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

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56 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES 00 00 63. :L: en:z' = :L: (e"T is a geometric series with r = e"', so the series converges {::} lrl < 1 {::} lex! < 1 {::} n =O n=O 1 - 1 < e"' < 1 0 < e"' < 1 {::} x < 0. In that case, the sum of the series is _a_ = - -. 1 - r l -ex 65. After defining J, We use convert (f, parfrac); in Maple, Apart in Mathematica, or Expand Rational and . . . 0 . fi d th h I . 3n2 + 3n + 1 1 1 s~mpl~fy 10 enve to n at t e genera term IS (n 2 + n)3 = n 3 ( )3 . So the nth partial sum is . n+1 n(1 1 ) ( 1) (1 1) (1 1 ) 1 Sn = ~.:'f1 k3 - (k + 1)3 = 1 - 23 + 23 - 33 + " . + n3 - (n + 1)3 = . 1 - (n + 1)3 The series converges to lim s,. = 1. This can be confirmed by directly computing the sum using n-oo s um (f , n=l.. i nfinity) ; (in Maple), Sum [f, {n , 1 , Inf inity}] (in Mathematica), or Calculus Sum (from 1 to oo) and Simplify (in Derive). 67. For n = 1, a1 = 0 since s1 = 0. For n > 1, . n- 1 (n - 1) - 1 (n - l)n- (n + l)(n- 2) 2 an = Sn - Sn- 1 = _n _+ _l - (n- 1) + 1 = (n + l)n = -n '(n- +:-1=-) I ~ I' 1' 1 - 1/n 1 A so, nL;:l a,.,. = n~~ Sn = n~ 1 + 1/n = . 69. (a) The quantity of the drug in the body after the first tablet is 150 m g. After the second tablet, there is 150 mg plus 5% of the first 150-mg tablet, that is, (150 + 150(0.05)] mg. After the third tablet, the quantity is (150 + 150(0.05) + 150(0.05) 2 ] = 157.875 mg. After n tablets, the quantity (in mg) is = W 1 - 0.05 .. n 1 . . . 150(1 - 0.05") 3000 ( ") 150 + 150 ( 0.05) + ... + 150 ( 0.05 ) - .·We can use Formula 3 to wnte this as 1 _ 0.0 5 _(b) The number of milligrams remaining in the body in the long run is lim ( 3 ~go (1 - 0.05")] = 3 ~go (1- 0) ~ 157.895, n.-.oo . only 0.02 mg more than the amount after 3 tablets. 71. (a) The first step in the chain occurs when the local government spends D dollars. The people who receive it spend a fraction e of those D dollars, that is, D e dollars. Those who receive the De dollars spend a fraction c of it, that is, D c 2 dollars. Continuing in U1is way, we see U1at the total spending after n transactions is D(1 en) Sn = D + De + De 2 + · · · + Dc" - 1 = - by (3). 1 -c (b) lim Sn = lim D( 1 - c" ) = __..!2_ lim (1 - c") = __..!2_ [since 0 < c < 1 => lim c" = o] n.-oo n.-+oo 1 - C 1 - C n -oo 1 - C n--.oo . = D [since c + s = 1] = kD [since k = 1/ s] 8 If c = 0.8, then s = 1 - c = 0.2 and the multiplier is k = 1/ s = 5. 73. f (1 + c)- n is a geometric series with a ,;, (1 + c)- 2 and r = (1 + c)-t, so the series converges when n = 2 IC1+ c)- 1 1< 1 {::} l1 +el > 1 {::} 1 +e>1or1+e< - 1 {::} c > Oore < 7 2.Wecalculatethesumofthe © 2012 Ccnsogc Learning. All Rights Rcscrvcd, Muy not be SC4JU1cd, copied, or duplicated, or posccd to a publicly ncccssiblc website, in whole or in pan.
. . (1+ c)- 2 scn es and set 1t equal to 2: ) 1 = 2
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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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CHAPTER 11 PROBLEMS PLUS D 107 l 9.
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CHAPTER 11 PROBLEMS PLUS 0 109 x x
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112 0 CHAPTER 12 VECTORS AND THE GE
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120 D CHAPTER 12 VECTORS AND THE GE
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136 D CHA~TER 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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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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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 16.8 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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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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