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APÊNDICES A79 13. f (x, y) � x2y3 fx (x, y) � 2xy3 fy (x, y) � 3x2y2 15. fx(x, y) � �3y, fy(x, y) � 5y4 �3x 17. fx(x, t) � �pe�t sen px, ft(x, t) � �e�t cos px 19. �z/�x � 20(2x � 3y)9, �z/�y � 30(2x � 3y)9 21. fx(x, y) � 1/y, fy(x, y) � �x/y2 23. fx(x, y) � , fy(x, y) � 25. tu(u, v) � 10uv(u2v � v3)4 gv(u, v) � 5(u2 � 3v2)(u2v � v3)4 27. �w/�a � cos a cos b, �w/�b � �sen a sen b 29. Fx(x, y) � cos(ex), Fy(x, y) � � cos(ey) 31. fx � z � 10xy3z4, fy � �15x2y2z4, fz � x � 20x2y3z3 33. �w/�x � 1/(x � 2y � 3z), �w/�y � 2/(x � 2y � 3z), �w/�z � 3/(x � 2y � 3z) 35. �u/�x � y sen�1(yz), �u/�y � x sen�1(yz) � xyz/√–––––1 � y2z2––, �u/�z � xy2/√ ––––– 1 � y2z2 –– 37. hx � 2xy cos(z/t), hy � x2 cos(z/t), hz � (�x2y/t) sen(z/t), ht � (x2yz/t2) sen(z/t) 39. �u/�xi � xi/√ ––––– x1 2 � x2 2 � ––––– . . . ––––– � xn 2 41. 43. 45. fx(x, y) � y2 � 3x2y, fy(x, y) � 2xy � x3 47. �� , � � 49. � , � 51. (a) f �(x), t�(y) (b) f �(x � y), f �(x � y) 53. fxx � 6xy5 � 24x2y, fxy � 15x2y4 � 8x3 � fyx, fyy � 20x3y3 55. wuu � v2/(u2 � v2)3/2, wuv � �uv/(u2 � v2)3/2 � wvu, wvv � u2/(u2 � v2)3/2 57. zxx � �2x/(1 �x2)2, zxy � 0 � zyx, zyy � �2y/(1 �y2)2 63. 24xy2 � 6y, 24x2y � 6x 65. (2x2y2z5 � 6xyz3 � 2z)exyz² 67. ueru(2 sen u � u cos u � ru sen u) 69. 4/(y � 2z)3, 0 71. 6yz2 73. � 12,2, � 16,8, � 23,25 83. R2/R12 87. �� , � � 93. Não 95. x � 1 � t, y � 2, z � 2 � 2t 99. �2 101. (a) (b) fx(x, y) � , fy(x, y) � (c) 0, 0 (e) Não, uma vez que fxy e fyx não são contínuas. EXERCÍCIOS 14.4 1. z � � 7x � 6y � 5 3. x � y � 2z � 0 5. x � y � z � 0 7. 9. 11. 6x � 4y � 23 13. x � y � 15. 1 � py 19. 6,3 21. x � y � z; 6,9914 23. 2T � 0,3H � 40,5; 44,4ºC 25. dz � �2e�2x cos 2pt dx � 2pe�2x sen 2pt dt 27. dm � 5p4q3dp � 3p5q2dq 29. dR � b2 cos g da � 2ab cos gdb � ab2 sen g dg 31. Δz � 0,9225, dz � 0,9 33. 5,4 cm2 35. 16 cm3 37. � �0,0165mg; decresce 39. � 0,059 ⏐ 41. 2,3% 43. e1 � Δx, e2 � Δy EXERCÍCIOS 14.5 1. (2x � y) cos t � (2y � x)et 3. [(x/t) � y sen t]/ √–––––1 � x2 � y2–––– 5. ey/z [2t � (x/z) � (2xy/z2)] 7. �z/�s � 2xy3 cos t � 3x2 y2 sen t, �z/�t � �2sxy3 sen t � 3sx2y2 cos t 9. �z/�s � t2 cos u cos f � 2st sen u sen f, �z/�t � 2st cos u cos f � s2 sen u sen f 11. � er(t cos u � sen u) � er(s cos u � sen u) 13. 62 15. 7, 2 �z �t t √ ––––– s2 � t2 �z �s s √ ––––– s2 � t2 1 17 3 7 2 7 6 7 1 9 2 9 2 3 400 200 0 y5 0 _5x 10 0 _10 z 0 2 x 0 2y _1 0z 1 x4y � 4x2y3 � y5 (x2� y2)2 x5 � 4x3y2 � xy4 (x2� y2)2 _0.2 0.2 0 _1 0 1 y 1 0 _1 x z �T �P V � nb nR �P �V 2n2a V3 nRT (V � nb)2 �z �x yz ez �xy �z �y xz ez �xy �z �x x 3y �z �y 2y 3z 1 5 1 4 (ad � bc)y ���(cx � dy)2 (bc � ad)x ���(cx � dy)2 _2 0 2x 0 2 y 0 20 40 _2 z _2 _20 2x 0 2 y _20 0 20 z _2 0 2x 0 2 y _20 0 20 z _2 apendices–res2:calculo7 5/25/13 11:55 AM Page A79
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