ISM_T11_C14_C

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 ßß Range: Nonnegative real numbers Level surfaces are ellipsoids with center (0 0 0). 7. Domain: All (x y z) such that (x y z) (0 0) ßß Á ß ! ß Range: Positive real numbers Level surfaces are spheres with center (0 0 0) and radius r 0. ± 8. Domain: All points (x y z) in space Range: (0 1] ß Level surfaces are spheres with center (0 0 0) and radius r 0. ± 9. lim e cos x e cos (2)( 1) 2 ÐßÑÄÐß Ñ xy ln2 1 yl n 2 œœ ² œ ² 1 10. lim 2 ÐßÑÄÐßÑ 00 2y xc o s y 0c o s 0 20 ± ±± ± 11. lim lim lim Á„ 11 x y (x y)(x y) x y 1 1 1 ²² ± ± ± # "" ## œ œ 12. lim lim lim x y xy 1 1 1 1 1 1 3 1 xy 1 xy 1 ( x y1 ) x y x $$ ² ²± ± # # ³ ³ œ ³ ³ œ ab †† 13. lim ln x y z ln 1 ( 1) e ln e 1 P1 1 e ÄÐ ß±ßÑ kk k k ³³ œ ³²³ œ œ 14. lim tan (x y z) tan (1 ( 1) ( 1)) tan ( 1) P 111 ÄÐ ß± ß± Ñ ²" ²" ²" ³²³² œ ²œ² 1 4 15. Let y kx , k 1. Then lim lim which gives different limits for œÁ œ œ # ² Á ßÄ Ð ß Ñ yx xkx # # y xk x 1k kx k # # # 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 œ ß ÑÄÐßÑ Á (xkx xy 0 x( k x ) xy x(kx) 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 œœ œ ÐßÑÄÐßÑ xy 00 xk x 1k x ## # # # # ± ²² ² ±± of k the limit does not exist so f(0 0) cannot be defined in a way that makes f continuous at the origin. Êß 18. Along the x-axis, y 0 and lim lim , so the limit fails to exist 1, x 0 , x 0 œ ± ²" ³ Ä x0 sin (x y) x sin x ± ² kk k k œ f is not continuous at (0 0). 19. cos sin , r sin r cos `` gg r œ´ œ ²´ )) ) ) ) 20. , `" `# ² ² ² ² ± ² ± f2 x x xx y x y x y x y 1 yx y œ ´ œ²œ Š‹ # ˆ‰ y x y x # ² ² ² ² ² ² fx y x y x y x y 2y y x y 1 œ ´ œ´œ # 1 x y x 21. , , ``` fff RRR "#$ ### œ² 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) z ßß œ ´ ² ßß œ² 11 1 1 23. , , , ```` P RT P nT P nR P nRT nV RV TV V V œœœœ ² # 24. f (r T w) , f (r T w) , f (r T w) r T ßjß ß œ "" " " " j# j j 2r w r w r TT w 2T ÉÉ 1 ÈÈ , f (r T w) w ß j ß ß œ ² œ ² " " " " jj # j # j ±$Î# 4r T w 4r T w r 4r w w TT T É É ˆ‰ ˆ 1 1 w 25. , 1 0, , ` ` ` ` ` ` " " g g y y x y yy x x y y x2 x ² Ê œ œ ² # $# # # 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 ßœ´ ßœ Ê ßœ² ßœ ßœ 27. 1 y 15x , x 30x , 0, 1 ` ± ` ` ` ` ` ` ` ` # ² x f f 2 2 x f f f x x 1 y x y x1 œ´² ´ œ Ê œ² ´ œ œ œ # # # # # # ab 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 ² ßœ ²² ´ Ê ßœ ßœ² ´ ßœ ß 3 29. y cos (xy ), x cos (xy ), e , " ² ww d x x y dt dt t 1 dy œ œ t [y cos (xy )]e [x cos (xy )] ; t 0 x 1 and y 0 Êœ ´ ´ ´ œ Ê œ œ dw dt t 1 t " ² 0 1 [1 ( 1)] 1 ´ ² œ ² ¸ˆ dw dt 0 1 t0 œ †† " ² 30. e , xe sin z, y cos z sin z, t , 1 , ` " ` ±"Î# w d x d z z d t d tt d t dy ´ œ ´ œ œ ´ œ 1 e t xe sin z 1 (y cos z sin z) ; t 1 x 2, y 0, and z ´ ´ ´ ´ ´ œ Ê œ dw dt t ±"Î# " 1 1 (2 1 0)(2) (0 0) 5 ´ ² ´ ´ œ ¸ dw dt t1 œ 1 31. 2 cos (2x y), cos (2x y), 1, cos s, s, r ` ` ` ` ` ` x x r s r s œ ² ² œ œ
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This homework help was uploaded on 09/23/2007 for the course MATH 1910 taught by Professor Berman during the Spring '07 term at Cornell.

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

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