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

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.

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.

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

HOMEWORK  Pg 172 #1-20 odds, 26

Ex 3: AP Practice

Review  Describe the Extreme Value Theorem.

 Describe Rolle’s Theorem.

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.

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

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 2 nd patrol car at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.

HOMEWORK  Pg 172 #27 – 38 odds, 53 - 56