46748j 062963k so cos1 vi ni vi ni 1123 611

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Unformatted text preview: t cos(x/c), zx = −(1/c)e−t sin(x/c), zxx = −(1/c2 )e−t cos(x/c); zt − c2 zxx = −e−t cos(x/c) − c2 (−(1/c2 )e−t cos(x/c)) = 0 75. ux = ω sin c ωt cos ωx, uxx = −ω 2 sin c ωt sin ωx, ut = c ω cos c ωt sin ωx, utt = −c2 ω 2 sin c ωt sin ωx; 1 1 uxx − 2 utt = −ω 2 sin c ωt sin ωx − 2 (−c2 )ω 2 sin c ωt sin ωx = 0 c c 76. (a) ∂u/∂x = ∂v/∂y = 2x, ∂u/∂y = −∂v/∂x = −2y (b) ∂u/∂x = ∂v/∂y = ex cos y , ∂u/∂y = −∂v/∂x = −ex sin y (c) ∂u/∂x = ∂v/∂y = 2x/(x2 + y 2 ), ∂u/∂y = −∂v/∂x = 2y/(x2 + y 2 ) 77. ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x so ∂ 2 u/∂x2 = ∂ 2 v/∂x∂y , and ∂ 2 u/∂y 2 = −∂ 2 v/∂y∂x, ∂ 2 u/∂x2 + ∂ 2 u/∂y 2 = ∂ 2 v/∂x∂y − ∂ 2 v/∂y∂x, if ∂ 2 v/∂x∂y = ∂ 2 v/∂y∂x then ∂ 2 u/∂x2 + ∂ 2 u/∂y 2 = 0; thus u satisfies Laplace’s equation. The proof that v satisfies Laplace’s equation is similar. Adding Laplace’s equations for u and v gives Laplaces’ equation for u + v . 78. ∂z/∂y = 6y , ∂z/∂y (2,1) =6 79. ∂z/∂x = −x 29 − x2 − y 2 −1/2 , ∂z/∂x](4,3) = −2 80. (a) ∂z/∂y = 8y , ∂z/∂y ](−1,1) = 8 (b) ∂z/∂x = 2x, ∂z/∂x](−1,1) = −2 81. (a) ∂V /∂r = 2πrh (b) ∂V /∂h = πr2 (c) ∂V /∂r]r=6, h=4 = 48π πsd2 82. (a) ∂V /∂s = √ 6 4s2 − d2 (c) ∂V /∂s]s=10, d=16 = 320π/9 (d) ∂V /∂h]r=8, h=10 = 64π (b) ∂V /∂d = (d) πd(8s2 − 3d2 ) √ 24 4s2 − d2 ∂V /∂d]s=10, d=16 = 16π/9 83. (a) P = 10T /V , ∂P/∂T = 10/V , ∂P/∂T ]T =80, V =50 = 1/5 lb/(in2 K) (b) V = 10T /P, ∂V /∂P = −10T /P 2 , if V = 50 and T = 80 then P = 10(80)/(50) = 16, ∂V /∂P ]T =80, P =16 = −25/8(in5 /lb3 ) 84. (a) ∂z/∂y = x2 , ∂z/∂y ](1,3) = 1, j + k is parallel to the tangent line so x = 1, y = 3 + t, z =3+t (b) ∂z/∂x = 2xy , ∂z/∂x](1,3) = 6, i + 6k is parallel to the tangent line so x = 1 + t, y = 3, z = 3 + 6t 85. 1+ ∂z ∂x cos(x + z ) + cos(x − y ) = 0, ∂z cos(x − y ) ∂z = −1 − ; cos(x + z ) − cos(x − y ) = 0, ∂x cos(x + z ) ∂y cos(x − y ) ∂ 2 z − cos(x + z ) sin(x − y ) + cos(x − y ) sin(x + z )(1 + ∂z/∂x) ∂z = ; = , ∂y cos(x + z ) ∂x∂y cos2 (x + z ) substitute for ∂z/∂x and simplify to get ∂2z cos2 (x + z ) sin(x − y ) + cos2 (x − y ) sin(x + z ) =− . ∂x∂y cos3 (x + z ) Exercise Set 15.4 86. ∂V /∂r = 538 21 2 πrh = ( πr2 h) = 2V /r 3 r3 87. (a) ∂T /∂x = 3x2 + 1, ∂T /∂x](1,2) = 4 (b) ∂T /∂y = 4y , ∂T /∂y ](1,2) = 8 2 2 2 2 88. ∂ 2 R/∂R1 = −2R2 /(R1 + R2 )3 , ∂ 2 R/∂R2 = −2R1 /(R1 + R2 )3 , 6 2 ∂ 2 R/∂R1 4 2 22 ∂ 2 R/∂R2 = 4R1 R2 / (R1 + R2 ) = 4/ (R1 + R2 ) 2 [R1 R2 / (R1 + R2 )] 4 = 4R2 / (R1 + R2 ) 89. 2(x + ∆x)2 − 3(x + ∆x)y + y 2 − (2x2 − 3xy + y 2 ) f (x + ∆x, y ) − f (x, y ) = = 4x + 2∆x − 3y, ∆x ∆x f (x + ∆x, y ) − f (x, y ) = lim (4x + 2∆x − 3y ) = 4x − 3y ; fx (2, −1) = 11 ∆x→0 ∆x fx = lim ∆x→0 2x2 − 3x(y + ∆y ) + (y + ∆y )2 − (2x2 − 3xy + y 2 ) f (x, y + ∆y ) − f (x, y ) = = −3x + 2y + ∆y, ∆y ∆y fy = lim f (x, y + ∆y ) − f (x, y ) = −3x + 2y ; fy (2, −1) = −8 ∆y 90. fx (x, y ) = 22 4x (x + y 2 )−1/3 (2x) = , (x, y ) = (0, 0); 3 3(x2 + y 2 )1/3 ∆y →0 fx (0, 0) = d [f (x, 0)] dx 91. (a) fy (0, 0) = = x=0 d [f (0, y )] dy d 4/3 [x ] dx = y =0 = x=0 d [y ] dy 4 1/3 x 3 = 0. x=0 =1 y =0 13 y2 (x + y 3 )−2/3 (3y 2 ) = 3 ; 3 (x + y 3 )2/3 fy (x, y ) does not exist where y = −x, x = 0. (b) If (x, y ) = (0, 0), then fy (x, y ) = EXERCISE SET 15.4 1. 42t13 2. 2(3 + t−1/3 ) 3(2t + t2/3 ) 3. 3t−2 sin(1/t) 4. 1 − 2t4 − 8t4 ln t √ 2t 1 + ln t − 2t4 ln t 5. − 10 7/3 1−t10/3 te 3 6. (1 + t)et cosh (tet /2) sinh (tet /2) 7. ∂z/∂u = 24u2 v 2 − 16uv 3 − 2v + 3, ∂z/∂v = 16u3 v − 24u2 v 2 − 2u − 3 8. ∂z/∂u = 2u/v 2 − u2 v sec2 (u/v ) − 2uv 2 tan(u/v ) ∂z/∂v = −2u2 /v 3 + u3 sec2 (u/v ) − 2u2 v tan(u/v ) 9. ∂z/∂u = − 2 sin u 2 cos u cos v , ∂z/∂v = − 3 sin v 3 sin2 v 10. ∂z/∂u = 3 + 3v/u − 4u, ∂z/∂v = 2 + 3 ln u + 2 ln v 539 Chapter 15 11. ∂z/∂u = eu , ∂z/∂v = 0 12. ∂z/∂u = − sin(u − v ) sin u2 + v 2 + 2u cos(u − v ) cos u2 + v 2 ∂ z/∂v = sin(u − v ) sin u2 + v 2 + 2v cos(u − v ) cos u2 + v 2 13. ∂T /∂r = 3r2 sin θ cos2 θ − 4r3 sin3 θ cos θ ∂T /∂θ = −2r3 sin2 θ cos θ + r4 sin4 θ + r3 cos3 θ − 3r4 sin2 θ cos2 θ 14. dR/dφ = 5e5φ 15. ∂t/∂x = x2 + y 2 / 4x2 y 3 , ∂t/∂y = y 2 − 3x2 / 4xy 4 16. ∂w/∂u = 2v 2 u2 v 2 − (u − 2v )2 2 [u2 v 2 + (u − 2v )2 ] , ∂w/∂v = u2 (u − 2v )2 − u2 v 2 17. −π 19. √ 2 [u2 v 2 + (u − 2v )2 ] 18. 351/2, −168 √ 3 3e √ √ , 2−4 3 e 3 21. F (x, y ) = x2 y 3 + cos y , 20. 1161 2xy 3 dy =− 2 2 dx 3x y − sin y 22. F (x, y ) = x3 − 3xy 2 + y 3 − 5, 23. F (x, y ) = exy + yey − 1, 24. F (x, y ) = x − (xy ) 1/2 3x2 − 3y 2 dy x2 − y 2 =...
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