CABChp4-2NTs - CalcAB Chp 4-2 Rolles Theorem and Mean Value...

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CalcAB Chp 4- 2 Rolle’s Theorem and Mean Value Theorem NOTES 1 ROLLE’S THEOEREM A set of conditions used to determine whether or not there is a turning point within the given interval. Let f be a function that satisfies the following three hypotheses: (1) f is CONTINUOUS on the (closed) interval [a, b] (2) f is DIFFERENTIABLE on the (open) interval (a, b) (3) f(a) = f(b) (the endpoints have the same y-value) Then there is at least one number c in the (open) interval (a, b) such that f’(c) = 0 Visual example: 2. Ex(Apply Rolle’s t heorem) Verify the function satisfies t he three hypotheses of Rolle’s t heorem on the given interval. If the conditions are met, find all numbers c that satisfy the conclusion of Ro lle’s t heorem. f (x) x x 6 [ 6, 0] 3. Ex(Look for contradiction with Rolle’s Th eorem) Prove that the equation 0 = x 3 + x 1 has exactly one root. <The nature of a proof in calculus is different than in geometry. This answer is done for you.> The equation is a polynomial; polynomials do not have cusps or sharp turns. If the polynomial were to have more than one root (x-intercept), then it would follow that it must have at least
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CABChp4-2NTs - CalcAB Chp 4-2 Rolles Theorem and Mean Value...

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