# 4.2 - 4.2 Mean Value Theorem ROLLE'S THEOREM Let f be differentiable on(a,b and continuous on[a,b If f(a = f(b = 0 then there is at least one point

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Calculus Section 4.1 Page5 4.2 Mean Value Theorem ROLLE'S THEOREM Let f be differentiable on (a,b) and continuous on [a,b]. If f (a) = f (b) = 0, then there is at least one point c in (a,b) where f c ( ) = 0 . (geometrically obvious) eg: (Set III #57) If c is the number defined by Rolle's Theorem, then for f x ( ) = 2x 3 - 6x on the interval 0 x 3 , then c is A. 1 B. -1 C. 2 D. 0 E. 3 eg: (Set III #56) If f (a) = f (b) = 0 and f(x) is continuous on [a,b], then A. f(x) must be identically zero B. f x ( ) may be different from zero for all x on [a,b] C. there exists at least one number c, a < c < b, such that f c ( ) = 0 D. f x ( ) must exist for every x on (a,b) E. none of the preceding is true MEAN VALUE THEOREM Let f be differentiable on (a,b) and continuous on [a,b]. There there is at least one point c in (a,b) where f c ( ) = f b ( ) - f a ( ) b - a

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Calculus Section 4.1 Page 6 (geometrically obvious) eg: Let f x ( ) = x 3 + 1. Show that f(x) satisfies the hypotheses of the
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## This note was uploaded on 02/04/2011 for the course MATH 116 taught by Professor John during the Spring '10 term at St. Michael.

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4.2 - 4.2 Mean Value Theorem ROLLE'S THEOREM Let f be differentiable on(a,b and continuous on[a,b If f(a = f(b = 0 then there is at least one point

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