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

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l CHAPTER 14 REVIEW 0 237 EXERCISES 1. ln(x + y + 1) is defined only when x + y + 1 > 0 ~ y > - x - 1, so the domain off is {(x, y) I y > - x - 1}, all those points above the line y = -x - 1. ', ' ' ' ' ' ' ' y 3. z = f(x, y) = 1 - y 2 , a parabolic cylinder 5. The level curves are J 4x2 + y2 = k or 4x 2 + y 2 = k 2 , k ::2::: 0, a family of e ll ipses. y X 7. y 9. f is a rational function, so it is continuous on its domain. 2 I (©J (©J (©J (©J 0 I 2 X Since f is defined at (1, 1), we use direct substitution to 1 th I" . lim 2x y 2(1)(1) 2 eva uate e 1m1t: 2 2 2 = 12 + ( ) 2 1 2 = - . (z,v)-(1 ,1) X + y 3 11 . (a) T.,(6 , 4) = lim T(B + h , 4 ~ - T (B, 4 ), so we can approximateT:c (6, 4) by considering h = ± 2 and h- o using the values given in the table: Tx(6, 4) ~ T( 8 , 4 ) ; T ( 6 , 4 ) = 86 ; 80 = 3, . T.,(6, 4) ~ T( 4 , 4 ) ~ 2 T(B, 4 ) = 72 ~ 2 80 = 4. Averaging these values, we estimate Tx(6, 4) to ~e approximately 3.'5°C/ m . Similarly, Tv (6, 4} = lim T(B, 4 + h~ - T ( 6 , 4 ) , which we can approximate with h = ±2: h - 0 ,.., (B ) _ T(6, 6} - T(6, 4) _ 75 80 _ _ ,.,., ( ) _ T(6, 2) - T(6, 4) _ 87- 80 _ 4 2 5 6 4 3 5 A . th .Lv ' - 2 - - - 2 - - · ' .L v , - _ 2 - ~- - .. veragmg ese values, we estimate Tv(6, 4) to be approximately - 3.0°C/ m. ® 2012 Ccnguge teaming. AU Rights Reserved. M:ty not be scanned. copiL.-d. or duplic:atcd, or posted to o. publicly acces!iblc webs he. in whole or in p01n.
238 0 CHAPTER 14 PARTIAL DERIVATIVES (b) Here u = ( 12• )?.). so by Equation 14.6.9, Du T(6,4) = VT(6,4) · u = T, (6,4) ~ + Tv(6,4) ~·Using our estimates from part (a), we have Du T(6, 4) l':;j (3.5) ~ + ( -3.0) ~ = ~ l':;j 0.35. This means that as we move through the point (6, 4) in the direction ofu , the temperature increases at a rate of approximately 0.35°C/m. . r(6+h~,4+h~)-T(6,4 ) Alternatively, we can use Definition 14.6.2: Du T( 6, 4) = lim , h-o h which ·we can estimate with h = ± 2 ,;2. Then Du T(6, 4) ~ T( 8 , 6 ) ~T( 6 , 4 ) = 80 - ~O = 0, 2 2 2v2 D T(6 4) T(4, 2) - T(6,4) 74-80 3 . ( ) 3 0 I - 2v2 -2v2 v2 u , ~ tn = ~ = tn' Averagmg these values, we have D u T 6, 4 ~ 27z ~ 1.1 C m. () T. ( ) _ !_ [T. ( )] - lim T,('x, y +h) - T:r.(x,y) T. ( 6 4 ) _lim T., (6,4 +h) - T.,(6, 4) 1 . h C :V!/ X 1 y - a x X 1 y - h , SO xy 1 - h W liC We Can . y h-o , h-o estimate with h = ±2. We have Tx (6, 4) ~ 3.5 from part (a), but we will also need values for T.,(6, 6) and T,(6, 2). If we use h = ±2 and the values given in the table, we have T,( 6 , 6 ) l':;j T(8, 6) - T(6, 6) = 80 - 75 = 2 . 5 , T, ( 6 , 6 ) ~ T(4, 6)- T (6, 6) = 68 - 75 = 3 . 5 . 2 2 -2 -2 Averaging these values, we estimate T.,(6, 6) ~ 3.0. Similarly, Tx( 6 , 2 ) l':;j T(8, 2) ~ Tx(6, 2) = 90 ; 87 = l.S, T.,( 6 , 2 ) ~ T(4, 2) ~ 2 T(6 , 2) = 74 ~ 87 = 6 . 5 . Averaging these values, we estimate T, (6, 2) ~ 4.0. Finally, we esti~ate T., 11 (6, 4): T. (6 4) ~ T.,(6, 6)- T.,(6, 4) _ 3.0 - 3.5 _ _ 0 25 T. ( 6 4 ) ~ T.,(6, 2)- T.,(6, 4) _ 4.0 - 3.5 __ 0 25 :vy ! ~ 2 - 2 - • > X)J 1 ~ -2 - -2 - • • Averaging these values, we have Txy (6, 4) ~ -0.25. 13. f(x, y) = (5y 3 + 2x 2 y) 8 => f.,= 8(5y 3 + 2x 2·yf(4xy) = 32xy(5y 3 + 2x 2 y) 7 , fv = 8(5y 3 + 2x 2 yf (15y 2 + 2x 2 ) = (16x 2 + 120y 2 )(5y 3 + 2x 2 y) 7 17. S(u,v,w) = uarctan(vyfw) S w = u · 1 + (:JW) 2 (v · ~ w - 1 1 2 ) 1 ufo S,. = arctan(vy'W), Su = u · 1 + (vfo)2 (fo) = 1 +v2w' = 2Vw (~: v2w) 19. f(x,y) = 4x 3 - x y 2 => f x = 12x 2 - y 2 , f v = -2xy, f.,., = 24x, fv 11 = - 2_:~: , fxv = fvx = -2y ® 2012 Ccngagc Lcnming. All Rights Reserved. May not be scanned, copied. or duplicoted, or posted to t1 publicly ncccssiblc website, in '"'hole or in part.
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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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CHAPTER 10 REVIEW 0 37 EXERCISES 1.
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CHAPTER 10 REVIEW 0 39 25. x = t +
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CHAPTER 10 REVIEW 0 41 2 2 . 45. ~
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0 PROBLEMS PLUS l lt sin u dx cost
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11 . D INFINITE SEQUENCES AND SERIE
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SECTION 11.1 SEQUENCES 0 47 35. a,.
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SECTION 11.1 SEQUENCES D 49 71. Sin
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SECTION 11.2 SERIES 0 51 ak+l- a1.:
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SECTION 11.2 SERIES 0 53 13. n s.,
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SECTION 11.2 SERIES 0 55 47. F~r th
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. . (1+ c)- 2 scn es and set 1t equ
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SECTION 11.3 THE INTEGRAL TEST AND
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, SECTION 11.3 THE INTEGRAL TEST AN
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SECTION 11.4 THE COMPARISON TESTS D
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SECTION 11.5 ALTERNATING SERIES 0 6
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SECTION 11.5 ALTERNATING SERIES D 6
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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17 lim I an+l I= SECTION 11.7 STRAT
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SECTION 11.8 POWER SERIES 0 75 l 00
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SECTION 11.8 POWER SERIES 0 n (b) I
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.10 TAYLOR AND MACLAURIN
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61 _ x_ x · sin x - x- tx 3 + 1~
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SECTION 11.11 APPLICATIONS OF TAYLO
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CHAPTER 11 REVIEW 0 97 J'(xn)(xn -
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CHAPTER 11 REVIEW 0 99 oo f (n) (O)
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CHAPTER 11 REVIEW 0 101 l 23. Consi
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49./- 1 - dx = -ln{4- x) + C and 4-
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D PROBLEMS PLUS 1. It would be far
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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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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 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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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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