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Chapter 4 Theorems

Chapter 4 Theorems. EXTREMELY important for the AP Exam. Rolle’s Theorem. In the first quadrant, mark “a” and “b” on the x axis. Plot points at f(a) and f(b) such that f(a) = f(b) Connect the two points however you would like but NOT in a horizontal line.

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Chapter 4 Theorems

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  1. Chapter 4Theorems EXTREMELY important for the AP Exam

  2. Rolle’s Theorem In the first quadrant, mark “a” and “b” on the x axis. Plot points at f(a) and f(b) such that f(a) = f(b) Connect the two points however you would like but NOT in a horizontal line. What do you notice?

  3. Rolle’s Theorem f(a)=f(b) a b What do you see between a and b on each of these graphs????

  4. Rolle’s Theorem Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). If f(a) = f(b) then there is at least one number c in (a, b) at which f’(c) = 0

  5. Mean Value Theorem This is a more generalized version of Rolle’s Theorem. First stated by Joseph-Louis Lagrange

  6. Mean Value Theorem Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). There is some point c in (a, b) at which

  7. Mean Value theorem a b

  8. Mean Value Theorem Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). There is some point c in (a, b) at which

  9. Mean Value theorem c a b

  10. Extreme Value Theorem If f is continuous on a closed interval [a, b] Then f attains both an absolute maximum M And an absolute minimum m In [a,b]

  11. Extreme Value Theorem (cont) That is, there are numbers x1 and x2 in [a, b] With f(x1) = m and f(x2) = M And m ≤ f(x) ≤ M for every other x in [a, b]

  12. y = f(x) M m b a

  13. y = f(x) M m a b

  14. y = f(x) M m a b

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