K Exam 2-solutions-1 - Version 020 K Exam 2 ben-zvi (55600)...

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Unformatted text preview: Version 020 K Exam 2 ben-zvi (55600) 1 This print-out should have 18 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 ) = 5 x 2 / 3 . Consider the following properties: A. concave up on ( , 0) (0 , ) B. has local maximum at x = 0 Which does f have? 1. both of them correct 2. B only 3. A only 4. neither of them Explanation: The graph of f is 2 4 2 4 2 4 On the other hand, after differentiation, f ( x ) = 2 3 x 1 / 3 , f ( x ) = 2 9 x 4 / 3 . Consequently, A. TRUE: f ( x ) > , x negationslash = 0 B. TRUE: see graph 002 10.0 points Determine the interval(s) where f ( x ) = x sin x is increasing on [0 , 2 ]. 1. bracketleftBig , bracketrightBig 2. bracketleftBig , 2 bracketrightBig correct 3. bracketleftBig , 1 2 bracketrightBig , bracketleftBig 3 2 , 2 bracketrightBig 4. bracketleftBig , 2 bracketrightBig 5. bracketleftBig 1 2 , 3 2 bracketrightBig Explanation: After differentiation, f ( x ) = 1 cos x. Now f will be increasing on an interval [ a, b ] if f ( x ) > 0 on ( a, b ). Determining where f ( x ) > 0 can be done graphically or alge- braically. But the graph 1 2 2 of 1 cos x shows that f ( x ) > 0 on the interval (0 , 2 ). Thus f will be increasing everywhere on [0 , 2 ] . On the other hand, we know that the inequal- ity cos x < 1 holds for all x on [0 , 2 ] except when cos x = 1, i.e. , when x = 0 , 2 . Thus 1 cos x > 0 holds everywhere on (0 , 2 ), showing algebraically that f will be increas- ing everywhere on [0 , 2 ] . Version 020 K Exam 2 ben-zvi (55600) 2 003 10.0 points Determine lim x 5 x 2 2 x + 7 5 + 5 x 7 x 2 . 1. limit = 2. limit = 0 3. limit = 5 14 4. limit = 5 7 correct 5. none of the other answers Explanation: Dividing the numerator and denominator by x 2 we see that 5 x 2 2 x + 7 5 + 5 x 7 x 2 = 5 2 x + 7 x 2 5 x 2 + 5 x 7 . On the other hand, lim x 1 x = lim x 1 x 2 = 0 . By Properties of limits, therefore, the limit = 5 7 . 004 10.0 points Which of the following is the graph of f ( x ) = x 2 x 1 ? 1. 2 4 6 2 4 6 2 4 6 2 4 6 2. 2 4 6 2 4 6 2 4 6 2 4 6 3. 2 4 6 2 4 6 2 4 6 2 4 6 4. 2 4 6 2 4 6 2 4 6 2 4 6 5. 2 4 6 2 4 6 2 4 6 2 4 6 correct Version 020 K Exam 2 ben-zvi (55600) 3 6. 2 4 6 2 4 6 2 4 6 2 4 6 Explanation: The graph of f will have y = 1 as a hori- zontal asymptote, and x 1 = 0 as vertical asymptote. Combining these with the fact that the y-intercept of the graph occurs at y = +2, we see that the the graph of f must be 2 4 6 2 4 6 2 4 6 2 4 6 005 10.0 points The figure below shows the graphs of three functions:...
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K Exam 2-solutions-1 - Version 020 K Exam 2 ben-zvi (55600)...

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