1 / 15

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem. 3.6. f(b). N. f(a). b. a. c. Intermediate Value Theorem (IVT).

gloria-baldwin
Télécharger la présentation

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6

  2. f(b) N f(a) b a c Intermediate Value Theorem (IVT) If f is continuous on [a, b] and N is a value between f(a) and f(b), then there is at least one point c between a and b where f takes on the value N.

  3. c a b Rolle’s Theorem If f is continuous on [a, b], if f(a) = 0, f(b) = 0, then there is at least one number c on (a, b) where f ‘ (c ) = 0 slope = 0 f ‘ (c ) = 0

  4. f(x+h) x x+h f(x) Given the curve:

  5. The Mean Value Theorem (MVT) aka the ‘crooked’ Rolle’s Theorem There is at least one number c on (a, b) at which f(b) a c b f(a) If f is continuous on [a, b] and differentiable on (a, b) Conclusion: Slope of Secant Line Equals Slope of Tangent Line

  6. f(0) = -1 f(1) = 2

  7. Find the value(s) of c which satisfy Rolle’s Theorem for on the interval [0, 1]. which is on [0, 1] Verify…..f(0) = 0 – 0 = 0 f(1) = 1 – 1 = 0

  8. Find the value(s) of c that satisfy the Mean Value Theorem for

  9. Find the value(s) of c that satisfy the Mean Value Theorem for Since has no real solution, there is no value of c on [-4, 4] which satisfies the Mean Value Theorem Note: The Mean Value Theorem requires the function to be continuous on [-4, 4] and differentiable on (-4, 4). Therefore, since f(x) is discontinuous at x = 0 which is on [-4, 4], there may be no value of c which satisfies the Mean Value Theorem

  10. Given the graph of f(x) below, use the graph of f to estimate the numbers on [0, 3.5] which satisfy the conclusion of the Mean Value Theorem.

  11. f(x) is continuous and differentiable on [-2, 2] On the interval [-2, 2], c = 0 satisfies the conclusion of MVT

  12. f(x) is continuous and differentiable on [-2, 1] On the interval [-2, 1], c = 0 satisfies the conclusion of MVT

  13. Since f(x) is discontinuous at x = 2, which is part of the interval [0, 4], the Mean Value Theorem does not apply

  14. f(x) is continuous and differentiable on [-1, 2] c = 1 satisfies the conclusion of MVT

  15. If , how many numbers on [-2, 3] satisfy the conclusion of the Mean Value Theorem. A. 0 B. 1 C. 2 D. 3 E. 4 CALCULATOR REQUIRED f(3) = 39 f(-2) = 64 For how many value(s) of c is f ‘ (c ) = -5? X X X

More Related