M408D Quest Homework 10-solutions

# M408D Quest Homework 10-solutions - hernandez(ah29758 –...

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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 10 – pascaleff – (54550) 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 10.0 points Find the value of f x at (3 , 2) when f ( x, y ) = 2 x 3- 6 x 2 y- 5 x + 6 y. Correct answer:- 23. Explanation: After differentiation, f x = ∂f ∂x = 6 x 2- 12 xy- 5 . At (3 , 2), therefore, f x vextendsingle vextendsingle vextendsingle (3 , 2) =- 23. 002 10.0 points From the contour map of f shown below de- cide whether f x and f y are positive, negative, or zero at P . 2 2 4 4 6 6 P x y 1. f x > , f y < 2. f x < , f y > 3. f x = 0 , f y > correct 4. f x = 0 , f y < 5. f x > , f y > 6. f x < , f y < Explanation: When we walk in the x-direction from P our elevation doesn’t change because we are walking along a contour, so f x = 0. On the other hand, when we walk in the y-direction from P we are walking uphill, so f y > 0. Consequently, at P f x = 0 , f y > . keywords: contour map, contours, partial derivative, slope, 003 10.0 points 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 =- 4 2. slope = 2 3. slope =- 2 4. slope = 0 5. slope =- 6 correct 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 = 6 e − xy- 3(2 xy + y 2 ) e − xy . Consequently, at P (0 , 2 , f (0 , 2)) slope =- 6 . hernandez (ah29758) – M408D Quest Homework 10 – pascaleff – (54550) 2 004 10.0 points Determine f xy when f ( x, y ) = 1 2 x tan − 1 parenleftBig y x parenrightBig . 1. f xy = x 2 y 2( x 2 + y 2 ) 2. f xy = x 2 y ( x 2 + y 2 ) 2 3. f xy =- xy 2 2( x 2 + y 2 ) 4. f xy = xy 2 ( x 2 + y 2 ) 2 correct 5. f xy =- xy 2 ( x 2 + y 2 ) 2 6. f xy =- x 2 y 2( x 2 + y 2 ) Explanation: Since we can choose whether to differentiate with respect to x or y first, for simplicity we will choose to differentiate first with respect to y because then the algebra is simpler. Indeed, by the Chain Rule, f y = 1 2 x ∂ ∂y parenleftBig tan − 1 parenleftBig y x parenrightBigparenrightBig = 1 2 parenleftBig 1 1 + ( y/x ) 2 parenrightBig = x 2 2( x 2 + y 2 ) . Thus by the Quotient Rule, f xy = 1 2 parenleftBig 2 x ( x 2 + y 2 )- 2 x ( x 2 ) ( x 2 + y 2 ) 2 parenrightBig = 1 2 parenleftBig 2 xy 2 ( x 2 + y 2 ) 2 parenrightBig . Consequently, f xy = xy 2 ( x 2 + y 2 ) 2 . keywords: partial differentiation, mixed par- tial derivative, Chain Rule, inverse sin, Par- tialDiffMV, PartialDiffMVExam, 005 10.0 points Determine f x and f y when f ( x, y ) = 3 x 2 + 3 y 2 + x 2 y ....
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M408D Quest Homework 10-solutions - hernandez(ah29758 –...

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