cal10 - choi(kc24547 HW10 BERG(56525 This print-out should...

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choi (kc24547) – HW10 – BERG – (56525) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 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 1. f x > 0 , f y > 0 2. f x < 0 , f y = 0 3. f x = 0 , f y = 0 correct 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 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 002 10.0 points Determine f x when f ( x, y ) = x cos( x + y ) - sin( x + y ) . 1. f x = - 2 x sin( x + y ) 2. f x = - x sin( x + y ) correct 3. f x = x cos( x + y ) 4. f x = 2 sin( x + y ) - x cos( x + y ) 5. f x = 2 x cos( x + y ) 6. f x = 2 sin( x + y ) + x cos( x + y ) 7. f x = 2 cos( x + y ) - x sin( x + y ) 8. f x = 2 cos( x + y ) + x sin( x + y ) Explanation: From the Product Rule we see that f x = cos( x + y ) - x sin( x + y ) - cos( x + y ) . Consequently, f x = - x sin( x + y ) . 003 10.0 points
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choi (kc24547) – HW10 – BERG – (56525) 2 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 = - 8 2. slope = - 4 correct 3. slope = - 12 4. slope = - 10 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 = 4 e - xy - 2(2 xy + y 2 ) e - xy . Consequently, at P (0 , 2 , f (0 , 2)) slope = - 4 . 004 10.0 points Determine f xy when f ( x, y ) = 2 x tan - 1 parenleftBig y x parenrightBig . 1. f xy = 4 xy 2 ( x 2 + y 2 ) 2 correct 2. f xy = 4 x 2 y ( x 2 + y 2 ) 2 3. f xy = xy x 2 + y 2 4. f xy = x 2 y x 2 + y 2 5. f xy = 4 xy x 2 + y 2 6. f xy = xy 2 ( x 2 + y 2 ) 2 Explanation: Differentiating f partially with respect to x , using also the Chain Rule and d dx tan - 1 x = 1 1 + x 2 , we see that f x = 2 tan - 1 parenleftBig y x parenrightBig - 2 x parenleftBig y x 2 parenrightBigparenleftBig 1 1 + ( y/x ) 2 parenrightBig . But after simplification, 2 x parenleftBig y x 2 parenrightBigparenleftBig 1 1 + ( y/x ) 2 parenrightBig = 2 xy x 2 + y 2 , and so f x = 2 tan - 1 parenleftBig y x parenrightBig - 2 xy x 2 + y 2 . Differentiating partially now with respect to y we see that f xy = 2 parenleftBig 1 x parenrightBigparenleftBig 1 1 + ( y/x ) 2 parenrightBig - 2 x x 2 + y 2 + 2 y parenleftBig 2 xy ( x 2 + y 2 ) 2 parenrightBig .
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