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Mastering Theorems for AP Exam Prep

Learn about Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem essential for AP exams. Understand these theorems through practical examples and their implications on functions. Prepare effectively with detailed explanations.

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Mastering Theorems for AP Exam Prep

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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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