|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.
|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.
|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.
|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.
|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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