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TANIMATE

Visual Representation of Rates of Change

Dirk A. DeLo and Gregory R. Somers

Abstract

The greatest potential for the TI-82 program TANIMATE, which is the focus of this module, may lie in its ability to help students visualize patterns in the rate of change of a function before they are formally introduced to the derivative of that function. We hope this tool will deepen students' understanding of the concept of a derivative and help them make some connections between the skills needed to find the derivative and what the derivative actually represents. We have included some suggestions for activities which use TANIMATE. We also encourage teachers to create new ways of using this tool to fit their needs and the needs of their students.

Directions for Using TANIMATE

TANIMATE (see attached program listing) is a collection of tools for the exploration and analysis of rates of change of functions. The program operates on a function which has been entered into Y1 and graphed on the function window. With this program, the user can display a single tangent line at a point on the curve, choose many points along the curve and display a tangent at each point, or create an animation sequence which will dynamically display the tangent lines to the curve along a predetermined interval. Once this has been done, TANIMATE stores the x-values that were sampled and the slope of the tangent lines at these points and will display the information graphically; which in effect gives a visual approximation of the derivative of a function. This "derivative" can be viewed by itself, or along with the original function.

In order to start the program, call the program after the function has been graphed in an appropriate window by accessing the program [PRGM] menu and executing [EXEC] the program TANIMATE. The first menu the user encounters is the MODE? menu:

MODE?

1:STATIC

2:DYNAMIC

3:PLOT CHANGE

4:PLOT CHNG W/Y1

5:GRAPH OPTIONS

6:QUIT

1:STATIC

This option will enable the user to draw tangent lines to the curve at selected x-values. The user will be prompted to enter the number of x-values desired. Valid x-values include the interval from 1 to 99. Once this number has been entered, the user will encounter the DISPLAY OPTIONS? menu:

DISPLAY OPTIONS?

1:TANGENT ONLY

2:TANGENT/POINTS

3:POINTS ONLY

1:TANGENT ONLY

This option draws tangent lines at the selected x-values.

2:TANGENT/POINTS

This option draws the tangent line and plots the coordinate (x-value, slope at x-value) in the same graphing window.

3:POINTS ONLY

This option displays the coordinate (x-value, slope at x-value) in the same graphing window.

After selecting one of these options, the function will appear in the graphing window with the flashing cursor on the curve. Using the right or left arrow keys, the user selects a point at which to draw the tangent and presses [ENTER]. The slope will appear in the upper left hand corner. Repeat this process until all points and/or tangents have been plotted. Pressing [ENTER] at the conclusion will return to the MODE? menu.

2:DYNAMIC

This option will enable the user to animate tangent lines to the curve along a predetermined interval of x-values. The user will encounter the SAMPLING RATE? menu:

SAMPLING RATE?

1:LOW (plot tangent every 5 pixels)

2:MEDIUM (plot tangent every 2 pixels)

3:HIGH (plot tangent every pixel)

The lower the SAMPLING RATE?, the faster the animation of the tangent lines will execute. Once this number has been entered, the user will encounter the DISPLAY OPTIONS? menu:

DISPLAY OPTIONS?

1:TANGENT ONLY

2:TANGENT/POINTS

3:POINTS ONLY

1:TANGENT ONLY

This option animates tangent lines at the selected interval of x-values.

2:TANGENT/POINTS

This option animates the tangent line and plots the coordinate (x-value, slope at x-value) in the same graphing window for the interval of x-values.

Note: The points plotted will provide a visual representation of the derivative over the selected interval if the sampling rate is high enough.

3:POINTS ONLY

This option displays the coordinate (x-value, slope at x-value) in the same graphing window for the interval of x-values.

Note: The points plotted will provide a visual representation of the derivative over the selected interval if the sampling rate is high enough.

After selecting one of these options, the function will appear in the graphing window with the flashing cursor on the curve. Using the right or left arrow keys, the user will be prompted to select the left endpoint for the x-values and press [ENTER]. Then the user will be prompted to select the right endpoint and press [ENTER]. The program will animate tangent lines along the curve between the end values at the interval specified in the SAMPLING RATE? menu. When the animation is complete, pressing [ENTER] at this point will return to the MODE? menu.

3:PLOT CHANGE

This option displays the coordinates (x-value, slope at x-value). However, this differs from the previous displays because it uses values which have been stored in the calculator as lists. This allows the user to view only the rate of change data without viewing the original function, and it displays the coordinates in a scatter plot using boxes instead of dots. The left and right arrow keys enable the user to identify coordinates of the rate of change points. Since the rate of change data is now in list form (L1 and L2,) it is possible to fit a curve to this data using the built in statistical data analysis of the TI-82. When PLOT CHANGE has been selected, the user will encounter the WINDOW? menu:

WINDOW?

1:SAME AS Y1

2:RESCALE

1:SAME AS Y1

This option preserves the windows setting used for the original function Y1 and plots the rate of change data as a scatter plot. Pressing [ENTER] returns the user to the MODE? menu.

2:RESCALE

This option sets new windows setting for optimum visibility of the rate of change data and plots these data points as a scatter plot. These windows settings are temporary and will be restored to the previous settings after you leave this option. Pressing [ENTER] returns the user to the MODE? menu.

4:PLOT CHNG W/Y1

This option displays the coordinates (x-value, slope at x-value) in addition to the original function. This option also differs from the previous displays because it uses values which have been stored in the calculator as lists. This allows the user to view only the rate of change data while viewing the original function. The arrow keys enable the user to identify coordinates of the rate of change points and points on the original curve. Pressing [ENTER] returns the user to the MODE? menu.

5:GRAPH OPTIONS

This option allows the user to change Y1, window settings, and access zooming options. The menu will display the current graph in its window settings and then present the user with the following GRAPH OPTIONS? menu:

GRAPH OPTIONS?

1:NEW Y1

2:NEW WINDOW

3:ZBox

4:Zoom In

5:Zoom Out

6:MODE MENU

1:NEW Y1

This option will prompt the user for a new function, Y1. The function must be entered between quotations, for example, "sin 3x" would be a valid entry. If the quotations are missing, the program will break and needs to be started again.

2:NEW WINDOW

This option allows the user to set the window settings manually. The user will be prompted to specify Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl. The function will then be displayed with these new window settings. Pressing [ENTER] will return the user to the GRAPH OPTIONS? menu.

3:ZBox

This option allows the user to zoom to a specific region using the arrow keys. When the user has reached the upper left corner of the box, he/she should press [ENTER] and continue to outline to the lower right hand corner of the zoom box and press [ENTER]. The function will be displayed with its new window settings. Pressing [ENTER] will return the user to the GRAPH OPTIONS? menu.

4:Zoom In

This option zooms in on the currently displayed graph using the TI-82 default zoom settings. The function will be displayed with its new window settings. Pressing [ENTER] will return the user to the GRAPH OPTIONS? menu.

5:Zoom Out

This option zooms out on the currently displayed graph using the TI-82 default zoom settings. The function will be displayed with its new windows settings. Pressing [ENTER] will return the user to the GRAPH OPTIONS? menu.

6:MODE MENU

This option returns the user to the MODE? menu and exits the GRAPH OPTIONS menu.

6:QUIT

Quits the program. We highly recommend using this option to quit the program. Exiting the program by other means may cause complications the next time you use the program.

Sample Activities for TANIMATE

The following are examples of activities which would incorporate TANIMATE into courses ranging from Algebra II to calculus. They introduce some of the concepts of calculus before students encounter then in a more formal sense. These are not meant to be student-ready worksheets (with the exception of the exploration summary sheet), but rather a discussion of possible ways to introduce a TANIMATE exploration or to prepare student explorations and teacher demonstrations.

When and why should students use TANIMATE?

As soon as students have been introduced to the concept of drawing a tangent line to a curve (representing the instantaneous rate of change) they are ready to use the TANIMATE program for the TI-82. The calculator has a built-in function which allows the user to draw a tangent line to a curve. TANIMATE, however, enables the student to draw many tangents without having to repeatedly access this function and it also computes the slopes of these tangent lines and stores those values for later use.

Sample activity:

Rates of Change of Polynomial Functions

Student instructions could take the following form:

1. Enter a quadratic function into Y1 and graph it.

2. Run TANIMATE several times in STATIC mode, drawing tangent lines at various x-values.

3. Based on the slope of the tangent line at each x-value, try to envision what the rate of change plot will look like.

4. Proceed to the DYNAMIC mode and look at the rate of change plot which has been generated in this mode. Does it look like you thought it would?

5. View the rate of change plot together with the original function. Sketch this picture on the exploration summary sheet. Try to understand the connection between them.

6. Repeat this process using several other quadratic functions.

7. Look for patterns and make conjectures.

8. Now proceed to cubics, quartics, etc. Continue to look for patterns and make conjectures about the observations.

Here are examples of quadratics that the students might explore:

f(x) = x2 f(x) = 3x2 f(x) = x2 + x - 4

f(x) = x2 + 4 f(x) = .5x2 f(x) = x2 - 3x + 2

f(x) = x2 - 3 f(x) = -2x2 f(x) = 2x2 + 6x - 7

f(x) = -x2 f(x) = -.125x2 f(x) = .5x2 - 2x

f(x) = -x2 + 2 f(x) = -3.5x2 f(x) = .5x2 - 2x + 3

Student conjectures from this activity might take the following form:

• The rate of change of a quadratic function is linear.

• The rate of change of a cubic function is a quadratic.

• Etc. ...

It is now possible to use the definition of the derivative to justify the results that the students have already discovered. By presenting the definition of the derivative in this way, students may connect it to the visual representation which they have already encountered.

Note: When TANIMATE computes the slopes of the tangent lines it stores the x-values in list 1 (L1) and the corresponding slopes in list 2 (L2). This enables the user to fit a function to this data after exiting the program. If students use the data analysis capabilities of the TI-82 to fit functions to the rate of change data, more specific conjectures may be obtained.

Similar types of explorations can be used when introducing the derivatives of other types of functions. For example:

Before introducing the derivative of other functions, students could explore groups of functions like these using TANIMATE:

• 1/x, 3/x,-5/x...10/(x+1), -25/(x-4), 4/x2, -8/x3, -8/(x-2)3

• 2x/(x-3), (x-3)/(x-5), (x-4)/(x-8)2,...

• sin(x), sin(2x), sin(5x), ..., 3sin(x), 4sin(2x),...,-sin(x), sin2(x).

• cos(x)+x, cos(x) + x2, cos(2x) + 5x,...

• ln(x), ln(x-4), ln(2x), 5ln(x), ln(x2),...

• 2x, .94x, 2000(1.06)x, 10000(.85)x, ex, ex+2, e2x, ...

TANIMATE:

Exploration Summary Sheet

Sketch of function and rate of change plot

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Observations/Conjectures:

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