hw 13 - rhodes (ajr2283) – HW13 – Gilbert – (56380) 1...

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Unformatted text preview: rhodes (ajr2283) – HW13 – Gilbert – (56380) 1 This print-out should have 26 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine f x + f y when f ( x, y ) = 2 x 2 + xy − 4 y 2 + x + 3 y . 1. f x + f y = 5 x + 9 y + 4 2. f x + f y = 3 x − 7 y − 2 3. f x + f y = 3 x + 9 y − 2 4. f x + f y = 5 x − 7 y − 2 5. f x + f y = 3 x + 9 y + 4 6. f x + f y = 5 x − 7 y + 4 correct Explanation: After differentiation we see that f x = 4 x + y + 1 , f y = x − 8 y + 3 . Consequently, f x + f y = 5 x − 7 y + 4 . 002 10.0 points From the contour map of f shown below decide whether f x , f y are positive, negative, or zero at P . 2 2 4 4 6 6 P x y 1. f x < , f y < 2. f x < , f y > 3. f x > , f y < 4. f x < , f y = 0 5. f x > , f y > 6. f x > , f y = 0 correct Explanation: When we walk in the x-direction from P we are walking uphill, so f x > 0. On the other hand, when we walk in the y-direction from P our elevation doesn’t change because we are walking along a contour, so f y = 0. Consequently, at P f x > , f y = 0 . keywords: contour map, slope, partial deriva- tive, 003 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 rhodes (ajr2283) – HW13 – Gilbert – (56380) 2 1. f x < , f y = 0 2. f x < , f y < 3. f x = 0 , f y > correct 4. f x < , f y > 5. f x = 0 , f y < 6. f x > , f y > 7. f x = 0 , f y = 0 8. f x > , 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 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 > . keywords: surface, partial derivative, first or- der partial derivative, graphical interpreta- tion 004 10.0 points Determine f x when f ( x, y ) = x + 2 y 2 x + y . 1. f x = − 4 x (2 x + y ) 2 2. f x = 3 x (2 x + y ) 2 3. f x = − 3 y (2 x + y ) 2 correct 4. f x = − 5 x (2 x + y ) 2 5. f x = 4 y (2 x + y ) 2 6. f x = 5 y (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 10.0 points 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 ) = 4( y 2 − x 2 ) ln( x + y ) ....
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This note was uploaded on 09/27/2010 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.

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hw 13 - rhodes (ajr2283) – HW13 – Gilbert – (56380) 1...

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