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Fall 2006 - Hall's Class - Practice Exam 1

Fall 2006 - Hall's Class - Practice Exam 1 - Math 20E...

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Math 20E Practice Exam 1 Solutions October 15, 2006 1. Let f ( x, y ) = ( x 2 y, e - 2 xy , 2 x - y ). (a) (10 points) Compute the derivative matrix D f(x,y). D f ( x, y ) = ∂f 1 ∂x ∂f 1 ∂y ∂f 2 ∂x ∂f 2 ∂y ∂f 3 ∂x ∂f 3 ∂y (pg. 135) = 2 xy x 2 - 2 ye - 2 xy - 2 xe 2 xy - 2 ( x - y ) 2 2 ( x - y ) 2 (b) (10 points) Suppose g : R 2 R 2 satisfies g (1 , 1) = (0 , 2), and D g (1 , 1) = 2 1 - 1 3 . Compute D ( f g )(1 , 1) . We use the form of the Chain Rule provided on the exam reference sheet. D ( f g )(1 , 1) = D f (0 , 2) D g (1 , 1) = 2(0)(2) 0 2 - 2(2) e - 2(0)(2) - 2(0) e 2(0)(2) - 2 (0 - 2) 2 2 (0 - 2) 2 2 1 - 1 3 = 0 0 - 4 0 - 1 2 1 2 2 1 - 1 3 = 0 0 - 8 - 4 - 3 2 1 1
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2. The height h in kilometers of a mountain is given by h ( x, y ) = 2 - x 2 - 1 4 y 2 , where x and y are the east-west and north-south distances from the top of the mountain in kilometers, respectively. (a) (10 points) A hiker is at the point (1,2) and is moving in the direction of steepest descent. Find the unit vector giving her direction.
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