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Unformatted text preview: 1. (a) (4 points) Consider the function f : R 2 → R 2 deﬁned f ± r θ ² = ± r cos θ r sin θ ² Compute the Jacobian [ Df ] = h df i dx j i i,j . Solution : [ Df ] = ³ df i dx j ´ i,j = " df 1 dx 1 df 1 dx 2 df 2 dx 1 df 2 dx 2 # = ³ cos θr sin θ sin θ r cos θ ´ (b) (4 points) Suppose the inputs are h r, θ i = h 1 , π 2 i and the rates of change of the inputs are h ˙ r, ˙ θ i = h , 4 π i . Find the rates of change of the outputs. Solution : ³ Df ± 1 π 2 ²´± ˙ r ˙ θ ² = ³1 1 ´± 4 π ² = ³4 π ´ Let x = r cos θ and y = r sin θ . Then ˙ x =4 π 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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 Spring '07
 Cremins
 Calculus, Linear Algebra, Algebra, Derivative

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