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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

00:25 GeoGebra 5.0.481.0 hyphen d.
00:31 To follow this tutorial, you should be familiar with:
00:34 GeoGebra interface


00:39 For relevant tutorials, please visit our website.
00:43 Differentiation: First Principles
00:47 Let us understand differentiation using first principles for the function f of x.
00:54 f of x is equal to x squared minus x
00:58 f prime x is the derivative of f of x.
01:04 Consider 2 points, A and B.
01:08 A is x comma f of x and B is x plus j comma f of x plus j
01:18 I have opened the GeoGebra interface.
01:22 In the input bar, type the following line.
01:26 For the caret symbol, hold the Shift key down and press 6.

Press Enter.

01:33 Observe the equation and the parabolic graph of function f.
01:40 Clicking on the Point on Object tool, create point A at 2 comma 2 and B at 3 comma 6.
01:53 Click on Line tool and click on points B and A to draw line g.
02:04 As shown earlier in this series, make this line g blue and dashed.
02:11 Under Perpendicular Line, click on Tangents.
02:16 Click on A and then on the parabola.
02:21 This draws a tangent h at point A to the parabola.
02:27 Let us make tangent h a red line.
02:31 Click on the Point tool and click anywhere in Graphics view.

This creates point C.

02:41 In Algebra view, double-click on C and change its coordinates to the following.
02:49 Now C has the same x coordinate as point B and the same y coordinate as point A.
02:58 Let us use the Segment tool to draw segments B C and A C.
03:08 We will make AC and BC purple and dashed segments.
03:14 With Move highlighted, drag B towards A on the parabola.
03:22 Observe lines g and h and the value of j (length of AC).
03:29 As j approaches 0, points B and A begin to overlap.
03:37 Lines g and h also begin to overlap.
03:42 Slope of line g is the ratio of length of BC to length of AC.
03:50 Derivative of the parabola is the slope of the tangent at each point on the curve.
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.
04:14 Differentiation: First Principles, the Algebra
04:18 Slope of line AB equals the ratio of the lengths of BC to AC.
04:25 Line AB becomes the tangent at point A as distance j between A and B approaches 0.
04:35 BC is the difference between y coordinates, f of x plus j and f of x, for A and B.
04:43 AC is the difference between the x-coordinates, x plus j and x.
04:50 Let us rewrite f of x plus j and f of x in terms of x squared minus x.
04:58 We will expand the terms in the numerator.
05:02 After expanding the terms in the numerator, we will cancel out similar terms with opposite signs.
05:10 We will pull out j from the numerator and cancel it.
05:15 Note that as j approaches 0, j can be ignored. So that 2x plus j minus 1 approaches 2x minus 1.
05:25 As we know, derivative of x squared minus x is 2x minus 1.
05:32 Let us look at derivative graphs for some functions.
05:37 Differentiation of a Polynomial Function
05:41 Consider g of x.
05:44 Derivative g prime x is the sum and difference of derivatives of the individual components.
05:53 g prime x' is calculated by applying these rules.
05:59 Let us differentiate g of x in GeoGebra.
06:04 Open a new GeoGebra window.
06:07 In the input bar, type the following line and press Enter.
06:13 As shown earlier in the series, zoom out to see function g properly.
06:24 Right-click in Graphics view and select xAxis is to yAxis option.

Select 1 is to 5.

06:35 I will zoom out again.
06:42 As shown earlier, draw point A on curve g and a tangent f at this point.
06:50 Under Angle, click on Slope and on tangent line f.
06:58 Slope of tangent line f appears as m value in Graphics view.
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
07:20 With Move tool highlighted, move point A on curve.
07:31 Observe the curve traced by point B.
07:35 Let us check whether we have the correct derivative graph.
07:39 In the input bar, type d e r i.

From the menu that appears, select Derivative Function option.

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

08:04 Compare the calculations in the previous slide with the equation of g prime x
08:11 Let us find the maxima and minima of the function g of x.
08:16 Derivative curve g prime x remains above the x axis (is positive) as long as g of x is increasing.
08:27 g prime x remains below the x axis is negative as long as g of x is decreasing.
08:37 2 and -2 are the values of x when g prime x equals 0.
08:45 Slope of the tangent at the corresponding point on g of x is 0.
08:52 Such points on g of x are maxima or minima.
08:58 Hence, for g of x, -2 comma -11 is the minimum and 2 comma 21 is the maximum.
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.
09:35 Press Tab to go to the next argument.
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
10:51 Let us draw the cardboard:
10:54 This is the volume function here.
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.
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.
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.
11:38 Observe the graph that is plotted for the volume function in Graphics view.
11:44 Drag the background to see the maximum.
11:48 Note that the maximum is on the top of this broad peak.
11:53 The length of the square side is plotted along the x axis.
11:58 Volume of the box is plotted along the y axis.
12:02 As before, let us find the maximum of this function.
12:13 This maps the maximum, point A, on the curve.
12:18 Its coordinates 3 comma 486 appear in Algebra view.
12:24 Thus, we have to cut out 3 inch squares from all corners.
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.
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13:44 This is Vidhya Iyer from IIT Bombay, signing off.

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