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Unformatted text preview: Version 004 Exam 2 gualdani (56410) 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine dy/dx when 6 cos x sin y = 1 . 1. dy dx = tan xy 2. dy dx = tan x 3. dy dx = cot x tan y 4. dy dx = cot x cot y 5. dy dx = tan x tan y correct Explanation: Differentiating implicitly with respect to x we see that 6 braceleftBig cos x cos y dy dx sin y sin x bracerightBig = 0 . Thus dy dx cos x cos y = sin x sin y . Consequently, dy dx = sin x sin y cos x cos y = tan x tan y . 002 10.0 points Find the slope of the tangent line to the graph of x 3 2 y 3 + xy = 0 at the point P (1 , 1). 1. slope = 5 4 2. slope = 3 2 3. slope = 4 5 correct 4. slope = 2 3 5. slope = 5 4 6. slope = 4 5 Explanation: Differentiating implicitly with respect to x we see that 3 x 2 6 y 2 dy dx + y + x dy dx = 0 . Consequently, dy dx = 3 x 2 + y 6 y 2 x . Hence at P (1 , 1) slope = dy dx vextendsingle vextendsingle vextendsingle P = 4 5 . 003 10.0 points Find the differential dy when y = 3 + sin x 5 sin x . 1. dy = 5 sin x (5 sin x ) 2 dx 2. dy = 3 cos x 5 sin x dx 3. dy = 3 sin x (5 sin x ) 2 dx 4. dy = 8 cos x (5 sin x ) 2 dx 5. dy = 8 cos x 5 sin x dx 6. dy = 8 cos x (5 sin x ) 2 dx correct Version 004 Exam 2 gualdani (56410) 2 Explanation: After differentiation of y = 3 + sin x 5 sin x using the quotient rule we see that dy = (5 sin x ) cos x + cos x (3 + sin x ) (5 sin x ) 2 dx . Consequently, dy = 8 cos x (5 sin x ) 2 dx . 004 10.0 points If f is the function whose graph is given by 2 4 6 2 4 6 which of the following properties does f have? A. local maximum at x = 4 , B. differentiable at x = 2 , C. f ( x ) > 0 on ( 1 , 2) . 1. none of them 2. C only 3. A and C only 4. all of them 5. A and B only 6. A only correct 7. B only 8. B and C only Explanation: The given graph has a removable disconti nuity at x = 4 and a critical point at x = 2. On the other hand, recall that f has a local maximum at a point c when f ( x ) f ( c ) for all x near c . Thus f could have a local max imum even if the graph of f has a removable discontinuity at c ; similarly, the definition of local minimum allows the graph of f to have a local minimum at a removable disconitu ity. So it makes sense to ask if f has a local extremum at x = 4. Inspection of the graph now shows of the three properties A. f has , B. f does not have , C. f does not have . 005 10.0 points Find the absolute minimum value of f ( x ) = 1 + 2 cos 2 x on [ , ]. 1. abs minimum value = 1 correct 2. abs minimum value = 0 3. abs minimum value = 2 4. abs minimum value = 4 5. abs minimum value = 2 6. abs minimum value = 3 Explanation: The absolute minimum value of f on [ , ] occurs (a) either at an endpoint x = or x = , (b) or at a critical point of f in ( , ). Version 004 Exam 2 gualdani (56410)...
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 Fall '09
 Gualdani
 Slope

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