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miniquiz_10_02_solns - MINIQUIZ REVISED SOLUTIONS Name...

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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 ) · 0 = 0 . 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 0 f ( x ) = f (0), or in our case that lim x 0 | x | = 0. We compute the left- and right-hand limits and show they are equal: lim x 0+ | x | = lim x 0+ x = 0 , lim x 0 - | x | = lim x 0 - ( - x ) = 0 .
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