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HW08-solutions.pdf

HW08-solutions.pdf - young(toy68 HW08 villafuerte...

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young (toy68) – HW08 – villafuerte altu – (53615) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points From the contour map of f shown below de- cide whether f x and f y are positive, negative, or zero at P . 0 0 2 2 4 4 6 6 P x y 1. f x = 0 , f y > 0 correct 2. f x > 0 , f y > 0 3. f x > 0 , f y < 0 4. f x = 0 , f y < 0 5. f x < 0 , f y < 0 6. f x < 0 , f y > 0 Explanation: When we walk in the x -direction from P our elevation doesn’t change because we are walking along a contour, so f x = 0. On the other hand, when we walk in the y -direction from P we are walking uphill, so f y > 0. Consequently, at P f x = 0 , f y > 0 . keywords: contour map, contours, partial derivative, slope, 002 10.0points 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 < 0 2. f x = 0 , f y > 0 correct 3. f x < 0 , f y = 0 4. f x < 0 , f y < 0 5. f x > 0 , f y = 0 6. f x > 0 , f y > 0 7. f x = 0 , f y = 0 8. f x < 0 , f y > 0 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 y -direction. Thus the sign of f x indicates whether f is increasing or decreasing in the x -direction, or whether the tangent line in that direction at P is horizontal. Similarly, the value of f y at P is the slope
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young (toy68) – HW08 – villafuerte altu – (53615) 2 of the tangent line at P in the y -direction, and so the sign of f y indicates whether f is increasing or decreasing in the y -direction, 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 . keywords: surface, partial derivative, first or- der partial derivative, graphical interpreta- tion 003 10.0points Determine f x - f y when f ( x, y ) = 3 x 2 + xy - 4 y 2 - 2 x + 3 y . 1. f x - f y = 5 x - 7 y - 5 2. f x - f y = 5 x + 9 y + 1 3. f x - f y = 7 x - 7 y - 5 4. f x - f y = 7 x + 9 y + 1 5. f x - f y = 5 x + 9 y - 5 correct 6. f x - f y = 7 x - 7 y + 1 Explanation: After differentiation we see that f x = 6 x + y - 2 , f y = x - 8 y + 3 . Consequently, f x - f y = 5 x + 9 y - 5 . 004 10.0points Determine f x when f ( x, y ) = x + 2 y 2 x + y . 1. f x = 4 y (2 x + y ) 2 2. f x = 5 y (2 x + y ) 2 3. f x = - 5 x (2 x + y ) 2 4. f x = - 4 x (2 x + y ) 2 5. f x = - 3 y (2 x + y ) 2 correct 6. f x = 3 x (2 x + y ) 2 Explanation: From the Quotient Rule we see that f x = (2 x + y ) - 2( x + 2 y ) (2 x + y ) 2 . Consequently, f x = - 3 y (2 x + y ) 2 . 005 10.0points Find the slope in the x -direction at the point P (0 , 2 , f (0 , 2)) on the graph of f when f ( x, y ) = 2(2 x + y ) e xy . 1. slope = 0 2. slope = - 8 3. slope = - 2 4. slope = - 6 5. slope = - 4 correct Explanation: The graph of f is a surface in 3-space and the slope in the x -direction 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 .
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young (toy68) – HW08 – villafuerte altu – (53615) 3 Consequently, at P (0 , 2 , f (0 , 2)) slope = - 4 .
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