1. 4.2 - The Mean Value Theorem

2. Theorems
If the conditions (hypotheses) of a theorem are satisfied, the conclusion is known to be true.

3. Rolle’s Theorem
Let f be a function that satisfies the following three hypotheses:
f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).
f (a) = f (b)
Then there is a number c in (a, b) such that
f ′(c) = 0.

4. Rolle’s Theorem

5. Example: Rolle’s Theorem
Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.
f(x) = (x - 3)(x + 1)2 on [-1,3]

6. Example: Rolle’s Theorem
f(x) = (x - 3)(x + 1)2 on [-1,3]
Check to see if Rolle’s Theorem applies:
f(x) is continuous on [-1,3]
f(x) is differentiable on (-1,3)
f(-1)=0 AND f(3)=0

7. Check to see if Rolle’s Theorem applies:
f(x) is continuous on [0,3]
f(x) is differentiable on (0,3)
f(0)=2 AND f(3)=2

8. The Mean Value Theorem
Let f be a function that satisfies the following two hypotheses:
f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).
Then there is a number c in (a, b) such that

9. Tangent parallel to chord.
Slope of tangent:
Slope of chord:

10. Apply the MVT to on [-1,4]. Here’s the idea behind the MVT:
There must be some value of c in the interval [-1,4] where the slope of the tangent line at c is the same as the slope of the line connecting the endpoints (i.e. the slope of the secant line… or the AVERAGE rate of change).

11. MVT applies! 1. Apply the MVT to on [-1,4].
f(x) is continuous on [-1,4].
MVT applies!
f(x) is differentiable on [-1,4].

12. 2. Apply the MVT to on [-1,2].

13. MVT does not apply! 2. Apply the MVT to on [-1,2].
f(x) is continuous on [-1,2].
f(x) is not differentiable at x = 0.
MVT does not apply!

14. Example: Mean Value Theorem
Verify that the function satisfies the two hypotheses of Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Mean Value Theorem.

17. WHY ? Application: Mean Value Theorem
You are driving on I-95 at 55 mph when you pass a police car with radar. Five minutes later, 6 miles down the road, you pass another police car with radar and you are still going 55 mph. She pulls you over and gives you a ticket for speeding citing the mean value theorem as proof.
WHY ?

19. Application: Mean Value Theorem
You are driving on I-95 at 55 mph when you pass a police car with radar. Five minutes later, 6 miles down the road you pass another police car with radar and you are still going 55mph. He pulls you over and gives you a ticket for speeding citing the mean value theorem as proof.
Let t = 0 be the time you pass PC1. Let s = distance traveled. Five minutes later is 5/60 hour = 1/12 hr. and 6 mi later, you pass PC2. There is some point in time c where your average velocity is defined by
72 mph

21. AP QUESTION