of 520 nanometers wavelength is reflected and reaches the eye.

Here, it activates the '''M''' or '''gamma''' type '''cones'''.

This visual information is sent from '''cones''' to retinal ganglia via the '''optic''' nerve.

The two '''optic''' nerves meet and cross over each other at the '''optic chiasma'''.

Now called '''optic''' tracts, they enter and '''synapse''' in the thalamus.

They then continue to the '''primary visual cortex''' in the '''occipital lobe''' in the back of the brain.
|-
| |
| | Let us return to the '''simulation'''.
|-
| | Point to the ellipses above the person’s head.
| | Observe the ellipses above the person’s head.

They will be filled with the colour that the person sees.
|-
| | Point to the '''Play/Pause button''' at the bottom and the '''Step button''' next to it.

Point to the '''Reset button'''.
| | Note the '''Play/Pause button''' at the bottom and the '''Step button''' next to it.

The '''Reset button''' will take us back to the start.
|-
| | Point to the torch or flashlight in front of the person’s eyes.

Point to the red '''button''' on the flashlight.
| | Observe the torch or flashlight in front of the person’s eyes.

The red '''button''' on the flashlight will switch it off and on.
|-
| | Click on the first bulb above the flashlight.

Show white light.

Click on the red '''button''' of the flashlight.
| | Click on the first bulb above the flashlight.

This is white light.

Click on the red '''button''' of the flashlight.
|-
| | Point to the first highlighted image below the flashlight.

Point to the beam of light.
| | The first image below the flashlight is highlighted by default

This will show the light as a beam.
|-
| | Click on the second image below the flashlight.

Show the '''photons'''.
| | Click on the second image below the flashlight.

This will show the light in the form of '''photons'''.
|-
| | Click on the '''Pause button'''.

Point to the '''Step button''' that is now active.
| | Click on the '''Pause button'''.

Observe that the '''Step button''' becomes active.
|-
| | Click on the '''Step button'''.
| | Click on the '''Step button '''to see the''' photons '''move in a stepwise manner.
|-
| | Click on the first beam image below the flashlight.
| | Click on the first image to return to the beam from the flashlight.
|-
| | Point to the white ellipses above the person’s head.
| | Observe the white ellipses above the person’s head.

This means that the person sees white light coming out of the flashlight.
|-
| | Click on the second bulb above the flashlight.

Point to the '''slider Bulb Color'''.
| | Click on the second bulb above the flashlight.

Observe the '''slider Bulb Color '''containing all colours of the visible '''spectrum'''.
|-
| | Drag the '''Bulb Color slider''' from end to end.

Point to the colour on which the handle is placed and the colour of the light from the flashlight.

Point to the ellipses above the person’s head.

Drag the '''Bulb Color slider''' to red at the right end.
| | Drag the '''Bulb Color slider''' from end to end.

The colour on which the handle is placed indicates the colour of the light from the flashlight.

Observe that the ellipses above the person’s head fill with the same colour.

You can change the colour that the person sees by dragging the '''Bulb Color slider'''.

Drag the '''Bulb Color slider''' to red at the right end.
|-
| | Point to the '''toggle switch''' in front of the person.

Point to the connection between the '''toggle switch''' and the '''Filter Color slider'''.

Point to the '''Filter Color slider'''.
| | Note that there is a '''toggle switch''' in front of the person.

It is connected to the '''Filter Color slider'''.

Observe that the '''Filter Color slider''' also contains the colours of the visible '''spectrum'''.
|-
| | Click on the '''toggle switch'''.

Point to the '''filter''' in the path of the beam from the flashlight.
| | Click on the '''toggle switch'''.

Observe that a '''filter''' appears in the path of the beam from the flashlight.
|-
| | Drag the '''Filter Color slider''' from end to end.

Point to the colour on which the handle is placed and the '''filter'''.

Drag '''Filter Color slider''' to red at the right end.
| | Drag the '''Filter Color slider''' from end to end.

The colour on which the handle is placed indicates the colour of the '''filter'''.

Drag '''Filter Color slider''' to red at the right end.
|-
| | Point to the red '''filter''' and the red beam.

Point to the red ellipses above the person’s head.
| | The '''filter''' becomes red and transmits a red beam.

The person also sees red light.
|-
| | Drag the '''Bulb Color slider''' to violet.

Point to the violet light beam from the flashlight.

Point to the absence of any beam from the filter to the person’s eyes.

Point to the empty ellipses above the person’s head.
| | Drag the '''Bulb Color slider''' to violet.

Violet light comes out from the flashlight.

But with red as the '''Filter Color''', no light is transmitted to the person.

The person does not see any light.
|-
| | Drag the '''Filter Color slider''' to violet at the left end.

Point to the violet '''filter''' and violet light transmitted to the person.

Point to the violet ellipses above the person’s head.
| | Drag the '''Filter Color slider''' to violet at the left end.

Note the '''filter''' becomes violet and violet light is transmitted to the person.

The person now sees violet light.
|-
| | Drag the '''Bulb Color slider''' to red at the right end.

Point to the violet '''filter''' and the red light beam.

Point to the violet '''filter''' and the absence of the red light from the filter.

Point to the empty ellipses above the person’s head.
| | Drag the '''Bulb Color slider''' to red at the right end.

Observe how the '''filter''' remains violet but the light beam is now red.

But the violet '''filter''' does not transmit the red light.

The person sees no light.
|-
| | Drag both '''sliders''' to different colours to see whether light is transmitted to the person.
| | Drag both '''sliders''' to different colours to see whether light is transmitted to the person.
|-
| | Point to both '''sliders'''.
| | Observe how light is only transmitted when both '''sliders''' are set at the same or nearly same colour.

The '''filter''' subtracts all wavelengths and only transmits the wavelength of its own colour.

Transmission is weaker for wavelengths very close to and maximum for the filter’s colour.
|-
| | '''Slide Number 9'''

'''Assignment'''

White light from flashlight

Observe light transmission from filter
| | As an assignment,

Choose white light to come from the flashlight and observe the transmission of light from the filter.
|-
| | Click on the '''RGB Bulbs screen''' at the bottom of the '''interface'''.
| | Now, let us click on the '''RGB Bulbs screen''' at the bottom of the '''interface'''.
|-
| | Show the person facing the right.

Click on the second image which allows you to see the cross-section of the person’s brain.
| | Here, too, you can see the person facing the right.

Click on the second image which allows you to see the cross-section of the person’s brain.
|-
| | Point to the '''Play/Pause, Step and Reset buttons''' at the bottom of the '''interface'''.
| | The '''Play/Pause, Step and Reset buttons''' are all seen at the bottom of the '''interface'''.
|-
| | Show 3 '''sliders''' and three flashlights on the '''screen'''.

All '''sliders''' are set at the minimum levels.

Point to the red '''slider''', then the green one and then the blue '''slider'''.
| | Three '''sliders''' and three flashlights are seen on the '''screen'''.

All '''sliders''' are set at the minimum levels.

The uppermost one is a red '''slider''', the second green and the lowermost one is a blue '''slider'''.
|-
| | Drag the red '''slider''' to the maximum level.

Show red '''photons''' coming out of the flashlight next to the red '''slider'''.

Show the '''photons''' reaching the person’s eye.

Point to the red ellipses above the person’s head.
| | Drag the red '''slider''' to the maximum level.

Note how red '''photons''' come out of the flashlight next to the red '''slider'''.

The '''photons''' reach the person’s eye and the person sees red light.
|-
| | Drag the green '''slider''' to the maximum level.

Show red and green '''photons''' coming out of the flashlights next to the red and green '''sliders'''.

Point to the yellow ellipses above the person’s head.
| | Drag the green '''slider''' to the maximum level.

Note how red and green '''photons''' come out of the flashlights next to the red and green '''sliders'''.

Red and green add to give yellow.

The '''cones''' sensitive to red and green light are activated so that the person sees yellow.
|-
| | Click the '''Pause button'''.
| | Click the '''Pause button'''.
|-
| | Drag the blue '''slider''' to the maximum level.
| | Drag the blue '''slider''' to the maximum level.
|-
| | Click on '''Play button'''.

Point to the three beams and the white ellipses above the person’s head.
| | Click on '''Play button'''.

Observe how all three colours mix so that the person sees white.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 10'''

'''Summary'''

We have demonstrated,

'''Color Vision PhET simulation'''
| | In this '''tutorial''', we have demonstrated how to use the '''Color Vision PhET simulation'''.
|-
| | '''Slide Number 11'''

'''Summary'''

White light

Light of different colours from visible '''spectrum'''

Light and filters for different colours of visible '''spectrum'''

Red, green and blue light, separately or in combination
| | Using this '''simulation''', we looked at colour vision when the human eye sees:

White light

Light of different colours from the visible '''spectrum'''

Light and filters for different colours of the visible '''spectrum'''

Red, green and blue light, separately or in combination
|-
| | '''Slide Number 12'''

'''Assignment'''

[[Image:]]

'''RGB'''?

'''Color wheel'''

Colors for schemes: '''complementary, analogous, triadic, rectangle'''
| | As an '''assignment''',

Adjust the red, blue and green '''sliders''' to get these 6 colours.

Which '''RGB''' combination would you pick to design these schemes?

Look for an image of the '''colour wheel''' on the '''Internet'''.

Pick colour combinations for the following schemes.
|-
| |
| | Remember to look at the '''Additional material''' provided with this '''tutorial'''.
|-
| | '''Slide Number 13'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it

|-
| | '''Slide Number 14'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 15'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 16'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 17'''

'''Acknowledgemen'''t
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Curve-Fitting/English

2018-09-12T13:42:18Z

Vidhya:

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on ''' Curve Fitting'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

Demonstrate an interactive '''PhET simulation'''
| | In this tutorial, we will demonstrate '''Curve Fitting PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | The learner should be familiar with topics in high school mathematics.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Lines '''y=ax + b'''

'''Quadratic polynomials y = ax2+bx+c'''

'''Cubic polynomials y= ax3 + bx2 + cx + d'''

'''Quartic polynomials y =''' '''ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi squared statistic χr2''' and '''correlation coefficient r2'''
| | Using this '''simulation''' we will look at,

Lines

'''Quadratic polynomials'''

'''Cubic polynomials'''

'''Quartic polynomials'''

'''Reduced chi squared statistic''' and '''correlation coefficient r squared'''
|-
| |
| | Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Show the '''Downloads''' folder.
| | I have already downloaded '''Curve Fitting simulation''' to my '''Downloads''' folder.
|-
| | Press Ctrl+Alt+T to the terminal.

Type '''cd Downloads''' >> press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Point to the opened '''file''' format.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space curve hyphen fitting underscore en period jar'''.

'''File''' opens in the '''browser''' in html '''format'''.
|-
| | '''Cursor''' on the '''interface'''.
| | This is the '''interface''' for the '''Curve Fitting simulation'''.
|-
| | Point to the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

Point to '''Linear''' and '''Best Fit''' default selections.
| | Observe the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

In '''Functions''' box, '''Linear''' and '''Best Fit radio buttons''' are default selections.
|-
| | Click the '''Help button'''.
| | Let us click the '''Help button'''.
|-
| | Point to the legend for '''draggable error bars''' in the first quadrant.

Point to the data point bucket.
| | A legend for '''draggable error bars''' appears in the first quadrant.

The data points can be pulled out or put in the bucket.
|-
| | Point to the '''Best Fit equation''' in the 4th quadrant.

Point to the '''display''' boxes for '''a''' and '''b'''.

Point to the equation '''y = a + bx'''.

Point to '''r2'''.
| | In the fourth quadrant, '''Best Fit equation''' is seen with the '''display''' boxes for '''a''' and '''b'''.

The equation is '''y equals a plus bx'''.

Below the '''display''' boxes is the '''correlation coefficient r squared'''.
|-
| | Point to the '''χr2''' scale.

Point to the formula in the '''Help''' box.
| | In the 2nd and 3rd quadrants is a scale for the '''reduced chi squared statistic'''.

The formula for the '''chi squared statistic''' is given in the '''Help''' box.
|-
| | Point to the '''conditions for fit''' in the '''Help''' box.
| | Below the formula, we see the '''conditions for fit'''.

Good or very good fit of data with the equation is seen with a '''chi squared statistic''' of or below 1.
|-
| | Click on '''Hide Help'''.
| | Let us click on '''Hide Help''' to hide these boxes.
|-
| | Drag three data points out of the bucket.

Place them at '''(-10, -4)''', '''(-4, 4)''' and '''(5, 10)'''.

Place the mouse on the '''co-ordinates''' to show them.
| | Drag three data points out of the bucket.

Place them at '''-10 comma -4''', -'''4 comma 4''', and '''5 comma 10'''.

Placing the mouse on them will show their '''co-ordinates'''.
|-
| | Point to the equation '''y = 6.07 + 0.912 x'''.

Point to '''r2 '''<nowiki>= 0.9616. </nowiki>

| | Note that the equation for the best fit line drawn is '''y equals 6.07 plus 0.912 x'''.

The '''correlation coefficient r squared''' for the '''best fit line''' is 0.9616.

The closer the '''r squared''' value is to 1, the better is the prediction of '''variance''' in '''y''' from '''x'''.
|-
| | Point to '''χr2''' = 6.74 and the red bar.

Click on '''Help''' and to conditions for a poor fit.
| | Note that the '''reduced chi statistic''' is 6.74 but the bar is red.

Click on '''Help''' and note that this means that the fit is poor.
|-
| | Drag another data point and place it at '''(0, 11)''' on the '''y axis'''.

Point to the '''best fit line''', '''y = 7.51 + 1.004 x'''.
| | Let us drag another data point and place it at '''0 comma 11''' on the '''y axis'''.

Note that the '''best fit line''' becomes '''y equals 7.51 plus 1.004 x'''.
|-
| | Point to the slope of the '''best fit line''', 1.004.

Point to the '''y intercept''' of 7.51.

Point to the data point '''(0, 11)'''.

Point to '''r2'''<nowiki>= 0.8529. </nowiki>

| | The slope of the '''best fit line''' has increased slightly from 0.912 to 1.004.

The '''y intercept''' has also increased from 6.07 to 7.51.

The data point '''0 comma 11''' is further away from the '''best fit line''' than the other points.

Note how the '''r squared''' value decreases from 0.9616 to 0.8529.

The prediction of '''variance''' in '''y''' from '''x''' with this equation has become less reliable.
|-
| | Point to the '''χr2''' of 18.66.
| | Note also how the '''reduced chi squared statistic''' has increased from 6.74 to 18.66.
|-
| | Drag the data point from '''(0, 11)''' to '''(0, 6''').

Point to the equation '''y = 6.05 + 0.911 x'''.

Point to '''r2''' = 0.9635 and''' χr2''' of 3.37.
| | Drag the data point from '''0 comma 11''' to '''0 comma 6'''.

Note how the equation becomes '''y equals 6.05 plus 0.911 x'''.

The '''r squared''' value increases to 0.9635 and the '''reduced chi squared statistic''' falls to 3.37.
|-
| | Drag the data point from '''(-4, 4)''' to '''(-4, 3.5)'''.

The '''r2''' value increases to 0.9772 and '''χr2 '''falls to 2.12.

Point to the green bar.

Click on '''Help''' and to the green zone indicating good fit.

Click on '''Hide Help'''.
| | Drag the data point from '''-4 comma 4''' to '''-4 comma 3.5'''.

The '''r squared''' value increases to 0.9772.

The '''reduced chi squared statistic''' falls to 2.12.

The bar now becomes green.

Click on '''Help'''<nowiki>; the green zone shows good fit. </nowiki>

Click on '''Hide Help'''.

A true '''best fit line''' explains all the data and gives a good prediction of '''y''' values from '''x''' values.
|-
| | Click '''Adjustable Fit'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Show space where erased line was seen.

Point to the line that is now parallel to the '''x axis'''.

Point to the '''display''' boxes for '''a''' and '''b''' and to '''sliders a''' and '''b'''.

Point to the data points.

Point to the red bar and '''χr2'''.

Point to the '''r2''' value of 0.
| | Click '''Adjustable Fit radio button'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Observe how this erases the line drawn earlier.

A line parallel to the '''x axis''' is seen.

'''Slider a''' and '''b''' values will be displayed in the boxes.

The data points are still where we placed them.

But the '''reduced chi square statistic''' is very high and in the red zone.

And the '''r squared''' value is 0, meaning poor correlation.
|-
| | Click '''Best Fit '''again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit'''.

Now drag '''sliders a''' and '''b''' and point to the line.

Point to the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

Point to the line, '''r2''' (0.9709) and '''χr2''' (2.23).
| | Click '''Best Fit radio button''' again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit radio button'''.

Now drag '''sliders a''' and '''b''' from end to end.

Observe the effects of these changes on the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

The line looks like the '''best fit line''' we saw earlier.

Note '''r squared''' and the '''reduced chi squared statistic'''.
|-
| | Check '''Show deviations''' and click '''Best Fit'''.

Point to the vertical lines from the data points to the '''best fit line'''.
| | Check '''Show deviations''' and click '''Best Fit'''.

The vertical lines from the data points to the '''best fit line''' show the deviations from the line.
|-
| | Drag the data points at '''(-4, 3.5)''' and '''(0, 6)''' into the bucket.

Point to the line and the two points.

Point to '''r2 '''and '''χr2'''.

| | Drag the data points at '''-4 comma 3.5''' and '''0 comma 6''' into the bucket.

Note how the line now passes through the two points.

'''R squared''' approaches 1 and the '''reduced chi squared statistic''' becomes 0.

The fit has become too good because a line is defined by two points.

Without a third point, there is no question of the line being anything but the '''best fit line'''.
|-
| |
| | Now, we will look at some information for you to graph a '''quadratic polynomial'''.
|-
| | '''Slide Number 7'''

'''Quadratic polynomials'''

FIGURE

'''y = a + bx + cx2'''

Degree = 2; '''quadratic'''

Maximum 2 roots

'''(-9, 10), (-7, 2), (2.5, -2.5), (5, 10)'''

'''a''' = -7.89, '''b''' = 1.495, '''c''' = 0.396

'''r2''' = ?, '''χr2''' = ?

Adjustable Fit
| | '''Quadratic polynomials'''

'''Quadratic polynomials''' are of the form '''y equals a plus bx plus c x squared'''.

The degree of the '''polynomial''' is 2, hence, it is called '''quadratic'''.

The '''function''' can have a maximum of 2 roots.

Drag and place data points at the following '''co-ordinates'''.

'''-9 comma 10''', '''-7 comma 2''', '''2.5 comma -2.5''' and '''5 comma 10'''

Note the '''r squared''' and '''reduced chi squared statistic''' values. (0.9794, 4.23)

Also, click '''Adjustable Fit''' and see effects of '''a''', '''b''', '''c''' on the fit.
|-
| | Show the '''best fit'''graph for the '''quadratic polynomial'''.
| | This is what the '''best fit''' graph for this '''quadratic polynomial''' will look like.
|-
| | '''Slide Number 8'''

'''Cubic polynomials'''

FIGURE

'''y = a + bx + cx2 + dx3'''

Degree = 3; '''cubic'''

Maximum 3 roots

'''(-9, 10), (-7, 2), (-6, -4), (5, 10), (13, 2)'''

'''r2''' = ?,''' χr2''' = ?

Adjustable Fit
| | '''Cubic polynomials'''

Now, we will look at some information for you to graph a '''cubic polynomial'''.

Note the '''r squared''' and '''reduced chi squared statistic''' values.
|-
| | Show the '''best fit''' graph for the '''cubic polynomial'''.
| | This is what the '''best fit''' graph for this '''cubic polynomial''' will look like.
|-
| | '''Slide Number 9'''

'''Quartic polynomials'''

FIGURE

'''y = a + bx + cx2 + dx3 + ex4'''

Degree = 4; '''quartic'''

Maximum 4 roots

'''(-9, 10), (-7, 2), (-6, -4), (5, 10), (9, 3) (13, 2)'''

'''r2''' = ?, '''χr2''' = ?

Adjustable Fit
| | '''Quartic polynomials'''

Now, we will look at some information for you to graph a '''quartic polynomial'''.

Note the '''r squared''' and '''reduced chi squared statistic''' values.
|-
| | Show the '''best fit''' graph for the '''quartic polynomial'''.
| | This is what the '''best fit''' graph for this '''quartic polynomial''' will look like.
|-
| | '''Slide Number 10'''

'''Assignment'''
| | As an '''assignment''',

Change the data points and their number.

Follow the steps shown earlier to get '''best fit''' graphs for all the '''polynomials'''.
|-
| | '''Slide Number 11'''

'''Summary'''
| | In this '''tutorial''', we have demonstrated the

'''Curve Fitting PhET simulation'''
|-
| | '''Slide Number 12'''

'''Summary'''

Lines '''y=ax + b'''

'''Quadratic polynomials y= ax2+bx+c'''

'''Cubic polynomials y = ax3 + bx2 + cx + d'''

'''Quartic polynomials y = 'ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi square statistic χr2''' and '''correlation coefficient r2'''
| | Using this '''simulation''', we have looked at:

Lines

'''Quadratic polynomials'''

'''Cubic polynomials'''

'''Quartic polynomials'''

'''Reduced chi square statistic χr2''' and '''correlation coefficient r2'''
|-
| | '''Slide Number 13'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it

| | The video at the following link summarizes the '''Spoken Tutorial project'''.''' '''

Please download and watch it
|-
| | '''Slide Number 14'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificate courses to learn the use of open source software.

For more details, please write to us.
|-
| | '''Slide Number 15'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 16'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT''', MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay'''.

Thank you for joining.
|-
|}

PhET/C3/Natural-Selection/English

2018-09-06T05:37:24Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Natural Selection''', an '''interactive..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Natural Selection''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Natural Selection PhET simulation'''
| | In this '''tutorial''', we will demonstrate, '''Natural Selection, an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux 'OS ''' version 16.04

'''Java ''' version 1.8.0

'''Firefox Web Browser ''' version 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java ''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with biology and ecology.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

We will look at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
| |
Using this '''simulation''', we will look at a population of rabbits for effects,

Of mutations and selection factors

Of environment

On pedigree
|-
| |
| | Let us begin.
|-
| | '''Slide Number 6'''

'''Mutations'''

Mutations

Changes in phenotype may be visible

Dominant

Recessive
| |

'''Mutations'''

Mutations are alterations in the '''nucleotide''' sequence of any genetic element.

They may arise due to errors in '''DNA''' replication or due to '''DNA''' damage.

They are passed onto offspring.

Mutations may or may not change the observable traits or phenotype of an organism.

Inheritance of mutations can be dominant or recessive.
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Point to the '''file''' in '''Downloads folder'''.
| | I have already downloaded the '''Natural Selection simulation''' to my '''Downloads folder'''.
|-
| | Open the '''terminal''' by pressing '''Ctrl+Alt+T simultaneously'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space Natural-Selection_en.jar'''.

Point to the '''browser''' address.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space natural-selection_en.jar'''.

'''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Natural Selection simulation'''.
|-
| | Point to the rabbit hopping in the '''simulation''' panel.

Click '''Pause button''' at the bottom of the '''interface'''.
| | Observe the rabbit hopping in the '''simulation''' panel.

Click the '''Pause button''' at the bottom of the '''interface'''.
|-
| |
| | Now we will explore the '''interface'''.
|-
| | Point to each section in the '''interface'''.

Point to the '''Reset All button'''.
| | The '''interface''' has :

'''Simulation''' panel

On the left side,

'''Add Mutation'''

'''Edit Genes'''

In the middle,

'''Graph'''

'''Time until next generation progress bar'''

'''Play/Pause button, Step button'''

On the right side,

'''Selection Factor'''

'''Environment'''

'''Chart'''

Clicking on the '''Reset All button''' takes you back to the starting point.
|-
| | Under '''Selection Factor''', point to the default selection '''None'''.

Under '''Environment''', point to the default selection of '''Equator'''.

Under '''Chart''', point to the default selection of '''Population'''.
| | Let us keep the default settings:

'''None''' for '''Selection Factor'''

'''Equator''' for '''Environment'''

'''Population''' for '''Chart'''
|-
| | Point to the '''Population vs Time''' graph.

| | Observe the '''Population versus Time''' graph.

It shows the number of rabbits plotted on the '''y axis''' and time on the '''x axis'''.
|-
| | Click '''Play button''' at the bottom of the '''interface'''.
| | Click '''Play button''' at the bottom of the '''interface'''.
|-
| | Point to the black line in the '''Population''' graph.
| | In the graph, observe how the black line moves to the right.
|-
| | Keep clicking on '''Step button'''.
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
|-
| | Point to the '''simulation''' panel and the '''Game Over pop-up box'''.

Point to the text “'''All of the bunnies died!'''
| | If there are no rabbits left, you will see a '''Game Over pop-up box''' like this.

The text will read “'''All of the bunnies died!”'''
|-
| | Click on '''Play Again button'''.
| | Click on the '''Play Again button''' to resume the '''simulation'''.
|-
| | Click on '''Add a Friend''' to add another rabbit to the '''simulation'''.
| | Click on '''Add a Friend''' to add another rabbit to the '''simulation'''.
|-
| | Click on '''Pause''' at the bottom of the interface.
| | Click on '''Pause''' at the bottom of the interface.
|-
| | '''Slide Number 8'''

'''Generations of Progeny'''

[[Image:]]
| | '''Generations of Progeny'''

Observe the labels on the right and roman numerals on the left of each row.

Generation '''P''' is shown in row I (one).

The '''F1''' generation in row II (two) is made up of the progeny or children of generation P.

And so on until row V (five).
|-
| | Point to the graph.
| | We will allow the population to grow until F3.

This would be three steps after the mating pair.
|-
| | Click '''Play''' at the bottom of the interface.
| | Click '''Play''' at the bottom of the interface.
|-
| | Point to the '''progress bar'''.
| | Note how the '''progress bar''' is full when a new generation begins.

The '''progress bar''' starts moving to the left as a generation starts growing.
|-
| | Click '''Pause button'''.
| | Click '''Pause button'''.
|-
| | Click the second button at the top left corner of the graph to '''zoom out'''.
| | Click the second button at the top left corner of the graph to '''zoom out'''.

You can now see the height of the next step.
|-
| | Point to the black line showing the total number of rabbits.
| | The black line shows the total number of rabbits in the graph.
|-
| | Under '''Selection Factor''', click the '''Food radio button'''.
| | Under '''Selection Factor''', click the '''Food radio button'''.
|-
| | Under '''Add Mutation''', click the '''Long Teeth button'''.
| | Under '''Add Mutation''', click the '''Long Teeth button'''.
|-
| | Point to the '''text-box''', “'''Mutation coming'''” at the bottom of the '''simulation''' panel.

Point to the yellow triangle with the lightning to the left of the text.

Point to the picture of long teeth in the text-box.
| | A '''text-box''', “'''Mutation coming'''” appears at the bottom of the '''simulation''' panel.

Note the yellow triangle with the lightning inside.

This indicates a mutation.

You can see a picture of long teeth in the text-box.
|-
| | Under '''Edit Genes''', point to the rows next to the '''Teeth''' label.
| | Under '''Edit Genes''', the rows next to the '''Teeth''' label have become active.

The '''radio buttons''' can now be clicked.
|-
| | Point to each row with two options appearing under the '''Dominant''' and '''Recessive''' columns.

Point to the upper row with a picture of short teeth to the left.

Point to the lower row with a picture of long teeth to the left.
| | For each row, two options appear under the '''Dominant''' and '''Recessive columns'''.

To the left of each row are the pictures of long and short teeth.
|-
| | Point to default selections of dominant mutation for long teeth and recessive for short teeth.
| | Default selections are dominant mutation for long teeth and recessive for short teeth.
|-
| | Point to the '''progress bar''' and the graph.
| | Observe the '''progress bar''' and the graph after the mutation has been added.
|-
| | Point to the graph.

Keep clicking on the '''Step button'''.

Point to the interval between two narrow steps after the mutation in the graph.
| | Let us allow the population to grow for another three generations after the mutation.

Keep clicking on the '''Step button'''.

The interval between two narrow steps in the graph corresponds to a generation.
|-
| | Click on the second '''Zoom Out button''' until you see the steps.
| | Click on the second '''Zoom Out button''' in the graph until you see the steps.
|-
| | Point to differently colored lines appearing in the graph after mutation and food selection.
| | Observe how differently colored lines appear in the graph after mutation and food selection.
|-
| |
Point to the legend below the graph.
| | Note that the timing of mutations and selection factors will affect population growth.

The legend below the graph gives the different colors and what they mean.
|-
| | Point to dominant '''radio button''' selection next to long teeth.

Point to the magenta line for long teeth, the olive line for short teeth.
| | We introduced a dominant mutation for long teeth.

Let us look at the magenta line for long teeth, the olive line for short teeth.
|-
| | Point to upper olive line and a lower magenta line.
| | Initially, the olive line is above the magenta line.

The number of short-toothed rabbits is higher than that of the long-toothed rabbits.
|-
| | Point to upper magenta line and a lower olive line.
| | Later, the magenta line is above the olive line.

The number of long-toothed rabbits has increased relative to the short-toothed ones.
|-
| |
| | This means that long teeth help rabbits survive by eating the available food.
|-
| | '''Slide Number 9'''

'''Fur Mutation'''

Set up conditions to study effects of a fur mutation on survival

Brown fur dominant, white fur recessive

Selection factor = Wolves

Three generations after mutation
| | '''Fur Mutation'''

Set up conditions to study effects of a fur mutation on survival of rabbits.

Keep brown fur as the dominant mutation and white fur as the recessive mutation.

Choose wolves as the selection factor.

Allow the population to grow for another 3 generations after the mutation.
|-
| | Show the wolves move in and out killing the rabbits.

Show the brown rabbits that begin to appear after the mutation.
| | Observe how the wolves move in and out, killing the rabbits.

Brown rabbits begin to appear after the mutation was introduced.
|-
| | Show the '''radio button''' selections for dominant mutations for long teeth and brown fur.

Show the graph.
| | Note that we are still looking at effects of the dominant long teeth mutation besides brown fur.
|-
| | Point to the graph and '''simulation''' panel.
| | Compare numbers of rabbits having white and brown fur in the graph and '''simulation''' panel.
|-
| | Point to the red line above the cyan one.
| | Initially, there are more white rabbits than brown rabbits.
|-
| | Point to later steps showing the cyan line above the red one.
| | Later, the number of brown rabbits has increased relative to white ones.
|-
| |
| | At the equator, with wolves killing the rabbits, brown fur is an advantage for survival.

This strategy to blend with the environment is called '''camouflage'''.
|-
| |
| | What can you say about the numbers of long- and short-toothed rabbits?

Sometimes a mutation changes the phenotype of all rabbits.

If so, the graph will not compare the mutation versus the wild type (unmutated) phenotypes.

Here, there are more long-toothed rabbits than short-toothed ones.
|-
| | '''Slide Number 10'''

'''Tail Mutation'''

Set up conditions to study effects of a tail mutation on survival

Long tail dominant, short tail recessive

Selection factor = Wolves

Three generations after mutation
| | '''Tail Mutation'''

Set up conditions to study effects of a tail mutation on survival of rabbits.

Keep long tail as the dominant mutation and short tail as the recessive mutation.

Choose wolves as the selection factor.

Allow the population to grow for another 3 generations after the mutation.
|-
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
| | Keep clicking on the '''Step button''' to move the '''simulation''' along faster.
|-
| | Click on the 2nd '''Zoom Out button'''.
| | Click on the 2nd '''Zoom Out button'''.
|-
| | Point to the differently colored lines in the graph.
| | Note the number of rabbits with brown fur, long teeth and short tails.

It is higher than that of rabbits with white fur, short teeth and long tails.
|-
| |
| | Brown fur and short tails help escape from wolves.

Long teeth help survival by making it easier to eat vegetation.
|-
| | Under '''Chart''', select '''Pedigree'''.
| | Under '''Chart''', click '''Pedigree'''.
|-
| | Point to the text, “'''Click a Bunny'''” at the top.

Click on the rabbit at the left bottom corner of the '''simulation''' panel.

Point to the rabbit framed inside a blue rectangle.
| | Note the text, “'''Click a Bunny'''” at the top.

Let us click on the rabbit at the left bottom corner of the '''simulation''' panel.

Observe how the selected rabbit is framed inside a blue rectangle.
|-
| | Point to the '''pedigree chart''' for the rabbit framed in the blue rectangle.
| | The '''pedigree chart''' appears for the rabbit framed in the blue rectangle.
|-
| | Click on the top right '''button''' in the '''Pedigree''' window.
| | Click on the top right '''button''' in the '''Pedigree''' window.
|-
| | Point to the '''Pedigree''' window that has separated.

Point to the '''Population''' chart behind it.

Resize the '''Pedigree''' window.
| | The '''Pedigree''' window is separated and the '''Population''' chart appears behind it.

We can now resize the '''Pedigree''' window.
|-
| | Point to the color of the previous generations of rabbits.

Point to the red crosses on the rabbits.

Point to the yellow triangle with the lightning symbol inside.
| | Note the color of the previous four generations of rabbits above the selected rabbit.

Red crosses on the rabbits indicate that they are dead.

The yellow triangle with the lightning symbol inside indicates a mutation.

It shows that that rabbit underwent a mutation so its genotype and phenotype changed.
|-
| |
| | '''Pedigree''' analysis allows study of inheritance of genes based on data about phenotypes.
|-
| | Click repeatedly on the '''Step button'''.

Show the window where rabbits can be seen all over the planet Earth.
| | Click repeatedly on the '''Step button'''.

Rabbits can be seen all over the continents on the planet Earth.
|-
| | Point to the caption, “'''Bunnies have taken over the world!'''”
| | Observe the caption, “'''Bunnies have taken over the world!'''”

These are the long-term effects of the '''simulation''' under these conditions.
|-
| |
| | Do refer to '''Additional material''' provided with this '''tutorial'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 11'''

'''Summary'''

We have demonstrated,

'''Natural Selection PhET simulation'''
| | In this '''tutorial''', we have demonstrated how to use the '''Natural Selection PhET simulation'''.
|-
| | '''Slide Number 12'''

'''Summary'''

We looked at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
| | Using this '''simulation''', we looked at a population of rabbits for effects:

Of mutations and selection factors

Of environment

On pedigree
|-
| | '''Slide Number 13'''

'''Assignment'''

Observe the rabbits:

For effects of mutations and selection factors in '''Arctic''' environment

After switching mutations from dominant to recessive and ''vice versa''

For changes in '''pedigree''' under different conditions
| | As an '''assignment''', observe the rabbits:

For effects of mutations and selection factors in '''Arctic''' environment

After switching mutations from dominant to recessive and ''vice versa''.

For changes in '''pedigree''' under different conditions
|-
| | '''Slide Number 14'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 15'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 16'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

PhET/C3/Curve-Fitting/English

2018-09-04T08:23:41Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on ''' Curve Fitting'''. |- | | '''Slide Num..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on ''' Curve Fitting'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

Demonstrate an interactive '''PhET simulation'''
| | In this tutorial, we will demonstrate '''Curve Fitting PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | The learner should be familiar with topics in high school mathematics.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Lines '''y=ax + b'''

'''Quadratic polynomials y = ax2+bx+c'''

'''Cubic polynomials y= ax3 + bx2 + cx + d'''

'''Quartic polynomials y =''' '''ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi squared statistic χr2''' and '''correlation coefficient r2'''
| | Using this '''simulation''' we will look at,

Lines of the form '''y=ax + b'''

'''Quadratic polynomials y equals ax squared plus bx plus c'''

'''Cubic polynomials y equals ax cubed plus bx squared plus cx plus d'''

'''Quartic polynomials y equals ax raised to 4 plus bx cubed plus cx squared plus dx plus e'''

'''Reduced chi squared statistic''' and '''correlation coefficient r squared'''
|-
| | '''Slide Number 6'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' ℝ, index ''n'' is a positive '''integer''', ''0 ≤ r ≤n, then,''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

'''Reminder:''''' nC1 = n!/[1! (n-1)!]''
| | '''Binomial Theorem'''

'''''a''''' and '''''b''''' are '''real numbers''', '''index''' '''''n''''' is a '''positive integer'''.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''a''' plus '''b''' raised to '''n''' can be expanded as shown.
|-
| |
| | Let us begin.
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Show the '''Downloads''' folder.
| | I have already downloaded '''Curve Fitting simulation''' to my '''Downloads''' folder.
|-
| | Press Ctrl+Alt+T to the terminal.

Type '''cd Downloads''' >> press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Point to the opened '''file''' format.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space curve hyphen fitting underscore en period jar'''.

'''File''' opens in the '''browser''' in html '''format'''.
|-
| | '''Cursor''' on the '''interface'''.
| | This is the '''interface''' for the '''Curve Fitting simulation'''.
|-
| | Point to the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

Point to '''Linear''' and '''Best Fit''' default selections.
| | Observe the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

In '''Functions''' box, '''Linear''' and '''Best Fit radio buttons''' are default selections.
|-
| | Click the '''Help button'''.
| | Let us click the '''Help button'''.
|-
| | Point to the legend for '''draggable error bars''' in the first quadrant.

Point to the data point bucket.
| | A legend for '''draggable error bars''' appears in the first quadrant.

The data points can be pulled out or put in the bucket.
|-
| | Point to the '''Best Fit equation''' in the 4th quadrant.

Point to the '''display''' boxes for '''a''' and '''b'''.

Point to the equation '''y = a + bx'''.

Point to '''r2'''.
| | In the fourth quadrant, '''Best Fit equation''' is seen with the '''display''' boxes for '''a''' and '''b'''.

The equation is '''y equals a plus bx'''.

Below the '''display''' boxes is the '''correlation coefficient r squared'''.
|-
| | Point to the '''χr2''' scale.

Point to the formula in the '''Help''' box.
| | In the 2nd and 3rd quadrants is a scale for the '''reduced chi squared statistic'''.

The formula for the '''chi squared statistic''' is given in the '''Help''' box.
|-
| | Point to the '''conditions for fit''' in the '''Help''' box.
| | Below the formula, we see the '''conditions for fit'''.

Good or very good fit of data with the equation is seen with a '''chi squared statistic''' of or below 1.
|-
| | Click on '''Hide Help'''.
| | Let us click on '''Hide Help''' to hide these boxes.
|-
| | Drag three data points out of the bucket.

Place them at '''(-10, -4)''', '''(-4, 4)''' and '''(5, 10)'''.

Place the mouse on the '''co-ordinates''' to show them.
| | Drag three data points out of the bucket.

Place them at '''-10 comma -4''', -'''4 comma 4''', and '''5 comma 10'''.

Placing the mouse on them will show their '''co-ordinates'''.
|-
| | Point to the equation '''y = 6.07 + 0.912 x'''.

Point to '''r2 '''<nowiki>= 0.9616. </nowiki>

| | Note that the equation for the best fit line drawn is '''y equals 6.07 plus 0.912 x'''.

The '''correlation coefficient r squared''' for the '''best fit line''' is 0.9616.

The closer the '''r squared''' value is to 1, the better is the prediction of '''variance''' in '''y''' from '''x'''.
|-
| | Point to '''χr2''' = 6.74 and the red bar.

Click on '''Help''' and to conditions for a poor fit.
| | Note that the '''reduced chi statistic''' is 6.74 but the bar is red.

Click on '''Help''' and note that this means that the fit is poor.
|-
| | Drag another data point and place it at '''(0, 11)''' on the '''y axis'''.

Point to the '''best fit line''', '''y = 7.51 + 1.004 x'''.
| | Let us drag another data point and place it at '''0 comma 11''' on the '''y axis'''.

Note that the '''best fit line''' becomes '''y equals 7.51 plus 1.004 x'''.
|-
| | Point to the slope of the '''best fit line''', 1.004.

Point to the '''y intercept''' of 7.51.

Point to the data point '''(0, 11)'''.

Point to '''r2'''<nowiki>= 0.8529. </nowiki>

| | The slope of the '''best fit line''' has increased slightly from 0.912 to 1.004.

The '''y intercept''' has also increased from 6.07 to 7.51.

The data point '''0 comma 11''' is further away from the '''best fit line''' than the other points.

Note how the '''r squared''' value decreases from 0.9616 to 0.8529.

The prediction of '''variance''' in '''y''' from '''x''' with this equation has become less reliable.
|-
| | Point to the '''χr2''' of 18.66.
| | Note also how the '''reduced chi squared statistic''' has increased from 6.74 to 18.66.
|-
| | Drag the data point from '''(0, 11)''' to '''(0, 6''').

Point to the equation '''y = 6.05 + 0.911 x'''.

Point to '''r2''' = 0.9635 and''' χr2''' of 3.37.
| | Drag the data point from '''0 comma 11''' to '''0 comma 6'''.

Note how the equation becomes '''y equals 6.05 plus 0.911 x'''.

The '''r squared''' value increases to 0.9635 and the '''reduced chi squared statistic''' falls to 3.37.
|-
| | Drag the data point from '''(-4, 4)''' to '''(-4, 3.5)'''.

The '''r2''' value increases to 0.9772 and '''χr2 '''falls to 2.12.

Point to the green bar.

Click on '''Help''' and to the green zone indicating good fit.

Click on '''Hide Help'''.
| | Drag the data point from '''-4 comma 4''' to '''-4 comma 3.5'''.

The '''r squared''' value increases to 0.9772.

The '''reduced chi squared statistic''' falls to 2.12.

The bar now becomes green.

Click on '''Help'''<nowiki>; the green zone shows good fit. </nowiki>

Click on '''Hide Help'''.

A true '''best fit line''' explains all the data and gives a good prediction of '''y''' values from '''x''' values.
|-
| | Click '''Adjustable Fit'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Show space where erased line was seen.

Point to the line that is now parallel to the '''x axis'''.

Point to the '''display''' boxes for '''a''' and '''b''' and to '''sliders a''' and '''b'''.

Point to the data points.

Point to the red bar and '''χr2'''.

Point to the '''r2''' value of 0.
| | Click '''Adjustable Fit radio button'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Observe how this erases the line drawn earlier.

A line parallel to the '''x axis''' is seen.

'''Slider a''' and '''b''' values will be displayed in the boxes.

The data points are still where we placed them.

But the '''reduced chi square statistic''' is very high and in the red zone.

And the '''r squared''' value is 0, meaning poor correlation.
|-
| | Click '''Best Fit '''again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit'''.

Now drag '''sliders a''' and '''b''' and point to the line.

Point to the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

Point to the line, '''r2''' (0.9709) and '''χr2''' (2.23).
| | Click '''Best Fit radio button''' again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit radio button'''.

Now drag '''sliders a''' and '''b''' from end to end.

Observe the effects of these changes on the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

The line looks like the '''best fit line''' we saw earlier.

Note '''r squared''' and the '''reduced chi squared statistic'''.
|-
| | Check '''Show deviations''' and click '''Best Fit'''.

Point to the vertical lines from the data points to the '''best fit line'''.
| | Check '''Show deviations''' and click '''Best Fit'''.

The vertical lines from the data points to the '''best fit line''' show the deviations from the line.
|-
| | Drag the data points at '''(-4, 3.5)''' and '''(0, 6)''' into the bucket.

Point to the line and the two points.

Point to '''r2 '''and '''χr2'''.

| | Drag the data points at '''-4 comma 3.5''' and '''0 comma 6''' into the bucket.

Note how the line now passes through the two points.

'''R squared''' approaches 1 and the '''reduced chi squared statistic''' becomes 0.

The fit has become too good because a line is defined by two points.

Without a third point, there is no question of the line being anything but the '''best fit line'''.
|-
| |
| | Now, we will look at some information for you to graph a '''quadratic polynomial'''.
|-
| | '''Slide Number 9'''

Quadratic polynomials

FIGURE

y = a + bx + cx2

Degree = 2; quadratic

Maximum 2 roots

(-9, 10), (-7, 2), (2.5, -2.5), (5, 10)

a = -7.89, b = 1.495, c = 0.396

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Quadratic polynomials'''

'''Quadratic polynomials''' are of the form '''y equals a plus bx plus c x squared'''.

The degree of the '''polynomial''' is 2, hence, it is called '''quadratic'''.

The '''function''' can have a maximum of 2 roots.

Drag and place data points at the following '''co-ordinates'''.

'''-9 comma 10''', '''-7 comma 2''', '''2.5 comma -2.5''' and '''5 comma 10'''

Note the '''r squared''' and '''reduced chi squared statistic''' values. (0.9794, 4.23)

Also, click '''Adjustable Fit''' and see effects of '''a''', '''b''' and '''c''' on the fit.
|-
| | Show the '''best fit'''graph for the '''quadratic polynomial'''.
| | This is what the '''best fit''' graph for this '''quadratic polynomial''' will look like.
|-
| |
| | Now, we will look at some information for you to graph a '''cubic polynomial'''.
|-
| | '''Slide Number 10'''

Cubic polynomials

FIGURE

y = a + bx + cx2 + dx3

Degree = 3; cubic

Maximum 3 roots

(-9, 10), (-7, 2), (-6, -4), (5, 10), (13, 2)

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Cubic polynomials'''

'''Cubic polynomials''' are of the form '''y equals a plus bx plus c x squared plus d x cubed'''.

The degree of the '''polynomial''' is 3, hence, it is called '''cubic'''.

The '''function''' can have a maximum of 3 roots.

Drag and place data points at the following '''co-ordinates'''.

'''9 comma 10, -7 comma 2, -6 comma -4, 5 comma 10''' and '''13 comma 2'''

Note the '''r squared''' and '''reduced chi squared statistic''' values.

Also, click '''Adjustable Fit''' and see effects of '''a, b, c''' and '''d''' on the fit.
|-
| | Show the '''best fit''' graph for the '''cubic polynomial'''.
| | This is what the '''best fit''' graph for this '''cubic polynomial''' will look like.
|-
| |
| | Now, we will look at some information for you to graph a '''quartic polynomial'''.
|-
| | '''Slide Number 11'''

Quartic polynomials

FIGURE

y = a + bx + cx2 + dx3 + ex4

Degree = 4; quartic

Maximum 4 roots

(-9, 10), (-7, 2), (-6, -4), (5, 10), (9, 3) (13, 2)

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Quartic polynomials'''

'''y equals a plus bx plus c x squared plus d x cubed plus e x raised to 4''' is a '''quartic polynomial'''.

The degree of the '''polynomial''' is 4, hence, it is called '''quartic'''.

The '''function''' can have a maximum of 4 roots.

Drag and place data points at the following '''co-ordinates'''.

'''-9 comma 10, -7 comma 2, -6 comma -4, -3 comma -8, 5 comma 10, 9 comma 3 and 13 comma 2'''

Note the '''r squared''' and '''reduced chi squared statistic''' values.

Also, click '''Adjustable Fit''' and see effects of '''a, b, c, d''' and '''e''' on the fit.
|-
| | Show the '''best fit''' graph for the '''quartic polynomial'''.
| | This is what the '''best fit''' graph for this '''quartic polynomial''' will look like.
|-
| | '''Slide Number 12'''

'''Assignment'''
| | As an '''assignment''',

Change the data points and their number.

Follow the steps shown earlier to get '''best fit''' graphs for all the '''polynomials'''.
|-
| | '''Slide Number 13'''

'''Summary'''
| | In this '''tutorial''', we have demonstrated the

'''Curve Fitting PhET simulation'''
|-
| | '''Slide Number 14'''

'''Summary'''
| | Using this '''simulation''', we have looked at:

Lines of the form '''y=ax + b'''

'''Quadratic polynomial functions''' of the form '''y= ax2+bx+c'''

'''Cubic polynomial functions''' of the form '''y = ax3 + bx2 + cx + d'''

'''Quartic polynomial functions''' of the form '''y = 'ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi square statistic χr2''' and '''correlation coefficient r2'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it

| | The video at the following link summarizes the '''Spoken Tutorial project'''.''' '''

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificate courses to learn the use of open source software.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT''', MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay'''.

Thank you for joining.
|-
|}

PhET/C2/Equation-Grapher/English

2018-09-03T06:21:31Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on ''' Equation Grapher'''. |- | | '''Slide N..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on ''' Equation Grapher'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate '''PhET simulation,'''

'''Equation Grapher'''
| | In this tutorial, we will demonstrate '''Equation Grapher PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learner should be familiar with topics in high school mathematics.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Lines '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y = ax2 + bx + c'''

| | Using this '''simulation''' we will look at,

Lines of the form '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y equals ax squared plus bx plus c'''
|-
| | '''Slide Number 6'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' ∈ ℝ, index ''n'' is a positive '''integer''', ''0 ≤ r ≤n, then,''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

'''Reminder:''''' nC1 = n!/[1! (n-1)!]''
| | '''Binomial Theorem'''

'''''a''''' and '''''b''''' are '''real numbers''', '''index''' '''''n''''' is a '''positive integer'''.

'''''r''''' lies between 0 and '''''n'''''. Then,

'''Binomial theorem''' states that '''a''' plus '''b''' raised to '''n''' can be expanded as shown.
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the simulation.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| |
| | I have already downloaded '''Equation Grapher''' simulation to my '''Downloads''' folder.
|-
| | Press Ctrl+Alt+T to open the terminal.

Type '''cd Downloads''' >> press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Point to the opened '''file format'''.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Press '''Enter'''.

'''File''' opens in the '''browser''' in '''html format'''.
|-
| | Cursor on the '''interface'''.
| | This is the '''interface''' for the '''Equation Grapher''' simulation.
|-
| | Point to the '''interface'''.

Point to the first quadrant.

Point to the quadratic function, '''y = ax2 + bx + c'''.

Point to the '''sliders''' and '''display boxes'''.

Point to the red '''Zero button'''.

Point to the green '''Save button'''.

Point to the equation.
| | The interface shows '''Cartesian co-ordinate system''' of '''x''' and '''y axes'''.

The first '''quadrant''' contains:

The red-colored''' quadratic equation''', '''y equals ax squared plus bx plus c'''

Three '''sliders''' and '''display boxes''' under '''ax2, bx''' and '''c'''

The '''sliders''' allow you to change the values of the '''coefficients, a, b''' and '''c'''.

The '''display boxes''' show these values and can be used to enter values.

A red '''Zero button''' to set all '''sliders''' at '''0'''

A green '''Save button''' to save the '''equation'''

The updated '''equation''' in red is shown below the '''sliders'''.
|-
| | Point to the fourth '''quadrant'''.

Point to the '''quadratic equation'''.

Point to the '''check boxes'''.

Point to the violet, green and blue terms.

| | The fourth '''quadrant''' contains the

'''quadratic''' equation '''y = ax2+bx+c'''

three '''check boxes''' under '''ax2, bx''' and '''c'''

Note that the '''ax squared''' term is violet, '''bx''' is green and '''c''' is blue.
|-
| | In the first '''quadrant''', in the '''display box''' below '''ax2''', type 1.

Point to the '''slider''' under '''ax2'''.
| | In the first '''quadrant''', in the '''display box''' below '''ax squared''', type 1.

Observe how the '''slider''' under '''ax squared''' also moves to 1.
|-
| | Point to the red parabola and origin '''(0,0)'''.
| | A red parabola with vertex at origin '''0 comma 0''' appears in the window.

It opens upwards.
|-
| | In the first '''quadrant''', in the '''display box''' below '''bx''', type 1
| | In the first '''quadrant''', in the '''display box''' below '''bx''', type 1.
|-
| | Point to the parabola.
| | Observe how the parabola shifts downwards and to the left.
|-
| | In the first '''quadrant''', in the display '''box''' below '''c''', type 1.
| | In the first '''quadrant''', in the '''display box''' below '''c''', type 1.
|-
| | Point to the parabola.
| | Observe how the parabola moves upwards.
|-
| | In the fourth '''quadrant''', check the box below the violet colored '''ax2''' term.
| | In the fourth '''quadrant''', check the box below the violet colored '''ax squared''' term.
|-
| | Point to the violet and red parabolas.

Point to the equation.
| | A violet parabola appears next to the red parabola.

This violet parabola corresponds to the '''y equals ax squared''' part of the red equation.
|-
| | Point to the equation, '''y = x2''', in the first '''quadrant'''.
| | The equation for the violet parabola is '''y equals x squared'''.
|-
| | Now, check the box below the green '''bx''' term in the fourth '''quadrant'''.
| | Now, in the fourth '''quadrant''', check the box below the green '''bx''' term.
|-
| | Point to the green line.

Point to the origin '''(0,0)'''.

Point to the equation, '''y = x''', in the first '''quadrant'''.
| | Observe how a green line appears in the '''Cartesian plane'''.

It passes through the origin '''0 comma 0'''.

It corresponds to the '''x''' term and its equation is '''y equals x'''.
|-
| | Now, check the box below the blue '''c''' term in the fourth '''quadrant'''.
| | Now, check the box below the blue '''c''' term in the fourth '''quadrant'''.
|-
| | Point to the blue line.

Point to the equation, '''y=c'''.
| | Observe how a blue line appears in the '''Cartesian plane'''.

Its equation is '''y equals c''' and it corresponds to the constant term of the equation.
|-
| | Click on the green '''Save button'''.
| | Click on the green '''Save button'''.
|-
| | Point to the blue saved parabola, '''y = x2+ x + 1'''.
| | This saves the equation '''y equals x squared plus x plus 1'''.
|-
| | Change values for '''a, b '''and''' c'''.

Point to the '''sliders''' and the '''display boxes''' below the terms.

Point to the graphs.
| | Change the values for '''a, b '''and''' c'''.

You can either use the '''sliders''' or type in the '''display boxes''' below the terms.

Observe the effects of these changes on the graphs.
|-
| | Point to the blue saved parabola, '''y = x2 + x + 1'''.
| | Note that as you change '''a, b '''and''' c''', you can still see the parabola '''y equals x squared plus x plus 1'''.

This is because we saved this equation.
|-
| |
| | Save other graphs that you want to compare to see the effects of '''a, b '''and''' c'''.

You can only save one equation at a time.
|-
| | Point to the blue '''Erase button'''.
| | Note that after you have saved an equation, a blue '''Erase button''' appears.

This will erase the saved equation.
|-
| | Click on the red '''Zero button'''.

| | Click on the red '''Zero button'''.

This resets all coefficients '''a, b '''and''' c''' to 0.
|-
| | '''Slide Number 8'''

'''Assignment'''
| | As an '''assignment''', compare the parabolas graphed for different combinations of:

'''a'''<0 and '''a'''>0

'''b'''<0 and '''b'''>0

'''c'''<0 and '''c'''>0
|-
| | '''Slide Number 9'''

'''Summary'''
| | In this '''tutorial''', we have demonstrated the

'''Equation Grapher PhET simulation'''
|-
| | '''Slide Number 10'''

'''Summary'''
| | Using this '''simulation''', we have looked at:

Lines of the form '''y = bx + c''' and '''y = c'''

'''Quadratic polynomials''' '''y = ax2 + bx + c'''
|-
| | '''Slide Number 11'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using spoken tutorials and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 14'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''
|-
| | '''Slide Number 15'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT''', MHRD, Government of India

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay'''.

Thank you for joining.
|-
|}

PhET/C2/Trig-tour/English

2018-08-31T13:57:47Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Trig Tour''', an '''interactive PhE..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Trig Tour''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

We will demonstrate,

'''Trig Tour PhET simulation'''
| | In this tutorial, we will demonstrate '''Trig Tour''', an '''interactive PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' v 1.8.0

'''Firefox Web Browser''' v 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' version 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | Learners should be familiar with trigonometry.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Construct right triangles for a point moving around a unit circle

Calculate trigonometric ratios, '''''cos, sin''''' and '''''tan''''', of angle '''ϴ''' (theta)

Graph ϴ versus '''''cos, sin''''' and '''''tan'' '''functions''' of '''ϴ''' along '''x''' and '''y axes'''
| |
Using this '''simulation''' we will learn how to,

Construct right triangles for a point moving around a unit circle

Calculate trigonometric ratios, '''''cos, sin''''' and '''''tan''''', of angle theta

Graph '''theta''' versus '''''cos, sin''''' and '''''tan''''' '''functions''' of '''theta''' along '''x''' and '''y axes'''
|-
| |
| | Let us begin.
|-
| | '''Slide Number 6'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Point to the file in '''Downloads folder'''.
| | I have already downloaded the '''Trig Tour simulation''' to my '''Downloads folder'''.
|-
| | Right click on '''trig-tour_en.html''' file.

Select '''Open With Firefox Web Browser''' option.

Point to the '''browser''' address.
| | To open the '''simulation''', right click on the '''trig-tour_en.html''' file.

Select the '''Open With Firefox Web Browser''' option.

The file opens in the '''browser'''.
|-
| | '''Cursor''' on the '''interface'''.
| | This is the '''interface''' for the '''Trig Tour''' simulation.
|-
| | Point to each box in the '''interface'''.

Point to the '''reset button'''.
| | The '''interface''' has four boxes:

'''Values'''

'''Unit circle'''

'''Functions, special angles, labels and grid'''

'''Graph'''

The '''reset button''' takes you back to the starting point.
|-
| | Check '''Special angles, Labels''' and '''Grid''' in '''Functions''' box.
| | In the '''Functions '''box,''' check '''Special angles, Labels, Grid''' and click '''cos'''.
|-
| | '''Slide Number 7'''

'''Cosine function'''

'''Cosine''' is ratio of lengths of adjacent side to hypotenuse.

'''Cosine''' is '''x co-ordinate''' of a point moving around unit circle.

Center of unit circle is origin (0,0).

'''cos(ϴ)''' = '''x'''/'''radius''' = '''x/1'''
| |

'''Cosine''' of an angle is the ratio of the lengths of the adjacent side to the hypotenuse.

'''Cosine''' value is the '''x co-ordinate''' of a point moving around a unit circle.

The center of this unit circle is the origin '''0 comma 0'''.

'''cosin theta''' is '''x''' divided by radius and hence, is '''x''' for the unit circle.
|-
| | Point to the '''Unit Circle''' box.
| | A unit circle is drawn in a '''Cartesian coordinate system''' with '''x''' and '''y axes''' in the '''Unit Circle''' box.
|-
| | Point to the red point.

Point to the blue arrow.
| | A red point is seen at the circumference of the circle on the '''x-axis'''.

A blue arrow is seen along the '''x-axis''' pointing to the red point.

This corresponds to a radius of 1 for the unit circle.
|-
| | Point to the '''Values''' box.
| | The '''Values''' box contains important values.
|-
| | Point to the '''degrees''' and '''radians radio buttons'''.
| | The '''angle ϴ''' (theta) can be given in '''degrees''' or '''radians'''.
|-
| | Check '''degrees radio button'''.
| | Click the '''degrees radio button'''.
|-
| | Point to '''(x,y) = (1,0)''' and '''angle = 0º''' in the '''Values box'''.

Point to the red point in '''Unit Circle''' box.
| | '''x comma y''' are '''co-ordinates 1 comma 0''' of the red point at '''angle theta equals''' 0 degrees .
|-
| | Point to '''cosϴ = x/1 = 1''' in '''Values''' box.
| | When '''angle theta equals''' 0 '''degrees''', '''x co-ordinate''' of the red point is 1.
|-
| | Point to the red point in the '''Graph''' box.
| | '''x-axis''' of the graph shows '''angle theta'''.

'''y-axis''' of the graph shows the amplitude of the '''cos theta''' function.

At an '''angle theta''' of 0 '''degrees''', '''cos theta''' is 1.

The red point is at the highest amplitude of 1.
|-
| | Check '''degrees radio button'''.
| | In the '''Values''' box, click the '''radians radio button'''.
|-
| | Point to the graph.
| | x axis of the '''theta''' vs '''cos theta''' graph is converted into '''radians'''.

Remember that '''pi radians''' are equal to '''180 degrees'''.

One full rotation of 360 '''degrees''' is equal to 2 '''pi radians'''.

Again, click the '''degrees radio button'''.
|-
| | Point to the empty circles.

In the '''Functions''' box, uncheck '''Special Angles'''.

| | You can see empty circles on the unit circle.

In the '''Functions''' box, uncheck '''Special Angles'''.

Observe how the empty circles disappear.
|-
| | Again, check '''Special Angles'''.
| | Again, check '''Special Angles'''.
|-
| | Point to the '''Special Angles'''.
| | These circles are angles made by the red point with the '''x-axis''' as it moves along the circle.

Important angles have been chosen as '''Special Angles'''.
|-
| | In the '''Unit Circle''', drag red point counter-clockwise (CCW) to the next '''special angle'''.

Point to '''angle = 30º''' in '''Values''' box and to the red point in the '''Unit Circle''' box.
| | In the '''Unit Circle''', drag the red point counter-clockwise (CCW) to the next '''special angle'''.

The red point has moved 30º in the counter-clockwise direction along the circle.
|-
| | Point to the '''Values''' box.

Point to the unit circle.
| | In the '''Values''' box, '''x comma y''' is the squareroot of 3 divided by 2 comma half.

In the unit circle, according to '''Pythagoras’ theorem''', '''x squared''' plus '''y squared''' is 1.
|-
| | Point to the unit circle.
| | Two square lengths in the '''Cartesian plane''' is equal to 1 as radius of unit circle is 1.

'''y''' covers only 1 square length and hence, is half.

'''x''' covers 1 full and almost three-fourths of a second square.
|-
| | Point to the '''Values''' box.
| | The squareroot of 3 divided by 2 is 0.866.

This is the value of '''x'''.
|-
| | Point to the graph.
| | Look at the graph.

The red point has moved to 30 degrees along the '''cos function'''.
|-
| | Check '''radians radio button''' in the '''Values''' box.

Point to the '''Values''' box and the Graph.
| | In the '''Values''' box, click '''radians radio button'''.

This converts 30 degrees into '''pi''' divided by 6 radians for '''theta''' in the '''Values''' box.
|-
| | '''Slide Number 8'''

'''Sine function'''

'''Sine''' is ratio of lengths of opposite side to hypotenuse.

'''Sine''' is '''y-co-ordinate''' of a point moving around unit circle.

'''sin(ϴ) = y/radius = y/1'''
| |

'''Sine function'''

'''Sine''' of an angle is the ratio of the lengths of the opposite side to the hypotenuse.

'''Sine''' value is the '''y-co-ordinate''' of the point moving around the same unit circle.

'''Sin theta''' is '''y''' divided by radius and hence, is '''y''' for the unit circle.
|-
| | Drag the red point back to the x axis.
| | Drag the red point back to the x axis.
|-
| | In the '''Functions''' box, click '''sin'''.
| | In the '''Functions''' box, click '''sin'''.
|-
| | Check '''degrees radio button'''.
| | Click the '''degrees radio button'''.
|-
| | Point to the '''Values''' box.

Point to the unit circle.
| | As seen earlier, '''x comma y''' are '''1 comma 0'''.

Note the definitions of '''sin theta''' given earlier.

When '''angle theta''' is 0 '''degrees''', the '''y co-ordinate''' of the red point is 0.
|-
| | Point to the graph.
| | The graph shows '''angle theta''' on the '''x-axis''' and the amplitude of the '''sin theta function''' on the '''y-axis'''.
|-
| | Point to the graph.
| | At '''angle theta''' of 0 '''degrees''', as '''sin theta''' is 0, the red point has amplitude 0.
|-
| | In the '''Unit Circle,''' drag red point CCW to the next '''special angle''' 30 '''degrees'''.
| | In the '''Unit Circle,''' drag the red point CCW to the next '''special angle''' 30 '''degrees'''.
|-
| | Point to the '''Values''' box.
| | In the '''Values''' box, note that '''x comma y''' is squareroot of 3 divided by 2 comma half.

Remember how you can calculate these.
|-
| | Point to the graph.
| | In the graph, the red point has moved to 30 '''degrees''' along the '''sine function'''.

Its amplitude is 0.5 or half.
|-
| | '''Slide Number 9'''

'''Tangent function'''

'''Tangent''' is ratio of lengths of opposite to adjacent sides.

'''tan(ϴ) = sinϴ/cosϴ = y/x'''
| |
'''Tangent function'''

'''Tangent''' of an angle is the ratio of the lengths of opposite side to adjacent side.

'''Tan theta''' is the ratio of '''sin theta''' to '''cos theta''' and to '''y''' divided by '''x'''.
|-
| | Drag the red point back to the '''x-axis''', that is to (1,0).
| | Drag the red point back to the '''x-axis''' that is to '''1 comma 0'''.
|-
| | Click '''tan''' in '''Functions''' box.
| | In the '''Functions''' box, click '''tan'''.
|-
| | Point to '''co-ordinates''' in '''Values''' box.
| | When angle '''theta''' is 0, '''tan theta''' is ratio of the '''y co-ordinate''' 0 to '''x co-ordinate''' 1 that is 0.
|-
| | Point to the graph.
| | The graph shows angle '''theta''' on the '''x-axis''' and the amplitude of the '''tan theta function''' on the '''y-axis'''.

At '''angle theta''' 0, as '''tan theta''' is 0, the red point has amplitude of 0.
|-
| | In the '''Unit Circle''', drag red point CCW to the '''special angle''' 90 '''degrees''' on the '''y-axis'''.
| | In the '''Unit Circle''', drag the red point CCW to the '''special angle''' 90 '''degrees''' on the '''y-axis'''.
|-
| | Point to the '''Values''' box.
| | In the '''Values''' box, '''x comma y''' has become '''0 comma 1'''.
|-
| | Point to the '''Values''' box.
| | Note that '''tan theta''' is '''plus or minus infinity''' in the '''Values''' box.
|-
| |
| | Now look at the graph.
|-
| | Point to the graph.
| | The red point has moved to 90 '''degrees''' where '''tan theta''' now falls on the vertical dotted line.

This dotted line is the '''vertical asymptote''' of the '''function'''.

It represents the value of '''x''' which the '''function''' approaches but never touches.

Here, the '''function''' increases without bound towards '''infinity''' in both directions.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 10'''

'''Summary'''
| | In this '''tutorial''', we have demonstrated how to use the '''Trig Tour Phet simulation'''.
|-
| | '''Slide Number 11'''

'''Summary'''

Construct right triangles for a point moving around unit circle

Calculate trigonometric ratios, '''''cos, sin''''' and '''''tan''''', of angle '''ϴ'''

Graph '''ϴ''' versus '''''cos, sin''''' and '''''tan''''' '''functions''' along '''x''' and '''y axes'''
| |

Using this '''simulation''', we have learnt to:

Construct right triangles for a point moving around a unit circle

Calculate trigonometric ratios, '''''cos, sin''''' and '''''tan''''', of angle '''theta'''

Graph '''theta''' versus '''''cos, sin''''' and '''''tan''''' '''functions''' of '''theta''' along '''x''' and '''y axes'''
|-
| | '''Slide Number 12'''

'''Assignment'''

Observe:

'''Cosine, sine''' and '''tangent''' values for all '''special angles'''

'''Cos, sin, tangent''' graphs

Relationship between ratios for supplementary angles (sum of 180 '''degrees''')
| | As an '''assignment''', observe:

'''Cosine, sine '''and''' tangent''' values for all '''special angles'''

'''Cosine, sine '''and''' tangent''' graphs

Relationship between ratios for supplementary angles

The sum of supplementary angles is 180 '''degrees'''.
|-
| | '''Slide Number 13'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it

| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it
|-
| | '''Slide Number 14'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using spoken tutorials and gives certificates on passing online tests.

For more details, please write to us.
|-
| | '''Slide Number 15'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 16'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 17'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT''', '''MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Hyperbola/English

2018-08-16T10:00:21Z

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Conic Sections - Hyperbola''' |- | | ''..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Conic Sections - Hyperbola'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will:

Study standard equations and parts of hyperbolae

Learn how to use '''GeoGebra''' to construct a hyperbola
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux OS''' version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this tutorial, you should be familiar with

'''GeoGebra''' interface

Conic Sections in geometry

For relevant tutorials, please visit our website.
|-
| | '''Slide Number 5'''

'''Hyperbola'''

[[Image:]]

Consider two fixed points '''F1'''and '''F2''' called foci.

A hyperbola is the locus of points whose difference of distances from these foci is constant.
| | Consider two fixed points '''F1''' and '''F2''' called foci.

A hyperbola is the locus of points whose difference of distances from these foci is constant.

In the image, observe that foci of a hyperbola lie along the transverse axis.''' '''

They are equidistant from the center which lies on the conjugate axis.

'''2b''' is the length of the conjugate axis.

'''c''' is the distance of each focus from the center.

The conjugate axis is perpendicular to the transverse axis.

The hyperbola intersects the transverse axis at the vertices''' A '''and''' B'''.

'''a''' is the distance of each vertex from the center.

The latus recti pass through the foci.

They are perpendicular to the transverse axis.
|-
| |
| | Be careful to distinguish lengths from letters used for '''sliders''', circles and hyperbolae.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct a hyperbola in '''GeoGebra'''.

I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click twice in '''Graphics''' view.

Point to '''A''' and '''B'''.
| | Click on '''Point''' tool and click twice in '''Graphics''' view.

This creates two points ''' A''' and '''B''', which will be the foci of our hyperbola.
|-
| | Right-click on '''A''' and choose '''Rename''' option.
| | Right-click on '''A''' and choose the '''Rename''' option.
|-
| | Type '''F1''' in the '''New Name''' field >> click '''OK '''button.
| | In the '''New Name''' field, type '''F1''' and click '''OK'''.
|-
| | This will be one of our foci, '''F1'''.
| | This will be one of our foci, '''F1'''.
|-
| |
| | Let us rename point''' B''' as '''F2'''.
|-
| | Click on '''Slider''' tool >> click in '''Graphics '''view.
| | Click on '''Slider''' tool and click in '''Graphics '''view.
|-
| | Point to the ''' Slider dialog box''' in '''Graphics''' view.
| | A '''Slider dialog-box''' appears in '''Graphics''' view.
|-
| | Stay with the default '''Number''' radio-button selection.

In the '''Name''' field, type '''k'''.
| | Stay with the default '''Number''' selection.

In the '''Name''' field, type '''k'''.
|-
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1, click '''OK'''.
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1, click '''OK'''.
|-
| | Point to '''slider k'''.
| | This creates a number '''slider''' named '''k'''.
|-
| | Drag the '''slider k''' from 1 to 10.
| | Using this '''slider''', '''k''' can be changed from 0 to 10.
|-
| | Point to''' slider k'''.
| | '''k''' will be the difference of the distances of any point on the hyperbola from the foci, '''F1''' and '''F2'''.
|-
| | Drag '''slider k''' to 4.
| | Drag '''slider k''' to 4.
|-
| |
| | We will create another '''number slider''' named “'''a'''”.
|-
| | Point to '''slider a'''.
| | Its '''Min''' value is 0, '''Max''' value is 25,'''increment''' is 0.1.
|-
| | Click on '''Circle with Center and Radius''' tool >> click on '''F1'''.
| | Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
|-
| | Point to '''text box'''; type '''a''' and click '''OK'''.
| | A '''text-box''' appears; type '''a''' and click '''OK'''.
|-
| | Drag '''a''' to a value between 2 and 3.
| | Drag '''a''' to a value between 2 and 3.
|-
| | Point to the circle '''c''' with center '''F1''' and '''radius a'''.
| | A circle '''c''' with center '''F1''' and radius '''a''' appears.
|-
| | Drag '''slider a''' to 5.
| | Drag '''slider a''' to 5.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
|-
| | Click in '''Graphics''' view.
| | Click twice in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool to move the background.
| | Click on '''Move Graphics View''' to move the background as required.
|-
| | Click again on '''Circle with Center and Radius''' tool.

Click on '''F2'''.
| | Click again on '''Circle with Center and Radius''' tool and click on ''' F2'''.
|-
| | Point to the '''text-box'''; type '''a-k''' and click '''OK'''.
| | In the '''text-box''', type '''a minus k''' and click '''OK'''.
|-
| | Point to the circle '''d''' with center '''F2''' and radius '''a-k'''.
| | Circle '''d''' with center '''F2''' and radius '''a minus k''' appears.
|-
| | Click again on '''Circle with Center and Radius''' tool.

Click on '''F2'''.
| | Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
|-
| | Point to the '''text box'''; type '''a+k''' and click '''OK'''.
| | In the '''text-box''', type '''a plus k''' and click '''OK'''.
|-
| | Point to the circle '''e''' with center '''F2''' and '''radius a+k'''.
| | Circle '''e''' with center '''F2''' and radius '''a plus k''' appears.
|-
| | Set '''slider k''' between 1 and 2, '''slider a''' between 3 and 4.
| | Set '''slider k''' between 1 and 2, '''slider a''' between 3 and 4.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on circles '''c'''and '''d'''and circles '''c'''and '''e'''.
| | Under '''Point''', click on '''Intersect'''.

Then click on circles '''c''' and '''d''' and circles '''c''' and '''e'''.
|-
| | Point to points '''A''', '''B''', '''C''' and '''D'''.
| | This creates points '''A''', '''B''', '''C''' and '''D'''.
|-
| | Under '''Line''' tool, click on '''Segment''' tool >> click on points '''A''' and '''F1'''

Click on points '''A''' and '''F2.'''
| | Under '''Line''', click on '''Segment''' and click on points '''A''' and '''F1''' to join them.

Then click on points '''A''' and '''F2''' to join them.
|-
| | Click on Segment tool >> join the points B and F1 >> join B and F2.
| | Similarly, using '''Segment''' tool, join '''B''' and '''F1''' as well as '''B''' and '''F2'''.
|-
| | Click on '''Move''' tool.
| | Click on '''Move'''.
|-
| | Double click on segment '''AF1''' >> '''Object Properties'''.
| | Double click on segment '''AF1''' and click on '''Object Properties'''.
|-
| | Point to the left panel and to highlighted segment '''AF1'''.
| | In the left panel, segment '''AF1''' is already highlighted.
|-
| | Holding '''Ctrl''' Key down, click and highlight segments '''AF2, BF1''' and '''BF2'''.
| | Holding '''Ctrl''' Key down, click and highlight segments '''AF2, BF1''' and '''BF2'''.
|-
| | Under '''Basic''' tab, make sure '''Show Label''' is checked.
| | Under the '''Basic''' tab, make sure '''Show Label''' is checked.
|-
| | Choose '''Name and Value''' from the dropdown menu next to it.
| | Choose '''Name and Value''' from the dropdown menu next to it.
|-
| | Under '''Color''' tab, select red.
| | Under the '''Color''' tab, select red.
|-
| | Under '''Style''' tab, select '''dashed line style'''.
| | Under the '''Style''' tab, select '''dashed line style'''.
|-
| | Close '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Click on '''Move''' tool >> move the labels properly in '''Graphics''' view.
| | Click on '''Move''' if it is not highlighted.

Move the labels to see them properly in '''Graphics''' view.
|-
| |
| | Now, let us carry out the same steps for segments '''CF1''', '''CF2''', '''DF1''' and '''DF2''' but make them blue.
|-
| | Click on '''Move''' tool >> move the labels properly in '''Graphics''' view.
| | Click on '''Move''' if it is not highlighted.

And move the labels to see them properly in '''Graphics''' view.
|-
| | Right-click on points '''A, B, C''' and '''D''' and select '''Trace On''' option.
| | Right-click on points '''A, B, C''' and '''D''' and select '''Trace On''' option.
|-
| | Set '''slider k''' at 1.

Drag '''slider a''' to both ends of '''slider'''.
| | Set '''slider k''' at 1.

Drag '''slider a''' to both ends of the '''slider'''.

Set
|-
| | Set '''slider k''' at 2 and drag '''slider a''' to both ends.
| | First '''k''' at 2.
|-
| | Set '''slider k''' at 3 and drag '''slider a''' to both ends.
| | Then at 3.
|-
| | Set '''slider k''' at 5 and drag '''slider a''' again to both ends.
| | At 5.
|-
| | Set '''slider k''' at 10 and drag '''slider a''' yet again to both ends.
| | And finally at 10.
|-
| | Point to the traces of hyperbolae for different values of '''a''' and '''k'''.
| | Observe the traces of hyperbolae for the different values of '''a''' and '''k'''.
|-
| |
| | Let us look at the equations of hyperbolae.
|-
| | In the '''input bar''', type '''(x-h)^2/a^2-(y-k)^2/b^2=1''' and press '''Enter'''.

To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
| | Open a new '''GeoGebra''' window.

In the '''input bar''', type the following line describing the difference of two fractions equal to 1.

To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

For the 1st '''fraction''', type the '''numerator''' as '''x minus h in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the '''denominator''' of the 1st '''fraction''' as '''a caret 2''' followed by '''minus'''.

For the 2nd fraction, type the '''numerator''' as '''y minus k in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the '''denominator''' of the 2nd '''fraction''' as '''b caret 2''' followed by '''equals sign 1'''.

Press '''Enter'''.
|-
| | Point to the popup window.
| | A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders a, b, h''' and '''k'''.
| | This creates number '''sliders''' for '''h, a, k''' and ''' b'''.
|-
| | Double-click on the '''sliders''' to see their properties.
| | By default, they go from minus 5 to 5 and are set at 1.

You can double-click on the '''sliders''' to see their properties.
|-
| | Point to the hyperbola in '''Graphics''' view.
| | A hyperbola appears in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and then in '''Graphics '''view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' and then in '''Graphics '''view.
|-
| | Click on '''Move Graphics View''' and drag '''Graphics''' view to see hyperbola properly.
| | Click on '''Move Graphics View''' and drag '''Graphics''' view to see the hyperbola properly.
|-
| | Point to the equation for hyperbola '''c''' in '''Algebra''' view.
| | In '''Algebra''' view, note the equation for hyperbola '''c'''.
|-
| | Drag boundary to left of '''Slider '''tool see equation properly.
| | Drag the boundary to see it properly.
|-
| | Point to the equations appearing in '''Algebra''' view.
| | Keep track of the equations appearing in '''Algebra''' view as you the drag the '''sliders''' from end to end.
|-
| | Point to hyperbola '''c'''.
| | You will see the effects on the shape of hyperbola '''c'''.
|-
| | Place the cursor over the equation in '''Algebra''' view.
| | Place the cursor over the equation in '''Algebra''' view.
|-
| | Point to '''slider a''' and hyperbola '''c'''.

Drag '''slider a'''.
| | Note that '''a''' is associated with the '''x minus h squared''' component of the equation.

It controls the horizontal movement of hyperbola '''c'''.
|-
| | Point to '''slider b''' and hyperbola '''c'''.

Drag '''slider b'''.
| | Associated with the '''y minus k squared''' component is '''b'''.

It controls the vertical movement of hyperbola '''c'''.
|-
| | Point to hyperbola '''c'''.
| | Note that the '''transverse axis''' of hyperbola '''c''' is horizontal like the '''x axis.'''
|-
| | Drag '''slider a''' to 2, leaving '''b''' at 1.
| | Drag '''slider a''' to 2, leaving '''b''' at 1.
|-
| | Point to hyperbola '''c'''.
| | When '''a''' is greater than '''b''', the arms of the hyperbola are closer to the '''x axis'''.
|-
| | Point to equation of hyperbola '''c'''.
| | Note the equation of the hyperbola.
|-
| | Drag boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Point to '''slider''' at 2.

Drag '''slider b''' to 3.
| | With '''slider a''' at 2, drag '''slider b''' to 3.
|-
| | Point to '''sliders a''' and '''b''', and hyperbola '''c'''.
| | When '''a''' is less than '''b''', the arms of the hyperbola stretch closer to the '''y axis'''.
|-
| | Point to '''equation''' of hyperbola '''c'''.
| | Note the equation of hyperbola '''c'''.
|-
| | Drag boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | With '''slider a''' at 2, drag '''slider b''' to 1.
| | With '''slider a''' at 2, drag '''slider b''' back to 1.
|-
| | Click in and drag '''Graphics''' view to see hyperbola properly.
| | Click in and drag '''Graphics''' view to see the hyperbola properly.
|-
| | Type '''Focus(c)''' in the '''input bar''' >> press '''Enter'''.
| | In the '''input bar''', type '''Focus c in parentheses''' and press '''Enter'''.
|-
| | Point to '''A''' and '''B''' in '''Graphics '''view'''.'''

Point to the '''co-ordinates''' in '''Algebra''' view.
| | Two foci,''' A''' and '''B''', are mapped in '''Graphics''' view.

Their '''coordinates''' appear in '''Algebra''' view.
|-
| | Type '''Center(c)''' in the '''input bar''' and press '''Enter'''.
| | In the '''input bar''', type '''Center c in parentheses''' and press '''Enter'''.
|-
| | Point to point '''C''' in '''Graphics''' view

Point the '''co-ordinates''' in '''Algebra''' view.
| | Center, point '''C''', appears in '''Graphics''' view.

Its '''co-ordinates''' appear in '''Algebra''' view.
|-
| | Point to the '''co-ordinates''' '''(h, k)''' of center '''C''' in '''Algebra''' view.
| | Note that the center has the '''coordinates h comma k'''.
|-
| | Drag '''sliders h''' and '''k''' from end to end.
| | Drag '''sliders h''' and '''k''' from end to end.
|-
| | Point to hyperbola '''c'''.
| | Note the effects on hyperbola '''c'''.
|-
| | Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
| | In the '''input bar''', type '''Vertex c in parentheses''' and press '''Enter'''.
|-
| | Point to '''D''' and '''E'''.

Let us drag '''slider a''' to ~0.5 so we can see the vertices clearly.
| | Vertices, '''D''' and '''E''', appear on hyperbola '''c'''.

Let us drag '''slider a''' so we can see the vertices clearly.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag the boundary to see '''Graphics''' view properly.
|-
| | Click in and drag '''Graphics''' view to see hyperbola.
| | Click in '''Graphics''' view and drag the background so you can see the hyperbola properly.
|-
| | Drag '''slider a''' back to 2.
| | Drag '''slider a''' back to 2.
|-
| | Click on '''Text''' tool under '''Slider''' tool and click in '''Graphics''' view.
| | Under '''Slider''', click on '''Text''' and click in '''Graphics''' view.
|-
| | Point to the '''text box'''.

In '''Edit''' field, type the following '''text'''.
| | A text-box opens up.

In the '''Edit''' field, type the following text.
|-
| | Show '''Slide Number 6'''.
| | Press '''Enter''' after each line to go to the next line.

Click '''OK'''.
|-
| | '''Slide Number 6'''

'''Text box for hyperbola c'''

transverse axis 2a = 4

c = 2.24

conjugate axis 2b = 2.018

e = 1.12

latus rectum = 1.018
| | '''Text box for hyperbola c'''

transverse axis '''2a''' equals 4

'''c''' equals 2.24

conjugate axis '''2b''' equals 2.018

'''e''' equals 1.12

latus rectum equals 1.018
|-
| |
| | Refer to '''additional material''' providedwith this '''tutorial''' for these calculations.
|-
| | Click on '''Move Graphics View''' and drag the background so you can see the hyperbola.
| | Click on '''Move Graphics View''' and drag the background so you can see the hyperbola.
|-
| | Uncheck equation '''c''' and all points and text generated for hyperbola '''c''' in '''Algebra''' view.
| | Uncheck equation '''c''' and all points and text generated for '''hyperbola c''' in '''Algebra''' view.
|-
| | Show screenshots of hyperbola '''d''' for '''a=2, b=1''' and '''a=2, b=3'''.
| | Follow the earlier steps to construct hyperbola '''d''' for these two conditions.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 8'''

'''Summary'''
| | In this tutorial, we have learnt how to use '''GeoGebra''' to:

Construct a hyperbola

Look at standard equations and parts of hyperbolae
|-
| | '''Slide Number 9'''

'''Assignment'''

Construct hyperbolae with:

Foci (± 3, 0) and vertices (± 2, 0)

Foci (0, ± 5) and vertices (0, ± 3)

Find their centres and equations.

Calculate eccentricity and length of latus recti, transverse and conjugate axes.
| | As an '''assignment,'''

Construct hyperbolae with the following foci and vertices.

Find all these values.
|-
| | '''Slide Number 10'''

'''Assignment'''

Find the coordinates of the foci, vertices and eccentricity for these hyperbolae.

Also calculate length of the latus rectum and transverse and conjugate axes.

'''x2/16 - y2/9 = 1'''

'''49y2 – 16x2 = 784'''
| | Find all these values for these hyperbolae.
|-
| | '''Slide Number 11'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 14'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project '''is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English

2018-08-13T06:59:25Z

Vidhya:

{|border=1
|| '''Visual Cue'''
|| '''Narration'''

|-
|| '''Slide Number 1'''

'''Title Slide'''
|| Welcome to this tutorial on '''Conic Sections - Ellipse'''
|-
|| '''Slide Number 2'''

'''Learning Objectives'''
|| In this tutorial, we will learn,

'''Standard equations''' and parts of an ellipse

To use '''GeoGebra''' to construct an ellipse
|-
|| '''Slide Number 3'''

'''System Requirement'''
|| Here, I am using:

'''Ubuntu Linux ''' OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
|| '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
|| To follow this '''tutorial''', you should be familiar with

'''GeoGebra''' interface

Conic sections in geometry

For relevant tutorials, please visit our website.
|-
|| '''Slide Number 5'''

'''Ellipse'''

An ellipse is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Point to the center '''O''', foci '''F1''' and '''F2'''.

Point to vertices '''A''' and '''B''' at the ends of the major axis '''AB.'''

Point to '''C''' and '''D''' at the ends of the minor axis '''CD'''.

Point to 2 latus recti passing through foci and the two axes.
||'''Ellipse'''

An ellipse is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Observe the centre '''O''', foci '''F1''' and '''F2'''.

Vertices '''A''' and '''B''' are at the ends of the major axis '''AB'''.

'''Co-vertices C''' and '''D''' are at the ends of the minor axis '''CD'''.

Two latus recti pass through the foci.

Axes lengths '''2a''' and '''2b''' and distance between the foci '''2c''' are shown in the figure.
|-
||
|| Be careful to distinguish length, from letters used for '''sliders''', circles and ellipses.
|-
|| Show the '''GeoGebra''' window.
|| Let us construct an ellipse in '''GeoGebra'''.

I have already opened the '''GeoGebra''' interface.
|-
|| Click on '''Point''' tool >> click twice in '''Graphics''' view.
|| Click on '''Point''' tool and click twice in '''Graphics''' view.
|-
|| Point to '''A''' and '''B'''.
|| This creates two points '''A''' and '''B''', which will be the foci of our ellipse.
|-
|| Right-click on '''A''' >> choose '''Rename''' option.
|| Right-click on '''A''' and choose the '''Rename''' option.
|-
|| Type '''F1''' in the '''New Name''' field >> click '''OK''' button.
|| In the '''New Name''' field, type '''F1''' and click '''OK'''.
|-
|| Point to the focus '''F1'''.
|| This will be one of our foci, '''F1'''.
|-
|| Point to '''B'''.
|| Let us rename '''B''' as '''F2'''.
|-
|| Click on '''Slider''' tool >> click in '''Graphics''' view.
|| Click on '''Slider''' tool and click in '''Graphics''' view.
|-
|| Point to the '''Slider dialog box''' in '''Graphics''' view.
|| '''Slider dialog box''' appears in '''Graphics''' view.
|-
|| Stay with the default '''Number''' selection >> in the '''Name''' field, type '''k'''.
|| Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
|-
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
|-
|| Point to '''slider k'''.
|| This creates a number '''slider''' named '''k'''.
|-
|| Drag '''slider k'''.
|| '''Slider k''' can be changed from 0 to 10.

'''k''' will be the sum of the distances of any point on the ellipse from the foci '''F1''' and '''F2'''.
|-
||point to '''a'''.
|| We will create another '''number slider''' named '''a'''.
|-
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
|| Its '''Min''' value is 0, '''Max''' value is 10, '''increment''' is 0.1."
|-
|| Click on '''Circle with Center and Radius''' tool >> click on '''F1'''.
|| Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
|-
|| Point to the '''text box'''; type '''a''' >> click '''OK'''.
|| A '''text-box''' appears; in the radius '''Name''' field, type '''a''' and click '''OK'''.
|-
|| Point to the circle with centre '''F1''' >> radius '''a'''.
|| A circle '''c''' with centre '''F1''' and radius '''a''' appears.
|-
|| Drag '''slider a''' to 2 >> '''slider k''' to 5.
|| Drag '''slider a''' to 2 and '''slider k''' to 5.
|-
|| Click on '''Circle with Center and Radius''' tool >> click on '''F2'''.
|| Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
|-
|| Point to the '''text box'''; type '''k-a''' and click '''OK'''.
|| In the '''text box''' that appears, type '''k minus a''' and click '''OK'''.
|-
|| Point to circle '''d''' with center '''F2''' >> radius '''k-a''' in '''Graphics''' view.
|| A circle '''d''' with center '''F2''' and radius '''k minus a''' appears in '''Graphics''' view.
|-
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|-
|| Click on '''Move Graphics View''' >> drag '''Graphics''' view.
|| Click on '''Move Graphics View''' and drag '''Graphics''' view.
|-
|| Click on '''Intersect''' tool under '''Point''' tool.

Click on the '''intersections''' of the two '''circles'''.
|| Under '''Point''', click on '''Intersect''' tool.

Click on the two circles '''c''' and '''d'''.
|-
|| Point to points '''A''' and '''B'''.
|| This creates points '''A''' and '''B'''.
|-
|| Click on '''Segment''' tool under '''Line''' tool.
|| Under '''Line''', click on '''Segment''' tool.
|-
|| Click on points '''F1''' >> '''A''' to join.
|| Click on points '''F1''' and '''A''' to join them.
|-
|| Click on points '''A''' >> '''F2''' to join.
|| Next, click on points '''A''' and '''F2''' to join them.
|-
|| Click on '''Move''' tool.
|| Click on '''Move'''.
|-
|| Double click on '''Segment AF1'''.
|| Double click on '''Segment AF1'''.
|-
|| Click '''Object Properties''' to open the '''Preferences''' dialog box.
|| Click on '''Object Properties''' to open the '''Preferences''' dialog box.
|-
|| Point to Segment '''AF1''' highlighted in the left panel.
|| Segment '''AF1''' is already highlighted in the left panel.
|-
|| Holding '''Ctrl''' key down, click >> highlight Segment '''AF2'''.
|| Holding '''Ctrl''' key down, click and highlight Segment '''AF2''' as well.
|-
|| Under '''Basic''' tab >> select '''Show Label'''.
|| Under the '''Basic''' tab, make sure that '''Show Label''' is selected.
|-
|| Pull down the '''drop down menu''' next to the '''Show Label''' check box.

Select '''Name and Value'''.
|| Pull down the '''drop down menu''' next to the '''Show Label''' check box.

Select '''Name and Value'''.
|-
|| Under '''Color''' tab >> select Red.
|| Under the '''Color''' tab, select red.
|-
|| Under '''Style''' tab >> choose '''dashed line style'''.
|| Under the '''Style''' tab, choose '''dashed line style'''.
|-
|| Close the '''Preferences''' dialog box.
|| Close the '''Preferences''' dialog box.
|-
|| Follow earlier steps to draw Segments '''BF1''' and '''BF2'''.

Make them '''dashed''' and blue.
|| Draw Segments '''BF1''' and '''BF2'''.

Make them '''dashed''' and blue.
|-
|| Point to '''Move''' tool.

If not highlighted, click on it.

Move the labels so you can see them properly.
|| Make sure that the '''Move''' tool is highlighted.

Move the labels so you can see them properly.
|-
|| Point to the length labels next to '''AF1, AF2, BF1, BF2''' and '''slider k'''.
|| Note that the sum of the segment lengths from both foci to each intersection point is equal to '''k'''.
|-
|| Right-click on '''A''' and '''B''' >> check '''Trace On''' option.
|| Right-click on '''A''' and '''B''' and check '''Trace On''' option.
|-
|| Uncheck circles '''c''' and '''d''' in '''Algebra''' view.
|| In '''Algebra''' view, uncheck circles '''c''' and '''d''' to hide the circles.
|-
|| Right-click on '''slider a''' >> check '''Animation On''' option.
|| Right-click on '''slider a''' and check '''Animation On''' option.
|-
|| Right-click on '''slider k''' >> check '''Animation On''' option.
|| Next, right-click on '''slider k''' and check '''Animation On''' option.
|-
|| Point to the traces of '''A''' and '''B''' and to '''A''' and '''B'''.
|| Note the locus of points traced by '''A''' and '''B'''.

These traced points are all equidistant from '''F1''' and '''F2''', the foci.

They lie on ellipses for which '''F1''' and '''F2''' are foci.
|-
|| Right-click on '''sliders a''' >> '''k''' >> uncheck '''Animation On''' option.
|| Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
|-
|| Drag '''sliders a''' and '''k'''.
|| Drag '''sliders a''' and '''k''' to different values to see more traces of ellipses.
|-
|| Set '''slider k''' between 9 and 10 >> '''slider a''' between 5 and 6.
|| Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
|-
|| Point to length labels next to '''AF1''' and '''AF2''' and '''sliders a''' and '''k'''.
|| Note that for a given value of '''k''', as '''a''' changes, lengths of '''Segments AF1''' and '''AF2''' change.

But their sum remains equal to the value of '''k'''.
|-
|| Point to '''BF1''' and '''BF2''' and '''sliders a''' and '''k'''.
|| Note the same fact for Segments '''BF1''' and '''BF2'''.
|-
|| Click in and move '''Graphics''' view slightly to erase the trace points.
|| Click in and move '''Graphics''' view slightly to erase the trace points.
|-
|| Click on the '''Move''' tool >> move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
|| Click on '''Move''' tool and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
|-
|| Point to '''sliders a''' and '''k'''.
|| Values can be changed on '''sliders a''' and '''k''' to see various ellipses.
|-
|| Open a new '''GeoGebra''' window.
|| Let us look at the equations of ellipses in a new '''GeoGebra''' window.
|-
|| In '''input bar''', type '''(x-h)^2/a^2+(y-k)^2/b^2=1''' >> press '''Enter'''.
|| In the '''input bar''', type the following line describing the sum of two fractions equal to 1.

To type the '''caret symbol''', hold '''Shift''' key down and press 6.

For the 1st fraction, type the numerator as '''x minus h in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the denominator of the 1st fraction as '''a caret 2''' followed by '''plus'''.

For the 2nd fraction, type the numerator as '''y minus k in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the denominator of the 2nd fraction as '''b caret 2''' followed by '''equals sign 1'''.

Press '''Enter'''.
|-
|| Point to the pop-up window.
|| A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
|-
|| Click on '''Create Sliders'''.
|| Click on '''Create Sliders'''.
|-
|| Point to sliders '''a, b, h''' and '''k'''.
|| This creates number '''sliders''' for '''h, a, k''' and '''b'''.

By default, they go from minus 5 to 5 and are set at 1.

You can double-click on the '''sliders''' to see their properties.
|-
|| Point to circle '''c''' and center '''(h, k)''' in '''Graphics''' view.
|| A circle '''c''', a special case of an ellipse, appears in '''Graphics''' view.

Center '''h comma k''' is at '''1 comma 1''' and radius is 1 unit.
|-
|| Point to equation for circle '''c''' in '''Algebra''' view.
|| In '''Algebra''' view, note the equation for circle '''c'''.
|-
|| Drag boundary to see it properly.
|| Drag the boundary to see it properly.
|-
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|-
|| Click on '''Move Graphics View''' tool >> drag '''Graphics''' view.
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
|-
|| Point to the equations in '''Algebra''' view as you change '''a''' and '''b''' on '''sliders'''.
|| Keep track of the equations in '''Algebra''' view as you change '''a''' and '''b''' on the '''sliders'''.
|-
|| Place '''cursor''' on equation '''c''' in '''Algebra''' view.
|| Place the '''cursor''' on equation '''c''' in '''Algebra''' view.
|-
|| Point to '''slider a'''.

Drag '''slider a''', leave between -2 and -3.
|| '''a''' is associated with the '''x minus h squared''' component of the equation.

Observe how '''a''' controls the horizontal axis of the ellipse.
|-
|| Point to '''slider b'''.

Drag '''slider b''', leave at 5.
|| Associated with the '''y minus k squared''' component is '''b'''.

Observe how '''b''' controls the vertical axis of the ellipse.
|-
|| Drag '''slider a''' to 2 and '''b''' to 1.
|| Drag '''slider a''' to 2 and '''b''' to 1.
|-
|| Point to major axis of the ellipse and along the '''x axis'''.

Point to equation of the ellipse.
|| When '''a''' is greater than '''b''', the major axis of the ellipse is horizontal.

Note the equation of the ellipse.
|-
|| Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.
|| In the '''input bar''', type '''Focus c in parentheses''' and press '''Enter'''.
|-
|| Point to '''A''' and '''B''' mapped in '''Graphics''' view and their '''co-ordinates''' in '''Algebra''' view.
|| Two foci, '''A''' and '''B''', are mapped in '''Graphics''' view and their '''coordinates''' appear in '''Algebra''' view.
|-
|| Type '''Center(c)''' in the '''input bar''' >> press '''Enter'''.
|| In the '''input bar''', type '''Center c in parentheses''' and press '''Enter.'''
|-
|| Point to point '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
|| Center '''C''' appears in '''Graphics''' view and its '''co-ordinates''' appear in '''Algebra''' view.
|-
|| Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
|| In the '''input bar''', type '''Vertex c in parentheses''' >> press '''Enter'''.

Vertices '''D''' and '''E''' appear at the ends of the major axis.

Co-vertices '''F''' and '''G''' appear at the ends of the minor axis.
|-
|| under '''Slider''' tool >> Click on '''Text''' tool >> click in '''Graphics''' view.
|| Under '''Slider''', click on '''Text''' tool and click in '''Graphics''' view.
|-
|| Point to the '''text-box'''.
|| A '''text-box''' opens up.
|-
|| '''Slide Number 6'''

'''Text box for ellipse c'''

'''Major axis 2a''' = 4

'''Minor axis 2b''' = 2

'''c''' = 1.732

'''e''' = 0.866

'''latus rectum''' = 1
|| '''Text box for ellipse c'''

In the '''Edit''' field, type the following text.

Press '''Enter''' after each line to go to the next line.

Click '''OK'''.
|-
||
|| Refer to additional material provided with this tutorial for these calculations.
|-
|| Leave '''slider a''' at 2, drag '''slider b''' to 3.
|| Leave '''slider a''' at 2, drag '''slider b''' to 3.
|-
|| Point to the ellipse '''c''' and major and minor axes.
|| Note the effects on the shape of ellipse '''c''' and the change in directions of major and minor axes.
|-
|| Point to the equation in '''Algebra''' view.
|| Note also the change in the equation in '''Algebra''' view.
|-
||Point to the elipse.
|| Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
|-
|| Show everything unchecked in '''Algebra''' view.
|| In '''Algebra''' view, uncheck ellipse '''c''' and all points and text generated for it to hide them.
|-
|| Show screenshots of ellipse '''d''' for '''a = 2, b =1''' and '''a =2, b = 3'''.
|| Follow the earlier steps to construct ellipse '''d''' for these two conditions.
|-
||
|| Let us summarize.
|-
|| '''Slide Number 7'''

'''Summary'''
|| In this tutorial, we have learnt how to:

Use '''GeoGebra''' to construct an ellipse

Look at standard equations and parts of an ellipse
|-
|| '''Slide Number 8'''

'''Assignment'''

Construct ellipses with:

'''Foci''' (± 4, 0) and '''vertices''' (± 5, 0)

'''Foci''' (0, ± 5) and '''vertices''' (0, ± 13)

Find their centres and equations.

Calculate eccentricity and length of latus recti, major and minor axes.
|| As an '''assignment''',

Construct ellipses with the following foci and vertices.

Find all these values.
|-
|| '''Slide Number 9'''

'''Assignment'''

Find the '''coordinates''' of the foci, vertices and co-vertices.

Eccentricity and length of major, minor axes and atus rectum for these ellipses:

'''x2/4 + y2/25 = 1'''

'''36x2 + 4y2 = 144'''
|| Find all these values for these ellipses.
|-
|| '''Slide Number 10'''

'''About Spoken Tutorial project'''
|| The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
|| '''Slide Number 11'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
|| '''Slide Number 12'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

|| Please post your timed queries on this forum.
|-
|| '''Slide Number 13'''

'''Acknowledgement'''
|| '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
||
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English

2018-07-23T06:31:03Z

Vidhya: Created page with "{|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Conic Sections - Ellipse''' |- | | '''Sl..."

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Conic Sections - Ellipse'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn,

'''Standard equations''' and parts of an ellipse

To use '''GeoGebra''' to construct an ellipse
|-
| | '''Slide Number 3'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux '''OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 4'''

'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with

'''GeoGebra''' interface

Conic sections in geometry

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 5'''

'''Ellipse'''

[[Image:]]

An ellipse is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Point to the center '''O''', foci '''F1''' and '''F2'''.

Point to vertices '''A''' and '''B''' at the ends of the major axis '''AB.'''

Point to '''C''' and '''D''' at the ends of the minor axis '''CD'''.

Point to 2 latus recti passing through foci and the two axes.
| | An ellipse is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Observe the center '''O''', foci '''F1''' and '''F2'''.

Vertices '''A''' and '''B''' are at the ends of the major axis '''AB.'''

'''Co-vertices C''' and '''D''' are at the ends of the minor axis '''CD'''.

Two latus recti pass through the foci.

Axes lengths '''2a''' and '''2b''' and distance between the foci '''2c''' are shown in the figure.
|-
| |
| | Be careful to distinguish length from letters used for '''sliders''', circles and ellipses.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct an ellipse in '''GeoGebra'''.

I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click twice in '''Graphics''' view.
| | Click on '''Point''' tool and click twice in '''Graphics''' view.
|-
| | Point to '''A''' and '''B'''.

| | This creates two points '''A''' and '''B''', which will be the foci of our ellipse.
|-
| | Right-click on '''A''' and choose '''Rename''' option.
| | Right-click on '''A''' and choose the '''Rename''' option.
|-
| | Type '''F1''' in the '''New Name''' field and click '''OK''' button.
| | In the '''New Name''' field, type '''F1''' and click '''OK'''.
|-
| | Point to the focus '''F1'''.
| | This will be one of our foci, '''F1'''.
|-
| | Point to '''B'''.
| | Let us rename '''B''' as '''F2'''.
|-
| | Click on '''Slider''' tool >> click in '''Graphics''' view.
| | Click on '''Slider''' tool and click in '''Graphics''' view.
|-
| | Point to the '''Slider dialog box''' in '''Graphics''' view.
| | '''Slider dialog box''' appears in '''Graphics''' view.
|-
| | Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
| | Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
|-
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
|-
| | Point to '''slider k'''.
| | This creates a number '''slider''' named '''k'''.
|-
| | Drag '''slider k'''.
| | '''Slider k''' can be changed from 0 to 10.

'''k''' will be the sum of the distances of any point on the ellipse from the foci '''F1''' and '''F2'''.
|-
| |
| | We will create another '''number slider''' named “'''a'''”.
|-
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
| | Its '''Min''' value is 0, '''Max''' value is 10, '''increment''' is 0.1."
|-
| | Click on '''Circle with Center and Radius''' tool >> click on '''F1'''.
| | Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
|-
| | Point to the '''text box'''; type '''a''' >> click '''OK'''.
| | A '''text-box''' appears; in the radius '''Name''' field, type '''a''' and click '''OK'''.
|-
| | Point to the circle with center '''F1''' and radius '''a'''.
| | A circle '''c''' with center '''F1''' and radius '''a''' appears.
|-
| | Drag '''slider a''' to 2 and '''slider k''' to 5.
| | Drag '''slider a''' to 2 and '''slider k''' to 5.
|-
| | Click again on '''Circle with Center and Radius''' tool >> click on '''F2'''.
| | Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
|-
| | Point to the '''text box'''; type '''k-a''' and click '''OK'''.
| | In the '''text box''' that appears, type '''k minus a''' and click '''OK'''.
|-
| | Point to circle '''d''' with center '''F2''' and radius '''k-a''' in '''Graphics''' view.
| | A circle '''d''' with center '''F2''' and radius '''k minus a''' appears in '''Graphics''' view.
|-
| | Drag '''slider k''' to 5 and '''slider a''' to 2.
| | Drag '''slider k''' to 5 and '''slider a''' to 2.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' and drag '''Graphics''' view.
| | Click on '''Move Graphics View''' and drag '''Graphics''' view.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the '''intersections''' of the two '''circles'''.
| | Under '''Point''', click on '''Intersect''' tool.

Click on the two circles '''c''' and '''d'''.
|-
| | Point to points '''A''' and '''B'''.
| | This creates points''' A''' and '''B'''.
|-
| | Click on '''Segment''' tool under '''Line''' tool.
| | Under '''Line''', click on '''Segment''' tool.
|-
| | Click on points '''F1''' and '''A''' to join them.
| | Click on points '''F1''' and '''A''' to join them.
|-
| | Next, click on points '''A''' and '''F2''' to join them.
| | Next, click on points '''A''' and '''F2''' to join them.
|-
| | Click on '''Move''' tool.
| | Click on '''Move'''.
|-
| | Double click on '''Segment AF1'''.
| | Double click on '''Segment AF1'''.
|-
| | Click '''Object Properties''' to open the '''Preferences''' dialog box.
| | Click on '''Object Properties''' to open the '''Preferences''' dialog box.
|-
| | Point to Segment '''AF1''' highlighted in the left panel.
| | Segment '''AF1''' is already highlighted in the left panel.
|-
| | Holding '''Ctrl''' key down, click and highlight Segment '''AF2'''.
| | Holding '''Ctrl''' key down, click and highlight Segment '''AF2''' as well.
|-
| | Under '''Basic''' tab >> select '''Show Label'''.
| | Under the '''Basic''' tab, make sure that '''Show Label''' is selected.
|-
| | Pull down the '''drop down menu''' next to the '''Show Label''' check box.

Select '''Name and Value'''.
| | Pull down the '''drop down menu''' next to the '''Show Label''' check box.

Select '''Name and Value'''.
|-
| | Under '''Color''' tab >> select Red.
| | Under the '''Color''' tab, select red.
|-
| | Under '''Style''' tab >> choose '''dashed line style'''.
| | Under the '''Style''' tab, choose '''dashed line style'''.
|-
| | Close the '''Preferences''' dialog box.
| | Close the '''Preferences''' dialog box.
|-
| | Follow earlier steps to draw Segments '''BF1''' and '''BF2'''.

Make them '''dashed''' and blue.
| | Draw Segments '''BF1''' and '''BF2'''.

Make them '''dashed''' and blue.
|-
| | Point to '''Move''' tool.

If not highlighted, click on it.

Move the labels so you can see them properly.
| | Make sure that the '''Move''' tool is highlighted.

Move the labels so you can see them properly.
|-
| | Point to the length labels next to '''AF1, AF2, BF1, BF2''' and '''slider k'''.
| | Note that the sum of the segment lengths from both foci to each intersection point is equal to '''k'''.
|-
| | Right-click on '''A''' and '''B''' >> check '''Trace On''' option.
| | Right-click on '''A''' and '''B''' and check '''Trace On''' option.
|-
| | Uncheck circles '''c''' and '''d''' in '''Algebra''' view.
| | In '''Algebra''' view, uncheck circles '''c''' and '''d''' to hide the circles.
|-
| | Right-click on '''slider a''' >> check '''Animation On''' option.
| | Right-click on '''slider a''' and check '''Animation On''' option.
|-
| | Right-click on '''slider k''' and check '''Animation On''' option.
| | Next, right-click on '''slider k''' and check '''Animation On''' option.
|-
| | Point to the traces of '''A''' and '''B''' and to '''A''' and '''B'''.
| | Note the locus of points traced by '''A''' and '''B'''.

These traced points are all equidistant from '''F1''' and '''F2''' (the foci).

They lie on ellipses for which '''F1''' and '''F2''' are foci.
|-
| | Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
| | Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
|-
| | Drag '''sliders a''' and '''k'''.
| | Drag '''sliders a''' and '''k''' to different values to see more traces of ellipses.
|-
| | Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
| | Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
|-
| | Point to length labels next to '''AF1''' and '''AF2''' and '''sliders a''' and '''k'''.
| | Note that for a given value of '''k''', as '''a''' changes, lengths of '''Segments AF1''' and '''AF2''' change.

But their sum remains equal to the value of '''k'''.
|-
| | Point to '''BF1''' and '''BF2''' and '''sliders a''' and '''k'''.
| | Note the same fact for Segments '''BF1''' and '''BF2'''.
|-
| | Click in and move '''Graphics''' view slightly to erase the trace points.
| | Click in and move '''Graphics''' view slightly to erase the trace points.
|-
| | Click on the '''Move''' tool and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
| | Click on '''Move''' tool and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
|-
| | Point to '''sliders a''' and '''k'''.
| | Values can be changed on '''sliders a''' and '''k''' to see various ellipses.
|-
| | Open a new '''GeoGebra''' window.
| | Let us look at the equations of ellipses in a new '''GeoGebra''' window.
|-
| | In '''input bar''', type '''(x-h)^2/a^2+(y-k)^2/b^2=1''' >> press '''Enter'''.
| | In the '''input bar''', type the following line describing the sum of two fractions equal to 1.

To type the '''caret symbol''', hold '''Shift''' key down and press 6.

For the 1st fraction, type the numerator as '''x minus h in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the denominator of the 1st fraction as '''a caret 2''' followed by '''plus'''.

For the 2nd fraction, type the numerator as '''y minus k in parentheses caret 2'''.

Then type '''division slash'''.

Now, type the denominator of the 2nd fraction as '''b caret 2''' followed by '''equals sign 1'''.

Press '''Enter'''.
|-
| | Point to the pop-up window.
| | A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to sliders '''a, b, h''' and '''k'''.
| | This creates number '''sliders''' for '''h, a, k''' and '''b'''.

By default, they go from minus 5 to 5 and are set at 1.

You can double-click on the '''sliders''' to see their properties.
|-
| | Point to circle '''c''' and center ''(h, k)''' in '''Graphics''' view.
| | A circle '''c''', a special case of an ellipse, appears in '''Graphics''' view.

Center '''h comma k''' is at '''1 comma 1''' and radius is 1 unit.
|-
| | Point to equation for circle '''c''' in '''Algebra''' view.
| | In '''Algebra''' view, note the equation for circle '''c'''.
|-
| | Drag boundary to see it properly.
| | Drag the boundary to see it properly.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
|-
| | Point to the equations in '''Algebra''' view as you change '''a''' and '''b''' on '''sliders'''.
| | Keep track of the equations in '''Algebra''' view as you change '''a''' and '''b''' on the '''sliders'''.
|-
| | Place '''cursor''' on equation '''c''' in '''Algebra''' view.
| | Place the '''cursor''' on equation '''c''' in '''Algebra''' view.
|-
| | Point to '''slider a'''.

Drag '''slider a''', leave between -2 and -3.
| | '''a''' is associated with the '''x minus h squared''' component of the equation.

Observe how '''a''' controls the horizontal axis of the ellipse.
|-
| | Point to '''slider b'''.

Drag '''slider b''', leave at 5.
| | Associated with the '''y minus k squared''' component is '''b'''.

Observe how '''b''' controls the vertical axis of the ellipse.
|-
| | Drag '''slider a''' to 2 and '''b''' to 1.
| | Drag '''slider a''' to 2 and '''b''' to 1.
|-
| | Point to major axis of the ellipse and along the '''x axis'''.

Point to equation of the ellipse.
| | When '''a''' is greater than '''b''', the major axis of the ellipse is horizontal.

Note the equation of the ellipse.
|-
| | Drag boundary to see the equation properly.
| | Drag the boundary to see it properly.
|-
| | Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.
| | In the '''input bar''', type '''Focus c in parentheses''' and press '''Enter'''.
|-
| | Point to '''A''' and '''B''' mapped in '''Graphics''' view and their '''co-ordinates''' in '''Algebra''' view.
| | Two foci, '''A''' and '''B''', are mapped in '''Graphics''' view and their '''coordinates''' appear in '''Algebra''' view.
|-
| | Type '''Center(c)''' in the '''input bar''' and press '''Enter'''.
| | In the '''input bar''', type '''Center c in parentheses''' and press '''Enter.'''
|-
| | Point to point '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
| | Center '''C''' appears in '''Graphics''' view and its '''co-ordinates''' appear in '''Algebra''' view.
|-
| | Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
| | In the '''input bar''', type '''Vertex c in parentheses''' and press '''Enter'''.

Vertices '''D''' and '''E''' appear at the ends of the major axis.

Co-vertices '''F''' and '''G''' appear at the ends of the minor axis.
|-
| | Click on '''Text''' tool under '''Slider''' tool and click in '''Graphics''' view.
| | Under '''Slider''', click on '''Text''' tool and click in '''Graphics''' view.
|-
| | Point to the '''text-box'''.
| | A '''text-box''' opens up.
|-
| | '''Slide Number 6'''

'''Text box for ellipse c'''

'''Major axis 2a''' = 4

'''Minor axis 2b''' = 2

'''c''' = 1.732

'''e''' = 0.866

'''latus rectum''' = 1
| | '''Text box for ellipse c'''

In the '''Edit''' field, type the following text.

Press '''Enter''' after each line to go to the next line.

Click '''OK'''.
|-
| |
| | Refer to additional material provided with this tutorial for these calculations.
|-
| | Leave '''slider a''' at 2, drag '''slider b''' to 3.
| | Leave '''slider a''' at 2, drag '''slider b''' to 3.
|-
| | Point to the ellipse '''c''' and major and minor axes.
| | Note the effects on the shape of ellipse '''c''' and the change in directions of major and minor axes.
|-
| | Point to the equation in '''Algebra''' view.
| | Note also the change in the equation in '''Algebra''' view.
|-
| |
| | Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
|-
| | Show everything unchecked in '''Algebra''' view.
| | In '''Algebra''' view, uncheck ellipse '''c''' and all points and text generated for it to hide them.
|-
| | Show screenshots of ellipse '''d''' for '''a = 2, b =1''' and '''a =2, b = 3'''.
| | Follow the earlier steps to construct ellipse '''d''' for these two conditions.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 8'''

'''Summary'''
| | In this tutorial, we have learnt how to:

Use '''GeoGebra''' to construct an ellipse

Look at standard equations and parts of an ellipse
|-
| | '''Slide Number 9'''

'''Assignment'''

Construct ellipses with:

'''Foci''' (± 4, 0) and '''vertices''' (± 5, 0)

'''Foci''' (0, ± 5) and '''vertices''' (0, ± 13)

Find their centres and equations.

Calculate eccentricity and length of latus recti, major and minor axes.
| | As an '''assignment''',

Construct ellipses with the following foci and vertices.

Find all these values.
|-
| | '''Slide Number 10'''

'''Assignment'''

Find the '''coordinates''' of the foci, vertices and co-vertices.

Eccentricity and length of major, minor axes and atus rectum for these ellipses:

'''x2/4 + y2/25 = 1'''

'''36x2 + 4y2 = 144'''
| | Find all these values for these ellipses.
|-
| | '''Slide Number 11'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 14'''

'''Acknowledgement'''
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English

2018-07-19T06:24:49Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
||'''Slide Number 1'''

'''Title Slide'''
||Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''.
|-
||'''Slide Number 2'''

'''Learning Objectives'''
||In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
* Study standard equations and parts of a parabola

* Construct parabolas.
|-
|| '''Slide Number 3'''

'''System Requirement'''
||Here I am using:
'''Ubuntu Linux''' Operating System version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
||'''Slide Number 4'''
'''Pre-requisites'''

www.spoken-tutorial.org
||To follow this '''tutorial''', you should have basic knowledge of
'''GeoGebra''' interface

'''Conic sections''' in geometry

For relevant tutorials, please visit our website.
|-
||'''Slide Number 5'''

'''Parabola'''

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.
||Parabola

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.

Observe the different features of the parabola in the image.

The '''Axis of Symmetry''' is perpendicular to the '''Directrix''' and passes through the Focus and '''Vertex'''.

'''Latus Rectum''' passes through the Focus and is perpendicular to the '''Axis of Symmetry'''.
|-
||Show the '''GeoGebra''' window.
||Let us construct a parabola in '''GeoGebra'''.

I have already opened '''GeoGebra''' interface.
|-
||Click on '''Point''' tool >> click in '''Graphics''' view.

Point to point '''A'''.
||Click on '''Point''' tool and click in '''Graphics''' view.

This creates point '''A'''.
|-
||Right-click on point '''A''' >> select the '''Rename''' option.
||Right-click on point '''A''' and select the '''Rename''' option.
|-
||In '''New Name''' text box, type '''Focus''' instead of '''A''' >> click '''OK'''.

Point to '''Focus'''.
||In the '''New Name''' text box, type '''Focus''' instead of '''A '''and click '''OK'''.

This renames point '''A''' as '''Focus'''.
|-
||Click on '''Line''' tool >> click in two places in '''Graphics''' view below '''Focus'''.

Point to line '''AB'''.
||Click on '''Line''' tool and click on two places in '''Graphics''' view, below '''Focus'''.

This creates line '''AB'''.
|-
||Right-click on line '''AB''' >> choose '''Rename''' option.
||Right-click on line '''AB''' and choose the '''Rename''' option.
|-
||Type '''directrix''' in '''New Name''' field >> click '''OK'''.

Point to '''directrix'''.
||In the '''New Name''' field, type '''directrix''' and click '''OK'''.

This renames line '''AB''' as the '''directrix'''.
|-
||Click on '''Perpendicular Line''' tool >> click on line '''AB'''.
||Click on '''Perpendicular Line''' tool, then click on line '''AB'''.
|-
||Drag the '''cursor''' until '''Focus''' >> click on point '''A'''.
||Drag the '''cursor''' until the resulting line passes through '''Focus''' and click on '''Focus'''.
|-
||Point to the perpendicular line through '''Focus'''.

Point to '''axis of symmetry'''.
||This draws a line perpendicular to line '''AB''', passing through '''Focus'''.

This line is the '''axis of symmetry'''.
|-
||Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field >> click '''OK'''.
||Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field.

Click '''OK'''.
|-
||Click on '''Parabola''' tool under '''Ellipse''' tool.

Click on '''Focus''' and line '''AB''' ('''directrix''').
||Under '''Ellipse''' tool, click on '''Parabola''' tool.

Then click on '''Focus''' and the '''directrix'''.
|-
||Point to the parabola.
||This creates a parabola with its focus at '''Focus''' and with line '''AB''' as the '''directrix'''.
|-
||Click on '''Intersect''' tool. >> Click on the '''parabola''' and '''axis of symmetry'''.
||Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and '''axis of symmetry'''.
|-
||Point to point '''C'''.
||This creates point '''C''' at the intersection.

It is the '''vertex''' of the parabola.
|-
||Right-click on point '''C''' >> choose the '''Rename''' option.
||Right-click on point '''C''' and choose the '''Rename ''' option.
|-
||Type '''Vertex''' in '''the '''New Name''' field >> click '''OK'''.
||In the '''New Name''' field, type '''Vertex''' and click '''OK'''.
|-
||Click on '''Perpendicular Line''' tool >> click on the '''axis of symmetry'''.
||Click on '''Perpendicular Line''' tool and click on the '''axis of symmetry'''.
|-
||Drag the '''cursor''' until the line passes through point '''A''' (Focus) >> click on point '''A'''.
||Drag the '''cursor''' until the line passes through the '''Focus''' and click on it.
|-
||Point to the parallel line.
||This results in a line parallel to the '''directrix''', passing through the '''Focus'''.
|-
||Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersections of the parabola and the newly drawn line through '''Focus'''.

||Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and the newly drawn line through '''Focus'''.
|-
||Point to points '''C''' and '''D'''.
||This creates points '''C''' and '''D'''.
|-
||Click on '''Segment''' tool under the '''Line''' tool >> click on points '''C'''and '''D'''.
||Under '''Line''' tool, click on '''Segment''' tool and click on points '''C''' and '''D'''.
|-
||Point to Segment '''CD'''.
||Resulting Segment '''CD''' is the '''latus rectum'''.
|-
||Right-click on Segment '''CD''' >> choose the '''Rename''' option.
||Right-click on Segment '''CD''' and choose the '''Rename''' option.
|-
||Type '''Latus Rectum''' in the '''New Name''' field >> click '''OK''' button.
||In the '''New Name''' field, type '''Latus Rectum''' and click '''OK'''.
|-
||Move the '''Latus''' label so you can see it properly.
||Move the '''Latus''' label so you can see it properly.
|-
||Click >> drag '''Graphics''' view to see the parabola properly.
||Click and drag '''Graphics''' view to see the parabola properly.
|-
||Point to '''Algebra''' view.

Drag boundary so you can see equation properly.

||In '''Algebra''' view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the '''axis of symmetry, directrix''' and '''latus rectum'''.
|-
||Drag boundary so you can see '''Graphics''' view properly.
||Drag boundary so you can see '''Graphics''' view properly again.
|-
||Click in '''Graphics''' view and drag background.
||Click in '''Graphics''' view and drag background.
|-
||Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersection of the '''axis of symmetry''' and the '''directrix'''.
||Under '''Point''' tool, click on '''Intersect''' tool.

Click on '''axis of symmetry''' and '''directrix'''.
|-
||Point to point '''E'''.
||This creates point '''E'''.
|-
||Click on '''Distance or Length''' tool under '''Angle''' tool.
||Under '''Angle''' tool, click on '''Distance or Length''' tool.
|-
||Click on '''Focus''' >> '''Vertex'''.
||Click on '''Focus''' and '''Vertex'''.
|-
||Point to the distance of '''FocusVertex''' appearing in '''Graphics''' view.
||Note the distance of '''FocusVertex''' appearing in '''Graphics''' view.
|-
||Click on '''Vertex''' >> point '''E'''.
||Click on '''Vertex''' and point '''E'''.
|-
||Point to the distance of '''Vertex E''' appearing in '''Graphics''' view.
||Note the distance of '''Vertex E''' appearing in '''Graphics''' view.

Both these distances are equal.
|-
||
||Let us look at the general equations of parabolas.
|-
||Show the new '''GeoGebra''' window.
||I have opened a new '''GeoGebra''' window.
|-
||Type '''(x-a)^2=4 p (y-b)''' in '''input bar''' >> press '''Enter'''.
||In '''input bar''', type '''x minus a in parentheses caret 2 equals 4 space p space y minus b''' in parentheses'''.

To type '''caret symbol''', hold '''Shift''' key down and press 6.

Note that the spaces denote multiplication.

Press '''Enter'''.
|-
||Point to '''Create Sliders''' window.
||'''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
|-
||Click on '''Create Sliders'''.
||Click on '''Create Sliders'''.
|-
||Point to '''sliders a, p''' and '''b'''.
||'''Sliders''' are created for '''a, p''' and '''b'''.

The '''default''' setting for all three '''coefficients''' is 1.
|-
||A parabola opening upwards appears in '''Graphics''' view.

Point to '''vertex''' of parabola.
||A parabola opening upwards appears in '''Graphics''' view.

'''a comma b''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
||Double click on parabola >> click on '''Object Properties''' and then on '''Color''' tab.
||Double click on the parabola, click on '''Object Properties''' and then on '''Color''' tab.
|-
||Select red >> close the '''Preferences''' box.
||Select red and close the '''Preferences''' box.
|-
||Point to the red parabola and its equation in '''Graphics''' and '''Algebra''' views.
||The parabola and its equation appear red in the '''Graphics''' and '''Algebra''' views.
|-
||Move boundary so you can see the equation properly.
||Move boundary so you can see the equation properly.
|-
||Right-click on '''slider a''' button >> check '''Animation On''' option.
||Right-click on '''slider a''' and check '''Animation On''' option.
|-
||Point to the parabola in '''Graphics''' view.
||Note the effects on the horizontal movement of the red parabola.
|-
||Right-click on '''slider a''' >> uncheck '''Animation On''' option.
||Right-click on '''slider a''' and uncheck '''Animation On''' option.
|-
||Right-click on '''slider p''' >> check '''Animation On''' option.
||Right-click on '''slider p''' and check '''Animation On''' option.
|-
||Point to parabola in '''Graphics view'''.
||Note the effects on the shape and orientation of the parabola.
|-
||Right-click on '''slider p''' >> uncheck '''Animation On''' option.
||Right-click on '''slider p''' and uncheck '''Animation On''' option.
|-
||Right-click on '''slider b''' >> check '''Animation On''' option.
||Right-click on '''slider b''' and check '''Animation On''' option.
|-
||Point to the parabola.
||Note the effects on the vertical movement of the parabola.
|-
||Right-click on '''slider b''' >> uncheck '''Animation On''' option.
||Right-click on '''slider b''' and uncheck '''Animation On''' option.
|-
||Point to '''sliders a, p''' and '''b''' (all = 1) and the red parabola '''c''' in '''Graphics '''view.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Point to equation '''c: (x2-2x-4y) = -5''' in '''Algebra''' view.
||Note that when '''a''', '''p''' and '''b '''are equal to 1, the red parabola '''c''' is described by equation '''c'''.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Equation '''c''' is given by x squared minus 2x minus 4y equals minus 5.
|-
||Type '''Focus(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''A''' in '''Graphics''' view.
||In '''input bar''', type '''Focus c''' in parentheses.

Press '''Enter'''.

'''Focus''' is drawn at point '''A ''' in '''Graphics''' view.
|-
||Point to the '''coordinates''' of point '''A''', the '''Focus''', in '''Algebra''' view.
||The coordinates of '''Focus''' of parabola '''c''', which is point '''A''', appear in '''Algebra''' view.
|-
||Type '''Vertex(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''B''' in '''Graphics''' view.
||In '''input bar''', type '''Vertex c''' in parentheses.

Press '''Enter'''.

'''Vertex''' is drawn at point '''B''' in '''Graphics''' view.
|-
||Point to the '''coordinates''' of point '''B''' in '''Algebra''' view.
||The '''coordinates''' of '''Vertex''' of parabola '''c''', which is point '''B''', appear in '''Algebra''' view.
|-
||Type '''Directrix(c)''' in '''input bar''' >> press '''Enter'''.

Point to '''Directrix''' in '''Graphics''' view.
||In '''input bar''', type '''Directrix c''' in parentheses.

Press '''Enter'''.

'''Directrix''' appears as a line along '''x axis''' in '''Graphics''' view.
|-
||Point to the equation, '''y=0''', in '''Algebra''' view.
||The equation for the '''Directrix''' of parabola '''c''', '''y equals 0''', appears in '''Algebra''' view.
|-
||Double click on '''Directrix''' in '''Graphics''' view >> '''Redefine''' text box >> '''Object Properties''' >> '''Color''' tab.
||Double click on '''Directrix''' in '''Graphics''' view.

In the '''Redefine''' text box, click on '''Object Properties''', then the '''Color''' tab.
|-
||In the left panel, point to highlighted '''Directrix''', identify '''Focus''' and ''' Vertex''' created for parabola '''c'''.
||In the left panel, note that the '''Directrix''' is highlighted.

Identify '''Focus''' and '''Vertex''' created for parabola '''c'''.
|-
||Click on each one to highlight while pressing the '''Control''' key.
||While pressing the '''Control''' key, click and highlight '''Focus''' and '''Vertex'''.
|-
||Click on red.
||Click on red.
|-
||Close the '''Preferences''' box.
||Close the '''Preferences''' box.
|-
||Point to '''Focus''', '''Vertex''' and '''Directrix''' and their '''co-ordinates''' and equation in '''Graphics''' and '''Algebra''' views.
||For parabola '''c''', '''Focus''', '''Vertex''' and '''Directrix''' and their '''coordinates''' and equation appear red.
|-
||Show '''Geogebra''' window with parabolas '''c''' and '''d'''.

In '''Algebra''' view, point to equation '''d''' and in '''Graphics''' view, point to parabola '''d'''.
||Follow the earlier steps to construct parabola '''d'''.
|-
||
||Let us summarize.
|-
||'''Slide Number 6'''

'''Summary'''
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
* Study the standard equations and parts of a parabola

* Construct parabolas
|-
||'''Slide Number 7'''
'''Assignment'''

Try these steps to construct parabolas with:

Focus (6,0) and '''directrix''' x = -6

Focus (0,-3) and '''directrix''' y = 3

Find their equations.
||As an assignment:
Try these steps to construct parabolas with these '''foci''' and '''directrices'''.

Find their equations.
|-
||'''Slide Number 8'''
'''Assignment'''

Find the coordinates of the '''foci''' >> length of '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.

y2 = 12x

x2 = -16y
||As an assignment:
Find the coordinates of the '''foci''' and length of the '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.
|-
||'''Slide Number 9'''
'''About Spoken Tutorial Project'''
||The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
||'''Slide Number 10'''
'''Spoken Tutorial workshops'''
||The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
||'''Slide Number 11'''
'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
||Please post your timed queries on this forum.
|-
||'''Slide Number 12'''
'''Acknowledgement'''
||'''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|-
||
||This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English

2018-07-13T06:48:02Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Study standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this '''tutorial''', you should have basic knowledge of
'''GeoGebra''' interface

'''Conic sections''' in geometry
|-
| s | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux '''OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''

'''Parabola'''

[[Image:]]

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.
| | A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.

Observe the different features of the parabola in the image.

The '''Axis of Symmetry''' is perpendicular to the '''Directrix''' and passes through the Focus and '''Vertex'''.

'''Latus Rectum''' passes through the Focus and is perpendicular to the '''Axis of Symmetry'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct a parabola in '''GeoGebra'''.

I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.

Point to point '''A'''.
| | Click on '''Point''' tool and click in '''Graphics''' view.

This creates point '''A'''.
|-
| | Right-click on point '''A''' and select the '''Rename''' option.
| | Right-click on point '''A''' and select the '''Rename''' option.
|-
| | In '''New Name''' text box, type '''Focus''' instead of '''A''' >> click '''OK'''.

Point to '''Focus'''.
| | In the '''New Name''' text box, type '''Focus''' instead of '''A '''and click '''OK'''.

This renames point '''A''' as '''Focus'''.
|-
| | Click on '''Line''' tool >> click in two places in '''Graphics''' view below '''Focus'''.

Point to line '''AB'''.
| | Click on '''Line''' tool and click on two places in '''Graphics''' view, below '''Focus'''.

This creates line ''' AB'''.
|-
| | Right-click on line '''AB''' >> choose '''Rename''' option.
| | Right-click on line '''AB''' and choose the '''Rename''' option.
|-
| | Type '''directrix''' in '''New Name''' field >> click '''OK'''.

Point to '''directrix'''.
| | In the '''New Name''' field, type '''directrix''' and click '''OK'''.

This renames line '''AB''' as the '''directrix'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on line '''AB'''.
| | Click on '''Perpendicular Line''' tool, then click on line '''AB'''.
|-
| | Drag the '''cursor''' until '''Focus''' >> click on point '''A'''.
| | Drag the '''cursor''' until the resulting line passes through '''Focus''' and click on '''Focus'''.
|-
| | Point to the perpendicular line through '''Focus'''.

Point to '''axis of symmetry'''.
| | This draws a line perpendicular to line '''AB''', passing through '''Focus'''.

This line is the '''axis of symmetry'''.
|-
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field >> click '''OK'''.
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field.

Click '''OK'''
|-
| | Click on '''Parabola''' tool under '''Ellipse''' tool.

Click on '''Focus''' and line '''AB''' ('''directrix''').
| | Under '''Ellipse''' tool, click on '''Parabola''' tool.

Then click on '''Focus''' and the '''directrix'''.
|-
| | Point to the parabola.
| | This creates a parabola with its focus at '''Focus''' and with line '''AB''' as the '''directrix'''.
|-
| | Click on '''Intersect''' tool. >> Click on the '''parabola''' and '''axis of symmetry'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and '''axis of symmetry'''.
|-
| | Point to point '''C'''.
| | This creates point '''C''' at the intersection.

It is the '''vertex''' of the parabola.
|-
| | Right-click on point '''C''' >> choose the '''Rename''' option.
| | Right-click on point '''C''' and choose the '''Rename ''' option.
|-
| | Type '''Vertex''' in '''the '''New Name''' field >> click '''OK'''.
| | In the '''New Name''' field, type '''Vertex''' and click '''OK'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on the '''axis of symmetry'''.
| | Click on '''Perpendicular Line''' tool and click on the '''axis of symmetry'''.
|-
| | Drag the '''cursor''' until the line passes through point '''A''' (Focus) >> click on point '''A'''.
| | Drag the '''cursor''' until the line passes through the '''Focus''' and click on it.
|-
| | Point to the parallel line.
| | This results in a line parallel to the '''directrix''', passing through the '''Focus'''.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersections of the parabola and the newly drawn line through '''Focus'''.

| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and the newly drawn line through '''Focus'''.
|-
| | Point to points '''C''' and '''D'''.
| | This creates points '''C''' and '''D'''.
|-
| | Click on '''Segment''' tool under the '''Line''' tool >> click on points '''C'''and '''D'''.
| | Under '''Line''' tool, click on '''Segment''' tool and click on points '''C''' and '''D'''.
|-
| | Point to Segment '''CD'''.
| | Resulting Segment '''CD''' is the '''latus rectum'''.
|-
| | Right-click on Segment '''CD''' and choose the '''Rename''' option.
| | Right-click on Segment '''CD'''and choose the '''Rename''' option.
|-
| | Type '''Latus Rectum''' in the '''New Name''' field >> click '''OK''' button.
| | In the '''New Name''' field, type '''Latus Rectum''' and click '''OK'''.
|-
| | Move the '''Latus''' label so you can see it properly.
| | Move the '''Latus''' label so you can see it properly.
|-
| | Click and drag '''Graphics''' view to see the parabola properly.
| | Click and drag '''Graphics''' view to see the parabola properly.
|-
| | Point to '''Algebra''' view.

Drag boundary so you can see equation properly.

| | In '''Algebra''' view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the '''axis of symmetry, directrix''' and '''latus rectum'''.
|-
| | Drag boundary so you can see '''Graphics''' view properly again.
| | Drag boundary so you can see '''Graphics''' view properly again.
|-
| | Click in '''Graphics''' view and drag background.
| | Click in '''Graphics''' view and drag background.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersection of the '''axis of symmetry''' and the '''directrix'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on '''axis of symmetry''' and '''directrix'''.
|-
| | Point to point '''E'''.
| | This creates point '''E'''.
|-
| | Click on '''Distance or Length''' tool under '''Angle''' tool.
| | Under '''Angle''' tool, click on '''Distance or Length''' tool.
|-
| | Click on '''Focus''' >> '''Vertex'''.
| | Click on '''Focus''' and '''Vertex'''.
|-
| | Point to the distance of '''FocusVertex''' appearing in '''Graphics''' view.

| | Note the distance of '''FocusVertex''' appearing in '''Graphics''' view.
|-
| | Click on '''Vertex''' >> point '''E'''.
| | Click on '''Vertex''' and point '''E'''.
|-
| | Point to the distance of '''Vertex E''' appearing in '''Graphics''' view.
| | Note the distance of '''Vertex E''' appearing in '''Graphics''' view.

Both these distances are equal.
|-
| |
| | Let us look at the general equations of parabolas.
|-
| | Show the new '''GeoGebra''' window.
| | I have opened a new '''GeoGebra''' window.
|-
| | Type '''(x-a)^2=4 p (y-b)''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses'''.

To type '''caret symbol''', hold '''Shift''' key down and press 6.

Note that the spaces denote multiplication.

Press '''Enter'''.
|-
| | Point to '''Create Sliders''' window
| | '''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders a, p''' and '''b'''.
| | '''Sliders''' are created for '''a, p''' and '''b'''.

The '''default''' setting for all three '''coefficients''' is 1.
|-
| | A parabola opening upwards appears in '''Graphics''' view.

Point to '''vertex''' of parabola.
| | A parabola opening upwards appears in '''Graphics''' view.

'''a comma b''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
| | Double click on parabola >> click on '''Object Properties''' and then on '''Color''' tab.
| | Double click on the parabola, click on '''Object Properties''' and then on '''Color''' tab.
|-
| | Select red and close the '''Preferences''' box.
| | Select red and close the '''Preferences''' box.
|-
| | Point to the red parabola and its equation in '''Graphics''' and '''Algebra''' views.
| | The parabola and its equation appear red in the '''Graphics''' and '''Algebra''' views.
|-
| | Move boundary so you can see the equation properly.
| | Move boundary so you can see the equation properly.
|-
| | Right-click on '''slider a''' button >> check '''Animation On''' option.
| | Right-click on '''slider a''' and check '''Animation On''' option.
|-
| | Point to the parabola in '''Graphics''' view.
| | Note the effects on the horizontal movement of the red parabola.
|-
| | Right-click on '''slider a''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider a''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider p''' >> check '''Animation On''' option.
| | Right-click on '''slider p''' and check '''Animation On''' option.
|-
| | Point to parabola in '''Graphics view'''.
| | Note the effects on the shape and orientation of the parabola.
|-
| | Right-click on '''slider p''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider p''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider b''' >> check '''Animation On''' option.
| | Right-click on '''slider b''' and check '''Animation On''' option.
|-
| | Point to the parabola.
| | Note the effects on the vertical movement of the parabola.
|-
| | Right-click on '''slider b''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider b''' and uncheck '''Animation On''' option.
|-
| | Point to '''sliders a, p''' and '''b''' (all = 1) and the red parabola '''c''' in '''Graphics '''view.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Point to equation '''c: (x2-2x-4y) = -5''' in '''Algebra''' view.
| | Note that when '''a''', '''p''' and '''b '''are equal to 1, the red parabola '''c''' is described by equation '''c'''.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Equation '''c''' is given by '''x squared minus 2x minus 4y equals minus 5'''.
|-
| | Type '''Focus(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''A''' in '''Graphics''' view.
| | In '''input bar''', type '''Focus c in parentheses'''.

Press '''Enter'''.

'''Focus''' is drawn at point '''A ''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''A''', the '''Focus''', in '''Algebra''' view.
| | The coordinates of '''Focus''' of parabola '''c''', which is point '''A''', appear in '''Algebra''' view.
|-
| | Type '''Vertex(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''B''' in '''Graphics''' view.
| | In '''input bar''', type '''Vertex c in parentheses'''.

Press '''Enter'''.

'''Vertex''' is drawn at point '''B''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''B''' in '''Algebra''' view.
| | The '''coordinates''' of '''Vertex''' of parabola '''c''', which is point '''B''', appear in '''Algebra''' view.
|-
| | Type '''Directrix(c)''' in '''input bar''' >> press '''Enter'''.

Point to '''Directrix''' in '''Graphics''' view.
| | In '''input bar''', type '''Directrix c''' in parentheses.

Press '''Enter'''.

'''Directrix''' appears as a line along '''x axis''' in '''Graphics''' view.
|-
| | Point to the equation, '''y=0''', in '''Algebra''' view.
| | The equation for the '''Directrix''' of parabola '''c''', '''y equals 0''', appears in '''Algebra''' view.
|-
| | Double click on '''Directrix''' in '''Graphics''' view >> '''Redefine''' text box >> '''Object Properties''' >> '''Color''' tab.
| | Double click on '''Directrix''' in '''Graphics''' view.

In the '''Redefine''' text box, click on '''Object Properties''', then on the '''Color''' tab.
|-
| | In the left panel, point to highlighted '''Directrix''', identify '''Focus''' and ''' Vertex''' created for parabola '''c'''.
| | In the left panel, note that the '''Directrix''' is highlighted.

Identify '''Focus''' and '''Vertex''' created for parabola '''c'''.
|-
| | Click on each one to highlight while pressing the '''Control''' key.
| | While pressing the '''Control''' key, click and highlight '''Focus''' and '''Vertex'''.
|-
| | Click on red.
| | Click on red.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to '''Focus''', '''Vertex''' and '''Directrix''' and their '''co-ordinates''' and equation in '''Graphics''' and '''Algebra''' views.
| | For parabola '''c''', '''Focus''', '''Vertex''' and '''Directrix''' and their '''coordinates''' and equation appear red.
|-
| |Show '''Geogebra''' window with parabolas '''c''' and '''d'''.

In '''Algebra''' view, point to equation '''d''' and in '''Graphics''' view, point to parabola '''d'''.
| |Follow the earlier steps to construct parabola '''d'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
Study the standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 7'''
'''Assignment'''

Try these steps to construct parabolas with:

Focus (6,0) and '''directrix''' x = -6

Focus (0,-3) and '''directrix''' y = 3

Find their equations.
| | As an assignment:
Try these steps to construct parabolas with these '''foci''' and '''directrices'''.

Find their equations.
|-
| | '''Slide Number 8'''
'''Assignment'''

Find the coordinates of the '''foci''' and length of '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.

y2 = 12x

x2 = -16y
| | As an assignment:
Find the coordinates of the '''foci''' and length of the '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.
|-
| | '''Slide Number 9'''
'''About Spoken Tutorial Project'''
| | The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
| | '''Slide Number 10'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 11'''
'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 12'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English

2018-07-13T06:44:08Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Study standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this '''tutorial''', you should have basic knowledge of
'''GeoGebra''' interface

'''Conic sections''' in geometry
|-
| s | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux '''OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''

'''Parabola'''

[[Image:]]

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.
| | A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.

Observe the different features of the parabola in the image.

The '''Axis of Symmetry''' is perpendicular to the '''Directrix''' and passes through the Focus and '''Vertex'''.

'''Latus Rectum''' passes through the Focus and is perpendicular to the '''Axis of Symmetry'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct a parabola in '''GeoGebra'''.

I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.

Point to point '''A'''.
| | Click on '''Point''' tool and click in '''Graphics''' view.

This creates point '''A'''.
|-
| | Right-click on point '''A''' and select the '''Rename''' option.
| | Right-click on point '''A''' and select the '''Rename''' option.
|-
| | In '''New Name''' text box, type '''Focus''' instead of '''A''' >> click '''OK'''.

Point to '''Focus'''.
| | In the '''New Name''' text box, type '''Focus''' instead of '''A '''and click '''OK'''.

This renames point '''A''' as '''Focus'''.
|-
| | Click on '''Line''' tool >> click in two places in '''Graphics''' view below '''Focus'''.

Point to line '''AB'''.
| | Click on '''Line''' tool and click on two places in '''Graphics''' view, below '''Focus'''.

This creates line ''' AB'''.
|-
| | Right-click on line '''AB''' >> choose '''Rename''' option.
| | Right-click on line '''AB''' and choose the '''Rename''' option.
|-
| | Type '''directrix''' in '''New Name''' field >> click '''OK'''.

Point to '''directrix'''.
| | In the '''New Name''' field, type '''directrix''' and click '''OK'''.

This renames line '''AB''' as the '''directrix'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on line '''AB'''.
| | Click on '''Perpendicular Line''' tool, then click on line '''AB'''.
|-
| | Drag the '''cursor''' until '''Focus''' >> click on point '''A'''.
| | Drag the '''cursor''' until the resulting line passes through '''Focus''' and click on '''Focus'''.
|-
| | Point to the perpendicular line through '''Focus'''.

Point to '''axis of symmetry'''.
| | This draws a line perpendicular to line '''AB''', passing through '''Focus'''.

This line is the '''axis of symmetry'''.
|-
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field >> click '''OK'''.
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field.

Click '''OK'''
|-
| | Click on '''Parabola''' tool under '''Ellipse''' tool.

Click on '''Focus''' and line '''AB''' ('''directrix''').
| | Under '''Ellipse''' tool, click on '''Parabola''' tool.

Then click on '''Focus''' and the '''directrix'''.
|-
| | Point to the parabola.
| | This creates a parabola with its focus at '''Focus''' and with line '''AB''' as the '''directrix'''.
|-
| | Click on '''Intersect''' tool. >> Click on the '''parabola''' and '''axis of symmetry'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and '''axis of symmetry'''.
|-
| | Point to point '''C'''.
| | This creates point '''C''' at the intersection.

It is the '''vertex''' of the parabola.
|-
| | Right-click on point '''C''' >> choose the '''Rename''' option.
| | Right-click on point '''C''' and choose the '''Rename ''' option.
|-
| | Type '''Vertex''' in '''the '''New Name''' field >> click '''OK'''.
| | In the '''New Name''' field, type '''Vertex''' and click '''OK'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on the '''axis of symmetry'''.
| | Click on '''Perpendicular Line''' tool and click on the '''axis of symmetry'''.
|-
| | Drag the '''cursor''' until the line passes through point '''A''' (Focus) >> click on point '''A'''.
| | Drag the '''cursor''' until the line passes through the '''Focus''' and click on it.
|-
| | Point to the parallel line.
| | This results in a line parallel to the '''directrix''', passing through the '''Focus'''.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersections of the parabola and the newly drawn line through '''Focus'''.

| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and the newly drawn line through '''Focus'''.
|-
| | Point to points '''C''' and '''D'''.
| | This creates points '''C''' and '''D'''.
|-
| | Click on '''Segment''' tool under the '''Line''' tool >> click on points '''C'''and '''D'''.
| | Under '''Line''' tool, click on '''Segment''' tool and click on points '''C''' and '''D'''.
|-
| | Point to Segment '''CD'''.
| | Resulting Segment '''CD''' is the '''latus rectum'''.
|-
| | Right-click on Segment '''CD''' and choose the '''Rename''' option.
| | Right-click on Segment '''CD'''and choose the '''Rename''' option.
|-
| | Type '''Latus Rectum''' in the '''New Name''' field >> click '''OK''' button.
| | In the '''New Name''' field, type '''Latus Rectum''' and click '''OK'''.
|-
| | Move the '''Latus''' label so you can see it properly.
| | Move the '''Latus''' label so you can see it properly.
|-
| | Click and drag '''Graphics''' view to see the parabola properly.
| | Click and drag '''Graphics''' view to see the parabola properly.
|-
| | Point to '''Algebra''' view.

Drag boundary so you can see equation properly.

| | In '''Algebra''' view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the '''axis of symmetry, directrix''' and '''latus rectum'''.
|-
| | Drag boundary so you can see '''Graphics''' view properly again.
| | Drag boundary so you can see '''Graphics''' view properly again.
|-
| | Click in '''Graphics''' view and drag background.
| | Click in '''Graphics''' view and drag background.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersection of the '''axis of symmetry''' and the '''directrix'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on '''axis of symmetry''' and '''directrix'''.
|-
| | Point to point '''E'''.
| | This creates point '''E'''.
|-
| | Click on '''Distance or Length''' tool under '''Angle''' tool.
| | Under '''Angle''' tool, click on '''Distance or Length''' tool.
|-
| | Click on '''Focus''' >> '''Vertex'''.
| | Click on '''Focus''' and '''Vertex'''.
|-
| | Point to the distance of '''FocusVertex''' appearing in '''Graphics''' view.

| | Note the distance of '''FocusVertex''' appearing in '''Graphics''' view.
|-
| | Click on '''Vertex''' >> point '''E'''.
| | Click on '''Vertex''' and point '''E'''.
|-
| | Point to the distance of '''Vertex E''' appearing in '''Graphics''' view.
| | Note the distance of '''Vertex E''' appearing in '''Graphics''' view.

Both these distances are equal.
|-
| |
| | Let us look at the general equations of parabolas.
|-
| | Show the new '''GeoGebra''' window.
| | I have opened a new '''GeoGebra''' window.
|-
| | Type '''(x-a)^2=4 p (y-b)''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses'''.

To type '''caret symbol''', hold '''Shift''' key down and press 6.

Note that the spaces denote multiplication.

Press '''Enter'''.
|-
| | Point to '''Create Sliders''' window
| | '''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders a, p''' and '''b'''.
| | '''Sliders''' are created for '''a, p''' and '''b'''.

The '''default''' setting for all three '''coefficients''' is 1.
|-
| | A parabola opening upwards appears in '''Graphics''' view.

Point to '''vertex''' of parabola.
| | A parabola opening upwards appears in '''Graphics''' view.

'''a comma b''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
| | Double click on parabola >> click on '''Object Properties''' and then on '''Color''' tab.
| | Double click on the parabola, click on '''Object Properties''' and then on '''Color''' tab.
|-
| | Select red and close the '''Preferences''' box.
| | Select red and close the '''Preferences''' box.
|-
| | Point to the red parabola and its equation in '''Graphics''' and '''Algebra''' views.
| | The parabola and its equation appear red in the '''Graphics''' and '''Algebra''' views.
|-
| | Move boundary so you can see the equation properly.
| | Move boundary so you can see the equation properly.
|-
| | Right-click on '''slider a''' button >> check '''Animation On''' option.
| | Right-click on '''slider a''' and check '''Animation On''' option.
|-
| | Point to the parabola in '''Graphics''' view.
| | Note the effects on the horizontal movement of the red parabola.
|-
| | Right-click on '''slider a''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider a''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider p''' >> check '''Animation On''' option.
| | Right-click on '''slider p''' and check '''Animation On''' option.
|-
| | Point to parabola in '''Graphics view'''.
| | Note the effects on the shape and orientation of the parabola.
|-
| | Right-click on '''slider p''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider p''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider b''' >> check '''Animation On''' option.
| | Right-click on '''slider b''' and check '''Animation On''' option.
|-
| | Point to the parabola.
| | Note the effects on the vertical movement of the parabola.
|-
| | Right-click on '''slider b''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider b''' and uncheck '''Animation On''' option.
|-
| | Point to '''sliders a, p''' and '''b''' (all = 1) and the red parabola '''c''' in '''Graphics '''view.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Point to equation '''c: (x2-2x-4y) = -5''' in '''Algebra''' view.
| | Note that when '''a''', '''p''' and '''b '''are equal to 1, the red parabola '''c''' is described by equation '''c'''.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Equation '''c''' is given by '''x squared minus 2x minus 4y equals minus 5'''.
|-
| | Type '''Focus(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''A''' in '''Graphics''' view.
| | In '''input bar''', type '''Focus c in parentheses'''.

Press '''Enter'''.

'''Focus''' is drawn at point '''A ''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''A''', the '''Focus''', in '''Algebra''' view.
| | The coordinates of '''Focus''' of parabola '''c''', which is point '''A''', appear in '''Algebra''' view.
|-
| | Type '''Vertex(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''B''' in '''Graphics''' view.
| | In '''input bar''', type '''Vertex c in parentheses'''.

Press '''Enter'''.

'''Vertex''' is drawn at point '''B''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''B''' in '''Algebra''' view.
| | The '''coordinates''' of '''Vertex''' of parabola '''c''', which is point '''B''', appear in '''Algebra''' view.
|-
| | Type '''Directrix(c)''' in '''input bar''' >> press '''Enter'''.

Point to '''Directrix''' in '''Graphics''' view.
| | In '''input bar''', type '''Directrix c''' in parentheses.

Press '''Enter'''.

'''Directrix''' appears as a line along '''x axis''' in '''Graphics''' view.
|-
| | Point to the equation, '''y=0''', in '''Algebra''' view.
| | The equation for the '''Directrix''' of parabola '''c''', '''y equals 0''', appears in '''Algebra''' view.
|-
| | Double click on '''Directrix''' in '''Graphics''' view >> '''Redefine''' text box >> '''Object Properties''' >> '''Color''' tab.
| | Double click on '''Directrix''' in '''Graphics''' view.

In the '''Redefine''' text box, click on '''Object Properties''', then on the '''Color''' tab.
|-
| | In the left panel, point to highlighted '''Directrix''', identify '''Focus''' and ''' Vertex''' created for parabola '''c'''.
| | In the left panel, note that the '''Directrix''' is highlighted.

Identify '''Focus''' and '''Vertex''' created for parabola '''c'''.
|-
| | Click on each one to highlight while pressing the '''Control''' key.
| | While pressing the '''Control''' key, click and highlight '''Focus''' and '''Vertex'''.
|-
| | Click on red.
| | Click on red.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to '''Focus''', '''Vertex''' and '''Directrix''' and their '''co-ordinates''' and equation in '''Graphics''' and '''Algebra''' views.
| | For parabola '''c''', '''Focus''', '''Vertex''' and '''Directrix''' and their '''coordinates''' and equation appear red.
|-
| |Follow the earlier steps to construct parabola '''d'''.
| |Follow the earlier steps to construct parabola '''d'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
Study the standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 7'''
'''Assignment'''

Try these steps to construct parabolas with:

Focus (6,0) and '''directrix''' x = -6

Focus (0,-3) and '''directrix''' y = 3

Find their equations.
| | As an assignment:
Try these steps to construct parabolas with these '''foci''' and '''directrices'''.

Find their equations.
|-
| | '''Slide Number 8'''
'''Assignment'''

Find the coordinates of the '''foci''' and length of '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.

y2 = 12x

x2 = -16y
| | As an assignment:
Find the coordinates of the '''foci''' and length of the '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.
|-
| | '''Slide Number 9'''
'''About Spoken Tutorial Project'''
| | The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
| | '''Slide Number 10'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 11'''
'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 12'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English

2018-06-30T16:16:08Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Study standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this '''tutorial''', you should have basic knowledge of
'''GeoGebra''' interface

'''Conic sections''' in geometry
|-
| s | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux '''OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''

'''Parabola'''

[[Image:]]

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.
| | A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.

Observe the different features of the parabola in the image.

The '''Axis of Symmetry''' is perpendicular to the '''Directrix''' and passes through the Focus and '''Vertex'''.

'''Latus Rectum''' passes through the Focus and is perpendicular to the '''Axis of Symmetry'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct a parabola in '''GeoGebra'''.

I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.

Point to point '''A'''.
| | Click on '''Point''' tool and click in '''Graphics''' view.

This creates point '''A'''.
|-
| | Right-click on point '''A''' and select the '''Rename''' option.
| | Right-click on point '''A''' and select the '''Rename''' option.
|-
| | In '''New Name''' text box, type '''Focus''' instead of '''A''' >> click '''OK'''.

Point to '''Focus'''.
| | In the '''New Name''' text box, type '''Focus''' instead of '''A '''and click '''OK'''.

This renames point '''A''' as '''Focus'''.
|-
| | Click on '''Line''' tool >> click in two places in '''Graphics''' view below '''Focus'''.

Point to line '''AB'''.
| | Click on '''Line''' tool and click on two places in '''Graphics''' view, below '''Focus'''.

This creates line ''' AB'''.
|-
| | Right-click on line '''AB''' >> choose '''Rename''' option.
| | Right-click on line '''AB''' and choose the '''Rename''' option.
|-
| | Type '''directrix''' in '''New Name''' field >> click '''OK'''.

Point to '''directrix'''.
| | In the '''New Name''' field, type '''directrix''' and click '''OK'''.

This renames line '''AB''' as the '''directrix'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on line '''AB'''.
| | Click on '''Perpendicular Line''' tool, then click on line '''AB'''.
|-
| | Drag the '''cursor''' until '''Focus''' >> click on point '''A'''.
| | Drag the '''cursor''' until the resulting line passes through '''Focus''' and click on '''Focus'''.
|-
| | Point to the perpendicular line through '''Focus'''.

Point to '''axis of symmetry'''.
| | This draws a line perpendicular to line '''AB''', passing through '''Focus'''.

This line is the '''axis of symmetry'''.
|-
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field >> click '''OK'''.
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field.

Click '''OK'''
|-
| | Click on '''Parabola''' tool under '''Ellipse''' tool.

Click on '''Focus''' and line '''AB''' ('''directrix''').
| | Under '''Ellipse''' tool, click on '''Parabola''' tool.

Then click on '''Focus''' and the '''directrix'''.
|-
| | Point to the parabola.
| | This creates a parabola with its focus at '''Focus''' and with line '''AB''' as the '''directrix'''.
|-
| | Click on '''Intersect''' tool. >> Click on the '''parabola''' and '''axis of symmetry'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and '''axis of symmetry'''.
|-
| | Point to point '''C'''.
| | This creates point '''C''' at the intersection.

It is the '''vertex''' of the parabola.
|-
| | Right-click on point '''C''' >> choose the '''Rename''' option.
| | Right-click on point '''C''' and choose the '''Rename ''' option.
|-
| | Type '''Vertex''' in '''the '''New Name''' field >> click '''OK'''.
| | In the '''New Name''' field, type '''Vertex''' and click '''OK'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on the '''axis of symmetry'''.
| | Click on '''Perpendicular Line''' tool and click on the '''axis of symmetry'''.
|-
| | Drag the '''cursor''' until the line passes through point '''A''' (Focus) >> click on point '''A'''.
| | Drag the '''cursor''' until the line passes through the '''Focus''' and click on it.
|-
| | Point to the parallel line.
| | This results in a line parallel to the '''directrix''', passing through the '''Focus'''.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersections of the parabola and the newly drawn line through '''Focus'''.

| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and the newly drawn line through '''Focus'''.
|-
| | Point to points '''C''' and '''D'''.
| | This creates points '''C''' and '''D'''.
|-
| | Click on '''Segment''' tool under the '''Line''' tool >> click on points '''C'''and '''D'''.
| | Under '''Line''' tool, click on '''Segment''' tool and click on points '''C''' and '''D'''.
|-
| | Point to Segment '''CD'''.
| | Resulting Segment '''CD''' is the '''latus rectum'''.
|-
| | Right-click on Segment '''CD''' and choose the '''Rename''' option.
| | Right-click on Segment '''CD'''and choose the '''Rename''' option.
|-
| | Type '''Latus Rectum''' in the '''New Name''' field >> click '''OK''' button.
| | In the '''New Name''' field, type '''Latus Rectum''' and click '''OK'''.
|-
| | Move the '''Latus''' label so you can see it properly.
| | Move the '''Latus''' label so you can see it properly.
|-
| | Click and drag '''Graphics''' view to see the parabola properly.
| | Click and drag '''Graphics''' view to see the parabola properly.
|-
| | Point to '''Algebra''' view.

Drag boundary so you can see equation properly.

| | In '''Algebra''' view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the '''axis of symmetry, directrix''' and '''latus rectum'''.
|-
| | Drag boundary so you can see '''Graphics''' view properly again.
| | Drag boundary so you can see '''Graphics''' view properly again.
|-
| | Click in '''Graphics''' view and drag background.
| | Click in '''Graphics''' view and drag background.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersection of the '''axis of symmetry''' and the '''directrix'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on '''axis of symmetry''' and '''directrix'''.
|-
| | Point to point '''E'''.
| | This creates point '''E'''.
|-
| | Click on '''Distance or Length''' tool under '''Angle''' tool.
| | Under '''Angle''' tool, click on '''Distance or Length''' tool.
|-
| | Click on '''Focus''' >> '''Vertex'''.
| | Click on '''Focus''' and '''Vertex'''.
|-
| | Point to the distance of '''FocusVertex''' appearing in '''Graphics''' view.

| | Note the distance of '''FocusVertex''' appearing in '''Graphics''' view.
|-
| | Click on '''Vertex''' >> point '''E'''.
| | Click on '''Vertex''' and point '''E'''.
|-
| | Point to the distance of '''Vertex E''' appearing in '''Graphics''' view.
| | Note the distance of '''Vertex E''' appearing in '''Graphics''' view.

Both these distances are equal.
|-
| |
| | Let us look at the general equations of parabolas.
|-
| | Show the new '''GeoGebra''' window.
| | I have opened a new '''GeoGebra''' window.
|-
| | Type '''(x-a)^2=4 p (y-b)''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses'''.

To type '''caret symbol''', hold '''Shift''' key down and press 6.

Note that the spaces denote multiplication.

Press '''Enter'''.
|-
| | Point to '''Create Sliders''' window
| | '''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders a, p''' and '''b'''.
| | '''Sliders''' are created for '''a, p''' and '''b'''.

The '''default''' setting for all three '''coefficients''' is 1.
|-
| | A parabola opening upwards appears in '''Graphics''' view.

Point to '''vertex''' of parabola.
| | A parabola opening upwards appears in '''Graphics''' view.

'''a comma b''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
| | Double click on parabola >> click on '''Object Properties''' and then on '''Color''' tab.
| | Double click on the parabola, click on '''Object Properties''' and then on '''Color''' tab.
|-
| | Select red and close the '''Preferences''' box.
| | Select red and close the '''Preferences''' box.
|-
| | Point to the red parabola and its equation in '''Graphics''' and '''Algebra''' views.
| | The parabola and its equation appear red in the '''Graphics''' and '''Algebra''' views.
|-
| | Move boundary so you can see the equation properly.
| | Move boundary so you can see the equation properly.
|-
| | Right-click on '''slider a''' button >> check '''Animation On''' option.
| | Right-click on '''slider a''' and check '''Animation On''' option.
|-
| | Point to the parabola in '''Graphics''' view.
| | Note the effects on the horizontal movement of the red parabola.
|-
| | Right-click on '''slider a''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider a''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider p''' >> check '''Animation On''' option.
| | Right-click on '''slider p''' and check '''Animation On''' option.
|-
| | Point to parabola in '''Graphics view'''.
| | Note the effects on the shape and orientation of the parabola.
|-
| | Right-click on '''slider p''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider p''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider b''' >> check '''Animation On''' option.
| | Right-click on '''slider b''' and check '''Animation On''' option.
|-
| | Point to the parabola.
| | Note the effects on the vertical movement of the parabola.
|-
| | Right-click on '''slider b''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider b''' and uncheck '''Animation On''' option.
|-
| | Point to '''sliders a, p''' and '''b''' (all = 1) and the red parabola '''c''' in '''Graphics '''view.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Point to equation '''c: (x2-2x-4y) = -5''' in '''Algebra''' view.
| | Note that when '''a''', '''p''' and '''b '''are equal to 1, the red parabola '''c''' is described by equation '''c'''.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Equation '''c''' is given by '''x squared minus 2x minus 4y equals minus 5'''.
|-
| | Type '''Focus(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''A''' in '''Graphics''' view.
| | In '''input bar''', type '''Focus c in parentheses'''.

Press '''Enter'''.

'''Focus''' is drawn at point '''A ''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''A''', the '''Focus''', in '''Algebra''' view.
| | The coordinates of '''Focus''' of parabola '''c''', which is point '''A''', appear in '''Algebra''' view.
|-
| | Type '''Vertex(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''B''' in '''Graphics''' view.
| | In '''input bar''', type '''Vertex c in parentheses'''.

Press '''Enter'''.

'''Vertex''' is drawn at point '''B''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''B''' in '''Algebra''' view.
| | The '''coordinates''' of '''Vertex''' of parabola '''c''', which is point '''B''', appear in '''Algebra''' view.
|-
| | Type '''Directrix(c)''' in '''input bar''' >> press '''Enter'''.

Point to '''Directrix''' in '''Graphics''' view.
| | In '''input bar''', type '''Directrix c''' in parentheses.

Press '''Enter'''.

'''Directrix''' appears as a line along '''x axis''' in '''Graphics''' view.
|-
| | Point to the equation, '''y=0''', in '''Algebra''' view.
| | The equation for the '''Directrix''' of parabola '''c''', '''y equals 0''', appears in '''Algebra''' view.
|-
| | Double click on '''Directrix''' in '''Graphics''' view >> '''Object Properties''' >> '''Color''' tab.
| | Double click on '''Directrix''' in '''Graphics''' view.

Choose '''Object Properties''', then the '''Color''' tab.
|-
| | In the left panel, point to highlighted '''Directrix''', identify '''Focus''' and ''' Vertex''' created for parabola '''c'''.
| | In the left panel, note that the '''Directrix''' is highlighted.

Identify '''Focus''' and '''Vertex''' created for parabola '''c'''.
|-
| | Click on each one to highlight while pressing the '''Control''' key.
| | While pressing the '''Control''' key, click and highlight '''Focus''' and '''Vertex'''.
|-
| | Click on red.
| | Click on red.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to '''Focus''', '''Vertex''' and '''Directrix''' and their '''co-ordinates''' and equation in '''Graphics''' and '''Algebra''' views.
| | For parabola '''c''', '''Focus''', '''Vertex''' and '''Directrix''' and their '''coordinates''' and equation appear red.
|-
| |'''(y-l)2=4 p (x-m)'''

'''''(y-l)(caret)2 = 4 (space) p (space) (x-m)'''''

'''''(l, m)''''' is vertex of parabola d

For '''''l = m = 2, p = 1, d = y2-4x-4y-12

Effects of '''''a, p, b, l''''' and '''''m''''' on parabolas '''c''' and '''d'''

'''Focus, vertex, directrix'''
| |Follow the earlier steps to construct parabola '''d'''.

In '''input bar''', type '''y minus l in parentheses caret 2 equals 4 space p space x minus m in parentheses'''.

'''l comma m''' correspond to the '''co-ordinates''' of the '''vertex'''.

Set '''sliders l''' and '''m''' at 2 and '''p''' at 1.

Equation '''d''' is given by '''y squared minus 4x minus 4y minus 12'''.

Note the effects of '''sliders a, p, b, l''' and '''m''' on the parabolas.

Find the '''focus, vertex''' and '''directrix''' for parabola '''d'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
Study the standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 7'''
'''Assignment'''

Try these steps to construct parabolas with:

Focus (6,0) and '''directrix''' x = -6

Focus (0,-3) and '''directrix''' y = 3

Find their equations.
| | As an assignment:
Try these steps to construct parabolas with these '''foci''' and '''directrices'''.

Find their equations.
|-
| | '''Slide Number 8'''
'''Assignment'''

Find the coordinates of the '''foci''' and length of '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.

y2 = 12x

x2 = -16y
| | As an assignment:
Find the coordinates of the '''foci''' and length of the '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.
|-
| | '''Slide Number 9'''
'''About Spoken Tutorial Project'''
| | The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
| | '''Slide Number 10'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 11'''
'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 12'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English

2018-06-28T13:17:07Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn to,
Plot graphs of''' '''quadratic '''functions'''

Calculate '''real''' and '''complex roots''' of quadratic '''functions'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this tutorial, you should be familiar with:
'''GeoGebra''' interface

Basics of quadratic equations, geometry and graphs

Previous tutorials in this series

If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''
'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux''' OS version 14.04

'''Geogebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''
'''Quadratic polynomials'''

2nd degree '''polynomial''' '''y =''' '''ax2+bx+c'''

Parabola

If parabola intersects '''x axis''', '''x intercepts''' are '''real roots'''.

If parabola does not intersect '''x axis''' at all, no '''real roots''', only '''complex'''
| | '''Quadratic polynomials'''

Let us find out more about a '''2nd degree polynomial'''.

'''y equals a x squared plus b x plus c'''

The '''function''' graphs as a parabola.

If the parabola intersects the''' x axis, '''the '''intercepts''' are real roots.

If the parabola does not intersect '''x axis''' at all, it has no '''real roots'''.

'''Roots''' are '''complex'''.

Let us look at '''complex''' numbers.
|-
| | '''Slide Number 6'''
'''Complex numbers, XY plane'''

As we know,

A '''complex number''' is expressed as '''''z = a + b i''''': where '''''a''''' is the '''real''' part, '''''b i''''' is '''imaginary'''part, and '''a''' and '''b''' are constants.

'''Imaginary number, ''i'' '''= sqrt(-1}

In the XY plane, '''''a + b i''''' corresponds to the point ('''a, b''').

In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
| | '''Complex numbers, XY plane'''
As we know,

A '''complex number''' is expressed as '''''z''''' equals a plus b i'''''.

'''''a''''' is the '''real''' part; '''''b i''''' is imaginary part<nowiki>; </nowiki>'''a''' and '''b''' are constants.

'''''i''''' is '''imaginary number''' and is equal to '''squareroot of minus 1'''.

In the XY plane, '''a plus b i''' corresponds to the point '''a comma b'''.

In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 7'''
'''Complex numbers, complex plane'''

[[Image:]]

In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' ‘'''a'''’ and '''imaginary axis coordinate''' ‘'''b'''’

Length of the '''vector ''z''''' = |'''''z'''''| =''' ''r'''''

'''''r'' = sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is ‘'''a'''’ and '''imaginary axis coordinate''' is ‘'''b'''’.

The length of the '''vector ''z''''' is equal to the absolute value''' '''of '''''z''''' and to '''''r'''''.

According to''' Pythagoras’ theorem, ''r'' '''is equal to '''squareroot of a squared plus b squared.'''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
| | Click on''' Slider '''tool and then click in '''Graphics''' view.
|-
| | Point to the dialog box.
| | '''Slider''' dialog-box appears.
|-
| | Point to '''Number''' radio button.
| | By default, '''Number''' radio-button is selected.
|-
| | Type Name as '''a.'''
| | In the '''Name '''field, type '''a'''.
|-
| | Point to''' Min, Max '''and''' Increment '''values.
| | Set '''Min '''value as 1, '''Max '''value as 5 and Increment as 1.
|-
| | Click '''OK''' button.
| | Click '''OK''' button.
|-
| | Point to the '''slider'''.
| | This creates a number '''slider''' named “'''a'''”.
|-
| | Drag to show the changing values.
| | Using the '''slider''', '''a''' can have values from 1 to 5, in increments of 1.
|-
| | Following the same steps, create '''sliders b''' and '''c'''.
| | Following the same steps, create '''sliders''' '''b''' and '''c'''.
|-
| | In '''input bar''', type '''f(x):=a x^2+b x+c'''>>press '''Enter'''.

Drag boundary to see '''Algebra''' view properly.
| | In '''input bar''', type the following line.

'''f x '''in parentheses '''colon equals a space x caret 2 plus b space x plus c'''.

Press '''Enter'''.

Drag boundary to see '''Algebra''' view properly.

Pay attention to the spaces''' '''indicating multiplication.
|-
| | Point to the equation for '''f(x)''' in '''Algebra view'''.
| | Observe the equation for '''f of x''' in '''Algebra view'''.
|-
| | On '''sliders''', move '''a''' to '''1''', '''b''' to '''-2''' and '''c''' to '''-3'''.
| | Set '''slider''' '''a '''at '''1''', '''slider''' '''b''' at minus''' 2''' and '''slider''' '''c '''at minus''' 3'''.
|-
| | Point to the equation '''f(x)=1x2-2x-3''' in '''Algebra view'''.
| | The equation '''f of x equals 1 x squared minus 2 x minus 3''' appears in '''Algebra view'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.
|-
| | Point to the parabola in '''Graphics View'''.
| | '''Function''' '''f''' is a parabola, intersecting '''x axis''' at '''minus 1 comma 0''' and '''3 comma 0'''.

Thus,''' root'''s of '''fx equals x squared minus 2x minus 3 '''are '''x equals minus 1''' and '''3'''.
|-
| | Type '''Root(f)''' in input bar>>press '''Enter'''.
| | In '''input bar''', type '''Root f '''in parentheses and press '''Enter'''.
|-
| | Point to the '''roots''' in '''Algebra view''' and the '''intercepts''' in '''Graphics view.'''
| | The '''roots''' appear in '''Algebra view'''.

They also appear as '''x-intercepts''' of the parabola in '''Graphics view'''.
|-
| | Type '''Extremum(f)''' in Input bar>>press '''Enter'''.
| | In '''input bar''', type '''Extremum f''' in parentheses and press '''Enter'''.
|-
| | Point to the '''extremum''' in the '''Algebra''' and '''Graphics views.'''
| | The '''minimum vertex''' appears in '''Algebra''' and '''Graphics views'''.
|-
| | Double click on point C ('''extremum''') in '''Graphics view'''>>Select '''Object Properties.'''
| | After double clicking on point '''C''' in '''Graphics View''', select '''Object Properties.'''
|-
| | Click on '''red''' color box.
| | From '''Color''' tab, change the color to red.
|-
| | Close the '''Preferences '''box.
| | Close the '''Preferences '''dialog-box.
|-
| | Point to '''C''' in '''Algebra''' and '''Graphics''' views.
| | '''Point C''' ('''extremum''' of '''f of x''') is red in '''Algebra''' and '''Graphics views'''.
|-
| | Click on '''Move''' tool, drag '''a''' to '''1''', '''b''' to '''5''', '''c''' to '''10'''.
| | Click on '''Move''' tool, set '''slider''' '''a''' at '''1''', '''slider''' '''b''' at '''5''', '''slider''' '''c''' at '''10'''.
|-
| | Point to the equation '''f(x)=1x2+5x+10''' in '''Algebra view.'''
| | The equation '''f of x equals 1 x squared plus 5x plus 10 '''appears in '''Algebra view'''.
|-
| | Click in and drag '''Graphics''' view to see this parabola.
| | Click in and drag '''Graphics''' view to see this parabola.
|-
| | Point to the parabola in '''Graphics View'''.
| | It does not intersect the '''x-axis'''.
|-
| | Point to the '''roots''', '''points A and B''' in the '''Algebra view'''.
| | '''Points A '''and''' B''' are undefined as the '''function''' does not intersect the '''x axis'''.
|-
| | Point to '''extremum''' point C in '''Algebra''' and '''Graphics views'''.
| | '''Extremum''' (point '''C''') is shown in red in '''Algebra''' and '''Graphics views'''.
|-
| |
| | '''Function f of x equals x squared plus 5x plus 10 '''has no '''real roots'''.

Let us see the '''complex root'''s of this equation.
|-
| | Click on '''View'''>>'''Spreadsheet'''.
| | Click on '''View''', then on '''Spreadsheet'''.

This opens a spreadsheet on the right side of the '''Graphics view'''.
|-
| | Click to close '''Algebra view'''.
| | Click to close '''Algebra view'''.
|-
| | Drag the boundary to see '''Spreadsheet''' view properly.
| | Drag the boundary to see '''Spreadsheet''' view properly.
|-
| | Point to the '''spreadsheet'''.
| | Type the following '''labels''' and formulae in the '''spreadsheet'''.
|-
| | Type “'''b^2-4ac'''” in cell '''A1'''>>press '''Enter'''.

Drag column to adjust width.
| | In '''cell A1''', type within quotes '''b caret 2 minus 4ac''' and press '''Enter.'''

Drag column to adjust width.

'''b squared minus 4ac '''is also called the''' discriminant'''.
|-
| | Type '''Root1''' and '''Root2''' in cells '''A4''' and '''A5'''>>press '''Enter'''.
| | In cells '''A4''' and '''A5''', type '''Root1''' and '''Root2, '''and press '''Enter.'''
|-
| | Type '''Complex root1''' and '''Complex root2''' in '''A9''' and '''A10'''>>press '''Enter'''.
| | In cells '''A9''' and '''A10, '''type '''Complex root1''' and '''Complex root2'''.

Press '''Enter'''.
|-
| | Drag column to adjust width.
| | Drag column to adjust width.
|-
| | Type '''b^2-4 a c''' in cell '''B1'''>>press '''Enter. '''
| | In '''cell B1''', type '''b caret 2 minus 4 space a space c''' and press '''Enter. '''
|-
| | Point to cell '''B1'''.
| | The value minus 15 appears in '''cell''' '''B1''' corresponding to '''b squared minus 4 a c''' for '''f x'''.

Note: '''Discriminant''' is always negative for quadratic '''functions''' without '''real roots'''.
|-
| | Type “'''-b/2a'''” in cell '''B3'''>>press '''Enter'''.
| | In '''cell B3''', type within quotes '''minus b divided by 2a'''.

Press '''Enter.'''
|-
| | Type '''–b/2 a''' in cell '''B4'''>>press '''Enter'''.
| | In '''cell B4''', type '''minus b divided by 2 space a'''.

Press '''Enter'''.

Note the value '''-2.5''' appear in cell '''B4'''.
|-
| | Type '''B4''' in cell '''B5'''>>press '''Enter'''.
| | In '''cell B5''', type '''B4''' and press '''Enter. '''

The value -'''2.5''' appears in cell '''B5 '''also.
|-
| | Type “'''+-sqrt(b^2-4ac)/2a'''” in cell '''C3'''>>press '''Enter.'''
| | In '''cell C3''', type the following '''line''' and press '''Enter'''.

Within quotes, '''plus minus sqrt D divided by 2a'''
|-
| | Type '''sqrt(B1)/2 a''' in cell '''C4'''>>press '''Enter.'''
| | In '''cell C4''', type '''sqrt B1''' in parentheses''' divided by 2 space a''' and press '''Enter. '''

Note that a question mark appears in '''cell C4'''.
|-
| | Type '''–C4''' in cell '''C5'''>>press '''Enter.'''
| | In '''cell C5''', type '''minus C4''' and press '''Enter. '''

Again, a question mark appears in '''cell''' '''C5'''.

There are no '''real''' solutions to the '''negative square root of the discriminant'''.
|-
| | Type '''(b4+c4,0)''' in the input bar>>press '''Enter.'''
| | In '''input bar''', type '''b4 plus c4 comma 0 in parentheses''' and press '''Enter.'''

This should '''plot''' the '''root''' corresponding to '''ratio of minus b plus squareroot of discriminant to 2a.'''
|-
| | Type '''(b5+c5,0)''' in the input bar>>press '''Enter. '''
| | In input bar, type '''b5 plus c5 comma 0''' '''in parentheses '''and press '''Enter. '''

This should plot the '''root''' corresponding to '''ratio of minus b minus squareroot of discriminant to 2a'''.
|-
| |
| | '''f x equals x squared plus 5x plus 10 '''has no '''real roots'''.

Hence, the points do not appear in '''Graphics view'''.
|-
| | Click in and drag''' Graphics '''view to see this properly.''' '''
| | Click in and drag''' Graphics '''view to see this properly.''' '''
|-
| | Type '''–b/2 a''' in cell '''B9'''>>press '''Enter'''.
| | In '''cell B9''', type '''minus b divided by 2 space a '''and press '''Enter.'''
|-
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
|-
| |
| | '''Discriminant''' is less than 0 for '''f x equals x squared plus 5x plus 10'''.

So the opposite sign will be taken to allow calculation of '''roots'''.
|-
| | Type '''sqrt(-B1)/2 a''' in cell '''C9'''>>press '''Enter.'''
| | In '''cell C9''', type '''sqrt minus B1''' in parentheses''' divided by 2 space a''' and press '''Enter.'''

1.94 appears in '''C9'''.
|-
| | Type '''–C9''' in cell '''C10'''>>press '''Enter.'''
| | In '''cell C10''', type '''minus C9''' and press '''Enter.'''

'''Minus''' 1.94 appears in '''C10'''.
|-
| | Click in and drag '''Graphics''' view to see both '''roots'''.
| | Click in and drag '''Graphics''' view to see the following '''complex''' '''roots'''.
|-
| | Type '''(b9,c9)''' in the input bar>>press '''Enter. '''
| | In '''input bar''', type '''b9 comma c9''' in parentheses and press '''Enter. '''

This '''complex root''' has '''real axis coordinate,''' '''minus b divided by 2a'''.

Imaginary axis co-ordinate is '''squareroot of negative discriminant divided by 2a'''.
|-
| | Type '''(b10,c10)''' in the input bar>>press '''Enter.'''
| | In input bar, type '''b10 comma''' '''c10 in parentheses''' and press '''Enter. '''

This complex root has '''real axis coordinate, minus b divided by 2a. '''

'''Imaginary axis''' co-ordinate is '''minus squareroot of negative discriminant divided by 2a'''.
|-
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.
|-
| | Drag the '''slider''' '''b''' to -2 and '''c''' to -3.
| | Drag the '''slider''' '''b''' to -2 and '''slider''' '''c''' to -3.
|-
| | Click in and drag '''Graphics''' view to see the parabola.
| | Click in and drag '''Graphics''' view to see the parabola.
|-
| | Point to the parabola in '''Graphics view'''.
| | Note how the parabola changes to the one seen for '''f x equals x squared minus 2x minus 3'''.
|-
| | Point to the '''roots''' appearing at x '''intercepts''' of parabola in '''Graphics view'''.
| | The '''real roots''' plotted earlier for '''f x equals x squared minus 2x minus 3''' appear now.
|-
| | Drag boundary to see '''Spreadsheet''' view.
| | Drag boundary to see '''Spreadsheet''' view.
|-
| | Point to the question marks appearing in '''C9''' and '''C10''' in the '''spreadsheet'''.
| | As '''roots''' are '''real''', calculations for '''complex roots''' become invalid.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 8'''
'''Summary'''
| | In this tutorial, we have learnt to:
Visualize quadratic '''polynomials''', their '''roots''' and '''extrema'''

Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''
|-
| | '''Slide Number 9'''
'''Assignment'''
| | As an assignment:
Drag '''sliders''' to graph different quadratic '''polynomials'''.

Calculate '''roots''' of the '''polynomials'''.
|-
| | '''Slide Number 10'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 11'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 12'''
'''Forum for specific questions:'''
Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 13'''
'''Acknowledgement'''
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is Vidhya Iyer from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Roots-of-Polynomials/English

2018-06-19T05:37:44Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Roots of Polynomials'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn:
*To plot graphs of '''polynomial''' equations

*About '''complex numbers''', '''real''' and '''imaginary roots'''

*To find '''extrema''' and '''inflection points'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this tutorial, you should be familiar with
*'''GeoGebra''' interface

*Basics of '''coordinate system'''

*'''Polynomials'''

*If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:

*'''Ubuntu Linux''' OS version 14.04

*'''GeoGebra 5.0.388.0-d'''
|-
| |
| | Let us begin with the '''binomial theorem'''.
|-
| | '''Slide Number 5'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' Єℝ, '''index''' ''n'' is a '''positive integer''',

*''0 ≤ r ≤n, then

*''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

Reminder: ''nC1 = n!/[1! (n-1)!]''
| | '''''a''''' and '''''b''''' are '''real numbers'''.

'''index''' '''''n''''' is a positive integer.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''''a''''' plus '''''b''''' raised to '''''n''''' can be expanded as shown.
|-
| | '''Slide Number 6'''

'''Quadratic Equations and Roots'''

*A second degree polynomial, '''y =''' '''''a''''' '''x2+''' '''''b''''' '''x+''' '''''c''''' has roots

*'''x=-''' '''''b''''' '''± sqrt{(''' '''''b''''' '''2-4''' '''''ac)/2a''''' '''}'''

*where '''▲=''' '''''b2-4ac'''''

*When ▲< 0, roots are complex

*When ▲=0, roots are real and equal

*When ▲>0, roots are real and unequal
| | '''Quadratic Equations and Roots'''

A '''2nd degree polynomial''', '''y equals''' '''''a''''' '''x squared plus''' '''''b''''' '''x plus''' '''''c''''' has '''roots''' given by values of '''''x'''''.

'''''x''''' is equal to '''ratio''' of minus '''''b''''' plus or minus '''squareroot''' of '''''b''''' '''squared''' minus 4 '''''a c''''' to 2 '''''a'''''.

Where '''discriminant''' '''Delta''' is equal to '''''b''''' '''squared''' minus 4 '''''a c'''''

When '''Delta''' is less than 0, '''roots''' are '''complex'''

When '''Delta''' is equal to 0, '''roots''' are '''real''' and equal

When '''Delta''' is greater than 0, '''roots''' are '''real''' and unequal
|-
| | '''Slide Number 7'''

'''Quadratic Equations and Roots'''

*When roots are real, '''''ax2+b''''' '''x+''' '''''c''''' '''=0''' has extremum '''(xv, yv)'''

*'''xv = -''' '''''b/2a''''' and '''yv=''' '''''a''''' '''xv2+''' '''''b''''' '''xv+''' '''''c'''''
| | '''Quadratic Equations and Roots'''

When '''roots''' are '''real''', '''''ax''''' '''squared plus''' '''''b x''''' '''plus''' '''''c''''' equals 0 has '''extremum''' '''''xv''''' '''comma''' '''''yv'''''

'''''xv''''' equals '''minus''' '''''b''''' '''divided by 2''' '''''a''''' and '''''yv''''' '''equals''' '''''axv''''' '''squared plus''' '''''bxv''''' '''plus''' '''''c'''''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' >> select '''CAS'''.

| | Click on '''View''' tool and select '''CAS''' to open the '''CAS''' view.
|-
| | In line 1 in '''CAS view''', type '''f(x):=x^2-2x-3''' >> press '''Enter'''.
| | In line 1 in '''CAS''' view, type the following line.

'''f x''' in parentheses '''colon equals x caret 2 minus 2 space x minus 3'''.

To type '''caret''' symbol, hold '''Shift''' key down and press 6.

The space indicates multiplication.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | Point to the '''equation f(x)''' appearing in '''Algebra''' view.

Point to '''exponent''' 2 in '''f(x)'''.
| | Observe the '''equation f of x''' in '''Algebra''' view.

The '''degree''' of this '''quadratic polynomial f of x''' is 2.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
|-
| | '''Move Graphics View'''>> click on '''Zoom Out'''
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
|-
| |Click in '''Graphics''' view >> minimum '''vertex''' of '''parabola f'''.

| |Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
|-
| | Click on '''Move Graphics View''' tool >> click in '''Graphics''' background.
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.
|-
| |Hand symbol appears >> drag '''Graphics''' view to see parabola '''f'''.
| |When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
|-
| | Drag boundaries
| | Drag boundaries to see '''CAS''' view properly.
|-
| | Type '''Root(f)''' in line 2 of '''CAS''' view >> press '''Enter'''.
| | In line 2 of '''CAS''' view, type '''Root f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''roots''' in '''CAS''' view.
| | The '''roots''' appear below, in the same box, in curly brackets.
|-
| | Point to '''roots''' in '''Graphics''' view.
| | Note that these are the '''x-intercepts''' of parabola '''f''' in '''Graphics''' view.
|-
| | Type '''Extremum(f)''' in line 3 of '''CAS''' view >> press '''Enter'''.
| | In line 3 of '''CAS''' view, type '''Extremum f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
|-
| | In line 4 in '''CAS''' view, type '''g(x):=x^2+5x+10''' >> press '''Enter'''.
| | In line 4 in '''CAS''' view, type the following line.

'''g x''' in parentheses '''colon equals x caret 2 plus 5 space x plus 10'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.

Point to the '''equation g(x)''' appearing in '''Algebra''' view.
| | Drag boundary to see '''Algebra''' view properly.

Observe the '''equation g of x''' in '''Algebra''' view.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
|-
| | Click and drag '''Graphics''' view so you can see parabola '''g'''.
| | Click in and drag '''Graphics''' view so you can see parabola '''g'''.
|-
| | Again, drag boundary to see '''CAS''' view properly.
| | Again, drag boundary to see '''CAS''' view properly.
|-
| | Type '''Root(g)''' in line 5 of '''CAS''' view >> press '''Enter'''.
| | In line 5 of '''CAS''' view, type '''Root g''' in parentheses.

Press '''Enter'''.
|-
| | Point to empty curly brackets for '''roots''' in '''CAS''' view.
| | Empty curly brackets appear below.

Parabola '''g''' does not have any '''real roots''' as it does not intersect '''x axis''' at all.

'''Roots''' are said to be '''complex'''.
|-
| | Type '''Extremum(g)''' in line 6 of '''CAS '''view >> press '''Enter'''.
| | In line 6 of '''CAS''' view, type '''Extremum g''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
|-
| | Point to '''Evaluate''' tool.

Point to '''extremum''' in form of '''fractions'''.
| | While '''Evaluate''' tool is highlighted in '''CAS View''' toolbar, the '''extremum''' appears as '''fractions'''.

'''Minus five divided by 2 comma 15 divided by 4'''.
|-
| | Click on the '''extremum''' in line 6 and click on '''Numeric''' tool.

Point to '''extremum''' in form of '''decimals'''.
| | In line 6, click on the '''extremum''' and click on '''Numeric''' tool.

The '''extremum '''now appears in '''decimal''' form.

'''Minus 2 point 5 comma 3 point 7 5'''.
|-
| |
| | Let us look at '''complex numbers'''.
|-
| | '''Slide Number 8'''

'''Complex numbers, XY plane'''

*A '''complex number''' is expressed as '''''z = a + bi''''': where ''''''a'''''' = real part, ‘'''''bi’''''' = '''imaginary '''part, and '''a''' and '''b''' are constants.

*'''Imaginary number, ''i'' '''= sqrt{-1}

*In the '''XY plane''', '''''a + bi''''' is point ('''a, b''').

*In the '''complex plane''', '''x axis''' = '''real axis''', '''y axis''' = '''imaginary axis'''.
| | '''Complex numbers, XY plane'''

A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''bi'''''.

'''''a''''' is the '''real''' part; '''''bi''''' is '''imaginary '''part;'''''a''''' and '''''b''''' are '''constants'''

'''''i''''' is '''imaginary number''' and is equal to '''square root of minus 1'''.

In the '''XY plane''', '''''a''''' '''plus''' '''''bi''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.

In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 9'''

'''Complex numbers, complex plane'''

*In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' '''''a''''' and '''imaginary axis coordinate''' '''''b'''''

*Length of the '''vector''' '''''z''''' = |'''''z'''''| = '''''r'''''

*'''''r''''' '''= sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is '''''a''''' and '''imaginary axis coordinate''' is '''''b'''''.

The length of the '''vector''' '''''z''''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

According to '''Pythagoras’ theorem''', '''''r''''' is equal to '''squareroot of''' '''''a''''' '''squared plus''' '''''b''''' '''squared'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us go back to the '''GeoGebra interface''' we were working on.

We will now use the '''input bar''' instead of '''CAS''' view.
|-
| | Click and close '''CAS''' view.
| | Click and close '''CAS''' view.
|-
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
|-
| | In '''input bar''', type the following line.

'''h(x):=x^3-4x^2+x+6''' >> press '''Enter'''.

| | In '''input bar''', type the following line.

'''h x''' in parentheses '''colon equals x caret 3 minus 4 space x caret 2 plus x plus 6'''.

Press '''Enter'''.
|-
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Point to the '''equation h(x)''' appearing in '''Algebra''' view.
| | Drag boundaries to see '''Algebra''' and '''Graphics''' views properly.

Observe equation '''h of x''' in '''Algebra''' view.

Function '''h of x''' is graphed in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
| | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
|-
| | In '''input bar''', type '''Root(h)''' and press '''Enter'''.
| | In '''input bar''', type '''Root h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') in '''Algebra''' view.
| | The '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') appear in '''Algebra''' view.
|-
| | Point to three '''roots''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The three '''roots''' are also mapped as '''x intercepts''' of the '''curve h of x''' in '''Graphics''' view.
|-
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of two '''extrema''' ('''D'''and '''E''') in '''Algebra''' view.
| | '''Co-ordinates''' of two '''extrema''' ('''D''' and '''E''') appear in '''Algebra''' view.
|-
| | Point to two '''extrema''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
|-
| | '''Slide Number 11'''

'''Point of inflection'''

'''Point of inflection''' ('''PoI''') on a curve is the point where '''curve''' changes direction.

*To find '''co-ordinates''' of '''PoI (x,y)''', we equate 2nd '''derivative''' of given '''function''' to 0

*Solve to get '''x''' ('''x co-ordinate''' of '''PoI''')

*Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''
| | '''Point of inflection'''

A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.

To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''',

We equate second '''derivative''' of the given '''function''' to 0.

Solution of this equation gives us '''x''' ('''x co-ordinate''' of '''PoI''').

Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''.
|-
| | Let us find the '''point of inflection''' on '''h(x)'''.
| | Let us find the '''point of inflection''' on '''h of x'''.
|-
| | In '''input bar''', type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In '''input bar''', type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''h''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''h''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | In '''Algebra''' view, '''point of inflection''' appears as point '''F''', below the two '''extrema'''.
|-
| | Point to '''F''' on '''h(x)''' in '''Graphics''' view.
| | '''F''' is mapped on '''h of x''' in '''Graphics''' view.
|-
| |
| | Correlate the '''degree''' of the '''polynomials''' and the number of '''roots''' seen so far.
|-
| | Point to '''CAS''', then '''Algebra''' and '''Graphics''' views.
| | Observe that '''functions''' entered in '''CAS''' appear in '''Algebra''' and '''Graphics''' views.
|-
| | Point to '''Algebra''' and '''Graphics''' views, then '''CAS''' view.
| | '''Functions''' entered in '''input bar''' appear in '''Algebra''' and '''Graphics''' views but not in '''CAS''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''
| | In this tutorial, we have learnt to:

*Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''

*Find '''real roots, extrema''' and '''inflection point(s)'''

*'''Complex roots''' will be covered in another tutorial
|-
| | '''Slide Number 13'''
'''Assignment'''

*d(x)=x2-6x+5

*e(x)=3x3-2x2+0.2x-1

*f(x)=-2x4-x3+3x2

*g(x)=x5-7x4+9x3+23x2-50x+24

*h(x)=(4x+3)/(x-1)

*i(x)=(3x2-2x-1)/(2x2+3x-2)
| | Assignment:
Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.
|-
| | '''Slide Number 14'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 15'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 16'''
'''Forum for specific questions:'''
Do you have questions in THIS Spoken Tutorial?
Please visit this site.
Choose the '''minute''' and '''second''' where you have the question.
Explain your question briefly.
Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 17'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English

2018-06-11T16:00:04Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Inverse Trigonometric Functions'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn to use '''GeoGebra''' to

Plot graphs of '''inverse trigonometric functions'''

Compare them to graphs of '''trigonometric functions'''

Create '''check-boxes''' to group and show or hide '''functions'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:
'''GeoGebra''' interface

'''Trigonometry'''

For relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 4'''
'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux OS version. 14.04'''

'''GeoGebra 5.0.388.0-d'''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | '''Switching x axis to radians'''

Double click on '''x axis''' in '''Graphics '''view >> '''Object Properties'''
| | Now let us change '''x Axis units''' to '''radians'''.

In '''Graphics''' view, double-click on the '''x axis''' and then on '''Object Properties'''.
|-
| | Click on '''Preferences-Graphics''' >> '''x axis'''.
| | In the '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''x Axis'''.
|-
| | Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
| | Check the '''Distance''' option, select '''pi''' divided by 2 and then the '''Ticks first option'''.
|-
| | Close the '''Preferences''' box.
| | Close the''' Preferences''' box.
|-
| | Point to '''x-axis'''.
| | Units of '''x-axis''' are in '''radians''' with interval of '''pi''' divided by 2 as shown.

'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.
|-
| | Point to the '''toolbar'''.
| | Note that the name appears when you place the mouse over any '''tool icon''' in the '''toolbar'''.
|-
| | Click on '''Slider''' tool >> click on '''Graphics''' view.
| | In the '''Graphics toolbar''', click on '''Slider''' and then in the top of '''Graphics''' view.
|-
| | Point to the '''Slider dialog box'''.
| | A '''slider dialog-box''' appears.
|-
| | Point to Number radio button.
| | By default, '''Number''' radio-button is selected.
|-
| | Type '''Name''' as '''symbol theta ϴ'''.
| | In the '''Name''' field, select '''theta''' from the '''Symbol menu'''.
|-
| | Point to''' Min, Max''' and '''Increment''' values.

Click '''OK'''.
| | Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.

Click '''OK'''.
|-
| | Point to slider '''ϴ'''.
| | This creates a '''number slider theta''' from minus 360 to plus 360.
|-
| | In '''input bar''', type '''α = (ϴ /180) π'''.

Point to space between the right parenthesis and '''pi''' for multiplication.

Press '''Enter'''
| | In the '''input bar''', type '''alpha is equal to theta divided by 180 in parentheses''', and then '''pi'''.

Note how '''GeoGebra''' inserts a space between the right parenthesis and '''pi''' for multiplication.

Press '''Enter'''.
|-
| | Drag '''slider ϴ''' to -360 and then back to 360.
| | Drag '''slider theta''' from minus 360 and then back to 360.
|-
| | Point to values of '''α''' in '''Algebra''' view.
| | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to 2 '''pi radians''' as you change '''theta'''.
|-
| | Drag '''slider ϴ''' to minus 360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Sine function'''

In '''input bar''', type''' f_S: = Function[sin(x), -2π, α]''' >> press '''Enter'''.
| | In the '''input bar''', type the following command:

'''f underscore S colon is equal to Function with capital F'''

Type the following words in square brackets.

'''sin, x in parentheses, comma minus 2 pi comma alpha'''.

Press '''Enter'''.
|-
| | Drag the boundary to see '''Algebra''' view properly.
| | Drag the boundary to see '''Algebra''' view properly.
|-
| | Point to '''fS''' in '''Algebra''' view.
| | Here, '''fS''' defines the '''sine function''' of '''x'''.

'''x''' is between -2 '''pi''' and '''alpha''' which can take a maximum value of 2 '''pi'''.

This is called the '''domain''' of the '''function'''.
|-
| | Drag the boundary to see '''Graphics''' View properly.
| | Drag the boundary to see '''Graphics''' View properly.
|-
| | Drag '''slider theta''' from minus 360 to 360.
| | Drag '''slider theta''' from minus 360 to 360.
|-
| | Point to '''fS sine function''' graph.
| | This graphs the '''sine function''' of '''x'''.
|-
| | In the toolbar, click on the bottom right triangle of the last button.

Point to '''Move Graphics View''' button.
| | In the toolbar, click on the bottom right triangle of the last button.

Note that this button is called '''Move Graphics View'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
| | In the menu that appears, click on '''Zoom Out'''.

Click in '''Graphics view''' to see 2 '''pi radians''' on either side of the '''origin'''.
|-
| | Again, click on '''Move Graphics View''' and drag the background to see the graph properly.
| | Again, click on '''Move Graphics View''' and drag the background to see the graph properly.
|-
| | Drag '''slider ϴ''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse Trigonometric Functions'''

e.g., If '''sin-1z''' (or '''arcsin z''') '''= w''', then '''z = sin w'''

Restrict '''domain''' of trigonometric function, define '''principal value'''

Interchange '''x''' and '''y''' axes

Change curvature of '''trigonometric function graph'''
| | '''Inverse Trigonometric Functions'''

For example, If '''inverse sine''' of '''z''' (also known as '''arcsin''' of '''z''') is '''w'''.

Then, '''z''' is '''sin w'''.

'''w''' can have multiple values.

So a '''principal value''' has to be defined and the '''domain''' has to be restricted.

To get the '''inverse function''' graph, interchange '''x''' and '''y''' axes

Next, change curvature of '''trigonometric function graph'''.

You can pause and refer to the example in the '''additional material''' provided for this '''tutorial'''.
|-
| |
| | Let us go back to the '''GeoGebra''' window.
|-
| | '''Inverse sine function'''

Type '''i_S: = Function[asin(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
| | In the '''input bar''', type the following command:

'''i underscore S colon is equal to Function with capital F'''

Type the following words in square brackets.

'''asin, x in parentheses, comma minus 1 comma 1”

Press '''Enter'''.
|-
| | Drag the boundary to see '''Algebra''' view properly.
| | Drag the boundary to see '''Algebra''' view properly.
|-
| | Point to''' iS function''' graph.
| | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.

Note that '''x''' and '''y axes''' are interchanged for this '''inverse sine function'''.

Its '''domain''' (set of '''x''' values) lies between minus 1 and 1.

Observe the graph.
|-
| | Drag the boundary to see '''Graphics''' view properly.
| | Drag the boundary to see '''Graphics''' view properly.
|-
| | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
| | In the '''input bar''', type the following command:

'''P underscore S colon is equal to'''

Type the following words in parentheses.

'''sin alpha in parentheses comma alpha'''

Press '''Enter'''.
|-
| | Point to''' PS'''.
| | This creates point '''PS''' on the '''inverse sine''' graph.

On the '''sine function''' graph, '''PS''' would be '''alpha comma sine alpha'''.
|-
| | In '''Algebra''' view, right-click on''' PS, check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PS''', check the '''Trace On''' option.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PS''', '''iS''' and '''FS'''.
| | Traces appear for the '''inverse sine function''' graph for '''alpha'''.

'''fs''' also appears in '''Graphics''' view.

Compare '''iS''' and traces of '''PS'''.

Note that the '''domain''' for the graph that '''PS''' traces is not restricted from -1 to 1.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
| | Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
|-
| | In '''Algebra''' view, uncheck '''fS''', '''iS''' and '''PS'''.
| | In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
|-
| | '''Cosine and Inverse Cosine Functions'''

'''Cosine function fC''' in '''domain [-2π, α]'''

'''Inverse cosine function iC''' in '''domain [-1,1]

'''PC (cos(α),α)'''
| | '''Cosine and Inverse Cosine Functions'''

Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.

Its '''domain''' should be from -2 '''pi''' to '''alpha'''.

Graph the '''inverse cosine function iC" in the '''domain''' from -1 to 1.

Create a point '''PC''' whose '''co-ordinates''' are '''cos alpha comma alpha'''.

The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond -1 and 1.
|-
| | Point to '''fC, iC and traces of PC in '''Graphics''' view.
| | The '''cosine''' and '''inverse cosine functions''' should look like this.
|-
| | In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.
| | In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Tangent and Inverse Tangent Functions'''
'''Tangent function fT''' in '''domain [-2π, α]'''

'''Inverse tangent function iT''' in '''domain [-∞, ∞]'''

'''PT (tan(α),α)'''
| | '''Tangent and Inverse Tangent Functions'''
Now graph the '''tangent function fT'''.

Its domain should also be from -2 '''pi''' to '''alpha'''.

We will look at the graph for the '''inverse tangent function iT'''.

Its domain will be from -infinity to infinity.

Create a point '''PT''' whose '''co-ordinates''' are '''tan alpha comma alpha'''.

The '''domain''' of the '''inverse tangent''' graph that '''PT''' traces will go beyond -1 and 1.
|-
| |
| | Let us look at the '''inverse tangent function''' graph in the domain from -1 to 1.
|-
| | '''Inverse tangent function'''

Type '''i_T: = Function[atan(x), -∞, ∞]''' in '''input bar''' >> press '''Enter'''.
| |
To type infinity, click in the '''input bar''' and on '''symbol alpha''' appearing at the right end of the bar.

In the '''symbol menu''', click on the '''infinity symbol''' in the third row and third column from the right.

In the '''input bar''', type the following command:

'''i underscore T colon is equal to Function with capital F'''

Type the following words in square brackets.

'''atan, x in parentheses, comma minus infinity comma infinity'''

Press '''Enter'''.
|-
| | Point to '''iT function''' graph.
| | This graphs the '''inverse tangent''' function of '''x'''.

'''x''' lies between minus '''infinity''' and '''infinity'''.

Observe the graph.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PT''' and '''iT'''.
| | Compare traces of '''PT''' and '''iT'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
| | In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
|-
| | In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
| | In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
|-
| | '''Check boxes'''

Under '''Slider''', click on check box tool.

Click on the top of the grid in '''Graphics''' view.
| | Under '''Slider''', click on '''Check-box'''.

Click on the top of the grid in '''Graphics''' view.
|-
| | Point to the '''dialog box'''.
| | '''Check-Box to Show/Hide Objects dialog-box''' appears.
|-
| | Type '''SIN''' as '''caption'''.
| | In the '''Caption''' field, type '''SIN.'''
|-
| | Click on '''Objects''' >> select '''fS, iS''' and '''PS''' >> '''apply'''
| | Click on '''Objects''' drop-down menu to select '''fS, iS''' and '''PS''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''SIN'''”.
| | A '''check-box''' “'''SIN'''” is created in '''Graphics''' view.

It gives us the option to display or hide '''sine, arcsine''' graphs and point '''PS'''.
|-
| | Click on '''Check Box'''.

Click on the top of the grid in '''Graphics''' view.
| | Again, click on '''Check Box'''.

Click on the top of the grid in '''Graphics''' view.
|-
| | Point to the '''dialog box'''.
| | A '''Check-Box to Show/Hide Objects dialog-box''' appears.
|-
| | Type '''TAN''' as '''caption'''.
| | In the '''Caption''' field, type '''TAN'''.
|-
| | Click on '''Objects''' >> select '''fT, iT''' and '''PT''' >> '''apply'''.
| | Click on '''Objects''' drop-down menu to select '''fT, iT''' and '''PT''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''TAN'''”.
| | A '''check-box''' “'''TAN'''” is created in '''Graphics''' view.

It gives us the option to display or hide '''tangent, arctangent''' graphs and point '''PT'''.
|-
| | Click on '''Move''' tool to uncheck all boxes.
| | In the '''toolbar''', click on the first '''Move''' button and uncheck all boxes.
|-
| | Check “'''SIN'''” box.
| | Check the “'''SIN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fS, iS''' and traces of '''PS''' in '''Graphics''' view.
| | Observe '''fS, iS''' and traces of '''PS''' appear in '''Graphics''' view.
|-
| | Uncheck '''SIN''' box.
| | Uncheck '''SIN''' box.
|-
| | Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Check “'''TAN'''” box.
| | Check “'''TAN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fT, iT''' and traces of '''PT''' in '''Graphics''' view.
| | Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.
|-
| | Drag '''slider theta''' back to minus 360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Check the '''SIN''' box.
| | Check the '''SIN''' box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to all the '''functions''' in '''Graphics''' view.
| | Observe all the '''functions''' appearing in '''Graphics''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Graph '''trigonometric functions'''

Graph '''inverse trigonometric functions'''

Create'''check-boxes''' to group and show/hide '''functions'''
|-
| | '''Slide Number 7'''

'''Assignment'''
| | As an assignment:

Plot graphs of

'''Secant''' and '''arcsecant'''

'''Cosecant''' and arccosecant'''

'''Cotangent''' and '''arccotangent'''

For hints, you can refer to the '''additional material''' provided.
|-
| | '''Slide Number 8'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 9'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 10'''

'''Forum for specific questions:'''

Do you have questions in THIS '''Spoken Tutorial'''?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 11'''

'''Acknowledgement'''
| | The '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Roots-of-Polynomials/English

2018-05-17T06:50:43Z

Vidhya:

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Roots of Polynomials'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn:
*To plot graphs of '''polynomial''' equations

*About '''complex numbers''', '''real''' and '''imaginary roots'''

*To find '''extrema''' and '''inflection points'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this tutorial, you should be familiar with *'''GeoGebra''' interface

*Basics of '''coordinate system'''

*'''Polynomials'''

If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:

*'''Ubuntu Linux''' OS version 14.04

*'''GeoGebra 5.0.388.0-d'''
|-
| |
| | Let us begin with the '''binomial theorem'''.
|-
| | '''Slide Number 5'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' Єℝ, '''index''' ''n'' is a '''positive integer''', ''0 ≤ r ≤n, then (a + b)n can be expanded as follows:''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

Reminder: ''nC1 = n!/[1! (n-1)!]''
| | '''''a''''' and '''''b''''' are '''real numbers'''.

'''index''' '''''n''''' is a positive integer.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''''a''''' plus '''''b''''' raised to '''''n''''' can be expanded as shown.
|-
| | '''Slide Number 6'''

'''Quadratic Equations and Roots'''

A second degree polynomial, '''y =''' '''''a''''' '''x2+''' '''''b''''' '''x+''' '''''c''''' has roots

'''x=-''' '''''b''''' '''± sqrt{(''' '''''b''''' '''2-4''' '''''ac)/2a''''' '''}'''

where '''▲=''' '''''b2-4ac'''''

When ▲< 0, roots are complex

When ▲=0, roots are real and equal

When ▲>0, roots are real and unequal
| | '''Quadratic Equations and Roots'''

A '''2nd degree polynomial''', '''y equals''' '''''a''''' '''x squared plus''' '''''b''''' '''x plus''' '''''c''''' has '''roots''' given by values of '''''x'''''.

'''''x''''' is equal to '''ratio''' of minus '''''b''''' plus or minus '''squareroot''' of '''''b''''' '''squared''' minus 4 '''''a c''''' to 2 '''''a'''''.

Where '''determinant''' '''Delta''' is equal to '''''b''''' '''squared''' minus 4 '''''a c'''''

When '''Delta''' is less than 0, '''roots''' are '''complex'''

When '''Delta''' is equal to 0, '''roots''' are '''real''' and equal

When '''Delta''' is greater than 0, '''roots''' are '''real''' and unequal
|-
| | '''Slide Number 7'''

'''Quadratic Equations and Roots'''

When roots are real, '''''ax2+b''''' '''x+''' '''''c''''' '''=0''' has extremum '''(xv, yv)'''

'''xv = -''' '''''b/2a''''' and '''yv=''' '''''a''''' '''xv2+''' '''''b''''' '''xv+''' '''''c'''''
| | '''Quadratic Equations and Roots'''

When '''roots''' are '''real''', '''''ax''''' '''squared plus''' '''''b x''''' '''plus''' '''''c''''' equals 0 has '''extremum''' '''''xv''''' '''comma''' '''''yv'''''

'''''xv''''' equals '''minus''' '''''b''''' '''divided by 2''' '''''a''''' and '''''yv''''' '''equals''' '''''axv''''' '''squared plus''' '''''bxv''''' '''plus''' '''''c'''''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' >> select '''CAS'''.

| | Click on '''View''' tool and select '''CAS''' to open the '''CAS''' view.
|-
| | In line 1 in '''CAS view''', type '''f(x):=x^2-2x-3''' >> press '''Enter'''.
| | In line 1 in '''CAS''' view, type the following line.

'''f x''' in parentheses '''colon equals x caret 2 minus 2 space x minus 3'''.

To type '''caret''' symbol, hold '''Shift''' key down and press 6.

The space indicates multiplication.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | Point to the '''equation f(x)''' appearing in '''Algebra''' view.

Point to '''exponent''' 2 in '''f(x)'''.
| | Observe the '''equation f of x''' in '''Algebra''' view.

The '''degree''' of this '''quadratic polynomial f of x''' is 2.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
|-
| | '''Move Graphics View'''>> click on '''Zoom Out'''
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
|-
| |Click in '''Graphics''' view >> minimum '''vertex''' of '''parabola f'''.

| |Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
|-
| | Click on '''Move Graphics View''' tool >> click in '''Graphics''' background.
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.
|-
| |Hand symbol appears >> drag '''Graphics''' view to see parabola '''f'''.
| |When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
|-
| | Drag boundaries
| | Drag boundaries to see '''CAS''' view properly.
|-
| | Type '''Root(f)''' in line 2 of '''CAS''' view >> press '''Enter'''.
| | In line 2 of '''CAS''' view, type '''Root f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''roots''' in '''CAS''' view.
| | The '''roots''' appear below, in the same box, in curly brackets.
|-
| | Point to '''roots''' in '''Graphics''' view.
| | Note that these are the '''x-intercepts''' of parabola '''f''' in '''Graphics''' view.
|-
| | Type '''Extremum(f)''' in line 3 of '''CAS''' view >> press '''Enter'''.
| | In line 3 of '''CAS''' view, type '''Extremum f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
|-
| | In line 4 in '''CAS''' view, type '''g(x):=x^2+5x+10''' >> press '''Enter'''.
| | In line 4 in '''CAS''' view, type the following line.

'''g x''' in parentheses '''colon equals x caret 2 plus 5 space x plus 10'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.

Point to the '''equation g(x)''' appearing in '''Algebra''' view.
| | Drag boundary to see '''Algebra''' view properly.

Observe the '''equation g of x''' in '''Algebra''' view.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
|-
| | Click and drag '''Graphics''' view so you can see parabola '''g'''.
| | Click in and drag '''Graphics''' view so you can see parabola '''g'''.
|-
| | Again, drag boundary to see '''CAS''' view properly.
| | Again, drag boundary to see '''CAS''' view properly.
|-
| | Type '''Root(g)''' in line 5 of '''CAS''' view >> press '''Enter'''.
| | In line 5 of '''CAS''' view, type '''Root g''' in parentheses.

Press '''Enter'''.
|-
| | Point to empty curly brackets for '''roots''' in '''CAS''' view.
| | Empty curly brackets appear below.

Parabola '''g''' does not have any '''real roots''' as it does not intersect '''x axis''' at all.

'''Roots''' are said to be '''complex'''.
|-
| | Type '''Extremum(g)''' in line 6 of '''CAS '''view >> press '''Enter'''.
| | In line 6 of '''CAS''' view, type '''Extremum g''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
|-
| | Point to '''Evaluate''' tool.

Point to '''extremum''' in form of '''fractions'''.
| | While '''Evaluate''' tool is highlighted in '''CAS View''' toolbar, the '''extremum''' appears as '''fractions'''.

'''Minus five divided by 2 comma 15 divided by 4'''.
|-
| | Click on the '''extremum''' in line 6 and click on '''Numeric''' tool.

Point to '''extremum''' in form of '''decimals'''.
| | In line 6, click on the '''extremum''' and click on '''Numeric''' tool.

The '''extremum '''now appears in '''decimal''' form.

'''Minus 2 point 5 comma 3 point 7 5'''.
|-
| |
| | Let us look at '''complex numbers'''.
|-
| | '''Slide Number 8'''

'''Complex numbers, XY plane'''

A '''complex number''' is expressed as '''''z = a + bi''''': where ''''''a'''''' = real part, ‘'''''bi’''''' = '''imaginary '''part, and '''a''' and '''b''' are constants.

'''Imaginary number, ''i'' '''= sqrt{-1}

In the '''XY plane''', '''''a + bi''''' is point ('''a, b''').

In the '''complex plane''', '''x axis''' = '''real axis''', '''y axis''' = '''imaginary axis'''.
| | '''Complex numbers, XY plane'''

A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''bi'''''.

'''''a''''' is the '''real''' part; '''''bi''''' is '''imaginary '''part;'''''a''''' and '''''b''''' are '''constants'''

'''''i''''' is '''imaginary number''' and is equal to '''square root of minus 1'''.

In the '''XY plane''', '''''a''''' '''plus''' '''''bi''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.

In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 9'''

'''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' '''''a''''' and '''imaginary axis coordinate''' '''''b'''''

Length of the '''vector''' '''''z''''' = |'''''z'''''| = '''''r'''''

'''''r''''' '''= sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is '''''a''''' and '''imaginary axis coordinate''' is '''''b'''''.

The length of the '''vector''' '''''z''''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

According to '''Pythagoras’ theorem''', '''''r''''' is equal to '''squareroot of''' '''''a''''' '''squared plus''' '''''b''''' '''squared'''.
|-
| | '''Slide Number 10'''

'''Complex numbers, complex plane'''

'''Argument '''''ϴ''''' '''= angle between '''real axis''' and '''line segment''' connecting '''''z''''' to O '''(0,0)'''; CCW

'''Polar form''' of '''''z = a + bi''''' is

'''''z = r (cosϴ + i sinϴ)'''''

where '''''a= r cosϴ, b=r sinϴ'''''
| | '''Argument ''theta''''' is angle between '''real axis''' and line segment connecting '''''z''''' to '''origin'''.

It is in counter-clockwise direction.

'''Polar form''' of '''''z''''' equals '''''a''''' '''plus''' '''''bi''''' is

'''''z''''' equals '''''r times cos theta plus i sin theta'''''

Where '''''a''''' is equal to '''''r cos theta''''' and '''''b is r sin theta'''''
|-
| | Show the '''GeoGebra''' window.
| | Let us go back to the '''GeoGebra interface''' we were working on.

We will now use the '''input bar''' instead of '''CAS''' view.
|-
| | Click and close '''CAS''' view.
| | Click and close '''CAS''' view.
|-
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
|-
| | In '''input bar''', type the following line.

'''h(x):=x^3-4x^2+x+6''' >> press '''Enter'''.

| | In '''input bar''', type the following line.

'''h x''' in parentheses '''colon equals x caret 3 minus 4 space x caret 2 plus x plus 6'''.

Press '''Enter'''.
|-
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Point to the '''equation h(x)''' appearing in '''Algebra''' view.
| | Drag boundaries to see '''Algebra''' and '''Graphics''' views properly.

Observe equation '''h of x''' in '''Algebra''' view.

Function '''h of x''' is graphed in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
| | Click on '''Move Graphics View''' and move '''Graphics''' background to see the graph.
|-
| | In '''input bar''', type '''Root(h)''' and press '''Enter'''.
| | In '''input bar''', type '''Root h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') in '''Algebra''' view.
| | The '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') appear in '''Algebra''' view.
|-
| | Point to three '''roots''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The three '''roots''' are also mapped as '''x intercepts''' of the '''curve h of x''' in '''Graphics''' view.
|-
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of two '''extrema''' ('''D'''and '''E''') in '''Algebra''' view.
| | '''Co-ordinates''' of two '''extrema''' ('''D''' and '''E''') appear in '''Algebra''' view.
|-
| | Point to two '''extrema''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
|-
| | '''Slide Number 11'''

'''Point of inflection'''

'''Point of inflection''' ('''PoI''') on a curve is the point where '''curve''' changes direction.

To find co-ordinates of PoI (x,y)

Equate 2nd derivative of given function to 0

Solve to get x (x co-ordinate of PoI)

Substitute this x in original function to get y co-ordinate
| | '''Point of inflection'''

A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.

To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''', we equate second '''derivative''' of the given '''function''' to 0.

Solution of this equation gives us '''x''' ('''x co-ordinate''' of '''PoI''').

Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''.
|-
| | Let us find the '''point of inflection''' on '''h(x)'''.
| | Let us find the '''point of inflection''' on '''h of x'''.
|-
| | In '''input bar''', type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In '''input bar''', type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''h''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''h''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | In '''Algebra''' view, '''point of inflection''' appears as point '''F''', below the two '''extrema'''.
|-
| | Point to '''F''' on '''h(x)''' in '''Graphics''' view.
| | '''F''' is mapped on '''h of x''' in '''Graphics''' view.
|-
| |
| | Let us open a new '''GeoGebra''' window to use '''CAS''' for a '''cubic polynomial'''.
|-
| | Click on '''View''' tool and click on '''CAS''' to show it.
| | Click on '''View''' tool and click on '''CAS''' to show it.
|-
| | Drag boundary to see '''CAS''' view properly.
| | Drag boundary to see '''CAS''' view properly.
|-
| | In line 1 of '''CAS''' view, type the following line.

'''i(x):=x^3-6 x^2+4 x+1''' >> press '''Enter'''.
| | In line 1 of '''CAS'' 'view, type the following line.

'''i x''' in parentheses '''colon equals x caret 3 minus 6 space x caret 2 plus 4 space x plus 1'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''CAS''' view properly.
| | Drag boundary to see '''CAS''' view properly.
|-
| | In line 2 of''' CAS''' view, type '''Root(i)''' >> press '''Enter'''.
| | In line 2 of''' CAS''' view, type '''Root i''' in parentheses and press '''Enter'''.
|-
| | Point to the three '''roots''' in '''CAS''' view.

Scroll to see them.

Point to '''Evaluate''' tool.
| | The three '''roots''' are shown below with '''square root notations'''.

Scroll to see them.

Note that the '''Evaluate''' tool is highlighted.
|-
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

Point to the three '''roots''' in decimal form.
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

The roots are now shown in '''decimal''' form in the next line.
|-
| | In line 4 of '''CAS''' view, type '''Extremum(i)''' >> press '''Enter'''.
| | In line 4 of '''CAS''' view, type '''Extremum i''' in parentheses and press '''Enter'''.
|-
| | Point to the two '''extrema''' in '''CAS''' view.

Scroll to see them.

Point to '''Numeric''' tool and to '''extrema''' in '''decimal''' form.
| | The two '''extrema''' points are shown below.

Scroll to see them.

As the '''Numeric''' tool was clicked, the points appear in '''decimal''' form.
|-
| | Click in and drag '''Graphics''' view so you can see '''i(x)'''.
| | Click in and drag '''Graphics''' view so you can see '''i of x'''.
|-
| | In line 5, type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In line 5, type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''i''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''i''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | '''Co-ordinates''' of '''point of inflection''' appear in curly brackets.
|-
| |
| | Correlate the '''degree''' of the '''polynomials''' and the number of '''roots''' seen so far.
|-
| | Point to '''CAS''', then '''Algebra''' and '''Graphics''' views.
| | Observe that '''functions''' entered in '''CAS''' appear in '''Algebra''' and '''Graphics''' views.
|-
| | Point to '''Algebra''' and '''Graphics''' views, then '''CAS''' view.
| | '''Functions''' entered in '''input bar''' appear in '''Algebra''' and '''Graphics''' views but not in '''CAS''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''
| | In this tutorial, we have learnt to:

*Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''

*Find '''real roots, extrema''' and '''inflection point(s)'''

*'''Complex roots''' will be covered in another tutorial
|-
| | '''Slide Number 13'''
'''Assignment'''
| | Assignment:

Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.

d(x)=x2-6x+5

e(x)=3x3-2x2+0.2x-1

f(x)=-2x4-x3+3x2

g(x)=x5-7x4+9x3+23x2-50x+24

h(x)=(4x+3)/(x-1)

i(x)=(3x2-2x-1)/(2x2+3x-2)
|-
| | '''Slide Number 14'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 15'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 16'''
'''Forum for specific questions:'''
Do you have questions in THIS Spoken Tutorial?
Please visit this site.
Choose the '''minute''' and '''second''' where you have the question.
Explain your question briefly.
Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 17'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English

2018-04-15T12:19:54Z

Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- |..."

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Inverse Trigonometric Functions'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn to plot graphs of '''inverse trigonometric functions''' in '''GeoGebra'''.
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:
'''GeoGebra''' interface

'''Trigonometry '''and related graphs

If not, for relevant '''tutorials''', please visit our website.
|-
| | '''Slide Number 4'''
'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux OS version. 14.04'''

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''
'''Inverse trigonometric functions'''
| | '''Arcsine, arccosine, arctangent''' etc are '''inverse trigonometric functions'''.

These ratios of right triangle lengths help to calculate the angle
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | '''Switching x axis to radians'''

Double click on '''x axis''' in '''Graphics '''view >> '''Object Properties'''
| | Now let us change '''x Axis units''' to '''radians'''.

Double-click on '''x axis''' in '''Graphics '''view and then on '''Object Properties'''.
|-
| | Click on '''Preferences-Graphics''' >> '''x axis'''.
| | In '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''x Axis'''.
|-
| | Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
| | Check the '''Distance''' option, select '''pi''' divided by 2 and then '''Ticks first option'''.
|-
| | Close the '''Preferences''' box.
| | Close the''' Preferences''' box.
|-
| | Point to '''x-axis'''.
| | Units of '''x-axis''' are in '''radians''' with interval of '''pi''' divided by 2 as shown.

'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.
|-
| | Click on '''Slider''' tool >> click on '''Graphics''' view.
| | Click on '''Slider''' tool and then click on '''Graphics''' view.
|-
| | Point to the '''Slider dialog box'''.
| | '''Slider dialog-box''' appears.
|-
| | Point to Number radio button.
| | By default, '''Number''' radio-button is selected.
|-
| | Type '''Name''' as '''symbol theta ϴ'''.
| | In the '''Name''' field, select '''theta''' from '''Symbol menu'''.
|-
| | Point to''' Min, Max''' and '''Increment''' values.

Click '''OK''' button.
| | Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.

Click '''OK''' button.
|-
| | Point to slider '''ϴ'''.
| | This creates a '''number slider theta''', from minus 360 to plus 360.
|-
| | In '''input bar''', type '''α = (ϴ /180) π'''.

Point to space between the right parenthesis and '''pi''' for multiplication.

Press '''Enter'''
| | In '''input bar''', type '''alpha is equal to theta divided by 180 in parentheses pi'''.

Note how '''GeoGebra''' inserts a space between the right parenthesis and '''pi''' for multiplication.

Press '''Enter'''.
|-
| | Drag '''slider ϴ''' to -360 and then to 360.
| | Drag '''slider theta''' from minus 360 to plus 360.
|-
| | Point to values of '''α''' in '''Algebra''' view.
| | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to plus 2 '''pi radians'''.
|-
| | Drag '''slider ϴ''' to minus 360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Sine function'''

In '''input bar''', type''' f_S: = Function[sin(x), -2π, α]''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''f underscore S colon is equal to Function with capital F'''

Type the following words in square brackets.

'''sin x in parentheses comma minus 2 pi comma alpha'''.

Press '''Enter'''.

Here, defines the '''sine function '''of''' x'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click on '''Graphics view''' to see 2 '''pi radians''' on either side of the '''origin'''.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' from minus 360 to plus 360.
|-
| | Point to''' fS sine function''' graph.
| | This graphs '''sine function''' of '''x'''.

'''Domain of x''' is between minus 2 '''pi''' and '''alpha'''.

That is,'''x''' lies between minus 2 '''pi''' and '''alpha'''.

Observe the graph of '''fS'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse sine function'''

Type '''i_S: = Function[asin(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''i underscore S colon is equal to Function with capital F'''

Type the following words in square brackets.

'''a sin x in parentheses comma minus 1 comma 1”

Press '''Enter'''.
|-
| | Point to''' iS function''' graph.
| | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.

'''x''' lies between minus 1 and plus 1.

Observe the graph.
|-
| | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
| | In '''input bar''', type the following command:

'''P underscore S colon is equal to'''

Type the following words in parentheses.

'''sin alpha in parentheses comma alpha'''

Press '''Enter'''.
|-
| | Point to''' PS'''.
| | This creates point '''PS'''.
|-
| | In '''Algebra''' view, right-click on''' PS, check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PS''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PS''', '''iS''' and '''FS'''.
| | Traces appear for '''inverse sine function''' graph for '''alpha'''.

'''fs''' also appears in '''Graphics''' view.

Compare '''iS''' and traces of '''PS'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click on and move '''Graphics''' view to erase traces of '''PS'''.
| | Click on and move '''Graphics''' view to erase traces of '''PS'''.
|-
| | In '''Algebra''' view, uncheck '''fS''', '''iS''' and '''PS'''.
| | In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
|-
| | '''Cosine function'''

Type '''f_C: = Function[cos(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''f underscore C colon is equal to Function with capital F'''

Type the following words in square brackets.
'''cos x in parentheses comma minus 2 pi comma alpha'''.

Press '''Enter'''.

Here, '''fC''' defines the '''cosine function'''.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to''' fC function''' graph.
| | This graphs the '''cos x function'''.

'''x''' lies between minus 2 '''pi''' and '''alpha'''.

Observe the graph.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse cosine function'''

Type '''i_C: = Function[acos(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''i underscore C colon is equal to Function with capital F'''

Type the following words in square brackets.

'''acos x in parentheses comma minus 1 comma 1'''.

Press '''Enter'''.

Here, '''IC''' defines the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.
|-
| | Point to '''iC function''' graph.
| | This graphs the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.

The '''domain''' of '''x''' is from minus 1 to plus 1.

Observe the graph.
|-
| | '''Point on cosine function'''.

Type '''P_C = (cos(α), α)''' in '''input bar''' >> press '''Enter'''
| | In '''input bar''', type the following command:

'''P underscore C colon is equal to'''

Type the following words in parentheses.

'''cos alpha in parentheses comma alpha'''

Press '''Enter'''.
|-
| | Point to '''Pc'''.
| | This creates a point '''PC'''.
|-
| | In '''Algebra''' view, right-click on '''PC''' check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PC''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' from 0 to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PC''', '''iC''' and '''FC'''.
| | Traces appear for '''inverse cosine function''' graph for '''alpha'''.

'''FC''' also appears in '''Graphics''' view.

Compare '''iC''' and traces of '''PC'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click on and move '''Graphics''' view to erase traces of '''PC'''.
| | Click on and move '''Graphics''' view to erase traces of '''PC'''.
|-
| | In '''Algebra''' view, uncheck '''fC''', '''iC''' and '''PC''' to hide them.
| | In '''Algebra''' view, uncheck '''fC, iC''' and '''PC''' to hide them.
|-
| | '''Tangent function'''

Type '''f_T: = Function[tan(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''f underscore T colon is equal to Function with capital F'''

Type the following words in square brackets.

'''Tan x in parentheses comma minus 2 pi comma alpha'''.

Press '''Enter'''.

Here, '''fT''' defines the '''tangent function''' of '''x'''.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fT tangent function''' graph.
| | This graphs '''tangent function''' of '''x''' in the '''domain''' from minus 2 '''pi''' to '''alpha'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse tangent function'''

Type '''i_T: = Function[atan(x), -∞, ∞]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''i underscore T colon is equal to Function with capital F'''

Type the following words in square brackets.

'''atan x in parentheses comma minus infinity comma infinity'''

Press '''Enter'''.

Here, '''IT''' defines the '''inverse tangent''' (or '''arctangent''') '''function''' of '''x'''.
|-
| | Point to '''iT function''' graph.
| | This graphs the '''inverse tangent''' function of '''x'''.

'''x''' lies between minus '''infinity''' and plus '''infinity'''.

Observe the graph.
|-
| | '''Point on tangent function'''

'''Type P_T = (tan(α), α)''' in '''input bar''' >> press '''Enter'''

| | In '''input bar''', type the following command:

'''P underscore T colon is equal to'''

Type the following words in parentheses.

'''Tan alpha in parentheses comma alpha'''

Press '''Enter'''.
|-
| | Point to '''PT'''.
| | This creates point '''PT'''.
|-
| | In '''Algebra''' view, right-click on '''PT''', check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PT''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PT''', '''iT''' and '''FT'''.
| | Traces appear for '''inverse tangent function''' graph for '''alpha'''.

'''FT''' also appears in '''Graphics''' view.

Compare '''iT''' and traces of '''PT'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''PT'''.
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''PT'''.
|-
| | In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
| | In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
|-
| | '''Check boxes'''

Under '''Slider''', click on check box tool.

Click on the top of the grid in '''Graphics view'''.
| | Under '''Slider''', click on '''Check-box''' tool.

Click on the top of the grid in '''Graphics view'''.
|-
| | Point to the '''dialog box'''.
| | '''Check-Box to Show/Hide Objects dialog-box''' appears.
|-
| | Type '''SIN''' as '''caption'''.
| | In the '''Caption''' field, type '''SIN.'''
|-
| | Click on '''Objects''' >> select '''fS, iS''' and '''PS''' >> '''apply'''
| | Click on '''Objects''' drop-down menu to select '''fS, iS''' and '''PS''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''SIN'''”.
| | A '''check-box''' “'''SIN'''” is created in '''Graphics''' view.

It gives us option to display or hide '''sine, arcsine''' graphs and point '''PS'''.
|-
| | Click on '''check box'''.

Click on the top of the grid in '''Graphics''' view.
| | Click on '''check box''' tool.

Click on the top of the grid in '''Graphics''' view.
|-
| | Point to the '''dialog box'''.
| | '''Check-Box to Show/Hide Objects dialog-box''' appears.
|-
| | Type '''COSIN''' as '''caption'''.
| | In the '''Caption''' field, type '''COSIN'''.
|-
| | Click on '''Objects''' >> select '''fS, iS''' and '''PS''' >> '''apply'''.
| | Click on '''Objects''' drop-down menu to select '''fc, ic''' and '''Pc''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''COSIN'''”.
| | A '''check-box''' “'''COSIN'''” is created in '''Graphics''' view.

It gives us option to display or hide '''cosine, arccosine''' graphs and point '''Pc'''.
|-
| | Click on '''check box'''.

Click on the top of the grid in '''Graphics''' view.
| | Click on '''check box''' tool.

Click on the top of the grid in '''Graphics''' view.
|-
| | Point to the '''dialog box'''.
| | '''Check-Box to Show/Hide Objects dialog-box''' appears.
|-
| | Type '''TAN''' as '''caption'''.
| | In the '''Caption''' field, type '''TAN'''.
|-
| | Click on '''Objects''' >> select '''fT, iT''' and '''PT''' >> '''apply'''.
| | Click on '''Objects''' drop-down menu to select '''fT, iT''' and '''PT''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''TAN'''”.
| | A '''check-box''' “'''TAN'''” is created on '''Graphics''' view.

It gives us option to display or hide '''tangent, arctangent''' graphs and point '''PT'''.
|-
| | Click on '''Move''' tool to uncheck all boxes.
| | Click on '''Move''' tool to uncheck all boxes.
|-
| | Check “'''SIN'''” box.
| | Check “'''SIN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fS, iS''' and traces of '''PS''' in '''Graphics''' view.
| | Observe '''fS, iS''' and traces of '''PS''' appear in '''Graphics''' view.
|-
| | Uncheck '''SIN''' box.
| | Uncheck '''SIN''' box.
|-
| | Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Check “'''COSIN'''” box.
| | Check “'''COSIN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fC, iC''' and traces of '''PC''' in '''Graphics''' view.
| | Observe '''fC, iC''' and traces of '''PC''' appear in '''Graphics''' view.
|-
| | Uncheck '''COSIN''' box.
| | Uncheck '''COSIN''' box.
|-
| | Click on and move '''Graphics''' view slightly to erase traces of '''PC'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''PC'''.
|-
| | Drag '''slider theta''' back to minus 360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Check “'''TAN'''” box.
| | Check “'''TAN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fT, iT''' and traces of '''PT''' in '''Graphics''' view.
| | Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.
|-
| | Drag '''slider theta''' back to minus 360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Check '''SIN''' and '''COSIN''' boxes.
| | Check '''SIN''' and '''COSIN''' boxes.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to all the '''functions''' in '''Graphics''' view.
| | Observe all the '''functions''' appearing in '''Graphics''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to graph:

'''Sine, cosine, tangent functions''' of '''alpha'''

Inverse '''sine, cosine, tangent functions''' of '''alpha'''

View or hide them using '''check-boxes'''
|-
| | '''Slide Number 7'''

'''Assignment'''
| | As an assignment:

Plot graphs of '''inverse functions''' of '''secant''', '''cosecant''' and '''cotangent'''.
|-
| | '''Slide Number 8'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial project'''.

Please download and watch it.
|-
| | '''Slide Number 9'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 10'''

'''Forum for specific questions:'''

Do you have questions in THIS '''Spoken Tutorial'''?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 11'''

'''Acknowledgement'''
| | The '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Trigonometric-Ratios-and-Graphs/English

2018-04-10T05:18:37Z

Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Trigonometric Ratios and Graphs'''. |- |..."

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Trigonometric Ratios and Graphs'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Calculate '''trigonometric ratios'''

Plot corresponding graphs
|-
| | '''Slide Number 3'''

'''Pre-requisites'''
| | To follow this '''tutorial''', you should be familiar with
'''GeoGebra''' interface

Previous '''tutorials''' in this series

If not, for relevant '''tutorials''', please visit our website '''www.spoken-tutorial.org'''.
|-
| >'''Slide Number 4'''

'''System Requirement'''
| | Here I am using
'''Ubuntu Linux OS version 14.04'''

'''GeoGebra 5.0.388.0-d'''
|-
| | Show the '''GeoGebra''' window.

Point to unit circle and right triangle '''ACB''''.
| | I have opened '''GeoGebra''' interface with a unit circle and a right triangle '''A C B prime.'''
|-
| | '''Slide Number 5'''

'''Sine function'''

'''Sine''' of an angle is the ratio of the lengths of the opposite side to the '''hypotenuse'''.

Angle B'AC = αº = βº

In triangle AB'C,

'''sin(α) = B'C/AB' = y(B')/radius'''

Here, '''sin(α) = y co-ordinate''' of point '''B''''
| | '''Sine''' of an angle is the ratio of the lengths of the opposite side to the '''hypotenuse'''.

'''Angle B prime A C''' is equal to '''alpha degrees''' and to '''beta degrees'''

In '''triangle A B prime C''', '''sine alpha''' equals ratio of the lengths '''B prime C''' to '''A B prime'''.

This is also equal to ratio of '''y co-ordinate''' of '''B prime''' to '''radius'''.

Here, '''sine alpha''' is '''y co-ordinate''' of point '''B prime'''.
|-
| | Click on '''Options''' menu >> select '''Rounding''' >> '''3 Decimal Places'''.
| | Click on '''Options''' menu.

Select '''Rounding''' and then '''3 Decimal Places'''.

All the ratios will now have 3 decimal places.
|-
| | '''Setting up the sine function'''

In '''input bar''', type '''SINE= y(B')/radius'''>> press '''Enter'''
| | Now let us show '''sine alpha''' values using the '''input bar'''.

In '''input bar''', type '''SINE is equal to y B prime in parentheses divided by radius'''.

Press '''Enter'''.
|-
| | Point to '''sine''' values in '''Algebra''' view.
| | '''Sine''' values are displayed in '''Algebra''' view.
|-
| | Drag '''slider α''' to 0 and then to 360º.
| | Drag '''alpha slider''' to 0 and then to 360 '''degrees'''.
|-
| |
| | Observe the change in '''sine''' values in '''Algebra''' view.

Observe that '''sine''' value remains positive as long as '''y axis''' values are positive.
|-
| | '''Graphing the sine function'''

Click on '''Point''' >> click on '''Graphics''' view.
| |
Click on '''Point''' tool.

Click on the screen outside the circle in '''Graphics view.'''
|-
| | Point to point '''D'''.
| | Point '''D''' appears outside the circle.
|-
| | Drag '''slider α''' to 0.
| | Set '''alpha''' to 0 '''degrees''' on the '''slider'''.
|-
| | Right click on '''D''' >> Select '''Object Properties''' >> '''Color''' tab >> red.
| | Right-click on '''D''' and click on '''Object Properties'''.

Select '''Color''' tab and choose red.
|-
| | Close the '''Preferences''' window.
| | Close the '''Preferences''' window.
|-
| | Again right-click on '''D''' and check '''Trace On''' option.
| | Again, right-click on '''D''' and check '''Trace On''' option.
|-
| | In '''Algebra''' view, double click on '''D'''.
| | In '''Algebra''' view, double click on '''D'''.
|-
| | Delete '''co-ordinates''' of '''D'''.
| | Delete '''co-ordinates''' of '''D'''.
|-
| | Select '''symbol α''' >> click on the letter '''α''' >> Insert '''α''' as '''x co-ordinate''' of '''D'''.
| | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''D'''.
|-
| | Type '''SINE''' as '''y co-ordinate''' of '''D''' >> press '''Enter'''
| | Type '''SINE''' as '''y co-ordinate''' of '''D''', and press '''Enter'''.
|-
| | Point to '''D (α, SINE)''' in the '''Algebra''' view.
| | '''D''' has been changed to '''alpha comma SINE'''.
|-
| |
| | '''GeoGebra''' will convert '''alpha''' into '''radians'''.

The '''alpha''' value in '''radians''' is the '''x co-ordinate''' of '''D'''.

Its '''y co-ordinate''' is the '''SINE''' value of '''alpha'''.

This will make '''D''' trace the '''sine function''' as you change '''angle alpha'''.
|-
| |
| | We want to see 2 '''pi radians''' along the positive side of the '''x axis'''.
|-
| | Under '''Move Graphics View''', click once on '''Zoom Out''' and then twice in '''Graphics''' view.
| | Under '''Move Graphics View''', click once on '''Zoom Out''' and then twice in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool.

Click on '''Graphics''' background and when hand '''symbol''' appears, move '''Graphics''' view.
| | Click on '''Move Graphics View''' tool.

Click on '''Graphics''' background and when hand '''symbol''' appears, move '''Graphics''' view.
|-
| | Point to circle and 2 '''pi radians''' on right side of origin on '''x axis'''.
| | You should see the circle and 2 '''pi radians''' along positive side of '''x axis'''.
|-
| | Drag '''slider α''' from 0º to 360º.

| | Increase '''alpha''' on the '''slider''' from 0 to 360 '''degrees''' 2 '''pi radians'''.
|-
| | Point to traces of '''D'''.
| | Point '''D''' will trace the '''sine function''' graph.
|-
| | Point to '''SINE''' values in '''Algebra''' view.
| | '''Sine''' values remain positive as long as '''y''' values are positive.
|-
| | In '''input bar''', type '''d(x) = sin(x)''' >> press '''Enter'''.
| | In '''input bar''', type '''d x in parentheses is equal to sin x in parentheses''' and press '''Enter'''.
|-
| | Point to the '''sine function''' graph beyond '''−2π''' and '''+2π''' '''radians'''.
| | '''Sine function''' will be graphed beyond '''minus 2 pi''' and '''plus 2 pi radians'''.
|-
| | Click on and move '''Graphics''' view to see '''d of x''' beyond '''minus''' and '''plus''' 2 '''pi radians'''.
| | Click on and move '''Graphics''' view to see '''d of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
| | Point to '''D'''.
| | Note that this will erase traces of '''D'''.
|-
| | Click on and move '''Graphics''' view to see circle and '''plus''' 2 '''pi radian''' along '''x axis'''.
| | Click on and move '''Graphics''' view to see circle and '''plus''' 2 '''pi radians''' along '''x axis'''.
|-
| | Drag '''slider α''' to 0 '''degrees''' to see traces of '''D'''.
| | Again drag '''slider alpha''' to 0 '''degrees''' to see traces of '''D'''.
|-
| | Point to '''d(x)''' and traces of '''D'''.
| | Compare '''d of x''' with traces of '''D'''.
|-
| | '''Slide Number 6'''

'''Cosine function'''

'''Cosine''' of an angle is the ratio of the lengths of the adjacent side to the hypotenuse.

cos(α) = AC/AB' = x(B')/radius

In this unit circle, cos(α) = x co-ordinate of point B'
| |
'''Cosine''' of an angle is the ratio of the lengths of the adjacent side to the '''hypotenuse'''.

'''Cos alpha''' is equal to the following ratios.

Length of '''AC to''' length of '''AB prime''' and '''x co-ordinate''' of '''B prime''' to '''radius'''.

In this '''unit circle, cos alpha''' corresponds to '''x co-ordinate''' of point '''B prime.'''
|-
| | Right-click on point '''D''' and uncheck '''Trace On''' option.
| | Right-click on point '''D''' and uncheck '''Trace On''' option.
|-
| | Click on and move '''Graphics''' view slightly to erase traces of '''D'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''D'''.
|-
| | In '''input bar''', type '''COSINE = x(B')/radius''' >> press '''Enter'''.

| | In '''input bar''', type the following line.

'''COSINE is equal to x B prime in parentheses divided by radius'''.

Press '''Enter'''.
|-
| | Point to '''cosine''' value in '''Algebra''' view.
| | '''Cosine''' value is displayed in '''Algebra''' view.
|-
| | Drag '''slider α''' from 0º to 360º.
| | Drag '''slider alpha''' from 0 to 360 '''degrees'''.
|-
| | Point to the '''cosine''' value in the '''Algebra''' view.
| | Observe how '''cosine''' values change in '''Algebra''' view.
|-
| | Point to positive side of '''x axis'''.
| | Note how '''cosine''' remains positive as long as '''x axis''' values are positive.
|-
| | '''Graphing the cosine function'''

Click on '''Point''' tool and click outside the circle.
| |
Click on '''Point''' tool and click outside the circle.

'''Point E''' appears outside the circle.
|-
| | Drag '''slider α''' to 0º.
| | Drag '''slider alpha''' to 0 '''degrees'''.
|-
| | Right click on '''E''' >> Select '''Object Properties'''>> '''Color''' tab >> Brown.
| | Right-click on '''E''', click on '''Object Properties'''.

Select '''Color''' tab and choose brown.
|-
| | Close the '''Preferences''' window.
| | Close the '''Preferences''' window.
|-
| | Right-click on '''E,''' check '''Trace On''' option.
| | Right-click on '''E''', check '''Trace On''' option.
|-
| | In '''Algebra''' view, double click on '''E'''.
| | In '''Algebra''' view, double click on '''E'''.
|-
| | Delete '''co-ordinates''' of '''E'''.
| | Delete '''co-ordinates''' of '''E'''.
|-
| | Select '''symbol α''' >> click on the letter '''α''' >> insert '''α''' as '''x co-ordinate''' of '''E'''
| | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''E'''.
|-
| | Type '''COSINE''' instead of the '''y co-ordinate''' of '''E''' >> press '''Enter'''
| | Type '''COSINE''' instead of '''y co-ordinate''' of '''E''', and press '''Enter'''.
|-
| | Point to '''E''' ('''α, COSINE''') in '''Algebra''' view.
| | '''E''' has been changed to '''alpha comma COSINE'''.
|-
| | Drag '''slider α''' from 0º to 360º.
| | Drag '''slider alpha''' from 0 to 360 '''degrees'''.
|-
| | Point to traces of '''E'''.
| | Point '''E''' will trace the '''cosine function''' graph.
|-
| | In '''input bar,''' type '''e(x) = cos(x)''' >> press '''Enter'''.
| | In input bar, type '''e x in parentheses is equal to cos x in parentheses'''.

Press '''Enter'''.
|-
| | Point to '''cosine function e(x)'''.
| | '''Cosine function e of x''' will be graphed beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
| | Click on and move '''Graphics''' view to see '''e(x''') beyond '''−2π''' and '''+2π radians'''.
| | Click and move '''Graphics''' view to see '''e of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
| | Point to '''E'''.
| | This will erase traces of '''E'''.
|-
| | Click on and move '''Graphics''' view to see +2 '''pi radians''' along '''x axis'''.
| | Click on and move '''Graphics '''view to see '''plus''' 2 '''pi radians''' along '''x axis'''.
|-
| | Drag '''slider α''' to 0 '''degrees''' to see traces of '''E'''.
| | Again drag '''slider alpha''' to 0 '''degrees''' to see traces of '''E'''.
|-
| | Point to '''e(x)''' and traces of '''E'''.
| | Compare the graph of '''e of x''' with traces of '''E'''.
|-
| | Right-click on point '''E''' >> Uncheck '''Trace on'''
| | Right-click on '''E''' and uncheck '''Trace On''' option.
|-
| | Click in and move '''Graphics''' view slightly to erase traces of '''E'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''E'''.
|-
| | '''Slide Number7'''

'''Tangent function'''

'''Tangent''' of an angle is the ratio of lengths of the opposite side to the adjacent side

tan(α) = sin(α)/cos(α) = B'C/AC

tan(α) = y(B')/x(B')
| |
'''Tangent''' of an angle is the ratio of lengths of the opposite side to the adjacent side.

'''Tan alpha''' is the ratio of '''sine alpha''' to '''cos alpha''' and the ratio of lengths of '''B prime C''' to '''AC'''.

'''Tan alpha''' is also the ratio of the '''y co-ordinate''' to '''x co-ordinate''' of '''B prime'''.
|-
| | In '''input bar''', type '''TANGENT = y(B')/x(B')''' >> press '''Enter'''.
| | In '''input bar''', type the following line.

'''TANGENT is equal to y B prime in parentheses divided by x B prime in parentheses'''

Press '''Enter'''.
|-
| | Point to the '''tangent''' value in '''Algebra''' view.
| | '''Tangent''' value is displayed in '''Algebra''' view.
|-
| | '''Setting up the tangent function'''

Drag '''alpha slider''' from 0º to 360º.
| | Drag '''alpha slider''' from 0 to 360 '''degrees'''.
|-
| | Point to the '''Tangent''' values in''' Algebra''' view.
| | Observe how '''tangent''' values change in '''Algebra''' view.
|-
| | Click on''' Point''' tool and click outside the circle.
| | Click on '''Point''' tool and click outside the circle.
|-
| | Point to point '''F'''.
| | Point '''F''' appears outside the circle.
|-
| | Drag '''α slider''' to 0.
| | Set '''alpha''' to 0 '''degrees''' on the '''slider'''.
|-
| | Right-click on '''F''' >> Select '''Object Properties''' >> '''Color''' tab >> green.
| | Right-click on '''F''' and select '''Object Properties'''.

Select '''Color''' tab and choose green.
|-
| | Close the '''Preferences''' window.
| | Close the '''Preferences''' window.
|-
| | Again right-click on '''F''', check '''Trace On''' option.
| | Again right-click on '''F''', check '''Trace On''' option.
|-
| | In '''Algebra''' view, scroll down and double click on '''F'''.
| | In '''Algebra''' view, scroll down and double click on '''F'''.
|-
| | Delete '''co-ordinates''' of '''F'''.
| | Delete '''co-ordinates''' of '''F'''.
|-
| | Select '''symbol α''' >> click on the letter '''α''' >> insert '''α''' as '''x co-ordinate''' of '''F'''
| | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''F'''.
|-
| | Type '''TANGENT''' as '''y co-ordinate''' of '''F''' >>press '''Enter'''
| | Type '''TANGENT''' as '''y co-ordinate''' of '''F''', and press '''Enter'''.
|-
| | Point to '''F''' ('''α, TANGENT''') in the '''Algebra''' view.
| | '''F''' has been changed to '''alpha comma TANGENT'''.
|-
| | Point to '''F'''.
| | Point '''F''' will trace the '''tangent function''' graph as '''alpha''' value changes.
|-
| | Drag '''α slider''' value from 0º to 360º.
| | Increase '''alpha''' on the '''slider''' from 0 to 360 '''degrees''' 2 '''pi radians'''.
|-
| | Point to traces of '''F''' from 0 to π/2 '''radians'''.
| | '''F''' increases from '''origin''' to '''infinity'''.

Note '''vertical asymptote''' at '''pi''' divided by 2 '''radians'''.
|-
| | Point to the graphs.
| | '''Tangent''' value is plus '''infinity''' at '''pi''' divided by 2 '''radians'''.

It is minus '''infinity''' at 3 '''pi''' divided by 2 '''radians'''.
|-
| | Type '''f(x) = tan(x)''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''f x in parentheses is equal to tan x in parentheses''' and press '''Enter'''.
|-
| | Point to '''f(x)'''.
| | The '''tangent function''' is graphed beyond minus 2 '''pi''' and plus 2 '''pi radians'''.
|-
| | Click on and move '''Graphics''' view beyond '''−2π''' and '''+2π radians'''.
| | Click on and move '''Graphics''' view to see graph of '''f of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
| | Click on and move '''Graphics''' background to see plus 2 '''pi radians''' along '''x axis'''.
| | Click on and move '''Graphics''' view to see '''plus 2''' '''pi radians''' along '''x axis'''.
|-
| | Drag '''α slider''' value from 360º to 0º.
| | Drag '''slider alpha''' back to 0 '''degrees''' to see traces of '''F'''.
|-
| | Point to '''f(x)''' and traces of '''F'''.
| | Also compare the '''tangent function f of x''' with traces of '''F'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 8'''

'''Summary'''
| | In this tutorial, we have learnt

how to use '''GeoGebra''' to calculate and graph '''sin alpha''', '''cos alpha''' and '''tan alpha'''
|-
| | '''Slide Number 9'''

'''Assignment'''
| | Assignment

Try these steps to graph '''secant, cosecant''' and '''cotangent functions'''.

Analyze the link between '''sine''' values for '''supplementary angles''' (angles whose sum is 180 '''degrees''').

Analyze the link between '''sine''' and '''cosine''' values for '''supplementary angles'''.
|-
| '''Slide Number 10'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
| | '''Slide Number 11'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 12'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 13'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Parabola/English

2018-04-03T09:36:36Z

Vidhya: Created page with " {|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''. |-..."

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this '''tutorial''' on '''Conic Sections – Parabola'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Study standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this '''tutorial''', you should have basic knowledge of
'''GeoGebra''' interface

'''Conic sections''' in geometry
|-
| s | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux '''OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''

'''Parabola'''

[[Image:]]

A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.
| | A parabola is the '''locus''' of points equidistant from the fixed point called the focus.

The points on the parabola are also equidistant from the fixed line called the '''directrix'''.

Observe the different features of the parabola in the image.

The '''Axis of Symmetry''' is perpendicular to the '''Directrix''' and passes through the Focus and '''Vertex'''.

'''Latus Rectum''' passes through the Focus and is perpendicular to the '''Axis of Symmetry'''.
|-
| | Show the '''GeoGebra''' window.
| | Let us construct a parabola in '''GeoGebra'''.

I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Point''' tool and click in '''Graphics''' view.

Point to point '''A'''.
| | Click on '''Point''' tool and click in '''Graphics''' view.

This creates point '''A'''.
|-
| | Right-click on point '''A''' and select the '''Rename''' option.
| | Right-click on point '''A''' and select the '''Rename''' option.
|-
| | In '''New Name''' text box, type '''Focus''' instead of '''A''' >> click '''OK'''.

Point to '''Focus'''.
| | In the '''New Name''' text box, type '''Focus''' instead of '''A '''and click '''OK'''.

This renames point '''A''' as '''Focus'''.
|-
| | Click on '''Line''' tool >> click in two places in '''Graphics''' view below '''Focus'''.

Point to line '''AB'''.
| | Click on '''Line''' tool and click on two places in '''Graphics''' view, below '''Focus'''.

This creates line ''' AB'''.
|-
| | Right-click on line '''AB''' >> choose '''Rename''' option.
| | Right-click on line '''AB''' and choose the '''Rename''' option.
|-
| | Type '''directrix''' in '''New Name''' field >> click '''OK'''.

Point to '''directrix'''.
| | In the '''New Name''' field, type '''directrix''' and click '''OK'''.

This renames line '''AB''' as the '''directrix'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on line '''AB'''.
| | Click on '''Perpendicular Line''' tool, then click on line '''AB'''.
|-
| | Drag the '''cursor''' until '''Focus''' >> click on point '''A'''.
| | Drag the '''cursor''' until the resulting line passes through '''Focus''' and click on '''Focus'''.
|-
| | Point to the perpendicular line through '''Focus'''.

Point to '''axis of symmetry'''.
| | This draws a line perpendicular to line '''AB''', passing through '''Focus'''.

This line is the '''axis of symmetry'''.
|-
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field >> click '''OK'''.
| | Right-click on this line perpendicular to line '''AB'''.

Choose the '''Rename''' option.

Type '''axis of symmetry''' in '''New Name''' field.

Click '''OK'''
|-
| | Click on '''Parabola''' tool under '''Ellipse''' tool.

Click on '''Focus''' and line '''AB''' ('''directrix''').
| | Under '''Ellipse''' tool, click on '''Parabola''' tool.

Then click on '''Focus''' and the '''directrix'''.
|-
| | Point to the parabola.
| | This creates a parabola with its focus at '''Focus''' and with line '''AB''' as the '''directrix'''.
|-
| | Click on '''Intersect''' tool. >> Click on the '''parabola''' and '''axis of symmetry'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and '''axis of symmetry'''.
|-
| | Point to point '''C'''.
| | This creates point '''C''' at the intersection.

It is the '''vertex''' of the parabola.
|-
| | Right-click on point '''C''' >> choose the '''Rename''' option.
| | Right-click on point '''C''' and choose the '''Rename ''' option.
|-
| | Type '''Vertex''' in '''the '''New Name''' field >> click '''OK'''.
| | In the '''New Name''' field, type '''Vertex''' and click '''OK'''.
|-
| | Click on '''Perpendicular Line''' tool >> click on the '''axis of symmetry'''.
| | Click on '''Perpendicular Line''' tool and click on the '''axis of symmetry'''.
|-
| | Drag the '''cursor''' until the line passes through point '''A''' (Focus) >> click on point '''A'''.
| | Drag the '''cursor''' until the line passes through the '''Focus''' and click on it.
|-
| | Point to the parallel line.
| | This results in a line parallel to the '''directrix''', passing through the '''Focus'''.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersections of the parabola and the newly drawn line through '''Focus'''.

| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on the parabola and the newly drawn line through '''Focus'''.
|-
| | Point to points '''C''' and '''D'''.
| | This creates points '''C''' and '''D'''.
|-
| | Click on '''Segment''' tool under the '''Line''' tool >> click on points '''C'''and '''D'''.
| | Under '''Line''' tool, click on '''Segment''' tool and click on points '''C''' and '''D'''.
|-
| | Point to Segment '''CD'''.
| | Resulting Segment '''CD''' is the '''latus rectum'''.
|-
| | Right-click on Segment '''CD''' and choose the '''Rename''' option.
| | Right-click on Segment '''CD'''and choose the '''Rename''' option.
|-
| | Type '''Latus Rectum''' in the '''New Name''' field >> click '''OK''' button.
| | In the '''New Name''' field, type '''Latus Rectum''' and click '''OK'''.
|-
| | Move the '''Latus''' label so you can see it properly.
| | Move the '''Latus''' label so you can see it properly.
|-
| | Click and drag '''Graphics''' view to see the parabola properly.
| | Click and drag '''Graphics''' view to see the parabola properly.
|-
| | Point to '''Algebra''' view.

Drag boundary so you can see equation properly.

| | In '''Algebra''' view, you can see the equation describing the parabola.

Drag boundary so you can see the equation properly.

Also, you can see the equations for the '''axis of symmetry, directrix''' and '''latus rectum'''.
|-
| | Drag boundary so you can see '''Graphics''' view properly again.
| | Drag boundary so you can see '''Graphics''' view properly again.
|-
| | Click in '''Graphics''' view and drag background.
| | Click in '''Graphics''' view and drag background.
|-
| | Click on '''Intersect''' tool under '''Point''' tool.

Click on the intersection of the '''axis of symmetry''' and the '''directrix'''.
| | Under '''Point''' tool, click on '''Intersect''' tool.

Click on '''axis of symmetry''' and '''directrix'''.
|-
| | Point to point '''E'''.
| | This creates point '''E'''.
|-
| | Click on '''Distance or Length''' tool under '''Angle''' tool.
| | Under '''Angle''' tool, click on '''Distance or Length''' tool.
|-
| | Click on '''Focus''' >> '''Vertex'''.
| | Click on '''Focus''' and '''Vertex'''.
|-
| | Point to the distance of '''FocusVertex''' appearing in '''Graphics''' view.

| | Note the distance of '''FocusVertex''' appearing in '''Graphics''' view.
|-
| | Click on '''Vertex''' >> point '''E'''.
| | Click on '''Vertex''' and point '''E'''.
|-
| | Point to the distance of '''Vertex E''' appearing in '''Graphics''' view.
| | Note the distance of '''Vertex E''' appearing in '''Graphics''' view.

Both these distances are equal.
|-
| |
| | Let us look at the general equations of parabolas.
|-
| | Show the new '''GeoGebra''' window.
| | I have opened a new '''GeoGebra''' window.
|-
| | Type '''(x-a)^2=4 p (y-b)''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''x minus a in parentheses caret 2 equals 4 space p space y minus b in parentheses'''.

To type '''caret symbol''', hold '''Shift''' key down and press 6.

Note that the spaces denote multiplication.

Press '''Enter'''.
|-
| | Point to '''Create Sliders''' window
| | '''Create Sliders''' window pops up asking if you want to create '''sliders''' for '''a, b''' and '''p'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders a, p''' and '''b'''.
| | '''Sliders''' are created for '''a, p''' and '''b'''.

The '''default''' setting for all three '''coefficients''' is 1.
|-
| | A parabola opening upwards appears in '''Graphics''' view.

Point to '''vertex''' of parabola.
| | A parabola opening upwards appears in '''Graphics''' view.

'''a comma b''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
| | Double click on parabola >> click on '''Object Properties''' and then on '''Color''' tab.
| | Double click on the parabola, click on '''Object Properties''' and then on '''Color''' tab.
|-
| | Select red and close the '''Preferences''' box.
| | Select red and close the '''Preferences''' box.
|-
| | Point to the red parabola and its equation in '''Graphics''' and '''Algebra''' views.
| | The parabola and its equation appear red in the '''Graphics''' and '''Algebra''' views.
|-
| | Move boundary so you can see the equation properly.
| | Move boundary so you can see the equation properly.
|-
| | Right-click on '''slider a''' button >> check '''Animation On''' option.
| | Right-click on '''slider a''' and check '''Animation On''' option.
|-
| | Point to the parabola in '''Graphics''' view.
| | Note the effects on the horizontal movement of the red parabola.
|-
| | Right-click on '''slider a''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider a''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider p''' >> check '''Animation On''' option.
| | Right-click on '''slider p''' and check '''Animation On''' option.
|-
| | Point to parabola in '''Graphics view'''.
| | Note the effects on the shape and orientation of the parabola.
|-
| | Right-click on '''slider p''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider p''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider b''' >> check '''Animation On''' option.
| | Right-click on '''slider b''' and check '''Animation On''' option.
|-
| | Point to the parabola.
| | Note the effects on the vertical movement of the parabola.
|-
| | Right-click on '''slider b''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider b''' and uncheck '''Animation On''' option.
|-
| | Point to '''sliders a, p''' and '''b''' (all = 1) and the red parabola '''c''' in '''Graphics '''view.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Point to equation '''c: (x2-2x-4y) = -5''' in '''Algebra''' view.
| | Note that when '''a''', '''p''' and '''b '''are equal to 1, the red parabola '''c''' is described by equation '''c'''.

Click on parabola '''c''' in '''Graphics''' view and note highlighting of equation '''c''' in '''Algebra''' view.

Equation '''c''' is given by '''x squared minus 2x minus 4y equals minus 5'''.
|-
| | Type '''Focus(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''A''' in '''Graphics''' view.
| | In '''input bar''', type '''Focus c in parentheses'''.

Press '''Enter'''.

'''Focus''' is drawn at point '''A ''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''A''', the '''Focus''', in '''Algebra''' view.
| | The coordinates of '''Focus''' of parabola '''c''', which is point '''A''', appear in '''Algebra''' view.
|-
| | Type '''Vertex(c)''' in '''input bar'''>> press '''Enter'''.

Point to point '''B''' in '''Graphics''' view.
| | In '''input bar''', type '''Vertex c in parentheses'''.

Press '''Enter'''.

'''Vertex''' is drawn at point '''B''' in '''Graphics''' view.
|-
| | Point to the '''coordinates''' of point '''B''' in '''Algebra''' view.
| | The '''coordinates''' of '''Vertex''' of parabola '''c''', which is point '''B''', appear in '''Algebra''' view.
|-
| | Type '''Directrix(c)''' in '''input bar''' >> press '''Enter'''.

Point to '''Directrix''' in '''Graphics''' view.
| | In '''input bar''', type '''Directrix c''' in parentheses.

Press '''Enter'''.

'''Directrix''' appears as a line along '''x axis''' in '''Graphics''' view.
|-
| | Point to the equation, '''y=0''', in '''Algebra''' view.
| | The equation for the '''Directrix''' of parabola '''c''', '''y equals 0''', appears in '''Algebra''' view.
|-
| | Double click on '''Directrix''' in '''Graphics''' view >> '''Object Properties''' >> '''Color''' tab.
| | Double click on '''Directrix''' in '''Graphics''' view.

Choose '''Object Properties''', then the '''Color''' tab.
|-
| | In the left panel, point to highlighted '''Directrix''', identify '''Focus''' and ''' Vertex''' created for parabola '''c'''.
| | In the left panel, note that the '''Directrix''' is highlighted.

Identify '''Focus''' and '''Vertex''' created for parabola '''c'''.
|-
| | Click on each one to highlight while pressing the '''Control''' key.
| | While pressing the '''Control''' key, click and highlight '''Focus''' and '''Vertex'''.
|-
| | Click on red.
| | Click on red.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to '''Focus''', '''Vertex''' and '''Directrix''' and their '''co-ordinates''' and equation in '''Graphics''' and '''Algebra''' views.
| | For parabola '''c''', '''Focus''', '''Vertex''' and '''Directrix''' and their '''coordinates''' and equation appear red.
|-
| | Type '''(y-l)^2=4 p (x-m)''' in the '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''y minus l in parentheses caret 2 equals 4 space p space x minus m in parentheses'''.

Press '''Enter'''.
|-
| | Point to the pop-up window asking if you want to create '''sliders''' for '''l''' and '''m'''.
| | A window pops up asking if you want to create '''sliders''' for '''l''' and '''m'''.
|-
| | Click on '''Create Sliders'''.
| | Click on '''Create Sliders'''.
|-
| | Point to '''sliders l''' and '''m'''.
| | '''Sliders''' are created for '''l''' and '''m'''.

The '''default''' setting for both '''coefficients''' is 1.
|-
| | Point to the parabola opening to the right appears in '''Graphics''' view.

Point to the '''vertex''' of the parabola.
| | A parabola opening to the right appears in '''Graphics''' view.

'''l comma m''' correspond to the '''co-ordinates''' of the '''vertex'''.
|-
| | Double click on it, click on '''Object Properties''' >> '''Color''' tab >> select blue
| | Double click on the parabola, click on '''Object Properties'''.

Select '''Color''' tab and choose blue.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to parabola '''d''' and its equation '''d''' in the '''Graphics''' and '''Algebra''' views.
| | Parabola '''d''' and its equation '''d''' appear blue in '''Graphics''' and '''Algebra''' views.
|-
| | Drag '''sliders l''' and '''m''' to 2.
| | Drag '''sliders l''' and '''m''' to 2.
|-
| | Point to '''sliders l = m = 2''' and '''p = 1''', and the blue parabola '''d''' in '''Graphics''' view.

Click on parabola '''d''' in '''Graphics''' view and note highlighting of equation '''d''' in '''Algebra''' view.

Point to equation '''d: y2-4x-4y= -12''' in '''Algebra''' view.
| | Note that when '''l''' and '''m''' are equal to 2, and '''p''' equals 1, the blue parabola '''d''' is described by equation '''d'''.

Click on parabola '''d''' in '''Graphics''' view and note highlighting of equation '''d''' in '''Algebra''' view.

Equation '''d''' is given by '''y squared minus 4x minus 4y equals minus 12'''.
|-
| | Right-click on '''slider l''' >> check '''Animation On''' option.
| | Right-click on '''slider l''' and check '''Animation On''' option.
|-
| | Point to parabola in '''Graphics''' view.
| | Note the effects on the vertical movement of the parabola.
|-
| | Right-click on '''slider l''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider l''' and uncheck '''Animation On''' option.
|-
| | Right-click on '''slider m''' >> check '''Animation On''' option.
| | Right-click on '''slider m''' and check '''Animation On'''option.
|-
| | Point to parabola in '''Graphics''' view.
| | Note the effects on the horizontal movement of the parabola.
|-
| | Right-click on '''slider m''' >> uncheck '''Animation On''' option.
| | Right-click on '''slider m''' and uncheck '''Animation On''' option.
|-
| | Type '''Focus(d)''' in the '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''Focus d''' in parentheses.

Press '''Enter'''.
|-
| | Type '''Vertex(d)''' in the ''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''Vertex d''' in parentheses.

Press '''Enter'''.
|-
| | Type '''Directrix(d)''' in the '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type '''Directrix d''' in parentheses.

Press '''Enter'''.
|-
| | Double click on '''Focus''' in '''Graphics''' view >> '''Object Properties''' >> '''Color''' tab.
| | Double click on '''Focus''' in '''Graphics''' view.

Choose '''Object Properties''', then the '''Color''' tab.
|-
| | In the left panel, point to highlighted '''Focus''' (point '''C''').

Point to '''Vertex''' and '''Directrix''' for parabola '''d'''.
| | In the left panel, note that '''Focus''', point '''C''', is highlighted.

Identify '''Vertex''' and '''Directrix''' for parabola '''d'''.
|-
| | Click on each one to highlight while pressing the '''Control''' key.
| | While pressing the '''Control''' key, click and highlight '''Vertex''' and '''Directrix'''.
|-
| | Click on Blue.
| | Click on Blue.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Right-click on '''slider a''' >> choose '''Animation On''' option to check it.
| | Right-click on '''slider a''' and choose '''Animation On''' option to check it.

Notice that this only affects the red parabola.
|-
| | Right-click on '''slider a''' and uncheck '''Animation On''' option.

Drag '''slider a''' so we can see the red parabola better.
| | Right-click and uncheck '''Animation On''' option for '''slider a'''.

Let us drag '''a''' so we can see the red parabola better.
|-
| | Drag '''b''' so you can see the vertical movement of parabola '''c'''.
| | Drag '''b''' so you can see the vertical movement of parabola '''c'''.
|-
| | Point to '''vertex (a, b)''' for red parabola.
| | Note that as the '''vertex''' is '''a comma b''' for red parabola, the blue parabola does not move at all.
|-
| | Drag '''sliders l''' and '''m'''.

Point to blue parabola.
| | Similarly, let us drag '''l''' and '''m'''.

This only affects the blue parabola.
|-
| | Drag '''slider p''', point to both parabolas.
| | But when we move '''p''', both parabolas are affected.
|-
| | Drag '''sliders a, b, l''' and '''m''' to 0.

Drag '''slider p''' to 1.
| | Drag '''sliders a, p, b, l''' and '''m''' to 0.

Now let us move '''p''' to 1.
|-
| | Point to the red and blue parabolas in '''Graphics''' view.
| | Note the effects on the parabolas’ graphs and equations in '''Graphics''' and '''Algebra''' views.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 6'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
Study the standard equations and parts of a parabola

Construct parabolas
|-
| | '''Slide Number 7'''
'''Assignment'''

Try these steps to construct parabolas with:

Focus (6,0) and '''directrix''' x = -6

Focus (0,-3) and '''directrix''' y = 3

Find their equations.
| | As an assignment:
Try these steps to construct parabolas with these '''foci''' and '''directrices'''.

Find their equations.
|-
| | '''Slide Number 8'''
'''Assignment'''

Find the coordinates of the '''foci''' and length of '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.

y2 = 12x

x2 = -16y
| | As an assignment:
Find the coordinates of the '''foci''' and length of the '''latus recti''' for these parabolas.

Also, find the equations of the '''axes of symmetry''' and '''directrices'''.
|-
| | '''Slide Number 9'''
'''About Spoken Tutorial Project'''
| | The video at the following link summarizes the '''Spoken Tutorial Project'''.

Please download and watch it.
|-
| | '''Slide Number 10'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 11'''
'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 12'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications of GeoGebra

2018-03-27T08:58:45Z

Vidhya: Created page with "GeoGebra is a dynamic and interactive mathematics software for geometry, algebra, calculus, trigonometry and statistics. Tools in GeoGebra are helpful in various constructions..."

GeoGebra is a dynamic and interactive mathematics software for geometry, algebra, calculus, trigonometry and statistics. Tools in GeoGebra are helpful in various constructions and calculations. Entry of equations and mapping of various variables can be done using the tools, input bar, CAS and spreadsheet views. Interactive explorations can be done using the tools in 2D and 3D Graphics modes. GeoGebra is a very useful tool to learn and teach different branches of mathematics.
GeoGebra desktop application are available for Windows, macOS and Linus and tablet appls are available for Android, iPad and Windows. Its web app is based on HTML5 technology. GeoGebra was created by Markus Hohenwarter and started as part of his master’s thesis at the University of Salzburg, continuing at Florida Atlantic University, Florida State University, and then at the University of Linz with the help of open-source developers and translators all over the world. Bernard Parisses’ Giac was integrated into GeoGebra’s CAS view in 2013. Both commercial and not-for-profit entities work together to expand the software and cloud services for users.
Contributors, Content Editors and Reviewers
The Spoken Tutorial Effort for Applications for GeoGebra is being contributed by Dr. Vidhya Iyer from IIT Bombay. Ms. Madhuri Ganapathy from IIT Bombay created two tutorials and prepared the outline along with Ms. Shruti Arya and Ms. Kiran Eranki. Other contributors include Ms. Sandhya Punekar and Ms. Madhuri Ganapathy, who contributed as domain reviewers, Ms. Minal Sathaye, who was the novice, and Ms. Nancy Varkey, the administrative reviewer.

Contents
# Basic Level
# Intermediate Level

==Basic Level==
# Vectors and Matrices    
#* Define a vector
#* Change the Font Size using Options menu
#* Magnitude and direction of a vector
#* Relation between vectors and a parallelogram
#* Arithmetic operations on vectors
#* To create a matrix
#* Arithmetic operations on matrices
#* Transpose of a matrix
#* Determinant of a matrix
#* Inverse of a matrix
# Introduction to Trigonometry using GeoGebra    
#* Construct a circle of variable radius
#* Construct a right triangle inside a unit circle
#* Create a slider to change angle in right triangle
#* Change properties (labels, colors and styles) of the right triangle
#* Change x axis values to radians
# Trigonometric Ratios and Graphs    
#* The concept of a unit circle to find trigonometric ratios
#* Conversion of degrees into radians to look at periodicity of functions
#* Creation of a slider to change angle alpha to look at trigonometric ratios
#* Changing the appearance of graphs
#* Sine function
#* Cosine function
#* Tangent function
#* Effects of rotation of the point around the circle on periodicity of functions
#* Looking at co-ordinates of points tracing graphs of above trigonometric functions
#* Corresponding graphs for sine, cosine and tangent functions
# Inverse Trigonometric Functions    
#* Creation of a slider to change angle alpha to look at trigonometric ratios
#* Sine function
#* Cosine function
#* Tangent function
#* Inverse sine function
#* Inverse cosine function
#* Inverse tangent function
#* Fixing the domain to look at trigonometric and inverse trigonometric functions
#* Co-ordinates of points tracing graphs of trigonometric and inverse trigonometric functions
#* Creation of check boxes to show or hide function
# Roots of Polynomials    
#* Binomial theorem and polynomials
#* Quadratic polynomials: real roots and complex roots
#* Finding roots and determinants
#* Quadratic functions: parabolic graphs, extremum
#* Complex numbers in XY plane
#* Complex numbers in complex plane
#* Cubic polynomials: roots and extrema
#* Point of inflection
#* Plotting polynomials using input bar in GeoGebra
#* Plotting polynomials in Computer Algebra system (CAS) in GeoGebra
# Complex Roots of Quadratic Equations    
#* Complex numbers in XY plane
#* Complex numbers in complex plane
#* Quadratic polynomials and parabolic functions
#* Using sliders to look at different quadratic equations in GeoGebra
#* Effects of quadratic equation coefficients on parabolic functions
#* Finding roots using coefficients in quadratic equations
#* Finding vertex (extremum) from formula for roots of quadratic equations
#* Real roots of a quadratic polynomial using Spreadsheet view in GeoGebra
#* Complex roots of a quadratic polynomial using Spreadsheet view in GeoGebra
#* Changes in determinants for real and complex roots of quadratic equations
# Conic Sections-Parabola    
#* Definition, parts and properties of a parabola
#* Construction of a parabola in GeoGebra
#* Changing the appearance of a parabola in GeoGebra
#* The standard equation of a parabola, (x-a)2 = 4p (y-b) in GeoGebra
#* The standard equation of a parabola, (y-l)2 = 4p (x-m) in GeoGebra
#* Differentiating the two forms of standard equations in terms of direction of axes of symmetry
#* Creation of sliders to change coefficients in standard equations in GeoGebra
#* The effects of the coefficients in standard equations on the parabola in GeoGebra
#* Finding foci, vertex and equations of the axis of symmetry and directrix of a parabola
#* Calculating length of latus rectum of a parabola
# Conic Sections-Ellipse    
#* Definition, parts and properties of an ellipse
#* Construction of an ellipse in GeoGebra by adapting the Arcs of Circle method
#* Changing the appearance of an ellipse in GeoGebra
#* The standard equation of an ellipse, (x-h)2/a2+(y-k)2/b2=1, in GeoGebra
#* The standard equation of an ellipse, (x-p)2/b2+(y-q)2/a2=1, in GeoGebra
#* Differentiating the two forms of standard equations in terms of direction of major and minor axes
#* Creation of sliders to change coefficients in standard equations in GeoGebra
#* The effects of the coefficients in standard equations on the ellipse in GeoGebra
#* Finding foci, vertices, co-vertices, center and eccentricity of an ellipse
#* Calculating length of major and minor axes, and latus recti of an ellipse
# Conic Sections-Hyperbola    
#* Definition, parts and properties of a hyperbola
#* Construction of a hyperbola in GeoGebra
#* Changing the appearance of a hyperbola in GeoGebra
#* The standard equation of a hyperbola, (x-h)2/a2-(y-k)2/b2=1, in GeoGebra
#* The standard equation of a hyperbola, (x-p)2/b2-(y-q)2/a2=1, in GeoGebra
#* Differentiating the two forms of standard equations in terms of direction of transverse and conjugate axes
#* Creation of sliders to change coefficients in standard equations in GeoGebra
#* The effects of the coefficients in standard equations on the hyperbola in GeoGebra
#* Finding foci, vertices, center and eccentricity of a hyperbola
#* Calculating length of transverse and conjugate axes, and latus recti of a hyperbola

==Intermediate Level==
# 3D Geometry
#* The rectangular co-ordinate system
#* Drawing a line in 3D
#* Drawing a plane in 3D
#* Drawing a sphere in 3D
#* Drawing a pyramid in 3D
#* Drawing a double-napped cone in 3D
#* Visualizing intersection of line with a plane and sphere in 3D
#* Visualizing intersection of a double-napped cone with a plane in 3D as a parabola
#* Visualizing the solid obtained by rotation of a polynomial about the x axis
#* Trigonometric functions in 3D
# Limits and Continuity of Functions
#* The concept of limits
#* Evaluation of functions
#* Left hand limit
#* Right hand limit
#* Limits that DNE
#* Using a spreadsheet to look at limits
#* Limits at infinity
#* Limits of rational polynomial functions
#* Continuous functions, their limits
#* Discontinuous functions, their limits
# Differentiation using GeoGebra
#* Differentiation using first principles
#* Differentiation to find maxima of a function
#* Differentiation to find minima of a function
#* Derivative of a trigonometric function
#* Addition rule of differentiation
#* Subtraction rule of differentiation
#* Product rule of differentiation
#* Chain rule of differentiation
#* Quotient rule of differentiation
#* Practical application of differentiation
# Integration using GeoGebra
#* Indefinite integrals
#* Definite integrals
#* Concept of area under curve (AUC)
#* Upper, lower Riemann and trapezoidal sums to estimate AUC
#* Integration as an estimation of AUC
#* Relationship between number of rectangles/trapezoids and AUC
#* Double integrals as area between two functions
#* Using input bar to find area between two functions in GeoGebra
#* Using CAS to find area between two functions in GeoGebra
#* Integration as anti-differentiation
# Statistics using GeoGebra
#* Pasting data into Spreadsheet in GeoGebra
#* Measures of central tendency: arithmetic mean, median and mode
#* Coefficient of variation to compare data series
#* Measures of dispersion: range, quartiles, mean and standard deviation
#* Box plot = 5 member summary: minimum and maximum values, first and third quartiles, median
#* One Variable Analysis—bar chart, box plot, histogram representation
#* Least squares linear regression (LSLR)—coefficient of determination R2, regression coefficient
#* Two Variable Regression Analysis: Scatterplot and residual plot
#* Multiple Variable Analysis tool
#* Stacked box plots
# Probability and Distributions using GeoGebra
#* Pasting data into Spreadsheet in GeoGebra
#* Hypothesis testing: null and alternative hypotheses
#* z-, T- and F- tests
#* Multiple Variable Analysis tool
#* Stacked box plots
#* Analysis of Variance (ANOVA)
#* T-tests: Unpaired and Paired
#* Probability Calculator tool in GeoGebra
#* Probability
#* Distributions

Contributors and Content Editors
Vidhya Iyer, Madhuri Ganapathy, Sandhya Punekar

Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English

2018-03-16T08:55:20Z

Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Complex Roots of Quadratic Equations''' |-..."

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn to,
Plot graphs of''' '''quadratic '''functions'''

Calculate '''real''' and '''complex roots''' of quadratic '''functions'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''
| | To follow this tutorial, you should be familiar with:
'''GeoGebra''' interface

Basics of quadratic equations, geometry and graphs

Previous tutorials in this series

If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''
'''System Requirement'''
| | Here I am using:
'''Ubuntu Linux''' OS version 14.04

'''Geogebra 5.0.388.0-d'''
|-
| | '''Slide Number 5'''
'''Quadratic polynomials'''

Let us find out more about a 2nd degree '''polynomial''' '''y =''' '''ax2+bx+c'''

Parabola

If '''a > 0''', parabola opens upwards, minimum '''vertex '''('''extremum''')

If '''a < 0''', parabola opens downwards, maximum '''vertex'''
| | Let us find out more about a '''2nd degree polynomial'''.
'''y equals a x squared plus b x plus c'''

The '''function''' graphs as a parabola.

If '''a''' is greater than 0, the parabola opens upwards and has a '''minimum vertex''' or '''extremum'''.

If '''a''' is less than 0, it opens downwards and has a '''maximum vertex''' or '''extremum'''.
|-
| | '''Slide Number 6'''
'''Quadratic polynomials'''

If parabola intersects '''x axis''', '''x intercepts''' are '''real roots'''.

'''Real roots''' x = -b ± sqrt(b2-4ac)/2a

If parabola does not intersect '''x axis''' at all, no '''real roots''', only '''complex'''

Two types of '''roots''': '''real''' and '''complex'''
| | If the parabola intersects the''' x axis, '''the '''intercepts''' are real roots.

'''Real roots''' are given by values of x.

'''x''' is '''ratio''' '''of''' '''minus b plus or minus squareroot of b squared minus 4ac to 2a'''.

If the parabola does not intersect '''x axis''' at all, it has no '''real roots'''.

'''Roots''' are '''complex'''.

Let us look at '''complex''' numbers.
|-
| | '''Slide Number 7'''
'''Complex numbers, XY plane'''

As we know,

A '''complex number''' is expressed as '''''z'' = a + ''i''b''': where ‘'''a'''’ is the '''real''' part, ‘'''b''i''’''' is '''imaginary '''part, and '''a''' and '''b''' are constants.

'''Imaginary number, ''i'' '''= sqrt(-1}

In the XY plane, '''a + ''i''b '''corresponds to the point ('''a, b''').

In the '''complex plane''', '''x axis''' is called''' real axis, y axis''' is called '''imaginary axis'''.
| | '''Complex numbers, XY plane'''
As we know,

A '''complex number''' is expressed as '''''z'' equals a plus ''i''b.'''

‘'''a'''’ is the '''real''' part; ‘'''b''i'''''’ is imaginary part<nowiki>; </nowiki>'''a''' and '''b''' are constants.

‘'''''i''’''' is '''imaginary number''' and is equal to '''squareroot of minus 1'''.

In the XY plane, '''a plus ''i''b '''corresponds to the point '''a comma b'''.

In the '''complex plane''', '''x axis''' is called''' real axis''', '''y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 8'''
'''Complex numbers, complex plane'''

[[Image:]]

In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' ‘'''a'''’ and '''imaginary axis coordinate''' ‘'''b'''’

Length of the '''vector ''z''''' = |'''''z'''''| =''' ''r'''''

'''''r'' = sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is ‘'''a'''’ and '''imaginary axis coordinate''' is ‘'''b'''’.

The length of the '''vector ''z''''' is equal to the absolute value''' '''of '''''z''''' and to '''''r'''''.

According to''' Pythagoras’ theorem, ''r'' '''is equal to '''squareroot of a squared plus b squared.'''
|-
| | '''Slide Number 9'''
'''Complex numbers, complex plane'''

[[Image:]]

'''Argument ϴ''' = angle between''' real axis''' and line segment connecting '''''z''''' to O '''(0,0)''' in counter-clockwise direction

'''Polar form''' of '''''z'' = a + ''i''b''' is

'''''z'' = ''r'' (cosϴ + ''i'' sinϴ)'''

where '''a= ''r''cosϴ, b=''r''sinϴ'''
| | '''Argument theta''' is angle between '''real axis''' and line segment connecting '''''z''''' to origin.

It is in counter-clockwise direction.

'''Polar form''' of '''''z''''' equals '''a plus ''i''b''' is

'''''z'' '''equals''' ''r'' times cos theta plus ''i'' sin theta'''

where '''a '''is equal to''' ''r'' cos theta '''and''' b is ''r'' sin theta'''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened '''GeoGebra''' interface.
|-
| | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
| | Click on''' Slider '''tool and then click in '''Graphics''' view.
|-
| | Point to the dialog box.
| | '''Slider''' dialog-box appears.
|-
| | Point to '''Number''' radio button.
| | By default, '''Number''' radio-button is selected.
|-
| | Type Name as '''a.'''
| | In the '''Name '''field, type '''a'''.
|-
| | Point to''' Min, Max '''and''' Increment '''values.
| | Set '''Min '''value as 1, '''Max '''value as 5 and Increment as 1.
|-
| | Click '''OK''' button.
| | Click '''OK''' button.
|-
| | Point to the '''slider'''.
| | This creates a number '''slider''' named “'''a'''”.
|-
| | Drag to show the changing values.
| | Using the '''slider''', '''a''' can have values from 1 to 5, in increments of 1.
|-
| | Following the same steps, create '''sliders b''' and '''c'''.
| | Following the same steps, create '''sliders''' '''b''' and '''c'''.
|-
| | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
| | Click on''' Slider '''tool, click in '''Graphics''' view.
|-
| | Type Name as '''b.'''
| | In the '''Name '''field of dialog box, type '''b'''.
|-
| | Point to''' Min, Max '''and''' Increment '''values.
| | Set '''Min '''value as -5, '''Max '''value as 10 and Increment as 1.
|-
| | Click '''OK''' button.
| | Click '''OK''' button.
|-
| | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
| | Again, click on''' Slider '''tool, click in '''Graphics''' view.
|-
| | In '''Slider''' dialog-box, in the '''Name '''field, type '''c'''.
| | In the '''Name '''field of dialog box, type '''c'''.
|-
| | Point to''' Min, Max '''and''' Increment '''values.
| | Set '''Min '''value as -5, '''Max '''value as 10 and Increment as 1.
|-
| | Click '''OK''' button.
| | Click '''OK''' button.
|-
| | In '''input bar''', type '''f(x):=a x^2+b x+c'''>>press '''Enter'''.

Drag boundary to see '''Algebra''' view properly.
| | In '''input bar''', type the following line.

'''f x '''in parentheses '''colon equals a space x caret 2 plus b space x plus c'''.

Press '''Enter'''.

Drag boundary to see '''Algebra''' view properly.

Pay attention to the spaces''' '''indicating multiplication.
|-
| | Point to the equation for '''f(x)''' in '''Algebra view'''.
| | Observe the equation for '''f of x''' in '''Algebra view'''.
|-
| | On '''sliders''', move '''a''' to '''1''', '''b''' to '''-2''' and '''c''' to '''-3'''.
| | Set '''slider''' '''a '''at '''1''', '''slider''' '''b''' at minus''' 2''' and '''slider''' '''c '''at minus''' 3'''.
|-
| | Point to the equation '''f(x)=1x2-2x-3''' in '''Algebra view'''.
| | The equation '''f of x equals 1 x squared minus 2 x minus 3''' appears in '''Algebra view'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.
|-
| | Point to the parabola in '''Graphics View'''.
| | '''Function''' '''f''' is a parabola, intersecting '''x axis''' at '''minus 1 comma 0''' and '''3 comma 0'''.

Thus,''' root'''s of '''fx equals x squared minus 2x minus 3 '''are '''x equals minus 1''' and '''3'''.
|-
| | Type '''Root(f)''' in input bar>>press '''Enter'''.
| | In '''input bar''', type '''Root f '''in parentheses and press '''Enter'''.
|-
| | Point to the '''roots''' in '''Algebra view''' and the '''intercepts''' in '''Graphics view.'''
| | The '''roots''' appear in '''Algebra view'''.

They also appear as '''x-intercepts''' of the parabola in '''Graphics view'''.
|-
| | Type '''Extremum(f)''' in Input bar>>press '''Enter'''.
| | In '''input bar''', type '''Extremum f''' in parentheses and press '''Enter'''.
|-
| | Point to the '''extremum''' in the '''Algebra''' and '''Graphics views.'''
| | The '''minimum vertex''' appears in '''Algebra''' and '''Graphics views'''.
|-
| | Double click on point C ('''extremum''') in '''Graphics view'''>>Select '''Object Properties.'''
| | After double clicking on point '''C''' in '''Graphics View''', select '''Object Properties.'''
|-
| | Click on '''red''' color box.
| | From '''Color''' tab, change the color to red.
|-
| | Close the '''Preferences '''box.
| | Close the '''Preferences '''dialog-box.
|-
| | Point to '''C''' in '''Algebra''' and '''Graphics''' views.
| | '''Point C''' ('''extremum''' of '''f of x''') is red in '''Algebra''' and '''Graphics views'''.
|-
| | Click on '''Move''' tool, drag '''a''' to '''1''', '''b''' to '''5''', '''c''' to '''10'''.
| | Click on '''Move''' tool, set '''slider''' '''a''' at '''1''', '''slider''' '''b''' at '''5''', '''slider''' '''c''' at '''10'''.
|-
| | Point to the equation '''f(x)=1x2+5x+10''' in '''Algebra view.'''
| | The equation '''f of x equals 1 x squared plus 5x plus 10 '''appears in '''Algebra view'''.
|-
| | Click in and drag '''Graphics''' view to see this parabola.
| | Click in and drag '''Graphics''' view to see this parabola.
|-
| | Point to the parabola in '''Graphics View'''.
| | It does not intersect the '''x-axis'''.
|-
| | Point to the '''roots''', '''points A and B''' in the '''Algebra view'''.
| | '''Points A '''and''' B''' are undefined as the '''function''' does not intersect the '''x axis'''.
|-
| | Point to '''extremum''' point C in '''Algebra''' and '''Graphics views'''.
| | '''Extremum''' (point '''C''') is shown in red in '''Algebra''' and '''Graphics views'''.
|-
| |
| | '''Function f of x equals x squared plus 5x plus 10 '''has no '''real roots'''.

Let us see the '''complex root'''s of this equation.
|-
| | Click on '''View'''>>'''Spreadsheet'''.
| | Click on '''View''', then on '''Spreadsheet'''.

This opens a spreadsheet on the right side of the '''Graphics view'''.
|-
| | Click to close '''Algebra view'''.
| | Click to close '''Algebra view'''.
|-
| | Drag the boundary to see '''Spreadsheet''' view properly.
| | Drag the boundary to see '''Spreadsheet''' view properly.
|-
| | Point to the '''spreadsheet'''.
| | Type the following '''labels''' and formulae in the '''spreadsheet'''.
|-
| | Type “'''b^2-4ac'''” in cell '''A1'''>>press '''Enter'''.

Drag column to adjust width.
| | In '''cell A1''', type within quotes '''b caret 2 minus 4ac''' and press '''Enter.'''

Drag column to adjust width.

'''b squared minus 4ac '''is also called the''' determinant. '''
|-
| | Type '''Root1''' and '''Root2''' in cells '''A4''' and '''A5'''>>press '''Enter'''.
| | In cells '''A4''' and '''A5''', type '''Root1''' and '''Root2, '''and press '''Enter.'''
|-
| | Type '''Complex root1''' and '''Complex root2''' in '''A9''' and '''A10'''>>press '''Enter'''.
| | In cells '''A9''' and '''A10, '''type '''Complex root1''' and '''Complex root2'''.

Press '''Enter'''.
|-
| | Drag column to adjust width.
| | Drag column to adjust width.
|-
| | Type '''b^2-4 a c''' in cell '''B1'''>>press '''Enter. '''
| | In '''cell B1''', type '''b caret 2 minus 4 space a space c''' and press '''Enter. '''
|-
| | Point to cell '''B1'''.
| | The value minus 15 appears in '''cell''' '''B1''' corresponding to '''b squared minus 4 a c''' for '''f x'''.

Note: '''Determinant''' is always negative for quadratic '''functions''' without '''real roots'''.
|-
| | Type “'''-b/2a'''” in cell '''B3'''>>press '''Enter'''.
| | In '''cell B3''', type within quotes '''minus b divided by 2a'''.

Press '''Enter.'''
|-
| | Type '''–b/2 a''' in cell '''B4'''>>press '''Enter'''.
| | In '''cell B4''', type '''minus b divided by 2 space a'''.

Press '''Enter'''.

Note the value '''-2.5''' appear in cell '''B4'''.
|-
| | Type '''B4''' in cell '''B5'''>>press '''Enter'''.
| | In '''cell B5''', type '''B4''' and press '''Enter. '''

The value -'''2.5''' appears in cell '''B5 '''also.
|-
| | Type “'''+-sqrt(b^2-4ac)/2a'''” in cell '''C3'''>>press '''Enter.'''
| | In '''cell C3''', type the following '''line''' and press '''Enter'''.

Within quotes, '''plus minus sqrt D divided by 2a'''
|-
| | Type '''sqrt(B1)/2 a''' in cell '''C4'''>>press '''Enter.'''
| | In '''cell C4''', type '''sqrt B1''' in parentheses''' divided by 2 space a''' and press '''Enter. '''

Note that a question mark appears in '''cell C4'''.
|-
| | Type '''–C4''' in cell '''C5'''>>press '''Enter.'''
| | In '''cell C5''', type '''minus C4''' and press '''Enter. '''

Again, a question mark appears in '''cell''' '''C5'''.

There are no '''real''' solutions to the '''negative square root of the determinant'''.
|-
| | Type '''(b4+c4,0)''' in the input bar>>press '''Enter.'''
| | In '''input bar''', type '''b4 plus c4 comma 0 in parentheses''' and press '''Enter.'''

This should '''plot''' the '''root''' corresponding to '''ratio of minus b plus squareroot of determinant to 2a.'''
|-
| | Type '''(b5+c5,0)''' in the input bar>>press '''Enter. '''
| | In input bar, type '''b5 plus c5 comma 0''' '''in parentheses '''and press '''Enter. '''

This should plot the '''root''' corresponding to '''ratio of minus b minus squareroot of determinant to 2a'''.
|-
| |
| | '''f x equals x squared plus 5x plus 10 '''has no '''real roots'''.

Hence, the points do not appear in '''Graphics view'''.
|-
| | Click in and drag''' Graphics '''view to see this properly.''' '''
| | Click in and drag''' Graphics '''view to see this properly.''' '''
|-
| | Type '''–b/2 a''' in cell '''B9'''>>press '''Enter'''.
| | In '''cell B9''', type '''minus b divided by 2 space a '''and press '''Enter.'''
|-
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
|-
| |
| | '''Determinant''' is less than 0 for '''f x equals x squared plus 5x plus 10'''.

So the opposite sign will be taken to allow calculation of '''roots'''.
|-
| | Type '''sqrt(-B1)/2 a''' in cell '''C9'''>>press '''Enter.'''
| | In '''cell C9''', type '''sqrt minus B1''' in parentheses''' divided by 2 space a''' and press '''Enter.'''

1.94 appears in '''C9'''.
|-
| | Type '''–C9''' in cell '''C10'''>>press '''Enter.'''
| | In '''cell C10''', type '''minus C9''' and press '''Enter.'''

'''Minus''' 1.94 appears in '''C10'''.
|-
| | Click in and drag '''Graphics''' view to see both '''roots'''.
| | Click in and drag '''Graphics''' view to see the following '''complex''' '''roots'''.
|-
| | Type '''(b9,c9)''' in the input bar>>press '''Enter. '''
| | In '''input bar''', type '''b9 comma c9''' in parentheses and press '''Enter. '''

This '''complex root''' has '''real axis coordinate,''' '''minus b divided by 2a'''.

Imaginary axis co-ordinate is '''squareroot of negative determinant divided by 2a'''.
|-
| | Type '''(b10,c10)''' in the input bar>>press '''Enter.'''
| | In input bar, type '''b10 comma''' '''c10 in parentheses''' and press '''Enter. '''

This complex root has '''real axis coordinate, minus b divided by 2a. '''

'''Imaginary axis''' co-ordinate is '''minus squareroot of negative determinant divided by 2a'''.
|-
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.
|-
| | Drag the '''slider''' '''b''' to -2 and '''c''' to -3.
| | Drag the '''slider''' '''b''' to -2 and '''slider''' '''c''' to -3.
|-
| | Click in and drag '''Graphics''' view to see the parabola.
| | Click in and drag '''Graphics''' view to see the parabola.
|-
| | Point to the parabola in '''Graphics view'''.
| | Note how the parabola changes to the one seen for '''f x equals x squared minus 2x minus 3'''.
|-
| | Point to the '''roots''' appearing at x '''intercepts''' of parabola in '''Graphics view'''.
| | The '''real roots''' plotted earlier for '''f x equals x squared minus 2x minus 3''' appear now.
|-
| | Drag boundary to see '''Spreadsheet''' view.
| | Drag boundary to see '''Spreadsheet''' view.
|-
| | Point to the question marks appearing in '''C9''' and '''C10''' in the '''spreadsheet'''.
| | As '''roots''' are '''real''', calculations for '''complex roots''' become invalid.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 10'''
'''Summary'''
| | In this tutorial, we have learnt to:
Visualize quadratic '''polynomials''', their '''roots''' and '''extrema'''

Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''
|-
| | '''Slide Number 11'''
'''Assignment'''
| | As an assignment:
Drag '''sliders''' to graph different quadratic '''polynomials'''.

Calculate '''roots''' of the '''polynomials'''.
|-
| | '''Slide Number 12'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 13'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 14'''
'''Forum for specific questions:'''
Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 15'''
'''Acknowledgement'''
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is Vidhya Iyer from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Roots-of-Polynomials/English

2018-03-09T10:14:52Z

Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Roots of Polynomials'''. |- | | '''Slide..."

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Roots of Polynomials'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn:
To plot graphs of '''polynomial''' equations

About '''complex numbers''', '''real''' and '''imaginary roots'''

To find '''extrema''' and '''inflection points'''
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this tutorial, you should be familiar with '''GeoGebra''' interface

Basics of '''coordinate system'''

'''Polynomials'''

If not, for relevant tutorials, please visit our website.
|-
| | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using:

'''Ubuntu Linux''' OS version 14.04

'''GeoGebra 5.0.388.0-d'''
|-
| |
| | Let us begin with the '''binomial theorem'''.
|-
| | '''Slide Number 5'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' Єℝ, '''index''' ''n'' is a '''positive integer''', ''0 ≤ r ≤n, then (a + b)n can be expanded as follows:''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

Reminder: ''nC1 = n!/[1! (n-1)!]''
| | '''''a''''' and '''''b''''' are '''real numbers''', '''index''' '''''n''''' is a positive integer.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''''a''''' plus '''''b''''' raised to '''''n''''' can be expanded as shown.
|-
| | '''Slide Number 6'''

'''Quadratic Equations and Roots'''

A second degree polynomial, '''y =''' '''''a''''' '''x2+''' '''''b''''' '''x+''' '''''c''''' has roots

'''x=-''' '''''b''''' '''± sqrt{(''' '''''b''''' '''2-4''' '''''ac)/2a''''' '''}'''

where '''▲=''' '''''b2-4ac'''''

When ▲< 0, roots are complex

When ▲=0, roots are real and equal

When ▲>0, roots are real and unequal
| | '''Quadratic Equations and Roots'''

A '''2nd degree polynomial''', '''y equals''' '''''a''''' '''x squared plus''' '''''b''''' '''x plus''' '''''c''''' has '''roots''' given by values of '''''x'''''.

'''''x''''' is equal to '''ratio''' of minus '''''b''''' plus or minus '''squareroot''' of '''''b''''' '''squared''' minus 4 '''''a c''''' to 2 '''''a'''''.

Where '''determinant''' '''Delta''' is equal to '''''b''''' '''squared''' minus 4 '''''a c'''''

When '''Delta''' is less than 0, '''roots''' are '''complex'''

When '''Delta''' is equal to 0, '''roots''' are '''real''' and equal

When '''Delta''' is greater than 0, '''roots''' are '''real''' and unequal
|-
| | '''Slide Number 7'''

'''Quadratic Equations and Roots'''

When roots are real, '''''ax2+b''''' '''x+''' '''''c''''' '''=0''' has extremum '''(xv, yv)'''

'''xv = -''' '''''b/2a''''' and '''yv=''' '''''a''''' '''xv2+''' '''''b''''' '''xv+''' '''''c'''''
| | '''Quadratic Equations and Roots'''

When '''roots''' are '''real''', '''''ax''''' '''squared plus''' '''''b x''''' '''plus''' '''''c''''' equals 0 has '''extremum''' '''''xv''''' '''comma''' '''''yv'''''

'''''xv''''' equals '''minus''' '''''b''''' '''divided by 2''' '''''a''''' and '''''yv''''' '''equals''' '''''axv''''' '''squared plus''' '''''bxv''''' '''plus''' '''''c'''''
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''View''' >> select '''CAS'''.

| | Click on '''View''' tool and select '''CAS''' to open the '''CAS''' view.
|-
| | In line 1 in '''CAS view''', type '''f(x):=x^2-2x-3''' >> press '''Enter'''.
| | In line 1 in '''CAS''' view, type the following line.

'''f x''' in parentheses '''colon equals x caret 2 minus 2 space x minus 3'''.

To type '''caret''' symbol, hold '''Shift''' key down and press 6.

The space indicates multiplication.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | Point to the '''equation f(x)''' appearing in '''Algebra''' view.

Point to '''exponent''' 2 in '''f(x)'''.
| | Observe the '''equation f of x''' in '''Algebra''' view.

The '''degree''' of this '''quadratic polynomial f of x''' is 2.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
| | Click in '''Graphics''' view to see '''Graphics View''' toolbar.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view to see the minimum '''vertex''' of '''parabola f'''.
|-
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.

When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
| | Click on '''Move Graphics View''' tool and click in '''Graphics''' background.

When hand symbol appears, drag '''Graphics''' view so you can see parabola '''f'''.
|-
| | Drag boundaries to see '''CAS''' view properly.
| | Drag boundaries to see '''CAS''' view properly.
|-
| | Type '''Root(f)''' in line 2 of '''CAS''' view >> press '''Enter'''.
| | In line 2 of '''CAS''' view, type '''Root f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''roots''' in '''CAS''' view.
| | The '''roots''' appear below, in the same box, in curly brackets.
|-
| | Point to '''roots''' in '''Graphics''' view.
| | Note that these are the '''x-intercepts''' of parabola '''f''' in '''Graphics''' view.
|-
| | Type '''Extremum(f)''' in line 3 of '''CAS''' view >> press '''Enter'''.
| | In line 3 of '''CAS''' view, type '''Extremum f''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''f''' in '''Graphics''' view.
|-
| | In line 4 in '''CAS''' view, type '''g(x):=x^2+5x+10''' >> press '''Enter'''.
| | In line 4 in '''CAS''' view, type the following line.

'''g x''' in parentheses '''colon equals x caret 2 plus 5 space x plus 10'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''Algebra''' view properly.

Point to the '''equation g(x)''' appearing in '''Algebra''' view.
| | Drag boundary to see '''Algebra''' view properly.

Observe the '''equation g of x''' in '''Algebra''' view.
|-
| | Drag boundary to see '''Graphics''' view properly.
| | Drag boundary to see '''Graphics''' view properly.
|-
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
| | Uncheck '''f of x''' in '''CAS''' view.

Note that this also unchecks it in '''Algebra''' view and hides parabola '''f''' in '''Graphics''' view.
|-
| | Click and drag '''Graphics''' view so you can see parabola '''g'''.
| | Click in and drag '''Graphics''' view so you can see parabola '''g'''.
|-
| | Again, drag boundary to see '''CAS''' view properly.
| | Again, drag boundary to see '''CAS''' view properly.
|-
| | Type '''Root(g)''' in line 5 of '''CAS''' view >> press '''Enter'''.
| | In line 5 of '''CAS''' view, type '''Root g''' in parentheses.

Press '''Enter'''.
|-
| | Point to empty curly brackets for '''roots''' in '''CAS''' view.
| | Empty curly brackets appear below.

Parabola '''g''' does not have any '''real roots''' as it does not intersect '''x axis''' at all.

'''Roots''' are said to be '''complex'''.
|-
| | Type '''Extremum(g)''' in line 6 of '''CAS '''view >> press '''Enter'''.
| | In line 6 of '''CAS''' view, type '''Extremum g''' in parentheses.

Press '''Enter'''.
|-
| | Point to '''extremum''' in '''CAS''' view.
| | The '''extremum''' appears below, in the same box, in curly brackets.
|-
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
| | Note that this is the minimum '''vertex''' of parabola '''g''' in '''Graphics''' view.
|-
| | Point to '''Evaluate''' tool.

Point to '''extremum''' in form of '''fractions'''.
| | While '''Evaluate''' tool is highlighted in '''CAS View''' toolbar, the '''extremum''' appears as '''fractions'''.

'''Minus five divided by 2 comma 15 divided by 4'''.
|-
| | Click on the '''extremum''' in line 6 and click on '''Numeric''' tool.

Point to '''extremum''' in form of '''decimals'''.
| | In line 6, click on the '''extremum''' and click on '''Numeric''' tool.

The '''extremum '''now appears in '''decimal''' form.

'''Minus 2 point 5 comma 3 point 7 5'''.
|-
| |
| | Let us look at '''complex numbers'''.
|-
| | '''Slide Number 8'''

'''Complex numbers, XY plane'''

As we know,
A '''complex number''' is expressed as '''''z = a + ib''''': where ''''''a'''''' is the real part, ‘'''''bi’''''' is '''imaginary '''part, and '''a''' and '''b''' are constants.

'''Imaginary number, ''i'' '''= sqrt{-1}

In the '''XY plane''', '''''a + ib''''' corresponds to the point ('''a, b''').

In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
| | '''Complex numbers, XY plane'''

As we know,
A '''complex number''' is expressed as '''''z''''' '''equals''' '''''a''''' '''plus''' '''''ib'''''.

'''''a''''' is the '''real''' part; '''''bi''''' is '''imaginary '''part;'''''a''''' and '''''b''''' are '''constants'''

'''''i''''' is '''imaginary number''' and is equal to '''square root of minus 1'''.

In the '''XY plane''', '''''a''''' '''plus''' '''''ib''''' corresponds to the point '''''a''''' '''comma''' '''''b'''''.

In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
|-
| | '''Slide Number 9'''

'''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector''' with '''real axis coordinate''' '''''a''''' and '''imaginary axis coordinate''' '''''b'''''

Length of the '''vector''' '''''z''''' = |'''''z'''''| = '''''r'''''

'''''r''''' '''= sqrt (a2+b2) (Pythagoras’ theorem)'''
| | '''Complex numbers, complex plane'''

In '''complex plane''', '''''z''''' is a '''vector'''.

Its '''real axis coordinate''' is '''''a''''' and '''imaginary axis coordinate''' is '''''b'''''.

The length of the '''vector''' '''''z''''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

According to '''Pythagoras’ theorem''', '''''r''''' is equal to '''squareroot of''' '''''a''''' '''squared plus''' '''''b''''' '''squared'''.
|-
| | '''Slide Number 10'''

'''Complex numbers, complex plane'''

'''Argument '''''ϴ''''' '''= angle between '''real axis''' and '''line segment''' connecting '''''z''''' to O '''(0,0)''' in counter-clockwise direction

'''Polar form''' of '''''z = a + ib''''' is

'''''z = r (cosϴ + i sinϴ)'''''

where '''''a= r cosϴ, b=r sinϴ'''''
| | '''Argument ''theta''''' is angle between '''real axis''' and line segment connecting '''''z''''' to '''origin'''.

It is in counter-clockwise direction.

'''Polar form''' of '''''z''''' equals '''''a''''' '''plus''' '''''ib''''' is

'''''z''''' equals '''''r times cos theta plus i sin theta'''''

where '''''a''''' is equal to '''''r cos theta''''' and '''''b is r sin theta'''''
|-
| | Show the '''GeoGebra''' window.
| | Let us go back to the '''GeoGebra interface''' we were working on.

We will now use the '''input bar''' instead of '''CAS''' view.
|-
| | Click and close '''CAS''' view.
| | Click and close '''CAS''' view.
|-
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
| | In '''Algebra''' view, uncheck '''g of x''' to hide it.
|-
| | In '''input bar''', type the following line.

'''h(x):=x^3-4x^2+x+6''' >> press '''Enter'''.

| | In '''input bar''', type the following line.

'''h x''' in parentheses '''colon equals x caret 3 minus 4 space x caret 2 plus x plus 6'''.

Press '''Enter'''.
|-
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Point to the '''equation h(x)''' appearing in '''Algebra''' view.
| | Drag boundaries to see '''Algebra''' and '''Graphics''' view properly.

Observe equation '''h of x''' in '''Algebra''' view.

Function '''h of x''' is graphed in '''Graphics''' view.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click in '''Graphics''' view.
|-
| | Click on '''Move Graphics View''' and move background to see the graph.
| | Click on '''Move Graphics View''' and move background to see the graph.
|-
| | In '''input bar''', type '''Root(h)''' and press '''Enter'''.
| | In '''input bar''', type '''Root h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') in '''Algebra''' view.
| | The '''co-ordinates''' of three '''roots''' ('''A, B''' and '''C''') appear in '''Algebra''' view.
|-
| | Point to three '''roots''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The three '''roots''' are also mapped as '''x intercepts''' of the '''curve h of x''' in '''Graphics''' view.
|-
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
| | In '''input bar''', type '''Extremum h''' in parentheses and press '''Enter'''.
|-
| | Point to '''co-ordinates''' of two '''extrema''' ('''D'''and '''E''') in '''Algebra''' view.
| | '''Co-ordinates''' of two '''extrema''' ('''D''' and '''E''') appear in '''Algebra''' view.
|-
| | Point to two '''extrema''' mapped on the '''curve h of x''' in '''Graphics''' view.
| | The two '''extrema''' are also mapped on '''curve h of x''' in '''Graphics''' view.
|-
| | '''Slide Number 11'''

'''Point of inflection'''

'''Point of inflection''' ('''PoI''') on a curve is the point where '''curve''' changes direction.

To find co-ordinates of PoI (x,y)

Equate 2nd derivative of given function to 0

Solve to get x (x co-ordinate of PoI)

Substitute this x in original function to get y co-ordinate
| | A '''point of inflection PoI''' on a curve is the point where the '''curve''' changes its direction.

To find the '''co-ordinates''' of '''PoI''' '''''x''''' comma '''''y''''':

We equate the second '''derivative''' of the given '''function''' to 0.

Solution of this equation gives us '''x''' ('''x co-ordinate''' of '''PoI''').

Substitute this '''x''' in original '''function''' to get '''y co-ordinate'''.
|-
| | Let us find the '''point of inflection''' on '''h(x)'''.
| | Let us find the '''point of inflection''' on '''h of x'''.
|-
| | In '''input bar''', type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In '''input bar''', type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''h''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''h''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | In '''Algebra''' view, '''point of inflection''' appears as point '''F''', below the two '''extrema'''.
|-
| | Point to '''F''' on '''h(x)''' in '''Graphics''' view.
| | '''F''' is mapped on '''h of x''' in '''Graphics''' view.
|-
| |
| | Let us open a new '''GeoGebra''' window to use '''CAS''' for a '''cubic polynomial'''.
|-
| | Click on '''View''' tool and click on '''CAS''' to show it.
| | Click on '''View''' tool and click on '''CAS''' to show it.
|-
| | In line 1 of '''CAS''' view, type the following line.

'''i(x):=x^3-6 x^2+4 x+1''' >> press '''Enter'''.
| | In line 1 of '''CAS'' 'view, type the following line.

'''i x''' in parentheses '''colon equals x caret 3 minus 6 space x caret 2 plus 4 space x plus 1'''.

Press '''Enter'''.
|-
| | Drag boundary to see '''CAS''' view properly.
| | Drag boundary to see '''CAS''' view properly.
|-
| | Drag boundary to see '''Algebra''' view properly.
| | Drag boundary to see '''Algebra''' view properly.
|-
| | In line 2 of''' CAS''' view, type '''Root(i)''' >> press '''Enter'''.
| | In line 2 of''' CAS''' view, type '''Root i''' in parentheses and press '''Enter'''.
|-
| | Point to the three '''roots''' in '''CAS''' view.

Scroll to see them.

Point to '''Evaluate''' tool.
| | The three '''roots''' are shown below with '''square root notations'''.

Scroll to see them.

Note that the '''Evaluate''' tool is highlighted.
|-
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

Point to the three '''roots''' in decimal form.
| | In line 2, click on the '''roots''' and click on '''Numeric''' tool.

The roots are now shown in '''decimal''' form in the next line.
|-
| | In line 4 of '''CAS''' view, type '''Extremum(i)''' >> press '''Enter'''.
| | In line 4 of '''CAS''' view, type '''Extremum i''' in parentheses and press '''Enter'''.
|-
| | Point to the two '''extrema''' in '''CAS''' view.

Scroll to see them.

Point to '''Numeric''' tool and to '''extrema''' in '''decimal''' form.
| | The two '''extrema''' points are shown below.

Scroll to see them.

As the '''Numeric''' tool was clicked, the points appear in '''decimal''' form.
|-
| | Click in and drag '''Graphics''' view so you can see '''i(x)'''.
| | Click in and drag '''Graphics''' view so you can see '''i of x'''.
|-
| | In line 5, type '''Inf''' >> choose '''InflectionPoint ( <Polynomial> )''' option from '''menu'''.
| | In line 5, type '''Inf''' and scroll down menu to choose '''InflectionPoint Polynomial''' option.
|-
| | Instead of highlighted '''Polynomial''', type '''i''' >> Press '''Enter'''.
| | Instead of highlighted '''Polynomial''', type '''i''' and press '''Enter'''.
|-
| | Point to the '''point of inflection''' in '''Algebra''' view.
| | '''Co-ordinates''' of '''point of inflection''' appear in curly brackets.
|-
| |
| | Correlate the '''degree''' of the '''polynomials''' and the number of '''roots''' seen so far.
|-
| | Point to '''CAS''', then '''Algebra''' and '''Graphics''' views.
| | Observe that '''functions''' entered in '''CAS''' appear in '''Algebra''' and '''Graphics''' views.
|-
| | Point to '''Algebra''' and '''Graphics''' views, then '''CAS''' view.
| | '''Functions''' entered in '''input bar''' appear in '''Algebra''' and '''Graphics''' views but not in '''CAS''' view.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 12'''

'''Summary'''
| | In this tutorial, we have learnt how to:

Plot graphs of '''polynomial functions''' using '''CAS''' view and '''input bar'''.

Find '''real roots, extrema''' and '''inflection point(s)'''.

'''Complex roots''' will be covered in another tutorial.
|-
| | '''Slide Number 13'''
'''Assignment'''
| | Assignment:

Plot '''graphs''' and find '''roots''', '''extrema''' and '''inflection points''' for the following '''polynomials'''.

d(x)=x2-6x+5

e(x)=3x3-2x2+0.2x-1

f(x)=-2x4-x3+3x2

g(x)=x5-7x4+9x3+23x2-50x+24

h(x)=(4x+3)/(x-1)

i(x)=(3x2-2x-1)/(2x2+3x-2)
|-
| | '''Slide Number 14'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 15'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 16'''
'''Forum for specific questions:'''
Do you have questions in THIS Spoken Tutorial?
Please visit this site.
Choose the '''minute''' and '''second''' where you have the question.
Explain your question briefly.
Someone from our team will answer them.
| | Please post your timed queries on this forum.
|-
| | '''Slide Number 17'''
'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Introduction-to-Trigonometry-Using-GeoGebra/English

2018-02-09T10:13:38Z

Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Introduction to Trigonometry using GeoGebra''..."

{|border=1
||'''Visual Cue'''
||'''Narration'''
|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on '''Introduction to Trigonometry using GeoGebra''' .
|-
| | '''Slide Number 2'''

'''Learning Objectives'''
| | In this tutorial, we will learn how to construct,

A unit circle

A right triangle inside the unit circle using '''GeoGebra'''.
|-
| | '''Slide Number 3'''

'''Pre-requisites'''
| | To follow this tutorial, you should be familiar with

the '''GeoGebra interface''',

basics of geometry, trigonometry and graphs.
|-
| | '''Slide Number 4'''

'''System Requirement'''
| | Here I am using,

Ubuntu Linux OS version 14.04

GeoGebra 5.0.388.0-d
|-
| | Show the '''GeoGebra''' window.
| | I have already opened the '''GeoGebra''' interface.
|-
| | Click on '''Move Graphics View''' tool.
| | Click on '''Move Graphics View''' tool.
|-
| | Drag the '''origin''' to the center .
| | Drag the '''origin''' to the center of the '''Graphics''' view.
|-
| | Click on '''Zoom In''' tool> click on screen in '''Graphics''' view.
| | Under '''Move Graphics View''', click on '''Zoom In''' tool.

Then click on screen in '''Graphics''' view.

This will magnify the '''Graphics''' view.

|-
| | Click on '''Slider''' tool >> click on screen in '''Graphics view'''.
| | Click on''' Slider''' tool and then click on the screen in '''Graphics''' view.
|-
| | Point to the dialog box.
| | '''Slider''' dialog-box appears in the '''Graphics''' view.
|-
| | Point to '''Number''' radio .
| | By default, the '''Number''' radio-button is selected.
|-
| | Type Name as '''radius''' .
| | In the '''Name''' field, type '''radius'''.
|-
| | Change '''Min''', '''Max''' and '''Increment''' values.
| | Set '''Min''' value as 1, '''Max''' value 5 and '''Increment''' of 0.1.
|-
| | Click '''OK''' button.
| | Click '''OK''' button.
|-
| | Point to the slider.

Drag to show the changing values.
| | This creates a '''number slider''' named '''radius'''.

Using this '''slider''', radius can be changed from 1 to 5 in increments of 0.1.
|-
| | Click on '''Slider''' tool >> click on screen in '''Graphics''' view.
| | Click on '''Slider''' tool and then click on the screen in '''Graphics''' view.
|-
| | Point to the dialog box.
| | '''Slider''' dialog box appears.
|-
| | Click on '''Angle''' radio button.
| | This time, select '''Angle''' radio button.
|-
| | Change '''Min''', '''Max''' and '''Increment''' values.
| | '''Min''', '''Max''' and '''Increment''' should be 0º, 360º and 1º, respectively.
|-
| | Click '''OK''' .
| | Click '''OK'''.
|-
| | Drag to show the changing values.
| | This sets up '''alpha''' slider on which angle '''alpha (α)''' can be changed from 0º to 360º .
|-
| | Click on '''Circle with center and radius''' tool.
| | Click on '''Circle with Center and Radius''' tool.
|-
| | Place the cursor on the '''origin''' and click on it.
| | Place the cursor on the '''origin''' (0,0) and click on it.
|-
| |
| | '''Circle with Centre and Radius''' text box appears.
|-
| | Type '''radius''' in the text box, click '''OK'''.
| | In text box, type '''radius''' and click '''OK''' .
|-
| | Point to the circle at the center.
| | A circle with center '''A''' at the '''origin''' is drawn.

Please note, we are using '''A''' for '''O'''(0,0).
|-
| | Drag the '''radius''' slider from 1 to 5 but leave it at 1 finally.
| | Drag the '''radius''' slider from 1 to 5 to change the radius of the circle.

Drag it to 1 to have a unit circle.
|-
| | Click on '''Segment''' tool, click on circumference of the circle at the '''x axis'''.
| | Click on '''Segment''' tool.

Click on the circumference of the circle at the '''x Axis'''.

This creates '''point B'''.
|-
| | Click on '''point A'''.

Point to '''segment AB'''.
| | Then click on '''point A''' to draw '''segment AB'''.
|-
| | Click on '''Angle with Given Size''' tool.
| | Now click on '''Angle with Given Size''' tool.
|-
| | Click on '''point B'''>>'''point A'''.
| | Click on '''point B''', then '''point A'''.
|-
| | Point to the '''Angle with Given Size'''.
| | '''Angle with Given Size''' text-box appears.
|-
| | Point to '''Angle with Given Size''' text box.
| | In the text box, delete '''45º''' and select '''alpha α''' from the symbol menu.
|-
| | Point to '''counter-clockwise''' radio button.
| | Leave direction at '''counter-clockwise''', click '''OK'''.
|-
| |
| | '''Angle B'AB (B prime A B)''' is created which is = '''beta β''' which is = '''alpha α'''.
|-
| | Drag the '''α''' slider.
| | Drag the '''alpha α''' slider from '''0º''' to '''360º'''.
|-
| |
| | '''B' (B prime)''' moves in counter clockwise direction around the circle as '''alpha α''' increases.
|-
| | (ideal angle is between 50-60 degree)
| | Now drag the '''alpha α''' slider so that '''beta β''' value is between '''50''' and '''60º'''.
|-
| | Click on '''Point''' tool>> click outside the circle>> Point to '''point C'''.
| | Click on '''Point''' tool and click outside the circle to create '''point C'''.
|-
| | Double click on C in the '''Algebra''' view.
| | In the '''Algebra''' view, double-click on point '''C''' to change its coordinates.
|-
| | Change co-ordinates of '''point C''' to '''(x(B'), 0)''' >> press enter.
| | Type '''x(B') (x B prime)''' as '''x-coordinate''' and '''y-coordinate''' as zero and press '''Enter'''.
|-
| | Move the cursor to point C.
| | This will shift point '''C''' right under '''B' (B prime)'''.
|-
| | Click on '''Segment''' tool>> click on points '''B'''' and '''A'''.
| | Click on '''Segment''' tool.

Click on points '''B' (B prime)''' and '''A''' to join them.
|-
| |
| | This forms the hypotenuse of the right triangle '''ACB' (A C B prime)'''.
|-
| | Click on '''Segment''' tool>>click on points '''B'''' and '''C'''.
| | Now using '''Segment''' tool join '''B' (B prime)''' and '''C'''.
|-
| | Point to the right angled triangle AB'C and its angles.
| | A right angle is formed at '''C'''- angle '''ACB' (A C B prime)'''.
|-
| |
| | Angle '''B'AC (B prime A C)''' <nowiki>=</nowiki> '''alpha αº''' which is <nowiki>= </nowiki>'''beta βº'''.
|-
| | Move the '''α''' slider from 0º to 360º .
| | Drag the '''alpha α''' slider from 0º to 360º to see how '''alpha α''' changes.
|-
| | '''Formatting the triangle'''
| | Let us enhance the visibility of the triangle.
|-
| | Double click on angle '''β'''>>Select '''Object Properties'''.
| | Double-click on angle '''beta β'''.

Click on '''Object properties'''.
|-
| | Click on '''color''' tab.
| | Click on the '''color''' tab; leave color as green.
|-
| | Drag the '''Opacity''' slider to 25.
| | Increase '''Opacity''' to 25.
|-
| | Click on '''Style''' tab>> drag '''size''' slider to 50.
| | Click on '''Style''' tab; increase '''size''' to 50.

|-
| | Change '''Decoration''' to '''arrow pointing counter-clockwise'''.
| | Change the '''Decoration''' to '''arrow pointing counter-clockwise'''.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' dialog-box.
|-
| |
| | I will now change the properties of the triangle segments.
|-
| | Double click on '''segment AB''''>>Select '''Object Properties'''.
| | To change the color of the segments, double-click on '''segment AB' (A B prime)'''.

Select '''Object Properties'''.
|-
| | Click on '''color''' tab and select '''blue'''.
| | Click on '''color''' tab; select '''blue'''.
|-
| | Double click on '''segment CB''''>>Select '''Object Properties'''.

Click on '''color''' tab and select '''red'''.

Double click on '''segment AB'''>>Select '''Object Properties'''.

Click on '''color''' tab and select '''orange'''.
| | Similarly, change the colors of '''CB' (C B prime)''' to red and of '''AB''' to orange.
|-
| | Right-click on '''segment AB''''>>Rename>>type '''c''' in '''name''' field>>click '''OK'''.
| | To '''rename''' the segments, right-click on '''segment AB' (A B prime)'''.

Choose '''Rename''' option.

Type '''c''' in the '''name''' field and click '''OK'''.
|-
| | Right-click on '''segment CB''''>>'''Rename'''>>type '''a''' in '''name''' field>>click '''OK'''.

Right-click on '''segment AC'''>>'''Rename'''>>type '''b''' in '''name''' field>>click '''OK'''.
| | Similarly, rename '''CB' (C B prime)''' to '''a''' and '''AC''' to '''b'''.
|-
| | Click on '''style''' tab>>point to '''line thickness''' and '''line style''' options.
| | If you wish, you may change the '''line thickness''' and the '''line style''' in the '''Style''' tab.
|-
| | In '''Algebra''' view, click and highlight segment '''a'''.
| | In '''Algebra''' view, click and highlight segment '''a'''.
|-
| | Holding '''Shift''' key down, drag and highlight '''b''' and '''c''' as well.
| | Holding the '''Shift''' key down, highlight all 3 segments.
|-
| | In '''Graphics''' view, click on '''Hidden''' option.
| | In '''Graphics''' view, click on '''Hidden''' option.
|-
| | Point to the segments in '''Graphics''' view.
| | All three labels are hidden.
|-
| | '''Switching x axis to radians'''
| | Now let us change '''x Axis''' units to '''radians'''.
|-
| | Double click on '''x axis''' in '''Graphics''' view>> '''Object properties'''.
| | Double click on '''x axis''' in '''Graphics''' view then on '''Object Properties'''.
|-
| | Click on '''Preferences-Graphics'''>>'''x axis'''.
| | In the '''Object Properties''' menu, click on '''Preferences-Graphics''' then on '''x Axis'''.
|-
| | Check the '''Distance''' option, select '''π/2'''>>select '''Ticks''' first option.
| | Check the '''Distance''' option, select '''pi over 2'''.

Select the '''Ticks''' first option.
|-
| | Close the '''Preferences''' box.
| | Close the '''Preferences''' box.
|-
| | Point to '''x-axis'''.
| | Units of '''x-axis''' are in '''radians''' with the intervals shown.

'''GeoGebra''' will convert degrees of angle '''alpha α''' to '''radians'''.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 5'''

'''Summary'''
| | In this tutorial, we have learnt how to use '''GeoGebra''' to,

Construct a unit circle and a right triangle inside it.
|-
| | '''Slide Number 6'''

'''Assignment'''
| | As an Assignment:

Try constructing circles with radius 2 and 3 units.

Draw right triangles in these circles.

Also try different styles, opacity & thickness.
|-
| | '''Slide Number 7'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
| | '''Slide Number 8'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 9'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site
Choose the minute and second where you have the question
Explain your question briefly
Someone from our team will answer them

| | Please post your timed queries on this forum.
|-
| | '''Slide Number 10'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is Vidhya Iyer from''' IIT Bombay''', signing off.

Thank you for joining.
|-
|}