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AP Calculus Section 3.2 Rolle’s Theorem and Mean Value Theorem

AP Calculus Section 3.2 Rolle’s Theorem and Mean Value Theorem. Theorems and Examples. Recall:. Theorem 3.1 The Extreme Value Theorem If f is continuous on the closed interval [a, b], then f has both a minimum and a maximum on the interval. Theorem 3.3 Rolle’s Theorem.

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AP Calculus Section 3.2 Rolle’s Theorem and Mean Value Theorem

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  1. AP Calculus Section 3.2Rolle’s Theorem and Mean Value Theorem Theorems and Examples

  2. Recall: Theorem 3.1 The Extreme Value Theorem If f is continuous on the closed interval [a, b], then f has both a minimum and a maximum on the interval.

  3. Theorem 3.3 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 exists at least one number c in (a, b) such that f’(c) = 0.

  4. Example 1 Find the two x-intercepts of and show that f’(x) = 0 at some point between the x-intercepts.

  5. Example 2 Let . Find allvalues of c in the interval (-2, 2) such that g’(c)=0.

  6. We need Rolle’s Theorem to prove 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 c in (a, b) such that

  7. Does look familiar?Where have you seen that before?

  8. The Mean Value Theorem is most often used to prove other theorems, including THE FUNDAMENTAL THEOREM OF CALCULUS. For this reason, some people consider the Mean Value Theorem to be the most important theorem in calculus.

  9. Example 3 #40 (Geometric Interpretation of the Mean Value Theorem) Given , find all values of c in the open interval (-1, 1) such that

  10. Read p. 175 Example 4

  11. Example 4 #60 (Instantaneous Rate of Change = Average Rate of Change) At 9:13 AM, a sports car is traveling 35 miles per hour. Two minutes later, the car is traveling 85 miles per hour. Prove that at some time during this two-minute interval, the car’s acceleration is exactly 1500 miles per hour squared.

  12. Alternative Form of the Mean Value Theorem If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that

  13. Remember:Polynomial functions, rational functions, and trigonometric functions are differentiable at all points in their domains.

  14. Assignment - Section 3.2:p.176-177 # 2, 18, 22, 26, 29, 33, 34, 35, 37, 42, 51, 58

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