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previous_final_answers - a) e 2 + 2 e 2 ( x-1) + 2 e 2 (...

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Math 237 Final F07 Answers 1. a) See text. b) Yes since Jacobian non-zero and continuous partial derivatives. c) ± 2 x sin y x 2 cos y y 2 2 xy ² . 2. a) z = 2 x - 2 y . b) 0.4 3. a) See text. b) θ = 3 π 4 , 5 π 4 , c) no. 4. a) R 2 . b) R 2 . c) ( x, y ) 6 = (0 , 0). 5. Verify. We need f differentiable. 6. Prove (max occurs at ( ± 1 3 , ± 1 3 , ± 1 3 )). 7. Max 1 2 at (1 , 1 / 2), Min 0 at ( x, 0) 0 x 2 and (0 , y ) 0 y 1. 8. 1 3 9. π 4 . 10. π 3 . Math 237 Final W08 Answers 1. a) See text. b) See text. c) Yes by the exterme value theorem. 2. a) Unit disc x 2 + y 2 1. c) NOT continuous. d) x 2 + y 2 < 1. 3.
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Unformatted text preview: a) e 2 + 2 e 2 ( x-1) + 2 e 2 ( y-1) 4. a) z xx = [2 sin v (2 x + y )+2 u cos v (3 x 2 y 3 )] (2 x + y )+[2 u cos v (2 x + y )-u 2 sin v (3 x 2 y 3 )] (3 x 2 y 3 ) + 4 u sin v + 6 xy 3 u 2 cos v . 5. (0,0) is a local min, ( 2 ,-1) are both saddle points. 6. Max 3, min-5 9 . 7. a)-2 u , c) e-1. 8. a) 35 2 , b) e 4-1 4 . 9. a), b) 1 3 . 10. 2 15 . 1...
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