Module 8  Derivative of a Function 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 8.3: The Derivative as a Function 
In Lesson 8.1 and Lesson 8.2 of this module you investigated the derivative of a function at a single point, which was defined to be the slope of the line tangent at that point. In this lesson you will download and use a program named tanimate to visualize derivatives at several sequential points along a curve. This will help to illustrate the concept of the derivative of a function as a function. The TI89 program called tanimate displays an animated sequence of tangent lines for a given function. It also plots points representing the slopes of those tangent lines. John Hanna, who is a teacher at Teaneck High School, New Jersey, wrote this program for the TI89. Before you can use this program, you will need to complete the following steps to install the program on your TI89 calculator. Downloading the Program to Your Computer
Transferring the Program to the TI89 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 y1 in the Y= Editor and then display its graph.
You could find the derivative at many more points as well. These points could then be plotted as a scatter plot. However, this is tedious, so let's use "tanimate" to automate this process. Running tanimate
The function y = x^{2} will be graphed again and the program's Main Menu dialog box should appear.
The next dialog box 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 Menu dialog box.
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 xvalue of the last tangent line that 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 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. The screen shows the graph at the end of the program. Finding Derivatives at Several Points The tanimate program can display the value of the derivative (slope of the tangent line) at specific points.
The derivative of f(x) = x^{2} at x = 0.9 is 1.8, as shown by the coordinates at the bottom of the screen below. 8.3.2 Find the derivative when x = 1.2 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 tangent lines. These points represent a function whose yvalues give the derivatives of the function at each point. Notice that the points on the graph of the derivative of f(x) = x^{2} appear to lie on a line. The derivative of f(x) = x^{2} is the linear function represented by the line that connects these points. You can use the derivative key to find the symbolic representation for the derivative function. The syntax for finding the derivative function is similar to the one for finding the value of the derivative at a point, but the "with" command that establishes the variable's value is omitted.
The derivative of f(x) = x^{2} is the function f'(x) = 2x.
8.3.3 Illustrate the derivative of f(x) = 3x^{2} + 4x using the tanimate program and then use the 
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