10Test2SampleSol - NAM E_Solution Math 121 Sample Test 2...

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NAME: ________Solution__________________ Math 121 Sample Test 2 Spring 2010 Instructions: Answer the multiple choice questions by circling the answers. Answer the free response in the space provided. TRUE OR FALSE: ( 3 pts each) Mark the following statements with “True” or “False” 1. d dx ( e x + 4 e x ) = 0 . T F Answer: T 2. Solving e x e 2 x = 4 , we get x = ln2 . T F Answer: T 3. d dx [(2 x + 1) e x ] = 2 e x T F Answer: F FREE RESPONSE: 1. (12 pts) Given f ( x ) = 1 + e x 1 e x and g ( x ) = 5 x + 1 . (a) Find f ( x ) . Solution: f ( x ) = (1 e x ) d dx (1 + e x ) (1 + e x ) d dx (1 e x ) (1 e x ) 2 = (1 e x ) d dx (1 + e x ) (1 + e x ) d dx (1 e x ) (1 e x ) 2 = (1 e x ) e x (1 + e x )( e x ) (1 e x ) 2 = (1 e x ) e x + (1 + e x )( e x ) (1 e x ) 2 = 2 e x (1 e x ) 2 (b) Use the chain rule to find d dx f ( g ( x )) . d dx f ( g ( x )) = f ( g ( x )) g ( x ) = 2 e 5 x + 1 (1 e 5 x + 1 ) 2 (5) = 10 e 5 x + 1 (1 e 5 x + 1 ) 2 2. (7 pts ) Differentiate f ( x ) = x x 2 + e 4 x Solution: f ( x ) = x d dx ( x 2 + e 4 x ) 1 2 + ( x 2 + e 4 x ) 1 2 d dx x = x ( 1 2 )( x 2 + e 4 x ) 1 2 (2 x + 4 e 4 x ) + ( x 2 + e 4 x ) 1 2 = x ( x + 2 e 4 x ) x 2 + e 4 x + x 2 + e 4 x
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3. (8 pts ) Find the equation of the tangent line to the curve e y + xy 2 = 1 at the point (0,0) . Solution: Slope of the tangent line is dy dx (0,0) . We apply Implicit Differentiation technique to find the derivative d dx ( e y + xy 2 ) = d dx 1 e y dy dx + x d dx y 2 + y 2 d dx x = 0 e y dy dx + x (2 y dy dx ) + y 2 = 0 ( e y + 2 xy ) dy dx =
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