This page intentionally left blank Student’s Solutions Manual to accompany Jon Rogawski’s Single Variable CALCULUS SECOND EDITION BRIAN BRADIE Christopher Newport University ROGER LIPSETT W.H.FREEMANANDCOMPANY NEWYORK ©2012byW.H.FreemanandCompany ISBN-13:978-1-4292-4290-5 ISBN-10:1-4292-4290-6 Allrightsreserved PrintedintheUnitedStatesofAmerica First Printing W.H.FreemanandCompany,41MadisonAvenue,NewYork,NY10010 Houndmills,BasingstokeRG216XS,England www.whfreeman.com CONTENTS Chapter 1 PRECALCULUS REVIEW 1 5.4 TheFundamentalTheoremofCalculus,PartII 290 5.5 NetChangeastheIntegralofaRate 296 1.1 RealNumbers,Functions,andGraphs 1 5.6 SubstitutionMethod 300 1.2 LinearandQuadraticFunctions 8 ChapterReviewExercises 307 1.3 TheBasicClassesofFunctions 13 1.4 TrigonometricFunctions 16 1.5 Technology:CalculatorsandComputers 23 Chapter 6 APPLICATIONS OF THE INTEGRAL 317 ChapterReviewExercises 27 6.1 AreaBetweenTwoCurves 317 6.2 SettingUpIntegrals:Volume,Density,AverageValue 328 Chapter 2 LIMITS 31 6.3 VolumesofRevolution 336 6.4 TheMethodofCylindricalShells 346 2.1 Limits,RatesofChange,andTangentLines 31 6.5 WorkandEnergy 355 2.2 Limits:ANumericalandGraphicalApproach 37 ChapterReviewExercises 362 2.3 BasicLimitLaws 46 2.4 LimitsandContinuity 49 2.5 EvaluatingLimitsAlgebraically 57 2.6 TrigonometricLimits 61 Chapter 7 EXPONENTIAL FUNCTIONS 370 2.7 LimitsatInfinity 66 2.8 IntermediateValueTheorem 73 7.1 Derivativeoff(x)=bx andtheNumbere 370 2.9 TheFormalDefinitionofaLimit 76 7.2 InverseFunctions 378 ChapterReviewExercises 82 7.3 LogarithmsandTheirDerivatives 383 7.4 ExponentialGrowthandDecay 393 7.5 CompoundInterestandPresentValue 398 Chapter 3 DIFFERENTIATION 91 7.6 ModelsInvolving y(cid:2)=k(y−b) 401 7.7 L’Hôpital’sRule 407 3.1 DefinitionoftheDerivative 91 7.8 InverseTrigonometricFunctions 415 3.2 TheDerivativeasaFunction 101 7.9 HyperbolicFunctions 424 3.3 ProductandQuotientRules 112 ChapterReviewExercises 431 3.4 RatesofChange 119 3.5 HigherDerivatives 126 3.6 TrigonometricFunctions 132 Chapter 8 TECHNIQUES OF INTEGRATION 446 3.7 TheChainRule 138 3.8 ImplicitDifferentiation 147 3.9 RelatedRates 157 8.1 IntegrationbyParts 446 8.2 TrigonometricIntegrals 457 ChapterReviewExercises 165 8.3 TrigonometricSubstitution 467 8.4 IntegralsInvolvingHyperbolicandInverseHyperbolic Chapter 4 APPLICATIONS OF THE DERIVATIVE 174 Functions 481 8.5 TheMethodofPartialFractions 485 4.1 LinearApproximationandApplications 174 8.6 ImproperIntegrals 503 4.2 ExtremeValues 181 8.7 ProbabilityandIntegration 520 4.3 TheMeanValueTheoremandMonotonicity 191 8.8 NumericalIntegration 525 4.4 TheShapeofaGraph 198 ChapterReviewExercises 537 4.5 GraphSketchingandAsymptotes 206 4.6 AppliedOptimization 220 4.7 Newton’sMethod 236 Chapter 9 FURTHER APPLICATIONS OF THE 4.8 Antiderivatives 242 INTEGRAL AND TAYLOR ChapterReviewExercises 250 POLYNOMIALS 555 Chapter 5 THE INTEGRAL 260 9.1 ArcLengthandSurfaceArea 555 9.2 FluidPressureandForce 564 5.1 ApproximatingandComputingArea 260 9.3 CenterofMass 569 5.2 TheDefiniteIntegral 274 9.4 TaylorPolynomials 577 5.3 TheFundamentalTheoremofCalculus,PartI 284 ChapterReviewExercises 593 iii iv CALCULUS CONTENTS Chapter 10 INTRODUCTION TO DIFFERENTIAL 11.5 TheRatioandRootTests 690 EQUATIONS 601 11.6 PowerSeries 697 11.7 TaylorSeries 710 10.1 SolvingDifferentialEquations 601 ChapterReviewExercises 727 10.2 GraphicalandNumericalMethods 614 10.3 TheLogisticEquation 621 Chapter 12 PARAMETRIC EQUATIONS, POLAR 10.4 First-OrderLinearEquations 626 COORDINATES, AND CONIC ChapterReviewExercises 637 SECTIONS 742 Chapter 11 INFINITE SERIES 646 12.1 ParametricEquations 742 12.2 ArcLengthandSpeed 759 11.1 Sequences 646 12.3 PolarCoordinates 766 11.2 SumminganInfiniteSeries 658 12.4 AreaandArcLengthinPolarCoordinates 780 11.3 ConvergenceofSerieswithPositiveTerms 669 12.5 ConicSections 789 11.4 AbsoluteandConditionalConvergence 683 ChapterReviewExercises 801 1 PRECALCULUS REVIEW 1.1 Real Numbers, Functions, and Graphs Preliminary Questions 1. Giveanexampleofnumbersaandbsuchthata|b|. solution Takea=−3andb=1.Thena1=|b|. 2. Whichnumberssatisfy|a|=a?Whichsatisfy|a|=−a?Whatabout|−a|=a? solution Thenumbersa≥0satisfy|a|=a and|−a|=a.Thenumbersa≤0satisfy|a|=−a. 3. Giveanexampleofnumbersaandbsuchthat|a+b|<|a|+|b|. solution Takea=−3andb=1.Then |a+b|=|−3+1|=|−2|=2, but |a|+|b|=|−3|+|1|=3 +1=4. Thus,|a+b|<|a|+|b|. 4. Whatarethecoordinatesofthepointlyingattheintersectionofthelinesx =9andy =−4? solution Thepoint(9,−4)liesattheintersectionofthelinesx =9andy =−4. 5. Inwhichquadrantdothefollowingpointslie? (a) (1,4) (b) (−3,2) (c) (4,−3) (d) (−4,−1) solution (a) Becauseboththex-andy-coordinatesofthepoint(1,4)arepositive,thepoint(1,4)liesinthefirstquadrant. (b) Becausethex-coordinateofthepoint(−3,2)isnegativebutthey-coordinateispositive,thepoint(−3,2)liesin thesecondquadrant. (c) Becausethex-coordinateofthepoint(4,−3)ispositivebutthey-coordinateisnegative,thepoint(4,−3)liesin thefourthquadrant. (d) Becauseboththex-andy-coordinatesofthepoint(−4,−1)arenegative,thepoint(−4,−1)liesinthethirdquadrant. 6. Whatistheradiusofthecirclewithequation(x−9)2+(y−9)2=9? solution Thecirclewithequation(x−9)2+(y−9)2=9hasradius3. 7. Theequationf(x)=5hasasolutionif(chooseone): (a) 5belongstothedomainoff. (b) 5belongstotherangeoff. solution Thecorrectresponseis(b):theequationf(x)=5hasasolutionif5belongstotherangeoff. 8. Whatkindofsymmetrydoesthegraphhaveiff(−x)=−f(x)? solution Iff(−x)=−f(x),thenthegraphoff issymmetricwithrespecttotheorigin. Exercises 1. Useacalculatortofindarationalnumberrsuchthat|r−π2|<10−4. solution r mustsatisfyπ2−10−4 0 1 1 solu(dt)io3na<|x3|b≤2 (e) −4a<−4b (f) < a b 5. (0,4) (−4,4) solution Themidpointoftheintervalisc = (0+4)/2 = 2,andtheradiusisr = (4−0)/2 = 2;therefore,(0,4) canbeexpressedas|x−2|<2. 7. [1,5] [−4,0] solution Themidpointoftheintervalisc = (1+5)/2 = 3,andtheradiusisr = (5−1)/2 = 2;therefore,the interval[1,5]canbeexpressedas|x−3|≤2. (−2,8) 1 LTSV SSM Second Pass June7,2011 2 CHAPTER 1 PRECALCULUSREVIEW InExercises9–12,writetheinequalityintheforma2} solution x−4>2orx −4<−2⇒x >6orx < 2⇒(−∞,2)∪(6,∞) 21. {x :{x|x:2|−2x1+|>4|2>} 3} √ √ solution x2−1>2orx2−1<−2⇒x2 >3orx2 <−1(thiswillneverhappen)⇒x > 3orx < − 3⇒ √ √ (−∞,− 3)∪( 3,∞). 23. Match(a)–(f)with(i)–(vi). {x :|x2+2x|>2} 1 (a) a>3 (b) |a−5|< (cid:2) (cid:2) 3 (cid:2) (cid:2) (c) (cid:2)(cid:2)a− 1(cid:2)(cid:2)<5 (d) |a|>5 3 (e) |a−4|<3 (f) 1≤a≤5 (i) aliestotherightof3. (ii) aliesbetween1and7. (iii) Thedistancefromato5islessthan 1. 3 (iv) Thedistancefromato3isatmost2. (v) aislessthan5unitsfrom 1. 3 (vi) alieseithertotheleftof−5ortotherightof5. solution (a) Onthenumberline,numbersgreaterthan3appeartotheright;hence,a>3isequivalenttothenumberstotheright of3:(i). (b) |a−5|measuresthedistancefromato5;hence,|a−5|< 1 issatisfiedbythosenumberslessthan 1 ofaunitfrom 3 3 5:(iii). (c) |a− 1|measuresthedistancefromato 1;hence,|a− 1|<5issatisfiedbythosenumberslessthan5unitsfrom 3 3 3 1:(v). 3 (d) Theinequality|a|>5isequivalenttoa>5ora< −5;thatis,eitheraliestotherightof5ortotheleftof−5:(vi). (e) Theintervaldescribedbytheinequality|a−4|<3hasacenterat4andaradiusof3;thatis,theintervalconsists ofthosenumbersbetween1and7:(ii). (f) Theintervaldescribedbytheinequality1sebt,othferneabl−n1um>baer−s1s,aptirsofvyiidnegd|xth−at3a|a=nd|xb−ha2v|e+th1eassamaheaslifg-nin.fiWnihteatinhtaeprpveanl.sifa>0andb<0? solution Case1a:Ifaandbarebothpositive,thena>b⇒1> b ⇒ 1 > 1. a b a Case1b:Ifaandbarebothnegative,thena>b⇒1< b (sinceaisnegative)⇒ 1 > 1 (again,sincebisnegative). a b a Case2:Ifa>0andb<0,then 1 >0and 1 <0so 1 < 1.(SeeExercise2fforanexampleofthis). a b b a 29. ShoWwhtihcahtxifs|aati−sfy5|b0} InExercises49–52,1determinewheref(x)isincreasing. g(t)=cos 49. f(x)=|x+1|t solution Agraphofthefunctiony =|x+1|isshownbelow.Fromthegraph,weseethatthefunctionisincreasing ontheinterval(−1,∞). y 2 1 x −3 −2 −1 1 51. f(x)=x4 f(x)=x3 solution Agraphofthefunctiony =x4 isshownbelow.Fromthegraph,weseethatthefunctionisincreasingon theinterval(0,∞). y 12 8 4 x −2 −1 1 2 InExercises53–58,fin1dthezerosoff(x)andsketchitsgraphbyplottingpoints.Usesymmetryandincrease/decrease informaftio(xn)w=hexre4a+ppxr2op+ri1ate. 53. f(x)=x2−4 solution Zeros:±2 Increasing:x >0 Decreasing:x <0 Symmetry:f(−x)=f(x)(evenfunction).So,y-axissymmetry. y 4 2 x −2 −1 1 2 −2 −4 55. f(x)=x3−4x f(x)=2x2−4 solution Zeros:0,±2;Symmetry:f(−x)=−f(x)(oddfunction).Sooriginsymmetry. y 10 5 x −2 −1−5 1 2 −10 LTSV SSM Second Pass June7,2011

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