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Unformatted text preview: mao (tm23477) – Hw 10 – gualdani – (57180) 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 If f is a continuous function on ( − 5 , 3) whose graph is 2 − 2 − 4 2 4 which of the following properties are satisfied? A. f has exactly 4 critical points, B. f ′′ ( x ) > 0 on ( − 2 , 1), C. f has exactly 2 local extrema. 1. all of them 2. A and C only correct 3. B only 4. C only 5. none of them 6. A only 7. B and C only 8. A and B only Explanation: A. True: f ′ ( x ) = 0 at x = − 3 , 1, while f ′ ( x ) does not exist at x = − 2; in addition, the graph of f has a vertical tangent at x = − 1. B. False: the graph changes concavity at the point x = − 1 in the interval ( − 2 , 1); in fact, f ′′ ( x ) > 0 on ( − 2 , − 1), while f ′′ ( x ) < 0 on ( − 1 , 1). C. True: f has a local minimum at x = − 2 and a local maximum at x = 1; the graph of f does have a horizontal tangent at ( − 3 , 1), but this is an inflection point. 002 10.0 points Which function could have 2 π π as its graph on [ 0 , 2 π ]? 1. f ( x ) = sin x 2. f ( x ) = sin x 2 − cos x correct 3. f ( x ) = − sin x 4. f ( x ) = sin x cos x − 2 5. f ( x ) = − sin x 2 + cos x 6. f ( x ) = sin x 2 + cos x Explanation: mao (tm23477) – Hw 10 – gualdani – (57180) 2 As f ( π/ 2) > 0, this already eliminates the three choices for f in which f ( π/ 2) < 0, leav ing only the possibilities sin x 2 − cos x , sin x , sin x 2 + cos x . To decide among these we check critical points because the graph has a local max imum in (0 , π/ 2) and a local minimum in (3 π/ 2 , 2 π ). Now sin x has critical points at x = π/ 2 , 3 π/ 2, eliminating this choice. On the other hand, by the Quotient Rule, d dx parenleftBig sin x 2 − cos x parenrightBig = cos x (2 − cos x ) − sin 2 x (2 − cos x ) 2 = 2 cos x − 1 (2 − cos x ) 2 , while d dx parenleftBig sin x 2 + cos x parenrightBig = cos x (2 + cos x ) + sin 2 x (2 + cos x ) 2 = 2 cos x + 1 (2 + cos x ) 2 . Thus f ( x ) = sin x 2 − cos x has critical points when cos x = 1 / 2, i.e. , at x = π/ 3 , 5 π/ 3, while f ( x ) = sin x 2 + cos x has critical points when cos x = − 1 / 2, i.e. , at x = 2 π/ 3 , 4 π/ 3 . Consequently, the graph can only be that of f ( x ) = sin x 2 − cos x . 003 10.0 points A function f is continuous and twice differentiable for all x negationslash = 1. Its derivatives have the properties (i) f ′ ( − 1) = 0, (ii) f ′′ > 0 on ( −∞ , − 2) uniondisplay (1 , ∞ ), (iii) f ′′ < 0 on ( − 2 , 1). If the lines x = 1 and y = 2 are asymptotes of the graph of f , which of the following could be the graph of f ?...
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 Fall '08
 schultz
 Derivative, Mathematical analysis

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