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ODDREV12 - R eview Exercises for(a Maximize g sin cos cos...

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Review Exercises for Chapter 12 45. (a) Maximize subject to the constraint (b) cos cos cos cos cos cos cos sin sin g cos cos cos α β 3 3 2 3 γ g 3 , 3 . 3 1 8 3 sin cos cos cos sin cos cos cos sin tan tan tan . cos cos cos g , , Review Exercises for Chapter 12 123 1. No, it is not the graph of a function. 3. The level curves are of the form The level curves are circles centered at the origin. x y Generated by Mathematica 2 2 c = 10 c = 1 2 2 ln c x 2 y 2 . c e x 2 y 2 f x , y e x 2 y 2 5. The level curves are of the form The level curves are hyperbolas. x y Generated by Mathematica 1 4 4 c = 12 c = 12 c = 2 c = 2 4 1 1 4 1 x 2 c y 2 c . c x 2 y 2 f x , y x 2 y 2 7. x y 3 3 3 3 3 3 z f x , y e x 2 y 2 9. Elliptic paraboloid y x 2 z 2 1 x y 3 2 3 z f x , y , z x 2 y z 2 1 11. Continuous except at 0, 0 . lim x , y 1, 1 xy x 2 y 2 1 2 13. For for For for Thus, the limit does not exist. Continuous except at 0, 0 . x 0 4 x 2 y x 4 y 2 0, y 0, x 0 4 x 2 y x 4 y 2 4 x 4 x 4 x 4 2, y x 2 , lim x , y 0, 0 4 x 2 y x 4 y 2
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124 Chapter 12 Functions of Several Variables 17. z y xe y e x z x e y ye x z xe y ye x 19. g y x x 2 y 2 x 2 y 2 2 y y 2 x 2 x 2 y 2 2 g x y x 2 y 2 xy 2 x x 2 y 2 2 g x , y xy x 2 y 2 21. f z arctan y x f y z 1 y 2 x 2 1 x xz x 2 y 2 f x z 1 y 2 x 2 y x 2 yz x 2 y 2 f x , y , z z arctan y x 23. u t cn 2 e n 2 t sin nx u x cne n 2 t cos nx u x , t ce n 2 t sin nx 31. Therefore, 2 z x 2 2 z y 2 0. 2 z y 2 2 z y 2 y 2 z x 2 2 z x 2 x z x 2 y 2 33. Therefore, 2 z x 2 2 z y 2 0. 2 y 3 x 2 y 2 x 2 y 2 3 2 z y 2 x 2 y 2 2 2 y 2 x 2 y 2 x 2 y 2 2 y x 2 y 2 4 z y x 2 y 2 2 y x 2 y 2 2 x 2 y 2 x 2 y 2 2 2 z x 2 2 y 4 x 2 x 2 y 2 3 1 x 2 y 2 2 2 y 3 x 2 y 2 x 2 y 2 3 z x 2 xy x 2 y 2 2 z y x 2 y 2 15. f y e x sin y f x e x cos y f x , y e x cos y 25. x y 3 3 1 3 z 27. f yx 1 f xy 1 f yy 12 y f xx 6 f y x 6 y 2 f x 6 x y f x , y 3 x 2 xy 2 y 3 29. h yx cos y sin x h xy cos y sin x h yy x sin y h xx y cos x h y x cos y cos x h x sin y y sin x h x , y x sin y y cos x 35. dz z x dx z y dy sin y x y x cos y x dx cos y x dy z x sin y x
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Review Exercises for Chapter 12 125 37. Percentage error: dz z 17 26 13 0.0503 5% dz x z dx y z dy 5 13 1 2 12 13 1 2 17 26 0.654 cm 2 z dx 2 x dx 2 y dy z 2 x 2 y 2 39. ± 5 6 ± 1 6 ± in. 3 dV 2 3 rh dr 1 3 r 2 dh 2 3 2 5 ± 1 8 1 3 2 2 ± 1 8 V 1 3 r 2 h 41.
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