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

2. Objectives Understand and use Rolle's Theorem
Understand and use the Mean Value Theorem

3. Rolle’s Theorem for Derivatives
Let f (x) be continuous on [a,b] and differentiable on (a,b). If f (a)= f (b) then there is at least one number c in (a,b) such that f '(c)=0.
Conditions:
f is continuous on [a,b]
f is differentiable on (a,b)
f(a) = f(b)

4. Does Rolle’s Theorem Apply?b
Continuous
Differentiable
f(a)=f(b)
Continuous
Not differentiable
The slope has to be zero somewhere between a and b.

5. Determine whether Rolle’s Theorem can be applied
Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.
1. Continuous?
2. Differentiable?
3. f(1)=f(2)
yes
yes
yes (=0)

6. Determine whether Rolle’s Theorem can be applied
Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.
1. Continuous?
2. Differentiable?
3. f(-2)=f(2)
yes
yes
yes

7. If f (x) is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that:
Mean Value Theorem

8. If f (x) is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that:
Mean Value Theorem
The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.
Or the instantaneous rate of change equals the average rate of change.

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

10. Example:
Find all the values of c in (1,4) such
that
Instantaneous rate of change
Average rate of change
So, c=2
(-2 is not in (1,4))

11. At time 0, the truck passes the 1st patrol car.
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 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit of 55 mph at some time during the four minutes.
At time 0, the truck passes the 1st patrol car.
At time 4 minutes (1/15 hr), the truck passes the 2nd patrol car.
5 miles
1/15 hour
0 miles
0 minutes
So, the truck must have been traveling at a rate of 75 mph sometime during the four minutes!

12. Homework
3.2 (page 172)
#1-15 odd
#19, 21, 25
#29-37 odd,
#43, 47