HW09-solutions - oh(myo92 HW09 mostovyi(54020 This...

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oh (myo92) – HW09 – mostovyi – (54020) 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 5. f x > 0 , f y = 0 6. f x < 0 , f y < 0 correct Explanation: When we walk in the x -direction from P we are walking downhill, so f x < 0. On the other hand, when we walk in the y -direction from P we are again walking downhill, so f y < 0 also. Consequently, at P f x < 0 , f y < 0 . keywords: contour map, slope, partial deriva- tive, 002 10.0points Determine f x when f ( x , y ) = ( x 2 - y )(2 y 2 - x ) . 1. f x = 2 y + 2 xy 2 + 3 x 2 2. f x = 2 xy 2 - 2 y + 3 x 2 3. f x = 2 y - 2 xy 2 + 3 x 2 4. f x = 4 xy 2 - y - 3 x 2 5. f x = y - 4 xy 2 - 3 x 2 6. f x = 4 xy 2 + y - 3 x 2 correct Explanation: From the Product Rule we see that f x = 2 x (2 y 2 - x ) - ( x 2 - y ) . Consequently, f x = 4 xy 2 + y - 3 x 2 . 003 10.0points Determine f y when f ( x, y ) = sin(2 x - y ) - y cos(2 x - y ) . 1. f y = 2 sin(2 x - y ) - y cos(2 x - y ) 2. f y = y sin(2 x - y ) 3. f y = - y sin(2 x - y ) 4. f y = - y cos(2 x - y ) 5. f y = y cos(2 x - y ) 6. f y = - 2 cos(2 x - y ) - y sin(2 x - y ) correct
oh (myo92) – HW09 – mostovyi – (54020) 2 7. f y = 2 cos(2 x - y ) + y sin(2 x - y ) 8. f y = - 2 sin(2 x - y ) + y cos(2 x - y ) Explanation: From the Product Rule we see that f y = - cos(2 x - y ) - cos(2 x - y ) - y sin(2 x - y ) . Consequently, f y = - 2 cos(2 x - y ) - y sin(2 x - y ) . 004 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 ) = 3(2 x + y ) e - xy . 1. slope = - 6 correct 2. slope = - 2 3. slope = - 10 4. slope = - 8 5. slope = - 4 On the other hand, differentiating f ( x, y ) with respect to y holding x fixed, we see that f y = 4 y + x 2 . 006 10.0points Find the value of f x at (3 , 2) when f ( x, y ) = 4 x 3 - 4 x 2 y - 7 x + 5 y. 2. f x = 8 x + 2 xy, f y = 4 y + x 2 correct 3. f x = 8 x + xy, f y = 4 y + x 2 4. f x = 4 y + x 2 , f y = 8 x + 2 xy 5. f x = 8 x + 2 x, f y = 4 y + x 2 Explanation: Differentiating f ( x, y ) with respect to x holding y fixed, we see that f x = 8 x + 2 xy .

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