Midterm2ReviewSolutions

# Midterm2ReviewSolutions - Math 016 Review Solutions for...

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Math 016 Review Solutions for Midterm II Thursday, April 17, 2008, 8-10PM at C20 PC. You need # 2 pencil and a picture ID (covers material up to (and including) section 5.5) The exam will be in the form of multiple choices as most of the university’s large classes. 1. Find the derivative of f ( x ) = - 3 - sin x. Since f ( x ) = - (3 - sin x ) 1 2 ,f 0 ( x ) = - 1 2 (3 - sin x ) - 1 2 ( - cos x ) = cos x 2 3 - sin x 2. Find the inﬂection points of 1 x 2 + 1 . By the quotient rule, the second derivative is - 2( x 2 + 1) 2 - ( - 2 x )(2)( x 2 + 1)(2 x ) ( x 2 + 1) 4 = - 2( x 2 + 1)( x 2 + 1 - 4 x 2 ) ( x 2 + 1) 4 = - 2(1 - 3 x 2 ) ( x 2 + 1) 3 = 0 when x = ± 1 3 Since the second derative changes sign at each of these x -values, we have inﬂection points ( ± 1 3 , 3 4 ). 3. What is the minimum value of x 3 + y 3 if x + y = 1 and both x and y are nonnegative? Substitute y = 1 - x into x 3 + y 3 to get x 3 + (1 - x ) 3 . The derivative 3 x 2 + 3(1 - x ) 2 ( - 1) = 3 x 2 - 3(1 - 2 x + x 2 ) = 3 x 2 - 3 + 6 x - 3 x 2 = 6 x - 3 = 0 when x = . 5 . The ﬁrst derivative or second derivative test shows there is a local minimum at x = . 5. Then, y = 1 - x = 1 - . 5 = . 5. The minimum value of x 3 + y 3 is . 5 3 + . 5 3 = . 25. 4. The second derivative of the function y = 1 1 + 2 x is The ﬁrst derivative is - (1 + 2 x ) - 2 (2) = - 2(1 + 2 x ) - 2 . The second deriative is ( - 2)( - 2)(1 + 2 x ) - 3 (2) = 8(1 + 2 x ) - 3 = 8 (1 + 2 x ) 3 .

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5. The derivative of f ( x ) = 9 e - 2 x is f 0 ( x ) = 9 e - 2 x ( - 2) = - 18 e - 2 x . 6. The derivative of
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## This note was uploaded on 05/09/2008 for the course CALC 22m:16 taught by Professor Wang during the Spring '08 term at University of Iowa.

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Midterm2ReviewSolutions - Math 016 Review Solutions for...

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