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Fa2010-Prelim2Solutions

Fa2010-Prelim2Solutions - Math 1110 Prelim 2 Answers Please...

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Math 1110 Prelim 2 Answers Please find below abbreviated answers to the questions on the second prelim. On several of the questions, more explanation would be necessary to receive full credit. 1. ( 20 points ) For each of the following functions, determine the derivative with respect to x . (a) ( 4 points ) f ( x ) = x 1 / 3 (b) ( 4 points ) f ( x ) = log 10 2 (c) ( 4 points ) f ( x ) = tan(sin(4 x )) (d) ( 4 points ) f ( x ) = xe - x + e 3 x (e) ( 4 points ) f ( x ) = 4 x ln x Solution (a) f 0 ( x ) = 1 3 x - 2 / 3 (b) f 0 ( x ) = 0 (c) f 0 ( x ) = 4 cos(4 x ) sec 2 (sin(4 x )) (d) f 0 ( x ) = e - x - xe - x + 3 e 3 x (e) f 0 ( x ) = 4 ln x - 4 (ln x ) 2 2. ( 15 points ) For each of the following functions, determine the derivative with respect to x . (a) ( 5 points ) f ( x ) = x tan( x ) (b) ( 5 points ) f ( x ) = e sin( x ) (c) ( 5 points ) f ( x ) = csc( x ) tan( x ) Solution: (a) f 0 ( x ) = x tan x (sec 2 ( x ) ln x + tan x x ) (b) f 0 ( x ) = 1 2 x cos( x ) e sin( x ) (c) f 0 ( x ) = sec( x ) tan( x ) 3. ( 15 points ) Find the slope of the tangent line to the curve y 2 + xy = sin( xy ) + 1 at the point (0 , 1). Solution: Taking the derivative of the equation we get 2 y 0 y + xy 0 + y = ( y + y 0 x ) cos( xy ).
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