Calculus 2nd Edition - Single Variable Solutions

816 Pages·2011·15.09 MB·English

by  Jon Rogawski

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<band|a|>|b|.
solution Takea=−3andb=1.Thena<bbut|a|=3 >1=|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 <r <π2+10−4,or9.869504<r <9.869705. r =9.8696= 12337 would
1250
beonesuchnumber.
InExercWishesic3h–o8f,(eax)p–r(efs)satrheetriunetefrovraali=nte−r3masnodfabn=in2e?qualityinvolvingabsolutevalue.
3. [(−a)2,a2<] b (b) |a|<|b| (c) ab>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
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2 CHAPTER 1 PRECALCULUSREVIEW
InExercises9–12,writetheinequalityintheforma<x <b.
9. |x|<8
solution −8<x <8
11. |2x+1|<5
|x−12|<8
solution −5<2x+1<5so−6 <2x <4and−3<x <2
InExerc|3isxes−134–|1<8,2expressthesetofnumbersxsatisfyingthegivenconditionasaninterval.
13. |x|<4
solution (−4,4)
15. |x−4|<2
|x|≤9
solution Theexpression|x−4|<2isequivalentto−2<x−4<2.Therefore,2<x <6,whichrepresentsthe
interval(2,6).
17. |4x−1|≤8
|x+7|<2
solution Theexpression|4x−1|≤8isequivalentto −8≤4x−1≤8or−7 ≤4x ≤9.Therefore,−7 ≤x ≤ 9,
4 4
whichrepresentstheinterval[−7,9].
4 4
InExerc|3isxes+195–|2<2,1describethesetasaunionoffiniteorinfiniteintervals.
19. {x :|x−4|>2}
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) Theintervaldescribedbytheinequality1<x <5hasacenterat3andaradiusof2;thatis,theintervalconsistsof
thosenumberslessthan2unitsfrom3:(iv).
25. Describe{x :(cid:3)x2+2x <3}a(cid:4)saninterval.Hint:Ploty =x2+2x−3.
x
solutiDonescrTibheeinxeq:uxal+ity1x<2+02xas<an3inisteerqvuailv.alenttox2+2x−3<0.Inthefigurebelow,weseethatthegraphof
y = x2+2x−3fallsbelowthex-axisfor−3 < x <1.Thus,theset {x : x2+2x < 3}correspondstotheinterval
−3<x <1.
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SECTION 1.1 RealNumbers,Functions,andGraphs 3
y
y = x2 + 2x − 3
10
8
6
4
2
x
−4 −3 −2 −2 1 2
27. ShoDwesthcaritbieftahe>sebt,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|b<oth12|xan−d|3b|−<82|a<nd12|,xth−en5||(<a+1?b)−13|<1.Hint:Usethetriangleinequality.
solution
|a+b−13|=|(a−5)+(b−8)|
≤|a−5|+|b−8| (bythetriangleinequality)
1 1
< + =1.
2 2
31. Supposethat|a−6|≤2and |b|≤3.
Supposethat|x−4|≤1.
(a) Whatisthelargestpossiblevalueof|a+b|?
(a) Whatisthemaximumpossiblevalueof|x+4|?
(b) Whatisthesmallestpossiblevalueof|a+b|?
(b) Showthat|x2−16|≤9.
solution |a−6|≤2guaranteesthat4 ≤a≤8,while|b|≤3guaranteesthat −3≤b≤3.Therefore1≤a+b≤11.
Itfollowsthat
(a) thelargestpossiblevalueof|a+b|is11;and
(b) thesmallestpossiblevalueof|a+b|is1.
33. ExpPrreosvser1th=at0|x.2|7−a|sya|f≤ra|cxtio−n.yH|.iHnti:nt1:0A0rp1pl−ytrh1eistraiannignlteegineer.qTuahleitnyetxopyreasnsdr2x=−0y..2666... asafraction.
solution Letr1=0.27.Weobservethat100r1=27.27.Therefore,100r1−r1=27.27−0.27=27and
27 3
r1= = .
99 11
Now,letr2=0.2666.Then10r2=2.666and100r2=26.666.Therefore,100r2−10r2=26.666−2.666=24and
24 4
r2= = .
90 15
35. Thetextstates:Ifthedecimalexpansionsofnumbersaandbagreetokplaces,then|a−b|≤10−k.Showthatthe
conversReeisprfeaslseen:tF1o/r7aallndk4th/e2r7eaasrerenpuematbienrgsdaeacnimdablsw.hosedecimalexpansionsdonotagreeatallbut|a−b|≤10−k.
solution Leta = 1andb = 0.9(seethediscussionbeforeExample1).Thedecimalexpansionsofaandbdonot
agree,but|1−0.9|<10−k forallk.
37. Findtheequationofthecirclewithcenter(2,4):
Ploteachpairofpointsandcomputethedistancebetweenthem:
(a) withradiusr =3.
(a) (1,4)and(3,2) (b) (2,1)and(2,4)
(b) thatpassesthrough(1,−1).
(c) (0,0)and(−2,3) (d) (−3,−3)and(−2,3)
solution
(a) Theequationoftheindicatedcircleis(x−2)2+(y−4)2=32=9.
(b) Firstdeterminetheradiusasthedistancefromthecentertotheindicatedpointonthecircle:
(cid:5)
√
r = (2−1)2+(4−(−1))2= 26.
Thus,theequationofthecircleis(x−2)2+(y−4)2=26.
39. Determinethedomainandrangeofthefunction
Findallpointswithintegercoordinateslocatedatadistance5fromtheorigin.Thenfindallpointswithinteger
coordinateslocatedatadistance5fromf(:2{,r3,)s.,t,u}→{A,B,C,D,E}
definedbyf(r)=A,f(s)=B,f(t)=B,f(u)=E.
solution ThedomainisthesetD={r,s,t,u};therangeisthesetR={A,B,E}.
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4 CHAPTER 1 PRECALCULUSREVIEW
InExercGisievse4a1n–e4x8a,mfipnldetohfeadfoumnacitnioannwdhroasnegdeoomfathineDfunhcatsiotnh.reeelementsandwhoserangeRhastwoelements.Doesa
41. ffu(nxc)ti=on−exxistwhosedomainDhastwoelementsandwhoserangeRhasthreeelements?
solution D:allreals;R:allreals
43. f(x)=x3
g(t)=t4
solution D:allreals;R:allreals
45. f(x)=|x|√
g(t)= 2−t
solution D:allreals;R:{y :y ≥0}
1
47. f(x)= 1
h(s)=x2
s
solution D:{x :x (cid:9)=0};R:{y :y >0}
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
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