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Unformatted text preview: tgo72 Homework 9 Gompf (58370) 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 Determine whether the partial derivatives f x , f y of f are positive, negative or zero at the point P on the graph of f shown in P x z y 1. f x = 0 , f y < 2. f x > , f y > 3. f x = 0 , f y = 0 correct 4. f x < , f y < 5. f x < , f y = 0 6. f x > , f y = 0 7. f x < , f y > 8. f x = 0 , f y > Explanation: The value of f x at P is the slope of the tangent line to graph of f at P in the x direction, while f y is the slope of the tangent line in the ydirection. Thus the sign of f x indicates whether f is increasing or decreasing in the xdirection, or whether the tangent line in that direction at P is horizontal. Similarly, the value of f y at P is the slope of the tangent line at P in the ydirection, and so the sign of f y indicates whether f is increasing or decreasing in the ydirection, or whether the tangent line in that direction at P is horizontal. From the graph it thus follows that at P f x = 0 , f y = 0 . 002 10.0 points Determine f x when f ( x, y ) = cos(4 y x ) x sin(4 y x ) . 1. f x = x sin(4 y x ) 2. f x = x cos(4 y x ) sin(4 y x ) 3. f x = x sin(4 y x ) 4. f x = x cos(4 y x ) 5. f x = cos(4 y x ) x sin(4 y x ) 6. f x = 2 sin(4 y x ) x cos(4 y x ) 7. f x = 2 sin(4 y x ) x cos(4 y x ) 8. f x = x cos(4 y x ) correct Explanation: From the Product Rule we see that f x = sin(4 y x ) sin(4 y x )+ x cos(4 y x ) . Consequently, f x = x cos(4 y x ) . 003 10.0 points Find the slope in the xdirection at the point P (0 , 2 , f (0 , 2)) on the graph of f when f ( x, y ) = 2(2 x + y ) e xy . tgo72 Homework 9 Gompf (58370) 2 1. slope = 10 2. slope = 12 3. slope = 6 4. slope = 4 correct 5. slope = 8 Explanation: The graph of f is a surface in 3space and the slope in the xdirection at the point P (0 , 2 , f (0 , 2)) on that surface is the value of the partial derivative f x at (0 , 2). Now f x = 4 e xy 2(2 xy + y 2 ) e xy . Consequently, at P (0 , 2 , f (0 , 2)) slope = 4 . 004 10.0 points Determine h = h ( x, y ) so that f x = h ( x, y ) ( x 2 + 4 y 2 ) 2 when f ( x, y ) = 2 x 2 y x 2 + 4 y 2 . 1. h ( x, y ) = 8 xy 3 2. h ( x, y ) = 16 xy 2 3. h ( x, y ) = 16 x 3 y 4. h ( x, y ) = 16 xy 3 correct 5. h ( x, y ) = 8 xy 2 6. h ( x, y ) = 8 x 3 y Explanation: Differentiating with respect to x using the quotient rule we obtain f x = 4 xy ( x 2 + 4 y 2 ) 4 x 3 y ( x 2 + 4 y 2 ) 2 ....
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 Spring '08
 RAdin
 Calculus, Slope

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