ISM_T11_C14_C - Chapter 14 Practice Exercises 6 Domain All...

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Chapter 14 Practice Exercises 923 6. Domain: All points (x y z) in space ß ß Range: Nonnegative real numbers Level surfaces are ellipsoids with center (0 0 0). ß ß 7. Domain: All (x y z) such that (x y z) (0 0) ß ß ß ß Á ß !ß Range: Positive real numbers Level surfaces are spheres with center (0 0 0) and ß ß radius r 0. 8. Domain: All points (x y z) in space ß ß Range: (0 1] ß Level surfaces are spheres with center (0 0 0) and ß ß radius r 0. 9. lim e cos x e cos (2)( 1) 2 Ð ß Ñ Ä Ð ß Ñ x y ln 2 1 y ln 2 œ œ œ 1 10. lim 2 Ð ß Ñ Ä Ð ß Ñ x y 0 0 2 y x cos y 0 cos 0 2 0 œ œ 11. lim lim lim Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Á „ Á „ x y 1 1 x y 1 1 x y 1 1 x y x y x y x y x y (x y)(x y) x y 1 1 1 # " " œ œ œ œ 12. lim lim lim x y xy 1 1 1 1 1 1 3 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ x y 1 1 x y 1 1 x y 1 1 x y 1 xy 1 xy 1 (xy 1) x y xy 1 # # # # œ œ œ œ a b a b 13. lim ln x y z ln 1 ( 1) e ln e 1 P 1 1 e Ä Ð ß ß Ñ k k k k œ œ œ 14. lim tan (x y z) tan (1 ( 1) ( 1)) tan ( 1) P 1 1 1 Ä Ð ß ß Ñ " " " œ œ œ 1 4 15. Let y kx , k 1. Then lim lim which gives different limits for œ Á œ œ # Ð ß Ñ Ä Ð ß Ñ Á ß Ä Ð ß Ñ x y 0 0 y x x kx 0 0 y x y x kx 1 k kx k a b different values of k the limit does not exist. Ê 16. Let y kx, k 0. Then lim lim which gives different limits for œ Á œ œ Ð ß Ñ Ä Ð ß Ñ ß Ñ Ä Ð ß Ñ Á x y 0 0 (x kx 0 0 xy 0 x y x (kx) xy x(kx) k 1 k
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924 Chapter 14 Partial Derivatives different values of k the limit does not exist. Ê 17. Let y kx. Then lim which gives different limits for different values œ œ œ Ð ß Ñ Ä Ð ß Ñ x y 0 0 x y x y x k x 1 k x k x 1 k of k the limit does not exist so f(0 0) cannot be defined in a way that makes f continuous at the origin. Ê ß 18. Along the x-axis, y 0 and lim lim , so the limit fails to exist 1, x 0 , x 0 œ œ œ " Ð ß Ñ Ä Ð ß Ñ Ä x y 0 0 x 0 sin (x y) x y x sin x k k k k œ f is not continuous at (0 0). Ê ß 19. cos sin , r sin r cos ` ` ` ` g g r œ œ ) ) ) ) ) 20. , ` " ` # f 2x x x x y x y x y x y 1 y x y œ œ œ Š Š ˆ y x y x ` " ` # f x y x y x y x y x y 2y y x y 1 œ œ œ Š Š ˆ 1 x y x 21. , , ` " ` " ` " ` ` ` f f f R R R R R R œ œ œ 22. h (x y z) 2 cos (2 x y 3z), h (x y z) cos (2 x y 3z), h (x y z) 3 cos (2 x y 3z) x y z ß ß œ ß ß œ ß ß œ 1 1 1 1 23. , , , ` ` ` ` ` ` ` ` P RT P nT P nR P nRT n V R V T V V V œ œ œ œ 24. f (r T w) , f (r T w) , f (r T w) r T ß jß ß œ ß jß ß œ ß jß ß œ " " " " " j # j # j j 2r w r w r T T w 2 T É É ˆ Š ‹ Š 1 1 1 È È , f (r T w) w œ œ ß jß ß œ œ " " " " " " j j # j # j $Î# 4r T w 4r T w r 4r w w T T T É É É É ˆ ˆ 1 1 1 1 w 25. , 1 0, , ` ` ` ` ` ` ` ` ` ` ` ` ` ` " " g g g g g g x y y y x y y y x x y y x 2x œ œ Ê œ œ œ œ 26. g (x y) e y cos x, g (x y) sin x g (x y) e y sin x, g (x y) 0, g (x y) g (x y) cos x x y xx yy xy yx x x ß œ ß œ Ê ß œ ß œ ß œ ß œ 27. 1 y 15x , x 30x , 0, 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` # f 2x f f 2 2x f f f x x 1 y x y y x x y x 1 œ œ Ê œ œ œ œ a b 28. f (x y) 3y, f (x y) 2y 3x sin y 7e f (x y) 0, f (x y) 2 cos y 7e , f (x y) f (x y) x y xx yy xy yx y y ß œ ß œ Ê ß œ ß œ ß œ ß 3 œ 29. y cos (xy ), x cos (xy ), e , ` ` " ` ` w w dx x y dt dt t 1 dy œ œ
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