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HW07-solutions - husain(aih243 HW07 Gilbert(56215 This...

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husain (aih243) – HW07 – Gilbert – (56215) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Let f be the function defined by f ( x ) = 3 + 2 x 1 / 3 . Consider the following properties: A. derivative exists for all x ; B. concave up on ( −∞ , 0) ; C. has vertical tangent at x = 0 . Which does f have? 1. B only 2. A only 3. All of them 4. A and B only 5. None of them 6. C only 7. B and C only correct 8. A and C only Explanation: The graph of f is 2 4 2 4 2 4 6 On the other hand, after differentiation, f ( x ) = 2 3 x 2 / 3 , f ′′ ( x ) = 4 9 x 5 / 3 . Consequently, A. not have: ( f ( x ) = (2 / 3) x 2 / 3 , x negationslash = 0); B. has: ( f ′′ ( x ) > 0 , x < 0); C. has: (see graph). 002 10.0 points When the graph of f is -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 6 2 4 6 2 4 6 which of the following is the graph of f ′′ ? 1. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 4 8 4 4 4 2. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 4 8 4 4 4
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husain (aih243) – HW07 – Gilbert – (56215) 2 3. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 4 8 4 4 4 4. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 4 8 4 4 4 correct 5. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1 4 8 4 4 4 Explanation: The graph of f has exactly one point at which it changes concavity, so the graph of f ′′ has exactly one x -intercept. This rules out the parabola. But the graph of f changes concavity at x = 2, so the graph of f ′′ must be one of the straight lines having x = 2 as x -intercept. This rules out two of the lines, leaving just two lines each with the same x - intercept but slopes of opposite sign. An inspection of the concavity of the graph of f to the left and right of x = 2 thus shows that -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 6 2 4 6 2 4 6 must be the graph of f ′′ . 003 10.0 points Find the interval(s) on which f ( x ) = x 3 x 2 8 x 1 is increasing. 1. bracketleftBig 2 , 8 3 bracketrightBig 2. bracketleftBig 8 3 , 2 bracketrightBig 3. parenleftBig −∞ , 2 bracketrightBig , bracketleftBig 8 3 , parenrightBig 4. parenleftBig −∞ , 4 3 bracketrightBig , bracketleftBig 2 , parenrightBig correct 5. parenleftBig − ∞ , 2 bracketrightBig , bracketleftBig 4 3 , parenrightBig 6. bracketleftBig 4 3 , 2 bracketrightBig Explanation: Since f is continuous, it will be increasing (i) on [ a, b ] when f ( x ) > 0 on ( a, b ), (ii) on ( −∞ , a ] when f ( x ) > 0 on ( −∞ , a ), (iii) on [ b, ) when f ( x ) > 0 on ( b, ).
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