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AP Calculus AB Chapter 3, Section 2

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

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AP Calculus AB Chapter 3, Section 2

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

  2. 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.

  3. Rolle’s Theorem • Let 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 .

  4. Determine whether Rolle’s Theorem can be applied… • [0, 5] • [-1, 1] • [-1, 1] • [2, 3]

  5. Illustrating Rolle’s Theorem • Find the two x-intercepts of and show that at some point between the two x-intercepts.

  6. Illustrating Rolle’s Theorem • Let . Find all values of c in the interval (-2, 2) such that .

  7. Rolle’s Theorem • Determine 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.

  8. The Mean Value Theorem • 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) that

  9. 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.

  10. Finding a tangent line • Given , find all values of c in the open interval (1, 4) such that

  11. Mean Value Theorem • Determine 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.

  12. Mean Value Theorem • Explain why the Mean Value Theorem does not apply to the function on the interval [0, 6].

  13. 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.

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

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