H12 - macias(ygm97 u2013 Homework 12 u2013 staron...

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macias (ygm97) – Homework 12 – staron – (53940) 1 This print-out should have 31 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 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 we are walking downhill, so f y < 0. Consequently, at P f x > 0 , f y < 0 . keywords: contour map, slope, partial deriva- tive, 002 10.0points Determine f x - f y when f ( x, y ) = 4 x 2 + 2 xy - 3 y 2 + 3 x + y . 1. f x - f y = 6 x + 8 y + 4 2. f x - f y = 10 x - 4 y + 2 3. f x - f y = 10 x + 8 y + 4 4. f x - f y = 6 x + 8 y + 2 correct 5. f x - f y = 10 x - 4 y + 4 6. f x - f y = 6 x - 4 y + 2 Explanation: After differentiation we see that f x = 8 x + 2 y + 3 , f y = 2 x - 6 y + 1 . Consequently, f x - f y = 6 x + 8 y + 2 . 003 10.0points Determine f x + f y when f ( x, y ) = x 2 + 4 xy - y 2 - 3 x + y . 1. f x + f y = 6 x + 6 y - 2 2. f x + f y = 6 x + 2 y - 2 correct 3. f x + f y = 6 x + 2 y - 4 4. f x + f y = - 2 x + 6 y - 4 5. f x + f y = - 2 x + 2 y - 4 6. f x + f y = - 2 x + 6 y - 2 Explanation: After differentiation we see that f x = 2 x + 4 y - 3 , f y = 4 x - 2 y + 1 .
macias (ygm97) – Homework 12 – staron – (53940) 2 Consequently, f x + f y = 6 x + 2 y - 2 . 004 10.0points Determine f x when f ( x, y ) = cos(3 y - x ) - x sin(3 y - x ) . 1. f x = - 2 sin(3 y - x ) - x cos(3 y - x ) 2. f x = - x cos(3 y - x ) 3. f x = - cos(3 y - x ) - x sin(3 y - x ) 4. f x = x cos(3 y - x ) correct 5. f x = - x sin(3 y - x ) 6. f x = 2 sin(3 y - x ) - x cos(3 y - x ) 7. f x = x sin(3 y - x ) 8. f x = x cos(3 y - x ) - sin(3 y - x ) Explanation: From the Product Rule we see that f x = sin(3 y - x ) - sin(3 y - x )+ x cos(3 y - x ) . Consequently, f x = x cos(3 y - x ) . 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 ) = 5( y 2 - x 2 ) ln( x + y ) . 1. slope = 10 correct 2. slope = 8 3. slope = 14 4. slope = 12 5. slope = 6 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 = 5 parenleftbigg - 2 x ln( x + y ) + y 2 - x 2 x + y parenrightbigg . Consequently, at P (0 , 2 , f (0 , 2)) slope = 2 × 5 = 10 . 006 10.0points Find the slope in the x -direction at the point (2, - 2, 0) on the graph of f when f ( x, y ) = xy 2 + x 2 y. Correct answer: - 4. Explanation: The graph of f is a surface in 3-space and the slope in the x -direction at the point (2, - 2, 0) on that surface is the value of the partial derivative f x at (2 , - 2). Now f x = y 2 + 2 xy, so f x vextendsingle vextendsingle vextendsingle (2 , 2) = - 4 . Thus slope = - 4. 007 10.0points Determine f y when f ( x, y ) = (2 x 2 + y )( x - y 2 ) .
macias (ygm97) – Homework 12 – staron – (53940) 3 1. f y = - 2 y 2 + x 2 y + x 2. f y = - 3 y 2 - 4 x 2 y + x correct 3. f y = 2 y 2 - x 2 y + 2 x 4. f y = 3 y 2 - x 2 y - 2 x 5. f y = - 3 y 2 + 4 x 2 y + x 6. f y = 2 y 2 - 4 x 2 y - x Explanation: Differentiating f ( x, y ) with respect to y holding x fixed, we see that f y = ( x - y 2 ) - 2 y (2 x 2 + y ) .

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