Module 10  The Derivative as a Function 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 10.3: The Derivative as a Function 
In Lesson 10.1 and Lesson 10.2 you investigated the derivative of a function at a single point. The derivative was defined to be the slope of the tangent line at that point. In this lesson you will use the derivative feature in the Graph screen's CALCULATE menu to find the derivative at a point on a curve. You will then download and use a program named TANIMATE to illustrate derivatives at several points along a curve. This will help illustrate the concept of the derivative as a function. Several notations are used to denote the derivative of a function. One commonly used notation is dy/dx, which is derived from the definition of the derivative as the limit of the difference of the yvalues divided by the difference of the xvalues. The dy/dx Feature The TI83 has a feature that allows you to find the approximate value of the derivative of a function at a given point on the graph of the function. Find the derivative of y = x^{2} at x = 1 using the dy/dx feature by following the procedure below.
The approximate value of the derivative of y = x^{2} at x = 1 is 2, as shown at the bottom of the screen. 10.3.1 Compute the derivative at x = 2, 1, 0, 1, 2. Click here for the answers. Plotting the Derivative versus the xvalues You could find the derivative at many more points using the dy/dx (derivative) command and create a table of values of the derivative at given xvalues. The corresponding points could then be displayed as a scatter plot. This would be tedious, but the program tanimate can be used to automate this process. The tanimate Program The program tanimate displays an animated sequence of tangent lines for a given function. It also plots points representing the slopes of those tangent lines at selected xvalues. Because the slope of a tangent line is the derivative of the function at that point, the program is actually plotting the values of the derivatives that correspond to specific xvalues. Downloading the Program to Your Computer
Transferring the Program to the TI83 Click here to get information about how to obtain the needed cable and to review the procedure to transfer the program from your computer to your calculator.
Animating Tangent Lines The program tanimate requires that you store the desired function into Y_{1} in the Y= editor and then graph the function in a selected viewing window.
Running tanimate
The function y = x^{2} will be graphed again and the tanimate program's Main menu should appear.
The next menu is used to set the sampling rate, which determines the number of points that will be used to create and display the tangent lines.
Now you should see the DISPLAY OPTIONS menu.
Setting the Left Endpoint Now you should see the graph and a prompt for the left endpoint. Tangent lines will be drawn beginning with the xvalue that you set as the left endpoint, and this value must be within the current window's coordinates.
Setting the Right Endpoint Next you should see a prompt for the right endpoint. This value determines the xcoordinate where the last tangent line will be drawn. Again, this value must be within the current window's coordinates.
Viewing Animated Tangent Lines and Plotted Slopes You should see an animation of tangent lines. As each tangent line is graphed, a corresponding point that represents the derivative of the function at that point is plotted. The xcoordinate of the plotted point is equal to the xcoordinate of the point of tangency, and the ycoordinate of the plotted point is the slope of the corresponding tangent line, i.e., the derivative at that point. The screen above shows the graph at the end of the program.
Finding Derivatives at Several Points The tanimate program can now display the value of the derivative (the slope of the tangent line) at specific points. In general, these values will be approximate values.
For example, the derivative of f(x) = x^{2} at x = 0.94 is 1.88, as shown by the coordinates at the bottom of the screen below. 10.3.2 Find the derivative when x = 1.22 and interpret its value. Click here for the answer. Exiting the Program
The Derivative as a Function The set of points plotted by tanimate represents the slopes of the corresponding tangent lines. These points represent a function whose yvalues give the derivative of the original function at each point. The points on the graph of the derivative of f(x) = x^{2} appear to lie on a line. Instead of using difference quotients and/or limits to compute the derivative at various points, if the equation for that line is known, then that equation can be used to find the value of the derivative at any xvalue. The x and ycoordinates of the points on the graph of the derivative were stored in L1 and L2 by the program tanimate. Linear regression can be used to find the equation of the line that contains these points and that equation can then be used to find derivatives of the function at other xvalues.
The derivative of f(x) = x^{2} is the function y = 2x. An Alternate Notation for a Derivative Function The notation f'(x) is another notation that is used to represent the derivative of f(x). For example, we have just shown that if f(x) = x^{2}, then f'(x) = 2x. f'(x) is read "f prime of x." 10.3.3 Illustrate the derivative of f(x) = 3x^{2} + 4x using the tanimate program and then use regression to find the derivative function. When prompted, select the options given below.
Mode: 2:DYNAMIC You may exit the program after it displays the animated tangent lines by selecting 6:QUIT. The data points for the derivative values are stored in L_{1} and L_{2}. Click here for the answer. 
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