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Unformatted text preview: MINIQUIZ 10/2/08 - REVISED SOLUTIONS Name: Topics covered: • § 4.1 - Maximum and minimum values • § 4.2 - The mean value theorem True-False: (3 pts. each) 1. If f is differentiable at x = a , then f is continuous at x = a . TRUE. The following proof, which you do NOT need to know, is taken from pg. 128 of Stewart 6e. Assume f is differentiable at x = a . Then f ( a ) = lim x → a f ( x )- f ( a ) x- a exists (and is finite). To show that f is continuous at x = a , we need to show that lim x → a f ( x ) = f ( a ), or equivalently, lim x → a ( f ( x )- f ( a )) = 0. Trivially, we have f ( x )- f ( a ) = f ( x )- f ( a ) x- a ( x- a ) and so lim x → a [ f ( x )- f ( a )] = lim x → a f ( x )- f ( a ) x- a ( x- a ) = lim x → a f ( x )- f ( a ) x- a · lim x → a ( x- a ) = f ( a ) · = . 2. If f is continuous at x = a , then f is differentiable at x = a . FALSE. A counterexample is f ( x ) = | x | . We claim that this function is continuous at x = 0. For this we need to show that lim x → f ( x ) = f (0), or in our case that lim x → | x | = 0. We compute the left- and right-hand limits and show they are equal:...
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This note was uploaded on 03/02/2010 for the course AST 00000 taught by Professor Edwardl.robinson during the Fall '10 term at University of Texas.
- Fall '10