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Unformatted text preview: Math 1151. Calculus 2009. S1. Test 2 Version 1b. (1) Determine all a, b R such that the function f ( x ) = ax + b if x 6 1 ln x if x > 1 is differentiable at x = 1. Solution. For f to be differentiable at x = 1, we require lim x 1- f ( x ) = lim x 1 + f ( x ) lim h - f (1 + h )- f (1) h = lim h + f (1 + h )- f (1) h . The first equation gives a + b = 0 and the second equation gives lim h - f (1 + h )- f (1) h = lim h - a (1 + h ) + b- ( a + b ) h = a = lim h + f (1 + h )- f (1) h = lim h + ln(1 + h ) h = lim h + 1 1 + h = 1 so ( a, b ) = (1 ,- 1). (2) State the mean value theorem and find a point which satisfies the conclu- sions of the MVT for f ( x ) = ( x- 1) 3 on [1 , 4]. Solution. The mean value theorem states that if f is a continuous function on [ a, b ] which is differentiable on ( a, b ), then there exists c ( a, b ) such that f ( c ) = f ( b )- f ( a ) b- a ....
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This document was uploaded on 03/19/2012.
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