2311lectureoutline4.2

# 2311lectureoutline4.2 - So, 3/2 is the value guaranteed by...

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MAC2311, Calculus I 4.2 The Mean Value Theorem Rolle’s Theorem : Let f be a function that satisfies the following: 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) . Then there is a number c in ( a, b) such that f’(c) = 0. Example: Does Rolle’s Theorem apply to the function 2 ( ) 3 2 f x x x = - + on the interval [1, 2]? If so, apply it. 1. Since f(x) is a polynomial function, it is continuous everywhere 2. Since f(x) is a polynomial function, it is differentiable everywhere. 3. f(1) = 0 and f(2) = 0, so f(1) = f(2). So there is a value c in the interval (1, 2) where f’(c) = 0. Let’s find c. f’(c) = 2c – 3 = 0 … c = 3/2 and 3/2 is in [1, 2]
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Unformatted text preview: So, 3/2 is the value guaranteed by Rolle’s Theorem. The Mean Value Theorem: Let f be a function that satisfies the following: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). Then there is a number c in ( a,b) such that ( ) ( ) ( ) f b f a f c b a-′ =-, or equivalently ( ) ( ) ( )( ) f b f a f c b a-= ′-Restated… …at some place c in [a, b], the instantaneous rate of change is equal to the average rate of change over the interval [a, b]. …at some time c between a and b, the instantaneous velocity is equal to the average velocity over the entire interval. Try exercises 2, 6, 14, 23 on pp 285-286 in your text....
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## This note was uploaded on 12/12/2011 for the course MAC 2311 taught by Professor Teague during the Spring '11 term at Santa Fe College.

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