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Section 4.2 Rolle’s Theorem & Mean Value Theorem Calculus Winter, 2010

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Calculus, Section 4.12 Rolle’s Theorem (or “What goes up must come down”) IF (condition) f is continuous on [a,b] f is differentiable on (a,b) f(a)=f(b) THEN (conclusion) There exist a number c in (a,b) such that f’(c)=0

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Calculus, Section 4.13 Rolle’s Theorem (or “What goes up must come down”) IF f is continuous on [a,b] f is differentiable on (a,b) f(a)=f(b) THEN There exist a number c in (a,b) such that f’(c)=0 a b c

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Calculus, Section 4.14 Rolle’s Theorem (or “What goes up must come down”) Since we know such a c exists, we now can solve from c with confidence. a b c

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Calculus, Section 4.15 Using Rolle’s Theorem Prove that the equation x 3 +x-1=0 has exactly one real root. Let f(x)=x 3 +x-1 continuous and differentiable everywhere Since f(-10) is a big negative number and f(10) is a big positive number, the Intermediate Value Theorem says that somewhere on (-10,10) f(x) = 0. Therefore there exists at least one root.

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Calculus, Section 4.16 Using Rolle’s Theorem Prove that the equation x 3 +x-1=0 has exactly one real root. Suppose there are two roots a and b If there are two roots, then f(a)=f(b)=0. Rolle’s Theorem says that somewhere there is c where f’(c) = 0, but we see the f’(x)=3x 2 +1 which is ALWAYS POSITIVE. Therefore our supposition must be false. Therefore there is exactly one root.

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Calculus, Section 4.17 Mean Value Theorem (or “someone’s got to be average”) Translation: On the interval (a,b) there is at least one place where the average slope is the instantaneous slope.

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Calculus, Section 4.18 Mean Value Theorem (or “someone’s got to be average”)

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Calculus, Section 4.19 Mean Value Theorem (or “someone’s got to be average”) There must be a place on (a,b) where f’(x) = -1

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Calculus, Section 4.110 Warnings! Don’t apply Rolle’s Theorem or The Mean Value Theorem unless the conditions are met Continuous on [a,b] Differentiable on (a,b)

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Calculus, Section 4.111 Assignment Section 4.2, # 1, 4, 6, 9, 11, 15, 17, 19, 21, 26, 29, 31, 40, 43

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