Problem:
Illustrate the Mean Value Theorem for the function

f(x) = x3 - 7 x2 + 6 x - 5

for x in the interval [-2, 7].

Visualization:
Using the TI-85 graphing calculator: [Press here to see animation again!]

1. If you are not familiar with the graphing of functions on the TI-85, then first read the Initial Setup page from Little's Basic Guide to the TI-85.
2. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
3. If necessary, keep pressing F4 key until only y1= appears on the screen.
4. At the prompt y1=, type in

x^3 - 7x^2 + 6x - 5

and press the ENTER key.

5. The range settings for the graph above is  [-2, 7] ×  b>[- 70, 50]: 6. Press the F5 key to choose GRAPH. 7. Press the EXIT key twice.
8. We now determine f(7) and f(-2). Type in

7 STO x : y1

followed by the ENTER key. Type in

-2 STO x : y1

followed by the ENTER key.

9. We see that f(7) = 37 and f(-2) = -53: 10. We now calculate the slope of the line passing through the points (-2, -53) and (7, 37). Type in (37 + 53) / (7 + 2) followed by the ENTER key. The slope is 10.
11. We now solve the equation f'(x) = 10, the solution of which is guaranteed by the Mean Value Theorem.
12. Press 2nd POLY; at the prompt order=, type in 2 followed by the ENTER key.
13. Enter the coefficients:
a2 = 3
a1 = -14
a0 = -4
14. Press the F5 key to choose SOLVE from the menu. The roots are: Note that both of the roots are in the interval [-2, 7].

15. We will now plot the graph of the straight line through the points (-2, -53) and (7, 37). Press GRAPH key followed by F1 key to choose y(x)=.
16. Press the ENTER key and, at the prompt y2=, type in 10 (x + 2) - 53 followed by the ENTER key.
17. Press the EXIT key and the F5 key to graph the function and the straight line: 18. We now evaluate f(-.27...) and f(4.936...). Press the EXIT key and type in

-.27 STO x : y1

and press the ENTER key. Type in

4.936 STO x : y1

and press the ENTER key. 19. We will now draw the tangent lines at the points (-.27..., -7.14...) and (4.936..., -25.67...). Press GRAPH key followed by F1 key to choose y(x)=.
20. Press the ENTER key twice and, at the prompt y3=, type in 10 (x + .27) - 7.15 followed by the ENTER key.
21. At the prompt y4=, type in 10 (x - 4.94) - 25.67 followed by the ENTER key, the EXIT key and the F5 key to get all of the graphs which are shown in the animation above.