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{|border=1
|| '''Time'''
|| '''Narration'''

|-
|| 00:01
|| Welcome to this tutorial on '''Differentiation using GeoGebra'''.

|-
|| 00:06
|| In this tutorial, we will learn how to use '''GeoGebra''' to:

|-
||00:11
|| Understand Differentiation

Draw graphs of derivative of functions.

|-
|| 00:18
|| Here I am using:

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

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||00:25
|| '''GeoGebra''' 5.0.481.0 hyphen d.

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|| 00:31
|| To follow this '''tutorial''', you should be familiar with:

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||00:34
|| '''GeoGebra''' interface

Differentiation

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||00:39
|| For relevant '''tutorials''', please visit our website.

|-
|| 00:43
||'''Differentiation: First Principles'''

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||00:47
|| Let us understand differentiation using '''first principles''' for the '''function f of x'''.

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||00:54
|| '''f of x''' is equal to '''x squared''' minus '''x'''

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||00:58
|| '''f prime x''' is the derivative of '''f of x'''.

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||01:04
|| Consider 2 points, '''A''' and '''B'''.

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||01:08
|| '''A''' is '''x''' comma '''f of x''' and '''B''' is '''x''' plus '''j''' comma '''f of x''' plus '''j'''
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|| 01:18
|| I have opened the '''GeoGebra''' interface.

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|| 01:22
|| In the '''input bar''', type the following line.
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||01:26
|| For the '''caret symbol''', hold the '''Shift''' key down and press 6.

Press '''Enter'''.
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|| 01:33
|| Observe the equation and the parabolic graph of '''function f'''.
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||01:40
|| Clicking on the '''Point on Object''' tool, create point A at 2 comma 2 and B at 3 comma 6.

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|| 01:53
|| Click on '''Line''' tool and click on points '''B''' and '''A''' to draw line '''g'''.
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|| 02:04
|| As shown earlier in this series, make this line '''g ''' blue and dashed.
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|| 02:11
|| Under '''Perpendicular Line''', click on '''Tangents'''.
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|| 02:16
|| Click on '''A''' and then on the parabola.
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|| 02:21
|| This draws a '''tangent h''' at point '''A''' to the parabola.
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|| 02:27
|| Let us make '''tangent h''' a red line.
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|| 02:31
|| Click on the '''Point''' tool and click anywhere in '''Graphics''' view.

This creates point '''C'''.
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||02:41
|| In '''Algebra''' view, double-click on '''C''' and change its '''coordinates''' to the following.
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||02:49
|| Now C has the same '''x coordinate''' as point '''B''' and the same '''y coordinate''' as point '''A'''.
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||02:58
|| Let us use the '''Segment''' tool to draw segments '''B C''' and '''A C'''.
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|| 03:08
|| We will make '''AC''' and '''BC''' purple and dashed segments.
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|| 03:14
|| With '''Move''' highlighted, drag '''B''' towards '''A''' on the parabola.
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|| 03:22
|| Observe lines '''g''' and '''h''' and the value of '''j''' (length of '''AC''').

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||03:29
|| As '''j''' approaches 0, points '''B''' and '''A''' begin to overlap.

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||03:37
|| Lines '''g''' and '''h''' also begin to overlap.
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|| 03:42
|| Slope of line '''g''' is the ratio of length of '''BC''' to length of '''AC'''.
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|| 03:50
|| Derivative of the parabola is the slope of the tangent at each point on the curve.
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|| 03:58
|| As '''B''' approaches '''A''' on '''f of x''', slope of '''AB''' approaches the slope of tangent at '''A'''.
|-
||04:08
|| Now let us look at the '''Algebra''' behind these concepts.
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|| 04:14
||'''Differentiation: First Principles, the Algebra'''

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||04:18
|| Slope of line '''AB''' equals the ratio of the lengths of '''BC''' to '''AC'''.

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||04:25
|| Line '''AB''' becomes the tangent at point '''A''' as distance '''j''' between '''A''' and '''B''' approaches 0.

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||04:35
|| '''BC''' is the difference between '''y coordinates''', '''f of x''' plus '''j''' and '''f of x''', for '''A''' and '''B'''.

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||04:43
|| '''AC''' is the difference between the '''x-coordinates''', '''x''' plus '''j''' and '''x'''.

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||04:50
|| Let us rewrite '''f of x''' plus '''j''' and '''f of x''' in terms of '''x squared''' minus '''x'''.

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||04:58
|| We will expand the terms in the numerator.
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|| 05:02
|| After expanding the terms in the numerator, we will cancel out similar terms with opposite signs.

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||05:10
|| We will pull out '''j''' from the numerator and cancel it.

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||05:15
|| Note that as '''j''' approaches 0, '''j''' can be ignored. So that '''2x''' plus '''j''' minus 1 approaches '''2x''' minus 1.

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||05:25
|| As we know, derivative of '''x squared''' minus<sup> '''</sup>x''' is '''2x''' minus 1.

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||05:32
|| Let us look at derivative graphs for some '''functions'''.
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|| 05:37
||'''Differentiation of a Polynomial Function'''

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||05:41
|| Consider '''g of x'''.

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||05:44
|| Derivative '''g prime x''' is the sum and difference of derivatives of the individual components.

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||05:53
|| ''g prime x''' is calculated by applying these rules.
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||05:59
|| Let us differentiate '''g of x''' in '''GeoGebra'''.
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|| 06:04
|| Open a new '''GeoGebra''' window.
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|| 06:07
|| In the '''input bar''', type the following line and press '''Enter'''.
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|| 06:13
|| As shown earlier in the series, zoom out to see '''function g''' properly.

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|| 06:24
|| Right-click in '''Graphics''' view and select '''xAxis''' is to '''yAxis''' option.

Select 1 is to 5.
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|| 06:35
|| I will zoom out again.
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||06:42
|| As shown earlier, draw point '''A''' on curve '''g''' and a tangent '''f''' at this point.
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|| 06:50
|| Under '''Angle''', click on '''Slope''' and on tangent line '''f'''.
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|| 06:58
|| Slope of tangent line '''f''' appears as '''m''' value in '''Graphics''' view.
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|| 07:04
|| Draw point '''B''' and change its '''coordinates''' to '''x A''' in parentheses comma '''m'''.
|-
|| 07:13
|| Right-click on '''B''' and select '''Trace On''' option
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||07:20
|| With '''Move''' tool highlighted, move point '''A''' on curve.

|-
||07:31
|| Observe the curve traced by point '''B'''.
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||07:35
|| Let us check whether we have the correct '''derivative''' graph.
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|| 07:39
|| In the '''input bar''', type '''d e r i'''.

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

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||07:49
|| Type '''g''' to replace the highlighted word ''' Function''.

Press '''Enter'''.
|-
||07:55
|| Note the equation of '''g prime x''' in '''Algebra''' view.

Drag the boundary to see it properly
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|| 08:04
|| Compare the calculations in the previous slide with the equation of '''g prime x'''
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||08:11
|| Let us find the maxima and minima of the '''function g of x'''.
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|| 08:16
|| Derivative curve '''g prime x''' remains above the '''x axis''' (is positive) as long as '''g of x''' is increasing.
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|| 08:27
|| '''g prime x''' remains below the '''x axis''' is negative as long as '''g of x''' is decreasing.
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|| 08:37
|| 2 and -2 are the values of '''x''' when '''g prime x''' equals 0.
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||08:45
|| Slope of the tangent at the corresponding point on '''g of x''' is 0.

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||08:52
||Such points on '''g of x''' are maxima or minima.
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||08:58
|| Hence, for '''g of x,''' -2 comma -11 is the minimum and 2 comma 21 is the maximum.
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|| 09:08
|| In '''GeoGebra''', we can see that the minimum value of '''g of x''' lies between '''x''' equals -3 and '''x''' equals -1.
|-
|| 09:20
|| In the '''input bar''', type '''M i n'''.

|-
||09:24
||From the menu that appears, select '''Min Function Start x Value, End x Value''' option.

|-
||09:31
||Type '''g''' for '''Function'''.

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||09:35
||Press '''Tab''' to go to the next argument.

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||09:38
||Type -4 and -1 as '''Start''' and '''End x-Values'''.

Press '''Enter'''.
|-
|| 09:47
|| In '''Graphics view''', we see the minimum on '''g of x'''.

|-
||09:52
||Its '''co-ordinates''' are -2 comma -11 in '''Algebra''' '''view'''.

|-
|| 09:58
|| In the '''input bar''', type '''Max'''.

|-
||10:02
||From the menu that appears, select '''Max Function Start x Value, End x Value''' option.

|-
||10:09
||Type '''g''', 1 and 4 as the arguments.

Press '''Enter.'''
|-
|| 10:17
|| We see the maximum on '''g of x''', 2 comma 21.
|-
||10:24
|| Finally, let us take a look at a practical application of differentiation.
|-
|| 10:31
|| We have a 24 inches by 15 inches piece of cardboard.

|-
||10:36
||We have to convert it into a box.

|-
||10:39
||Squares have to be cut from the four corners.

|-
||10:43
||What size squares should we cut out to get the maximum volume of the box?
|-
||10:49
|| '''A Sketch of the Cardboard'''

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||10:51
||Let us draw the cardboard:

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||10:54
||This is the volume '''function''' here.

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||10:58
||You could expand it into a '''cubic polynomial''' but we will leave it as it is.
|-
|| 11:05
|| Open a new '''GeoGebra''' window.
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|| 11:08
|| In the '''input bar''', type the following line and press '''Enter'''.
|-
|| 11:13
|| Drag the boundary to see the equation properly in '''Algebra''' view.
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|| 11:19
|| Right click in '''Graphics''' view and set '''xAxis''' is to '''yAxis''' to 1 is to 50.

|-
||11:29
||Now, zoom out to see the function properly.
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|| 11:38
|| Observe the graph that is plotted for the volume '''function''' in '''Graphics''' view.

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||11:44
||Drag the background to see the maximum.
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|| 11:48
|| Note that the maximum is on the top of this broad peak.
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|| 11:53
|| The length of the square side is plotted along the '''x axis'''.

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||11:58
||Volume of the box is plotted along the '''y axis'''.
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|| 12:02
|| As before, let us find the maximum of this '''function'''.

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||12:13
|| This maps the maximum, point '''A''', on the curve.

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||12:18
||Its '''coordinates''' 3 comma 486 appear in '''Algebra''' view.

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||12:24
||Thus, we have to cut out 3 inch squares from all corners.

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||12:30
||This will give the maximum possible volume of 486 cubic inches for the cardboard box.
|-
||12:39
|| Let us summarize.
|-
|| 12:41
|| In this tutorial, we have learnt how to use '''GeoGebra''' to:

Understand differentiation, Draw graphs of derivatives of '''functions'''
|-
|| 12:53
|| As an assignment:

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

|-
||13:00
||Find the derivatives of these '''functions''' independently and compare with '''GeoGebra''' graphs.
|-
|| 13:07
|| The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
|| 13:15
|| The '''Spoken Tutorial Project '''team:

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

|-
||13:24
||For more details, please write to us.
|-
|| 13:27
|| Please post your timed queries on this forum.
|-
|| 13:31
|| '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
||13:44
|| This is '''Vidhya Iyer''' from''' IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C3/Differentiation-using-GeoGebra/English-timed - Revision history

PoojaMoolya: Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this tutorial on '''Differentiation using GeoGebra'''. |- || 00:06 || In this tutorial, we will learn h..."