Mean Value Theorem

1. Mean Value Theorem

2. Do Now • When you’re finished, enter your answer in Socrative – Rm: # 487130

3. What gives? • After I started driving this morning, I passed a speed trap going 65 mph (the legal speed limit). • A half hour later, I passed another speed trap 40 miles down the same highway, still going 65 mph (the legal speed limit). • The cop pulls me over and issues me a ticket for speeding. What gives?

4. Mean Value Theorem • If a function f(x) is continuous on [a , b] and differentiable on (a , b), then, for some c on [a , b]:

5. Looking at the MVT graphically:

6. Why not the “Nice” Value Theorem?

7. A typical problem: • Find a number “c” on the interval [-1 , 3] that satisfies the Mean Value Theorem for the function

8. A typical problem:

9. A typical problem:

10. A typical problem:

11. Why doesn’t this work? • Can you find a value “c” on the interval [1 , 4] that satisfies the Mean Value Theorem for the function ?

12. Why doesn’t this work?

13. Why doesn’t this work?

14. Rolle’s Theorem • If f(x) is continuous on [a , b] and differentiable on (a , b) and f(a) = f(b), then for some c on [a , b], f’(c) = 0.

15. Rolle’s Theorem

16. Three Theorems: • The Intermediate Value Theorem (IVT) • The Extreme Value Theorem (EVT) • The Mean Value Theorem (MVT)

17. Practice AP Problems: Multiple Choice • Practice with the following 3 AP multiple choice problems

18. Answers: • 2003 #80 – B • 1998 #4 – B • 1998 #91 - E