AP Calculus AB Chapter 3, Section 2 Rolle’s Theorem and the Mean Value Theorem 2013 - 2014.

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Presentation transcript:

AP Calculus AB Chapter 3, Section 2 Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem Named after the French mathematician, Michel Rolle (1652 – 1719) Gives conditions that guarantee the existence of an extreme value in the interior of the closed interval The Extreme Value Theorem (as in section 1) stated the extrema could be inside the interval or include the endpoints.

Rolle’s Theorem

Determine whether Rolle’s Theorem can be applied…

Illustrating Rolle’s Theorem

Rolle’s Theorem

The Mean Value Theorem

In Plain English If ◦ a function is continuous for all x-values in the interval [a, b], and differentiable for all x- values between a and b, Then ◦ There is at least one point that exists between a and b where the instantaneous rate of change is equal to the average rate of change.

Finding a tangent line

Mean Value Theorem

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 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 4 minutes.

Ch 3.2 Homework Pg. 176 – 178: 7, 13, 19, 29, 37, 41, 25, 57, 61, 69 Total problems: 10