Practice Exam 2_answers

Practice Exam 2_answers - MATH 2401 Fall 2007 Practice Exam...

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Unformatted text preview: MATH 2401, Fall 2007 Practice Exam 2, Solutions Problem 1 . Calculations. (a) Find the directional derivative of f ( x, y, z ) = xy + yz + zx at P (1 ,- 1 , 1) in the direction of i + 2 j + k Solution: ∇ f = ( y + z ) i + ( x + z ) j + ( y + x ) k , ∇ f (1 ,- 1 , 1) = 2 j . u = √ 6 6 ( i + 2 j + k ), so f u (1 ,- 1 , 1) = ∇ f (1 ,- 1 , 1) • u = 2 3 √ 6 . (b) Find the rate of change of f ( x, y ) = xe y + ye- x along the curve r ( t ) = ( lnt ) i + t ( lnt ) j . Solution: ∇ f = ( e y- ye- x ) i + ( xe y + e- x ) j , ∇ f ( r ( t )) = ( t t- lnt ) i + ( t t lnt + 1 t ) j , df dt = ∇ f ( r ( t )) • r ( t ) = t t ( 1 t + lnt + ( lnt ) 2 ) + 1 t . (c) Find ∂u ∂s for u = x 2- xy , x = scost , y = tsins . Solution: ∂u ∂s = ∂u ∂x ∂x ∂s + ∂u ∂y ∂y ∂s = (2 x- y )( cost )+(- x )( tcoss ) = 2 scos 2 t- t sins cost- st coss cost. (d) Find dy dx if xcos ( xy ) + ycos ( x ) = 2....
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Practice Exam 2_answers - MATH 2401 Fall 2007 Practice Exam...

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