cal10 - choi(kc24547 – HW10 – BERG –(56525 1 This...

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Unformatted text preview: 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 > , f y > 2. f x < , f y = 0 3. f x = 0 , f y = 0 correct 4. f x < , f y < 5. f x > , f y = 0 6. f x = 0 , f y < 7. f x < , f y > 8. f x = 0 , f y > 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 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 ....
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas.

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cal10 - choi(kc24547 – HW10 – BERG –(56525 1 This...

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