# The origin is not such a point so assume that the

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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 &lt; 0; moving up increases f , so fy &gt; 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−...
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## This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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