(cid:2)c 2008byW.H.FreemanandCompany ISBN-13:978-0-7167-9866-8 ISBN-10:0-7167-9866-2 Allrightsreserved PrintedintheUnitedStatesofAmerica W.H.FreemanandCompany,41MadisonAvenue,NewYork,NY10010 Houndmills,BasingstokeRG216XS,England www.whfreeman.com Student’s Solutions Manual to accompany Jon Rogawski’s Single Variable CALCULUS BRIAN BRADIE Christopher Newport University With Chapter 12 contributed by GREGORY P. DRESDEN Washington and Lee University and additional contributions by Art Belmonte Cindy Chang-Fricke Benjamin G. Jones Kerry Marsack Katherine Socha Jill Zarestky Kenneth Zimmerman W. H. Freemanand Company New York This page intentionally left blank CONTENTS Chapter1 PrecalculusReview 1 5.6 SubstitutionMethod 276 ChapterReviewExercises 283 1.1 RealNumbers,Functions,andGraphs 1 1.2 LinearandQuadraticFunctions 8 Chapter6 Applicationsof the Integral 292 1.3 TheBasicClassesofFunctions 13 1.4 TrigonometricFunctions 16 6.1 AreaBetweenTwoCurves 292 1.5 Technology:CalculatorsandComputers 23 6.2 SettingUpIntegrals:Volume,Density,AverageValue 300 ChapterReviewExercises 26 6.3 VolumesofRevolution 308 6.4 TheMethodofCylindricalShells 316 Chapter2 Limits 31 6.5 WorkandEnergy 324 ChapterReviewExercises 331 2.1 Limits,RatesofChange,andTangentLines 31 2.2 Limits:ANumericalandGraphicalApproach 36 Chapter7 The ExponentialFunction 338 2.3 BasicLimitLaws 45 2.4 LimitsandContinuity 48 7.1 Derivativeofbx andtheNumbere 338 2.5 EvaluatingLimitsAlgebraically 55 7.2 InverseFunctions 344 2.6 TrigonometricLimits 59 7.3 LogarithmsandTheirDerivatives 349 2.7 IntermediateValueTheorem 65 7.4 ExponentialGrowthandDecay 357 2.8 TheFormalDefinitionofaLimit 68 7.5 CompoundInterestandPresentValue 362 ChapterReviewExercises 72 7.6 ModelsInvolving y(cid:3)=k(y−b) 365 7.7 L’Hoˆpital’sRule 369 Chapter3 Differentiation 79 7.8 InverseTrigonometricFunctions 377 7.9 HyperbolicFunctions 386 3.1 DefinitionoftheDerivative 79 ChapterReviewExercises 392 3.2 TheDerivativeasaFunction 90 3.3 ProductandQuotientRules 99 Chapter8 Techniquesof Integration 404 3.4 RatesofChange 105 3.5 HigherDerivatives 112 8.1 NumericalIntegration 404 3.6 TrigonometricFunctions 118 8.2 IntegrationbyParts 415 3.7 TheChainRule 123 8.3 TrigonometricIntegrals 426 3.8 ImplicitDifferentiation 132 8.4 TrigonometricSubstitution 435 3.9 RelatedRates 141 8.5 TheMethodofPartialFractions 449 ChapterReviewExercises 148 8.6 ImproperIntegrals 469 ChapterReviewExercises 487 Chapter4 Applicationsof the Derivative 159 Chapter9 FurtherApplicationsof the Integral 4.1 LinearApproximationandApplications 159 and TaylorPolynomials 503 4.2 ExtremeValues 165 4.3 TheMeanValueTheoremandMonotonicity 173 9.1 ArcLengthandSurfaceArea 503 4.4 TheShapeofaGraph 180 9.2 FluidPressureandForce 513 4.5 GraphSketchingandAsymptotes 188 9.3 CenterofMass 518 4.6 AppliedOptimization 202 9.4 TaylorPolynomials 526 4.7 Newton’sMethod 216 ChapterReviewExercises 538 4.8 Antiderivatives 221 ChapterReviewExercises 228 Chapter10 Introductionto DifferentialEquations 545 Chapter5 The Integral 237 10.1 SolvingDifferentialEquations 545 5.1 ApproximatingandComputingArea 237 10.2 GraphicalandNumericalMethods 558 5.2 TheDefiniteIntegral 250 10.3 TheLogisticEquation 563 5.3 TheFundamentalTheoremofCalculus,PartI 259 10.4 First-OrderLinearEquations 568 5.4 TheFundamentalTheoremofCalculus,PartII 266 ChapterReviewExercises 578 5.5 NetorTotalChangeastheIntegralofaRate 273 iv CALCULUS CONTENTS Chapter11 InfiniteSeries 587 Chapter12 Parametric Equations, Polar Coordinates,and 11.1 Sequences 587 Conic Sections 674 11.2 SumminganInfiniteSeries 598 11.3 ConvergenceofSerieswithPositiveTerms 608 12.1 ParametricEquations 674 11.4 AbsoluteandConditionalConvergence 622 12.2 ArcLengthandSpeed 690 11.5 TheRatioandRootTests 628 12.3 PolarCoordinates 698 11.6 PowerSeries 636 12.4 AreaandArcLengthinPolarCoordinates 711 11.7 TaylorSeries 646 12.5 ConicSections 721 ChapterReviewExercises 660 ChapterReviewExercises 732 PRECALCULUS 1 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|=aand|−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)isnegativebutthe y-coordinateispositive,thepoint(−3,2)liesin thesecondquadrant. (c) Becausethex-coordinateofthepoint(4,−3)ispositivebutthe y-coordinateisnegative,thepoint(4,−3)liesin thefourthquadrant. (d) Because both the x- and y-coordinates of the point (−4,−1) are negative, the point (−4,−1) lies in the third quadrant. 6. Whatistheradiusofthecirclewithequation(x−9)2+(y−9)2=9? SOLUTION Thecirclewithequation(x−9)2+(y−9)2=9hasradius3. 7. Theequation f(x)=5hasasolutionif(chooseone): (a) 5belongstothedomainof f. (b) 5belongstotherangeof f. SOLUTION Thecorrectresponseis(b):theequation f(x)=5hasasolutionif5belongstotherangeof f. 8. Whatkindofsymmetrydoesthegraphhaveif f(−x)=−f(x)? SOLUTION If f(−x)=−f(x),thenthegraphof f issymmetricwithrespecttotheorigin. Exercises 1. Useacalculatortofindarationalnumberr suchthat|r−π2|<10−4. SOLUTION r mustsatisfyπ2−10−4 0 1 1 SOL(UdT)IO3aN<|3xb|≤2 (e) −4a<−4b (f) < a b 5. (0,4) (−4,4) 2 CHAPTER 1 PRECALCULUS REVIEW 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. InExerc(i−se2s,98–)12,writetheinequalityintheforma2} SOLUTION x−4>2orx−4<−2⇒x >6orx <2⇒(−∞,2)∪(6,∞) 21. {x :{x|x:2|−2x1+|>4|2>} 3} √ √ SOLUTIO√N x2√−1>2orx2−1<−2⇒x2 >3orx2 <−1(thiswillneverhappen)⇒x > 3orx <− 3⇒ (−∞,− 3)∪( 3,∞). 23. Matchtheinequalities(a)–(f)withthecorrespondingstatements(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 3isequivalenttothenumberstothe rightof3:(i). (b) |a−5|measuresthedistancefroma to5;hence,|a−5| < 1 issatisfiedbythosenumberslessthan 1 ofaunit 3 3 from5:(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) Theintervaldescribedbytheinequality1b,(cid:3)thenb−1 >a−1(cid:4),providedthataandbhavethesamesign.Whathappensifa>0andb <0? x SOLUTIDOeNscriCbeastehe1ase:tIfaxa:nxd+ba1reb⇒1> ab ⇒ b1 > a1. Case 1b: If a and b are both negative, then a > b ⇒ 1 < b (since a is negative) ⇒ 1 > 1 (again, since b is a b a negative). Case2:Ifa>0andb <0,then 1 >0and 1 <0so 1 < 1.(SeeExercise2fforanexampleofthis). a b b a 27. ShoWwhtihcahtixfs|aat−isf5y||x<−123a|n0 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.

Bradie B. Single Variable Calculus: Student's Solutions Manual to accompany Jon Rogawski's

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