100points From the contour map of f shown below decide...

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bretting (nmb2278) – HW09 – meth – (54160) 1 This print-out should have 20 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 decide whether f x , 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 2. f x < 0 , f y = 0 3. f x < 0 , f y > 0 4. f x < 0 , f y < 0 correct 5. f x > 0 , f y > 0 6. f x > 0 , f y < 0 002 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 3. f x > 0 , f y = 0 4. f x < 0 , f y > 0 correct 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
bretting (nmb2278) – HW09 – meth – (54160) 2 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 when f ( x, y ) = 2 x - y 2 x + y . 1. f x = - 4 x (2 x + y ) 2 2. f x = - 5 x (2 x + y ) 2 3. f x = 5 y (2 x + y ) 2 4. f x = 3 x (2 x + y ) 2 5. f x = 4 y (2 x + y ) 2 correct 6. f x = - 3 y (2 x + y ) 2 Explanation: From the Quotient Rule we see that f x = 2(2 x + y ) - 2(2 x - y ) (2 x + y ) 2 . Consequently, f x = 4 y (2 x + y ) 2 . 004 10.0points Determine f x when f ( x, y ) = cos(2 y - x ) - x sin(2 y - x ) . 1. f x = - 2 sin(2 y - x ) - x cos(2 y - x ) 2. f x = - x cos(2 y - x ) 3. f x = - cos(2 y - x ) - x sin(2 y - x ) 4. f x = x cos(2 y - x ) - sin(2 y - x ) 5. f x = - x sin(2 y - x ) 6. f x = x cos(2 y - x ) correct 7. f x = 2 sin(2 y - x ) - x cos(2 y - x ) 8. f x = x sin(2 y - x ) Explanation: From the Product Rule we see that f x = sin(2 y - x ) - sin(2 y - x )+ x cos(2 y - x ) . Consequently, f x = x cos(2 y - x ) . 005 10.0points Determine f x when f ( x, y ) = (1 - 3 xy ) e - xy . 1. f x = y (3 xy - 4) e - xy correct 2. f x = x (2 - xy ) e - xy 3. f x = x (4 - xy ) e - xy 4. f x = x ( xy - 4) e - xy 5. f x = x (2 - 3 xy ) e - xy 6. f x = y (2 + 3 xy ) e - xy
bretting (nmb2278) – HW09 – meth – (54160) 3

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