1. Rolle’s Theorem and the Mean Value Theorem3.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

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

3. 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. Rolle’s Theorem for Derivatives Conditions: 1.f is continuous on [a,b] 2.f is differentiable on (a,b) 3.f(a) = f(b)

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

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

6. 1. Continuous? 2. Differentiable? 3. f(-2)=f(2) Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0. 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. Slope of chord: Slope of tangent: Tangent parallel to chord.

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

11. 5 miles 1/15 hour 0 miles 0 minutes 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 1 st patrol car. At time 4 minutes (1/15 hr), the truck passes the 2 nd patrol car. So, the truck must have been traveling at a rate of 75 mph sometime during the four minutes!

12. Homework 3.2 (page 176) #1-5 odd #9, 11, 15, 19 #23, 25, 29, 33, 35 #39-47 odd, #53, 59