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Rolle’s Theorem & the Mean Value Theorem (3.2)

Learn about Rolle's Theorem and the Mean Value Theorem, including their applications and how to find the necessary values for each theorem. Also, explore an example involving car acceleration.

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Rolle’s Theorem & the Mean Value Theorem (3.2)

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  1. November 16th, 2017 Rolle’s Theorem & the Mean Value Theorem(3.2)

  2. I. Rolle’s Theorem Thm. 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 is at least one number c in (a, b) such that f’(c)=0.

  3. Ex. 1: Determine whether Rolle’s Theorem can be applied to on the closed interval [1, 4]. If Rolle’s Theorem can be applied, find all values c in the open interval (1, 4) such that f’(c)=0.

  4. II. The Mean Value Theorem Thm. 3.4: 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) such that .

  5. Ex. 2: Determine whether the Mean Value Theorem can be applied to on the closed interval [-1, 1]. If the Mean Value Theorem can be applied, find all the values of c in the open interval (-1, 1) such that .

  6. You Try: Determine whether the Mean Value Theorem can be applied to on the closed interval [0, 1]. If the Mean Value Theorem can be applied, find all values of c in the open interval (0, 1) such that .

  7. Ex. 3: At 9:13am, 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.

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