HW07-solutions

HW07-solutions - husain (aih243) – HW07 – Gilbert –...

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Unformatted text preview: 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 ) > , x < 0); C. has: (see graph). 002 10.0 points When the graph of f is 2 4 6 8 10 − 2 − 4 − 6 2 4 6 − 2 − 4 − 6 which of the following is the graph of f ′′ ? 1. 4 8 − 4 4 − 4 2. 4 8 − 4 4 − 4 husain (aih243) – HW07 – Gilbert – (56215) 2 3. 4 8 − 4 4 − 4 4. 4 8 − 4 4 − 4 correct 5. 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 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 ) > on ( a, b ), (ii) on ( −∞ , a ] when f ′ ( x ) > on ( −∞ , a ), (iii) on [ b, ∞ ) when f ′ ( x ) > on ( b, ∞ )....
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This note was uploaded on 11/30/2010 for the course M 408c taught by Professor Mcadam during the Spring '06 term at University of Texas.

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HW07-solutions - husain (aih243) – HW07 – Gilbert –...

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