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Unformatted text preview: Version 062 Exam 2 gualdani (56455) 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 3 cos x sin y = 7 . 1. dy dx = cot x tan y 2. dy dx = tan x tan y correct 3. dy dx = tan xy 4. dy dx = tan x 5. dy dx = cot x cot y Explanation: Differentiating implicitly with respect to x we see that 3 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 2 x 3 + y 3 xy = 0 at the point P ( 1 , 1). 1. slope = 3 2 2. slope = 4 5 3. slope = 5 4 correct 4. slope = 5 4 5. slope = 2 3 6. slope = 4 5 Explanation: Differentiating implicitly with respect to x we see that 6 x 2 + 3 y 2 dy dx y x dy dx = 0 . Consequently, dy dx = 6 x 2 y 3 y 2 x . Hence at P ( 1 , 1) slope = dy dx vextendsingle vextendsingle vextendsingle P = 5 4 . 003 10.0 points Find the differential dy when y = 4 + sin x 2 sin x . 1. dy = 2 sin x (2 sin x ) 2 dx 2. dy = 6 cos x 2 sin x dx 3. dy = 6 cos x (2 sin x ) 2 dx correct 4. dy = 4 cos x 2 sin x dx 5. dy = 6 cos x (2 sin x ) 2 dx 6. dy = 4 sin x (2 sin x ) 2 dx Version 062 Exam 2 gualdani (56455) 2 Explanation: After differentiation of y = 4 + sin x 2 sin x using the quotient rule we see that dy = (2 sin x ) cos x + cos x (4 + sin x ) (2 sin x ) 2 dx . Consequently, dy = 6 cos x (2 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. differentiable at x = 2 , B. f ( x ) > 0 on (2 , 4) , C. local minimum at x = 4 . 1. C only 2. B and C only 3. A and B only 4. B only correct 5. A and C only 6. A only 7. none of them 8. all of them 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 does not have , B. f has , C. f does not have . 005 10.0 points Find the absolute maximum value of f ( x ) = 1 2 cos 2 x on [ , ]. 1. abs maximum value = 1 2. abs maximum value = 2 3. abs maximum value = 0 4. abs maximum value = 3 5. abs maximum value = 1 correct 6. abs maximum value = 4 Explanation: The absolute maximum value of f on [ , ] occurs (a) either at an endpoint x = or x = , (b) or at a critical point of f in (...
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This note was uploaded on 08/29/2010 for the course M 46455 taught by Professor Gualdani during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Gualdani

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