short_test_2_soln

# short_test_2_soln - Math 237 1. Short Answer Problems Short...

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Math 237 Short Test # 2 Solutions 1. Short Answer Problems [1] a) Change the equation x 2 + y 2 = 1 to polar coordinates. Solution: r = 1 [2] b) Change the equation z = r x 2 + y 2 into spherical coordinates. Solution: ρ cos φ = R ρ 2 sin 2 φ ρ cos φ = ρ sin φ tan φ = 1 φ = π 4 [3] c) State the method of Lagrange Multipliers. Solution: Evaluate f ( x, y ) at all points which satisfy one of the following: 1) f ( x, y ) = λ g ( x, y ), g ( x, y ) = k , 2) g ( x, y ) = (0 , 0), g ( x, y ) = k . 3) ( x, y ) is an end point.

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2 [4] 2. Find the image of the triangular region R with vertices (0 , 0), (0 , 2), and (2 , 2) under the mapping ( u, v ) = F ( x, y ) = ( x 2 y 2 , xy ). Solution: We have u = x 2 y 2 and v = xy . Mapping each line of the region gives LINE 1: y = 2, 0 x 2. This gives u = x 2 4 and v = 2 x x = v 2 . Thus u = v 2 4 4 and 0 v 4. LINE 2: x = 0, 0 y 2. This gives
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## This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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short_test_2_soln - Math 237 1. Short Answer Problems Short...

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