quiz7-1 - 1(a(4 points Consider the function f R 2 → R 2...

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Quiz 7 Name: Math 240 - Calculus III March 31, 2009 Note: In order to receive full credit, you must show work that justifies your answer. 1. (a) (4 points) Define a function f : R 3 R 3 as the cross product of the input vector with the vector h 1 , 1 , 0 i . That is, f ( x, y, z ) = h x, y, z i × h 1 , 1 , 0 i = h- z, z, x - y i . Compute the Jacobian [ Df ] = h df i dx j i i,j . Solution : [ Df ] = ± df i dx j ² i,j = df 1 dx 1 df 1 dx 2 df 1 dx 3 df 2 dx 1 df 2 dx 2 df 2 dx 3 df 3 dx 1 df 3 dx 2 df 3 dx 3 = 0 0 - 1 0 0 1 1 - 1 0 Note that the Jacobian [ Df ] does not depend on the inputs h x, y, z i . (b) (4 points) Suppose the inputs are h x, y, z i = h 0 , 1 , 1 i and the rates of change of the inputs are h ˙ x, ˙ y, ˙ z i = h 1 , 0 , 2 i . Find the rates of change of the outputs. Solution : Df 0 1 1 ˙ x ˙ y ˙ z = 0 0 - 1 0 0 1 1 - 1 0 1 0 2 = - 2 2 1 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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Quiz 7 Name: Math 240 - Calculus III April 2, 2009 Note: In order to receive full credit, you must show work that justifies your answer.
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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 θ-r 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 = ³-1 1 ´± 4 π ² = ³-4 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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quiz7-1 - 1(a(4 points Consider the function f R 2 → R 2...

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