1. Mean Value & Rolle’s Theorems
3.2
Mean Value & Rolle’s Theorems
Teddy Roosevelt National Park, North Dakota
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2002

2. Let f be continuous on the closed interval [a, b] and differentiable
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 number c in (a, b) such that f ’(c)=0
So as long as the endpoints are equal, this guarantees an
extreme value on the interior of the interval.

3. Ex. 1 Find the two x-intercepts of the function and show that
f’(x) = 0 at some point between the two intercepts.
Note that f(x) is differentiable and continuous over all real numbers.
So since f(x) is continuous over [1, 2] and diff. over (1, 2)
we can apply Rolle’s Thm. There must be a c in the interval
such that f’(c)=0.
How can we find c?

4. Ex. 2 Find all values of c in the interval (-2, 2) such that f’(c) = 0
First- Are the conditions for Rolle’s Thm. satisfied on the interval?
Yes- continuous on closed interval
differentiable on open interval
Are the endpoint values equal?
Yes: f(-2) = f(2) = 8

5. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives

6. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives
Differentiable implies that the function is also continuous.

7. If f (x) is continuous over [a,b], and differentiable over (a, b) then at some point between a and b:
Mean Value Theorem for Derivatives
The Mean Value Theorem only applies over a closed interval.

8. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives
The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

9. Illustration of Mean Value Theorem
Tangent parallel to chord.
Slope of tangent:
Slope of chord:

10. find all values of c in the open interval (1, 4) such that
Ex. 3 Given the function
find all values of c in the open interval (1, 4) such that
Remember, this is the slope of the secant line through the points
(1, f(1)) and (4, f(4)):
Note that f satisfies the condition of MVT: it’s continuous on [1, 4]
and differentiable on (1, 4).
So there is at least one point in the interval where f’(c) = 1.

11. Solve the equation: So
since that is in the interval.

12. Ex. 5 Two stationary patrol cars equipped with radar are 5 miles
apart on a highway. As a truck passes the first patrol car, its speed
is clocked at 55 miles per hour. Four minutes later, when the truck
passes the second patrol car, its speed is clocked at 50 miles per
hour. Prove that the truck must have exceeded the speed limit
(of 55 miles per hour) at some time during the four minutes.
Let t = 0 be the time (in hours) when the truck passes the
first patrol car.
When (in hours) does the truck pass the second patrol car?
Find the average velocity for the truck:
By the MVT, at some time on (0, 1/15) hr. the truck hit
75 mph, and was therefore speeding.