Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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88 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES • oo x2n+l 00 x2n+ 4 51 . arctanx = 2: (- 1)" - 2 1 for lxl < 1, so x 3 arctanx = 2: ( -1)"- - for Jxl < 1 and n = O n + n =O 2n+ 1 I oo x2n+5 x 3 arctanx dx = C + ,..;;:o(-1)"' ( 2 n + 1 )( 2 n + )' 5 Since ~ < 1, we have (1/2) 5 (1/2r (1/ 2) 9 · · {1/ 2) 11 1.5 - ~ + ~ ~ 0.0059 and subtracting Til ~ 6.3 x 10- 6 does not affect the fourth decimal place, so J112 0 x 3 arctan x dx ~ 0.0059 by the Alternating Series Estimation Theorem.' (1/ 2) I oo ( 1/ 2) x4n+1 n =O n , n =O n 4n + .1 53. v'1 + x4 = (1 +.x 4 ) 1 1 2 00 = 2: (x 4 )", so }1 + x 4 dx = C + L: --and hence, since 0.4 < 1, we have 1 I= } 1 +x 4 00 0.4 (1/ 2) (0.4) 4n+l dx = E o n = O n 4n + 1 (o.4) 1 ~ (o.4) 5 H-4) (o.4) 9 H-t)(-~) (o.4) 13 H-4)(-~)(-~) (o.4) 17 =(1)or+l!-5-+-2-,---9-+ 3! ~ + 4! ---rr- +· ·· (0.4) 5 (0.4) 0 (0.4) 13 5(0.4) 17 = 0.4 + ----w- - ----;:;2 + 208 - 2176 + ... Now (O~~)o ~ 3.6 x 10- 6 < 5 x 10- 6 , so by the Alternating Series Esti~ation Theorem, I ~ 0.4 + (0~~) 5 ~ 0.40102 (correct to five decimal places). x -ln(1 + x) x- (x - l x 2 + l x 3 - lx 4 + lx 5 - • • ·) lx 2 - lx 3 + l x 4 - l x 5 + · · · 55. lim = lim 2 3 ., 4 fi = lim 2 :l 4 5 x-0 x2 x-.0 x· .,_,o X 2 = lim(! - l x + lx 2 - ix3 +: .. ) = ! m-+0 2 3 4 5 2 since power series are continuous functions. sinx- x + l x 3 6 57. lim x-o x5 since power series are continuous functions. . 2 ~ ~ ~ ~ ~ _59. From Equation 1 I, we have e-x = 1 - I + - 21 - I + · · · and we know that cos x . = 1 - - 21 + - 41 - · · · from 1. . 3. . . Equation 16. Therefore, e-"' 2 cosx = (1 - x 2 + ~x 4 - ·- ·) (1 - ~x 2 + i4x 4 - - - • ). Writing only the tenns with - degree < 4 we get e-"' 2 cos x = 1 - lx 2 + .l.x 4 - x 2 + lx" + lx 4 + . ·. = 1 - !!.x 2 + ~x 4 +- ·. - ' 2 24 2 2 2 24 . © 2012 Cc.ngace Learning. All Rights Rcscn·cd. May not be SCOJ:mcd, oo picd~ or duplicat<strong>ed</strong>, or po.stcd too publiCly accessible website, in whole or in part.
61 _ x_ x · sin x - x- tx 3 + 1~ 0 x5 - · · · · x- *x3 + l ~o xs - .. . j x SECTION 11.10 TAYLOR AND MACLAURIN SERIES 0 89 1 I 2 7 4 + 6x +aGo X + · · · x - ~ x3 + rlo xs - .. . l x3 _ ....Lxs + .. . 6 120 1 3 1 5 oX - 36X + .. . From the long division above, ~ = 1 + ~x 2 + 3 ~x4 + · · ·. · I SlllX · I oo 4n oo ( - x ~~) n a 63. L; (- 1) "~ = L; - - 1 - = e- x· , by (I I). n=O n. n =O n. 3 65. ~(-1) " - J nn = ~(- 1 )" _ 1 ( 3 / 5 )" = ln( 1 + ~) [fromTable l ] = ln~ n=l n5 n = 1 n 5 5 n. (i ) ( ) 71. lfp is an n th-degree polynomial, then p
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- STUDENT SOLUTIONS MANUAL for STEW
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.. BROOKS/COLE ~ I ~~r CENGAGE Lear
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D ABBREVIATIONS AND SYMBOLS CD cu D
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viii o CONTENTS 12.4 The Cross Prod
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10 D PARAMETRIC EQUATIONS AND POLAR
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.2 CALCULUS WITH PARAMETR
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SECTION 10.3 POLAR COORDINATES 0 13
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SECTION 10 .~ POLAR COORDINATES 0 1
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SECTION 10.4 A~~S AND LENGTHS IN PO
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.5 CONIC SECTIONS 0 27 5.
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SECTION 10.5 CONIC SECTIONS 0 29 35
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x2 y2 y2 a:2 _ a2 b 61. ;_2 - - = 1
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SECTION 10.6 CONIC SECTIONS IN POLA
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CHAPTER 10 REVIEW 0 35 the length o
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Page 57 and 58: SECTION 11.1 SEQUENCES D 49 71. Sin
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Page 77 and 78: SECTION 11.6 ABSOLUTE CONVERGENCE A
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140 D CHAPTER 12 VECTORS AND THE GE
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142 D CHAPTER 12 VECTORS AND THE GE
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144 0 CHAPTER 12 VECTORS AND THE GE
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D PROBLEMS PLUS 1. Since three-dime
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l CHAPTER 12 PROBLEMS PLUS 0 149 Eq
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13 D VECTOR FUNCTIONS 13.1 Vector F
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 13.2 DERIVATIVES AND INTEGR
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SECTION 13.2 DERIVATIVES AND INTEGR
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SECTION 13.3 ARC LENGTH AND CURVATU
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SECTION 13.4 MOTION IN SPACE: VELOC
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2 SECTION 13.4 MOTION IN SPACE: VEL
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CHAPTER 13 REVIEW D 173 43. The tan
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5. f~(t 2 i +tcos 1rtj +sin 1rt k )
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CHAPTER 13 REVIEW 0 177 23. (a) r (
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180 0 CHAPTER 13 PROBLEMS PLUS (-2a
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182 0 CHAPTER 13 PROBLEMS PLUS vect
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184 0 CHAPTER 14 PARTIAL DERIVATIVE
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196 D CHAPTER 14 PARTIAL DERIVATIVE
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D PROBLEMS PLUS 1. The areas of the
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15 D MULTIPLE INTEGRALS 15.1 Double
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SECTION 15.2 ITERATED INTEGRALS D 2
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l SECTION 15.3 DOUBLE INTEGRALS OVE
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SECTION 15.3 DOUBLE INTEGRALS OVER
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SECTION 15.3 DOUBLE INTEGRALS OVER
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61 . Since m :S j(x, y) :S M, ffD m
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SECTION 15.4 DOUBLE INTEGRALS IN PO
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SECTION 15.5 APPLICATIONS OF DOUBLE
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SECTION 15.5 APPLICATIONS OF DOUBLE
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SECTION 1505 APPLICATIONS OF DOUBLE
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-I SECTION 15.6 SURFACE AREA 0 267
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SECTION 15.7 TRIPLE INTEGRALS 0 269
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SECTION 15.7 TRIPLE INTEGRALS D 271
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Therefore E = { (x, y, z) I -2 ~ X~
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43. I,. = foL foL foL k(y2 + z2)dz.
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SECTION 15.8 TRIPLE INTEGRALS IN CY
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M xv = I~1f I: I:2 6 - 3 r 2 (zK) r
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SECTION 15.9 TRIPLE INTEGRALS IN SP
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SECTION 15.9 TRIPLE INTEGRALS IN SP
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(b) The wedge in question is the sh
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SECTION 15.10 CHANGE OF VARIABLES I
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CHAPTER 15 REVIEW 0 289 15 Review C
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CHAPTER 15 REVIEW 0 291 l 9. The vo
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CHAPTER 15 REVIEW D 293 33. Using t
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49. Since u = x- y and v = x + y, x
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298 D CHAPTER 15 PROBLEMS PLUS To e
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300 0 CHAPTER 15 PROBLEMS PLUS 13.
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16 0 VECTOR CALCULUS 16.1 Vector Fi
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SECTION 16.2 LINE INTEGRALS 0 305 2
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SECTION 16.2 LINE INTEGRALS 0 307 (
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SECTION 16.2 LINE INTEGRALS D 309 3
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SECTION 16.3 THE FUNDAMENTAL THEORE
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SECTION 16.4 GREEN'S THEOREM D 313
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SECTION 16.4 GREEN'S THEOREM 0 315
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8 8 8 (b)clivF = 'V ·iF=- (x+yz) +
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SECTION 16.5 CURL AND DIVERGENCE 0
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SECTION 16.6 PARAMETRIC SURFACES AN
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that is, D = {( x, y) I x 2 + y 2 :
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SECTION 16.7 SURFACE INTEGRALS 0 32
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SECTION 16.7 SURFACE INTEGRALS 0 33
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SECTION 16.8 STOKES' THEOREM 0 333
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dS SECTION 16.9 THE DIVERGENCE THEO
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CHAPTER 16 REVIEW 0 337 27. JI 5 cu
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CHAPTER 16 REVIEW 0 339 TRUE-FALSE
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CHAPTER 16 REVIEW D 341 Alternate s
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344 0 CHAPTER 16 PROBLEMS PLUS Simi
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3.c6 0 CHAPTER 17 SECOND-ORDER DIFF
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348 0 CHAPTER 17 SECOND-ORDER DIFFE
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352 D CHAPTER 17 SECOND-ORDER DIFFE
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356 D CHAPTER 17 SECOND-ORDER DIFFE
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0 APPENDIX Appendix H Complex Numbe
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APPENDIX H COMPLEX NUMBERS 0 361 43
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Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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