Illustrate the Mean Value Theorem for the function

for **x** in the interval **[-2, 7]**.

Using the TI-85 graphing calculator:

[Press here to see animation again!]

- 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.
- Press the GRAPH key and then pick
**y(x)=**by pressing the F1 key. - If necessary, keep pressing F4 key until only
**y1=**appears on the screen. - At the prompt
**y1=**, type in**x^3 - 7x^2 + 6x - 5**and press the ENTER key.

- The range settings for the graph above is
**[-2, 7]**× b>[- 70, 50]: - Press the F5 key to choose
**GRAPH**. - Press the EXIT key twice.
- 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.

- We see that
**f(7) = 37**and**f(-2) = -53**: - 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**. - We now solve the equation
**f'(x) = 10**, the solution of which is guaranteed by the Mean Value Theorem. - Press 2nd POLY; at the prompt
**order=**, type in**2**followed by the ENTER key. - Enter the coefficients:
- a2 = 3

a1 = -14

a0 = -4 - 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]**. - 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)=**. - Press the ENTER key and, at the prompt
**y2=**, type in**10 (x + 2) - 53**followed by the ENTER key. - Press the EXIT key and the F5 key to graph the function and the straight line:
- 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.

- 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)=**. - Press the ENTER key twice and, at the prompt
**y3=**, type in**10 (x + .27) - 7.15**followed by the ENTER key. - 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.