short_test_2_soln (2)

short_test_2_soln (2) - Math 237 Short Test 2 Solutions 1...

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Unformatted text preview: Math 237 Short Test # 2 Solutions 1. Short Answer Problems [2] a) State the definition of a critical point of f : R 2 → R . Solution: A point ( a, b ) is a critical point of f if f x ( a, b ) = 0 = f y ( a, b ). [3] b) State the transformation from spherical coordinates to Cartesian coordinates. Solution: x = ρ cos θ sin φ y = ρ sin θ sin φ z = ρ cos φ [2] c) Convert z = r 2 sin θ from cylindrical coordinates to Cartesian coordinates. Solution: We have z = r 2 sin θ z = r ( r sin θ ) z = y radicalbig x 2 + y 2 [4] 2. Consider the two curves in polar coordinates r = sin θ and r = cos θ . Sketch both curves on the same axis and find the area inside both curves. Solution: Sketching the graphs we get the diagram below. Drawing sectors from the origin we see that from 0 ≤ θ ≤ π 4 we get the area is bounded by r = sin θ . From π 4 ≤ θ ≤ π 2 the area is bounded by r = cos θ . Thus we get A = integraldisplay π/ 4 1 2 (sin θ ) 2 dθ + integraldisplay π/ 2 π/ 4 1 2 (cos θ ) 2 dθ = bracketleftbigg 1 4 θ − 1 8 sin 2 θ bracketrightbigg π/ 4 + bracketleftbigg 1 4 θ + 1 8 sin 2 θ bracketrightbigg π/ 2 π/ 4 = π 8 − 1 4 NOTE: This could also have been done by using symmetry.. i.e. A = 2 integraltext π/ 4 1 2 (sin θ ) 2 dθ . 2 [4] 3.3....
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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 (2) - Math 237 Short Test 2 Solutions 1...

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