Difference between revisions of "Applications-of-GeoGebra/C3/Differentiation-using-GeoGebra/English"

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Click on '''Point''' tool and click on '''(3,6)'''.   
 
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|| Clicking on the '''Point on Object''', create point A at 2 comma 2 and B at 3 comma 6.
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|| Clicking on the '''Point on Object''' tool, create point A at 2 comma 2 and B at 3 comma 6.
  
 
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Latest revision as of 06:05, 15 January 2019

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 Operating System 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


f(x) = x2-x

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

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

Differentiation: First Principles

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 tool, create point A at 2 comma 2 and B at 3 comma 6.
Click on Line tool >> click on points B and A. Click on Line tool and click on points B and A to draw line g.
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 >> click 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.
Point to B and A. 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 C >> 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 slope of the tangent at each point on the 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 6

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)

Differentiation: First Principles, the Algebra

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 Number 7

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.


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 8

Differentiation of a Polynomial Function


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

Differentiation rules:

d(u±v)/dx = d(u)/dx ± d(v)/dx

d(5+12x-x3)/dx = d(5)/dx + d(12x)/dx - d(x3)/dx = 0 + 12 - 3x2 = -3x2 +12


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

Differentiation of a Polynomial Function

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 tangent line f 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 >> select Trace On option. Right-click on 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 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 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 equation in Algebra view.

Drag the boundary.

Note the equation of g prime x in Algebra view.

Drag the boundary to see it properly

Compare slides' 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.
Point to slope. Slope of the tangent at the corresponding point on g of x is 0.

Such points on g of x are maxima or 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. 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.

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 9

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 out from the four corners

What size squares should we cut out to get the maximum volume of the box?

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 10

A Sketch of the Cardboard

Let’s draw the cardboard:

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; but we will leave it as it is.

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 the 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 this broad peak.
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 11

Summary

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

Understand differentiation

Draw graphs of derivatives of functions

Slide Number 12

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 13

About Spoken Tutorial project

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 14

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Slide Number 15

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Slide Number 16

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.

Contributors and Content Editors

Madhurig, Snehalathak, Vidhya