AP Calculus ABChapter 3, Section 2Rolle’s Theorem and the Mean Value Theorem2013 - 2014

Rolle’s TheoremNamed 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 TheoremLet f be continuous on the closed

interval [a, b] and differentiable on the open interval (a, b). If then there is at least one number c in (a, b) such that .

Determine whether Rolle’s Theorem can be applied… [0, 5]

[-1, 1]

[-1, 1]

[2, 3]

Illustrating Rolle’s TheoremFind the two x-intercepts of

and show that at some point between the two x-intercepts.

Illustrating Rolle’s TheoremLet . Find all values of c in the

interval (-2, 2) such that .

Rolle’s TheoremDetermine whether Rolle’s Theorem

can be applied to f on the closed interval [a, b]. If so, find all values of c in the open interval.

The Mean Value TheoremIf 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) that

In Plain EnglishIf

◦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 lineGiven , find all values of c in the

open interval (1, 4) such that

Mean Value TheoremDetermine whether the Mean Value Theorem

can be applied to f on the closed interval. If so, find all the values of c in the open interval.

Mean Value TheoremExplain why the Mean Value Theorem

does not apply to the function on the interval [0, 6].

Finding an Instantaneous Rate of ChangeTwo 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 HomeworkPg. 176 – 178: 7, 13, 19, 29, 37,

41, 25, 57, 61, 69

Total problems: 10