14.7-15.2solns

14.7-15.2solns - 3(c min-√ n max √ n 4 Calculate the...

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MATH 231 Group Work Quiz Related 1. Find the absolute maximum and minimum values of he function f on the set D . (a) f ( x,y ) = 3 + xy - x - y 2 , where D is the triangle bounded by the points (0 , 0), (2 , 0), (1 , 1). (b) f ( x,y ) = 4 x + 6 y - x 2 - y 2 , where D = { ( x,y ) | 0 x 3 , | y | ≤ 1 } . (c) f ( x,y ) = 7 - x 2 y , where D = { ( x,y ) | x 0 ,y 0 ,x 2 + y 2 3 } . Answers: (a) max: 3 min: 1 (b) max: 9 min: - 7 (c) max: 9 min: 5 2. Find the volume of the solid enclosed by the surface z = e x sin y and the planes x = ± 1, y = 0, y = π , z = - 1. Answer: 2 π + 2 e - 2 e 3. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. (a) f ( x,y,z ) = ( xyz ) 2 , where x 2 + y 2 + z 2 = 1. (b) f ( x,y,z ) = x 2 + y 2 + z 2 , where x 4 + y 4 + z 4 = 1. (c) f ( x 1 ,x 2 ,...,x n ) = x 1 + x 2 + ··· + x n , where x 2 1 + x 2 2 + ··· + x 2 n = 1. Answers: (a) min: 0, max: 1 27 . (b) min: 1, max:
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Unformatted text preview: 3. (c) min:-√ n , max: √ n . 4. Calculate the double integral. (a) ZZ R 1 + x 2 1 + y 2 dA, R = [0 , 1] × [0 , 1] (b) ZZ R x x 2 + y 2 dA, R = [1 , 2] × [0 , 1] (c) ZZ R cos x + sin y dA, R = [0 ,π ] × [0 ,π/ 2] Answers: (a) π/ 3 (b) 1 2 ln(5 / 2) + 2 tan-1 (1 / 2)-π/ 4 1 (c) π 5. Find the average value of f ( x,y ) = e y √ x + e y over the rectangle [0 , 4] × [0 , 1]. Answer: ≈ 3 . 327 6. Find the maximum and minimum volumes of a rectangular box whose surface area is 1000 cm 3 and whose total edge length is 200 cm. Answer: 1416 . 5 and 3268 . 35 2...
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This homework help was uploaded on 04/17/2008 for the course MATH 231 taught by Professor Bociu during the Spring '08 term at UVA.

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14.7-15.2solns - 3(c min-√ n max √ n 4 Calculate the...

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