Section 2: Rolle’s Theorem & The Mean Value Theorem

1. Section 2: Rolle’s Theorem & The Mean Value Theorem

2. I. 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. • Rolle’s Theorem guarantees an _________ _________ inside of the interval where the Extreme Value Theorem can have them on the endpoints.

3. Ex. 1 Illustrating Rolle’s Theorem • Find the two x-intercepts of f(x) = x² - 3x + 2 and show that f’(x) = 0 at some point between the intercepts.

4. Ex 2 • Let f(x) = . Find all values of c on the interval (-2, 2) such that f’(c) = 0.

5. HOMEWORK • Pg 172 #1-20 odds, 26

6. Ex 3: AP Practice

7. Review • Describe the Extreme Value Theorem. • Describe Rolle’s Theorem.

8. II. Mean Value Theorem (MVT) • If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number, c, in (a, b) such that • The MVT says that the slope of a tangent line on a curve is equal to the slope of the secant line on the same curve at a particular point. 

9. Ex 1: Slope of the Tangent Line • What value of c in the open interval (0, 4) satisfies the MVT for ? • Given , find all values of c in the open interval (1,4) such that

10. Ex 2: Finding an Instantaneous Rate of Change • Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first car, its speed is clocked at 55 mph. Four minutes later, the truck passes the 2nd patrol car at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.

11. HOMEWORK • Pg 172 #27 – 38 odds, 53 - 56