4.2 Rolle's Theorem and Mean Value 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) = f (b), then there is at least one c in (a,b) such that f '(c) = 0.

Example 1: Find the two x-intercepts of and show that f '(x)= 0 at some point between the 2 intercepts.

1 Mean Value Theorem Exploration

1. For , there is a value x = c between 3 and 7 at which the tangent to the graph is parallel to the secant through (3, f (3)) and (7, f (7)).

The secant line is drawn in ______

The tangent line is drawn in ______

For the graph, c = ____

Is f differentiable on (3,7)?

Is f continuous on [3,7]?

2 2. The function f from problem 1 has two values of x = c between x = 1 and x = 7 at which f '(c) equals the slope of the corresponding secant line. In other words,

The tangent line is PARALLEL to the secant line. c = ____ and c = ____

Is f differentiable on (1,7)? ______

Is f continuous on [1,7]? ______

3 3. For ,

the secant line through (1, g(1)) and (5, g(5)) is drawn in green.

Is g differentiable on (1,5)? ______

Is g continuous on [1,5]? ______

Why is there no value of x = c between x = 1 and x = 5 at which g'(c) equals the slope of the secant line?

4 4. Function g from problem 3 does have a value x = c in (1, 4) for which g'(c) equals the slope of the secant line through (1, g(1)) and (4, g(4)).

From the graph, c = ____

Is g differentiable on (1,4)? ______

Is g continuous on [1,4]? ______

5 5. Piecewise function h is defined by

The secant line through (5, h(5)) and (7, h(7)) is drawn in green.

Is h differentiable on (5,7)? ______

Is h continuous on [5,7]? ______

Why is there no value x = c in (5,7) for which h'(c) equals the slope of the secant line?

6 6. The graph to the right is the function h from problem 5.

The secant line through (5, h(5)) and (9, h(9)) is drawn in green.

Is h differentiable on (5,9)? ______

Is h continuous on [5,9]? ______

There is a point x = c in (5,9) where h'(c) equals the slope of the secant line. The tangent line is drawn in red.

Estimate the value of c. ______

7 7. The graph to the right is the function h from problem 5.

The secant line through (1, h(1)) and (5, h(5)) is drawn in green.

Is there a value x = c in (1,5) such that h'(c) equals the slope of the secant line? ______

Is h differentiable on (1,5)? ______

Explain why h is continuous on [1,5] even though there is a step discontinuity at x = 5.

8 The Mean Value Theorem

If f (x) is continuous at every point on the closed interval [a,b] and differentiable at every point on the open interval (a,b), then there is at least one point c in (a,b) at which

The number x = c in the seven problems discussed is the "mean" value referred to in the name "mean value theorem". Explain why the hypotheses are sufficient conditions for the conclusion, but not necessary conditions.

Sufficient means if you meet the condition of the hypothesis, you are guaranteed the conclusion.

Not necessary means that if you do not meet the conditions of the hypothesis, you may still reach the conclusion, but you are not guaranteed to reach the conclusion.

9 The Mean Value Theorem can be explained in different ways.

*There is at least one tangent line in the interval that is parallel to the secant line that goes through the endpoints of the interval.

*The instantaneous rate of change is equal to the average rate of change at some time in the interval.

10 Example 2: Find c exactly for problem #1.

Steps:

1. State if f is continuous on [a,b] and differentiable on (a,b). If both of these are true, then continue with steps 2 through 4. If not, then the theorem does not apply and you are done with the problem!

2. Find on the interval (3, 7).

3. Find f '(x).

4. Solve for x in the equation *use your calculator to find the intersection of and .

11 Example 3: If Mean Value Theorem applies, find the values of c in the given interval.

a. ,

b. ,

12 Pg. 216 #17 Determine whether Rolle's Thm can be applied to f on the closed interval [a,b]. If Rolle's Thm can be applied, find all values of c in the open interval (a,b) such that f '(c) = 0. If Rolle's Thm cannot be applied, explain why not.

17. ,

13 Pg. 217 #45

Determine whether the Mean Value Thm can be applied to f on the closed interval [a,b]. If the Mean Value Thm can be applied, find all values of c in the open interval (a,b) such that . If the Mean Value Thm cannot be applied, explain why not.

45. ,

14 Pg. 217 #51

51. ,