The origin is not such a point so assume that the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = fyx = 8xy (x2 − y 2 )/(x2 + y 2 )3 √ √ √ 35. (a) 2x − 2z (∂z/∂x) = 0, ∂z/∂x = x/z = ±3/(2 6) = ± 6/4, ∂z/∂x = ± 6/4 √ (b) z = ± x2 + y 2 − 1, ∂z/∂x = ±x/ x2 + y 2 − 1 = ± 6/4 √ √ 36. (a) 2y − 2z (∂z/∂y ) = 0, ∂z/∂y = y/z = ±4/(2 6) = ± 6/3 √ (b) z = ± x2 + y 2 − 1, ∂z/∂y = ±y/ x2 + y 2 − 1 = ± 6/3 37. 32 x + y2 + z2 2 38. 4x − 2x2 − y + z 3 1 − 3z 2 (∂z/∂y ) 1 − 2x2 − y + z 3 ∂z ∂z 4x − 3z 2 (∂z/∂x) = = = 1, ; = 1, 2x2 + y − z 3 ∂x 3z 2 2x2 + y − z 3 ∂y 3z 2 39. 2x + z xy z xy 1/2 2x + 2z ∂z ∂x = 0, ∂z/∂x = −x/z ; similarly, ∂z/∂y = −y/z ∂z ∂z 2x + yz 2 cos xyz ∂z + yz cos xyz + sin xyz = 0, =− ; ∂x ∂x ∂x xyz cos xyz + sin xyz ∂z ∂z xz 2 cos xyz ∂z + xz cos xyz + sin xyz = 0, =− ∂y ∂y ∂y xyz cos xyz + sin xyz 40. exy (cosh z ) exy (cosh z ) ∂z z 2 − yexy sinh z ∂z ∂z + yexy sinh z − z 2 − 2xz = 0, = xy ; ∂x ∂x ∂x e cosh z − 2xz ∂z xexy sinh z ∂z ∂z + xexy sinh z − 2xz = 0, = − xy ∂y ∂y ∂y e cosh z − 2xz 535 Chapter 15 41. III is a plane, and its partial derivatives are constants, so III cannot be f (x, y ). If I is the graph of z = f (x, y ) then (by inspection) fy is constant as y varies, but neither II nor III is constant as y varies. Hence z = f (x, y ) has II as its graph, and as II seems to be an odd function of x and an even function of y , fx has I as its graph and fy has III as its graph. 42. Moving to the right from (x0 , y0 ) decreases f (x, y ), so fx < 0; moving up increases f , so fy > 0. 43. (a) 30xy 4 − 4 (b) 60x2 y 3 (c) 60x3 y 2 44. (a) 120(2x − y )2 (b) −240(2x − y )2 (c) 480(2x − y ) 45. (a) fxyy (0, 1) = −30 (b) fxxx (0, 1) = −125 (c) fyyxx (0, 1) = 150 46. (a) ∂3w ∂3w = −ey sin x, 2 ∂x ∂y ∂y 2 ∂x 3 (b) 47. (a) √ = −1/ 2 (π/4,0) √ = −1/ 2 3 ∂w ∂w = −ey cos x, 2 ∂y ∂x ∂x2 ∂y ∂3f ∂x3 (b) (π/4,0) ∂3f ∂y 2 ∂x (b) fxxxx 48. (a) fxyy (c) ∂4f ∂x2 ∂y 2 (d) (c) fxxyy ∂4f ∂y 3 ∂x (d) fyyyxx 49. (a) 2xy 4 z 3 + y (d) 2y 4 z 3 + y (b) 4x2 y 3 z 3 + x (e) 32z 3 + 1 (c) 3x2 y 4 z 2 + 2z (f ) 438 50. (a) 2xy cos z (d) 4y cos z (b) x2 cos z (e) 4 cos z (c) −x2 y sin z (f ) 0 51. fx = 2z/x, fy = z/y , fz = ln(x2 y cos z ) − z tan z 52. fx = y −5/2 z sec(xz/y ) tan(xz/y ), fy = −xy −7/2 z sec(xz/y ) tan(xz/y ) − (3/2)y −5/2 sec(xz/y ), fz = xy −5/2 sec(xz/y ) tan(xz/y ) 53. fx = −y 2 z 3 / 1 + x2 y 4 z 6 , fy = −2xyz 3 / 1 + x2 y 4 z 6 , fz = −3xy 2 z 2 / 1 + x2 y 4 z 6 √ z sinh x2 yz cosh x2 yz , fy = 2x2 z cosh √ z sinh x2 yz cosh x2 yz , √ √ fz = 2x2 y cosh z sinh x2 yz cosh x2 yz + (1/2)z −1/2 sinh z sinh2 x2 yz 54. fx = 4xyz cosh 55. ∂w/∂x = yzez cos xz , ∂w/∂y = ez sin xz , ∂w/∂z = yez (sin xz + x cos xz ) 2 56. ∂w/∂x = 2x/ y 2 + z 2 , ∂w/∂y = −2y x2 + z 2 / y 2 + z 2 , ∂w/∂z = 2z y 2 − x2 / y 2 + z 2 57. ∂w/∂x = x/ x2 + y 2 + z 2 , ∂w/∂y = y/ x2 + y 2 + z 2 , ∂w/∂z = z/ x2 + y 2 + z 2 58. ∂w/∂x = 2y 3 e2x+3z , ∂w/∂y = 3y 2 e2x+3z , ∂w/∂z = 3y 3 e2x+3z 59. (a) e (b) 2e (c) e 2 Exercise Set 15.3 536 √ 60. (a) 2/ 7 62. √ (b) 4/ 7 -2 -1 0 √ (c) 1/ 7 -2 y 1 z 63. (3/2) x2 + y 2 + z 2 + w2 and ∂w/∂z = −z/w 1 2x + 2w ∂w ∂x 2 6 2 1/2 y 0 -2 -1 0 x 1 0 2 1 z0 -1 2 -1 0 1 -1 x -2 = 0, ∂w/∂x = −x/w; similarly, ∂w/∂y = −y/w 64. ∂w/∂x = −4x/3, ∂w/∂y = −1/3, ∂w/∂z = (2x2 + y − z 3 + 3z 2 + 3w)/3 65. yzw cos xyz ∂w xzw cos xyz ∂w xyw cos xyz ∂w =− , =− , =− ∂x 2w + sin xyz ∂y 2w + sin xyz ∂z 2w + sin xyz 66. ∂w yexy sinh w ∂w xexy sinh w ∂w 2zw =2 , =2 , = xy xy cosh w ∂y ∂x z −e z − exy cosh w ∂z e cosh w − z 2 67. (a) fxy = 15x2 y 4 z 7 + 2y (b) fyz = 35x3 y 4 z 6 + 3y 2 (c) fxz = 21x2 y 5 z 6 (d) fzz = 42x3 y 5 z 5 (e) fzyy = 140x3 y 3 z 6 + 6y (f ) fxxy = 30xy 4 z 7 (g) fzyx = 105x2 y 4 z 6 (h) fxxyz = 210xy 4 z 6 68. (a) 160(4x − 3y + 2z )3 2 69. fx = ex , fy = −ey (b) −1440(4x − 3y + 2z )2 2 (c) −5760(4x − 3y + 2z ) 22 70. fx = yex 71. ∂w/∂xi = −i sin(x1 + 2x2 + . . . + nxn ) y 72. ∂w/∂xi = 22 , fy = xex 1 n y (1/n)−1 n xk k=1 73. (a) fx = 2x + 2y, fxx = 2, fy = −2y + 2x, fyy = −2; fxx + fyy = 2 − 2 = 0 (b) zx = ex sin y − ey sin x, zxx = ex sin y − ey cos x, zy = ex cos y + ey cos x, zyy = −ex sin y + ey cos x; zxx + zyy = ex sin y − ey cos x − ex sin y + ey cos x = 0 (c) zx = zy = 2x y 2x − 2y x2 − y 2 − 2xy 1 −2 2 =2 , zxx = −2 , x2 + y 2 x 1 + (y/x)2 x + y2 (x2 + y 2 )2 2y 1 2y + 2x y 2 − x2 + 2xy 1 +2 =2 , zyy = −2 ; x2 + y 2 x 1 + (y/x)2 x + y2 (x2 + y 2 )2 zxx + zyy = −2 x2 − y 2 − 2xy y 2 − x2 + 2xy −2 =0 2 + y 2 )2 (x (x2 + y 2 )2 537 Chapter 15 74. (a) zt = −e−t sin(x/c), zx = (1/c)e−t cos(x/c), zxx = −(1/c2 )e−t sin(x/c); zt − c2 zxx = −e−t sin(x/c) − c2 (−(1/c2 )e−t sin(x/c)) = 0 (b) zt = −e−...
View Full Document

This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

Ask a homework question - tutors are online