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Unformatted text preview: jiwani (amj566) Homework10 Fouli (58395) 1 This printout should have 17 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the increasing and decreasing properties of the function f ( x ) = ( x 1) 4 5 ( x + 2) 1 5 on its natural domain. 1. inc: [ 2 , 7 5 ] , dec: [ 7 5 , ) 2. inc: [ 2 , 7 5 ] [1 , ) , dec: [ 7 5 , 1] 3. inc: ( , 7 5 ] [1 , ) , dec: [ 7 5 , 1] cor rect 4. inc: [ 7 5 , 1] , dec: [ 2 , 7 5 ] [1 , ) 5. inc: ( , 2] [1 , ) , dec: [ 2 , 1] Explanation: The natural domain of f is ( , ) and f is differentiable everywhere on its domain except at x = 2 , 1. Now by the Product Rule, f ( x ) = 1 5 braceleftBig 4 parenleftBig x + 2 x 1 parenrightBig 1 5 + parenleftBig x 1 x + 2 parenrightBig 4 5 bracerightBig . Now 4 parenleftBig x + 2 x 1 parenrightBig 1 5 + parenleftBig x 1 x + 2 parenrightBig 4 5 = 4( x + 2) + x 1 ( x 1) 1 5 ( x + 2) 4 5 = 5 x + 7 ( x 1) 1 5 ( x + 2) 4 5 . Thus f ( x ) = x + 7 5 ( x 1) 1 5 ( x + 2) 4 5 . Since ( x + 2) 4 5 is positive everywhere except at x = 2, where it is zero, we have only to look at the sign chart + + 7 5 1 for x + 7 5 ( x 1) 1 5 to determine where f is increasing or decreas ing. Consequently, f is inc: ( , 7 5 ] [1 , ) , dec: [ 7 5 , 1] . 002 10.0 points Find all values of x at which the graph of y = x 2 + 4 sin x changes concavity on ( / 2 , / 2). 1. x = 6 , 6 2. x = 6 correct 3. x = 3 4. x = 6 5. x = 3 , 3 6. x = 3 7. there are no values of x Explanation: The graph changes concavity at x when d 2 y/dx 2 changes sign at x . Now after differentiating twice we see that d 2 y dx 2 = d dx parenleftBig 2 x + 4 cos x parenrightBig = 2 4 sin x . The sign chart jiwani (amj566) Homework10 Fouli (58395) 2 / 2 + / 6 / 2 for d 2 y/dx 2 thus shows that on ( / 2 , / 2) the graph changes sign at x = / 6 only. Con sequently, the graph changes concavity only at x = 6 . 003 10.0 points Let f be the function defined by f ( x ) = x cos 2 x, x . Determine all interval(s) on which f is de creasing. 1. [ 5 12 , 6 ] , [ 6 , 11 12 ] 2. [ 6 , 12 ] , [ 6 , 11 12 ] 3. [ 5 12 , 12 ] , [ 7 12 , 11 12 ] correct 4. [ , 5 12 ] , [ 7 12 , ] 5. [ 5 12 , 8 ] , [ 3 8 , 11 12 ] Explanation: After differentiation, f ( x ) = 1 + 2 sin2 x....
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 Spring '11
 Jensen

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