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Exam02-solutions

# Exam02-solutions - Version 074 Exam02 Gilbert(56215 This...

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Version 074 – Exam02 – Gilbert – (56215) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points ±ind the interval(s) where f ( x ) = x 3 + 9 x 2 + 6 is increasing. 1. ( −∞ , 3) , (3 , ) 2. ( −∞ , 6] , [0 , ) correct 3. [ 3 , 0] 4. ( −∞ , 3] , [0 , ) 5. [ 6 , 0] 6. ( −∞ , 6) , (6 , ) Explanation: The Function f will be increasing on an interval [ a, b ] (resp. ( −∞ , b ] or [ a, )) when f > 0 on ( a, b ) (resp. ( −∞ , b ) or ( a, )), i.e. , on those values oF x For which f ( x ) = 3 x 2 + 18 x = 3 x ( x + 6) > 0 . Consequently, f will be increasing on ( −∞ , 6] , [0 , ) . 002 10.0 points ±ind all asymptotes oF the graph oF y = 3 x 2 16 x + 16 x 2 5 x + 4 . 1. vertical: x = 1 , horizontal: y = 3 2. vertical: x = 4 , horizontal: y = 3 3. vertical: x = 4 , horizontal: y = 3 4. vertical: x = 1 , 4 , horizontal: y = 3 5. vertical: x = 1 , horizontal: y = 3 cor- rect 6. vertical: x = 1 , 4 , horizontal: y = 3 7. vertical: x = 1 , horizontal: y = 3 Explanation: AFter Factorization y = (3 x 4)( x 4) ( x 1)( x 4) . Thus y is not defned at x = 4, but For x n = 4 y = 3 x 4 x 1 ; notice, however, that lim x 4 3 x 4 x 1 = 8 3 exists, so the graph does not have a vertical asymptote at x = 4. Since v v v 3 x 4 x 1 v v v −→ ∞ as x 1 From the leFt and the right, the line x = 1 will, however, be a vertical asymptote. On the other hand, lim x →±∞ 3 x 4 x 1 = 3 , so y = 3 will be a horizontal aymptote. Hence the graph has only the Following asymptotes vertical: x = 1 , horizontal: y = 3. 003 10.0 points Determine which oF the Following graphs could be that oF f ( x ) = 3 + 2 x 2 1 + x 2 .

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Version 074 – Exam02 – Gilbert – (56215) 2 1. 2. 3. correct 4. Explanation: The line y = 2 is a horizontal asymptote because lim x →±∞ 3 + 2 x 2 1 + x 2 = 2 . On the other hand, f ( x ) = 2 x (1 + x 2 ) 2 . Thus the critical point of f occurs at x = 0.
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Exam02-solutions - Version 074 Exam02 Gilbert(56215 This...

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