1. Rolle’s Theorem/Mean-Value Theorem
Objective: Use and interpret the Mean-Value Theorem

2. Rolle’s Theorem
Rolle’s Theorem is a special case of the Mean-Value Theorem.
Theorem (Rolle’s Theorem)
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = 0 and f(b) = 0, then there is at least one point c in the interval (a, b) such that f /(x) = 0

3. Example 1 Find the two intercepts of the function
and confirm that f /(c) = 0 at some point between those intercepts.

4. Example 1 Find the two intercepts of the function
and confirm that f /(c) = 0 at some point between those intercepts.
If we factor the expression, we get (x – 1)(x – 4), so the zeros are x = 1 and x = 4. To confirm the hypothesis of Rolle’s Theorem, f(1) = f(4) = 0. We know that there must a point between these two values where f /(c) = 0.

5. Example 1 Find the two intercepts of the function
and confirm that f /(c) = 0 at some point between those intercepts.
If we factor the expression, we get (x – 1)(x – 4), so the zeros are x = 1 and x = 4. To confirm the hypothesis of Rolle’s Theorem, f(1) = f(4) = 0. We know that there must a point between these two values where f /(c) = 0.

6. Example 2
The differentiability requirement in Rolle’s Theorem is critical. If f fails to be differentiable at even one place in the interval (a, b), then the conclusion of the theorem may not hold. For example, the function
graphed below has f(-1) = f(1) = 0, yet there is no horizontal tangent to the graph of f over the interal (-1, 1).

7. Hypothesis of Rolle’s Theorem
The homework will ask you first to confirm the hypothesis of Rolle’s Theorem. They want you to say that:
the function is continuous and differentiable over the interval (a, b)
f(a) = f(b) = 0.

8. Example3
If f satisfies the conditions of Rolle’s Theorem on [a, b], then the theorem guarantees the existence of at least one point c in (a, b) at which f /(c) = 0. There may, however, be more than one such c. For example, the function f(x) = sinx is continuous and differentiable everywhere, so the hypothesis of Rolle’s Theorem are satisfied on the interval [0, 2p] whose endpoints are roots of f [f(0) = f (2p) = 0]. We can see that there are two points that satisfy the theorem, x = p/2 and x = 3p/2.

9. Mean-Value Theorem
Rolle’s Theorem is a special case of a more general result called the Mean-Value Theorem. Geometrically, this theorem states that between any two points A(a, f(a)) and B(b, f(b)) on the graph of a differentiable function f, there is at least one place where the tangent line to the graph is parallel to the secant line joining A and B.

10. Mean-Value Theorem
Note that the slope of the secant line joining A and B is
The slope of the tangent line at c is
This leads us to The Mean-Value Theorem:
Theorem 5.7.2
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there is at least one point c in (a, b) such that

11. Example 4
Show that the function satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 2], and find all values of c in the interval (0, 2) at which the tangent line to the graph of f is parallel to the secant line joining the points (0, f(0)) and (2, f(2)).

12. Example 4
Show that the function satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 2], and find all values of c in the interval (0, 2) at which the tangent line to the graph of f is parallel to the secant line joining the points (0, f(0)) and (2, f(2)).
The function is continuous and differentiable everywhere since it is a polynomial, so the hypothesis is satisfied.

13. Example 4
Show that the function satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 2], and find all values of c in the interval (0, 2) at which the tangent line to the graph of f is parallel to the secant line joining the points (0, f(0)) and (2, f(2)).
f(0) = 1
f(2) = 3
The only answer in the interval is

14. Example 4
Show that the function satisfies the hypothesis of the Mean-Value Theorem over the interval [0, 2], and find all values of c in the interval (0, 2) at which the tangent line to the graph of f is parallel to the secant line joining the points (0, f(0)) and (2, f(2)).

15. Velocity
There is a nice interpretation of the Mean-Value Theorem in the situation where x = f(t) is the position vs. time curve for a car moving along a straight road. In this case, the right side of the equation is the average velocity of the car over the interval from a to b and the left side is the instantaneous velocity at time t = c. Thus, the Mean-Value Theorem implies that at least once during the time interval the instantaneous velocity must equal the average velocity.

16. Example 5
You are driving on a straight highway on which the speed limit is 55 mi/h. At 8:05 AM a police car clocks your velocity at 50 mi/h and at 8:10 AM a second police car posted 5 mi down the road clocks your velocity at 55 mi/h. Explain why the police have a right to charge you with a speeding violation.

17. Example 5
You are driving on a straight highway on which the speed limit is 55 mi/h. At 8:05 AM a police car clocks your velocity at 50 mi/h and at 8:10 AM a second police car posted 5 mi down the road clocks your velocity at 55 mi/h. Explain why the police have a right to charge you with a speeding violation.
You traveled 5 miles in 5 minutes, or a mile per minute, or 60 mi/h. The Mean-Value Theorem guarantees the police that your instantaneous velocity was 60 mi/h at some point.

18. Homework
Section 4.8
1-7 odd
11, 12, 19