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Unformatted text preview: 1. (a) (4 points) Consider the function f : R 2 R 2 dened f r = r cos r sin Compute the Jacobian [ Df ] = h df i dx j i i,j . Solution : [ Df ] = df i dx j i,j = " df 1 dx 1 df 1 dx 2 df 2 dx 1 df 2 dx 2 # = cos -r sin sin r cos (b) (4 points) Suppose the inputs are h r, i = h 1 , 2 i and the rates of change of the inputs are h r, i = h , 4 i . Find the rates of change of the outputs. Solution : Df 1 2 r = -1 1 4 = -4 Let x = r cos and y = r sin . Then x =-4 and y = 0. 2. (2 points) Write at least one thing you learned from the midterm last week. Solution : Exams can be learning experiences just like homework and quizzes! What did you learn?...
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