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Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem. Guaranteeing Extrema. Rolle’s Theorem. Guarantees the existence of an extreme value in the interior of a closed interval. Rolle’s Theorem.

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Rolle’s Theorem and the Mean Value Theorem

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  1. Rolle’s Theorem and the Mean Value Theorem Guaranteeing Extrema

  2. Rolle’s Theorem • Guarantees the existence of an extreme value in the interior of a closed interval

  3. Rolle’s Theorem • Let f be a continuous function 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.

  4. Rolle’s Theorem • Three things that must be true for the theorem to hold: • (a) the function must be continuous • (b) the function must be differentiable • (c) f(a) must equal f(b)

  5. Rolle’s Theorem Examples • Book • On-line

  6. 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

  7. Mean Value Theorem • In other words, the derivative equals the slope of the line.

  8. Mean Value Theorem • Tutorial • Examples • More Examples

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