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

747 Pages·2008·7.06 MB·English

by  Jon Rogawski

(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<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|=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 <r <π2+10−4,or9.869504<r <9.869705.r =9.8696= 12337 would
1250
beonesuchnumber.
InExercLiseetsa3=–8−,e3xparnedsbst=he2in.tWerhviaclhinoftetrhmesfoolfloawniinngeqinueaqliutayliitniveoslavriengtraube?solutevalue.
3. [(−a)2,a2<] b (b) |a|<|b| (c) ab>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,writetheinequalityintheforma<x <bforsomenumbersa,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<8,2expressthesetofnumbersx satisfyingthegivenconditionasaninterval.
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} √ √
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<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 > 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) Theintervaldescribedbytheinequality1<x <5hasacenterat3andaradiusof2;thatis,theintervalconsistsof
thosenumberslessthan2unitsfrom3:(iv).
SECTION 1.1 RealNumbers,Functions,andGraphs 3
25. Showthatifa>b,(cid:3)thenb−1 >a−1(cid:4),providedthataandbhavethesamesign.Whathappensifa>0andb <0?
x
SOLUTIDOeNscriCbeastehe1ase:tIfaxa:nxd+ba1re<bo0thpasosaintivinet,etrhveanl.a>b⇒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|n<d|2ba−nd8||x<−125,|th<en1|?(a+b)−13|<1.Hint:Usethetriangleinequality.
SOLUTION
|a+b−13|=|(a−5)+(b−8)|
≤|a−5|+|b−8| (bythetriangleinequality)
1 1
< + =1.
2 2
29. Supposethat|x−4|≤1.
Supposethat|a|≤2and|b|≤3.
(a) Whatisthemaximumpossiblevalueof|x+4|?
(a) Whatisthelargestpossiblevalueof|a+b|?
(b) Showthat|x2−16|≤9.
(b) Whatisthelargestpossiblevalueof|a+b|ifaandbhaveoppositesigns?
SOLUTION
(a) |x−4|≤1guarantees3 ≤x ≤5.Thus,7≤x+4≤9,so|x+4|≤9.
(b) |x2−16|=|x−4|·|x+4|≤1 ·9=9.
31. ExpPrreosvser1th=at0|x.2|7−a|sya|≤fra|cxti−on.yH|.iHnti:nt1:0A0rp1pl−ytrh1eistraiannignlteegineer.qTuahleitnyetxopyreasnsdr2x=−0y..2666... asafraction.
SOLUTION Letr1=.27.Weobservethat100r1=27.27.Therefore,100r1−r1 =27.27−.27=27and
27 3
r = = .
1
99 11
Now,letr2=.2666.Then10r2=2.666and100r2=26.666.Therefore,100r2−10r2=26.666−2.666=24and
24 4
r = = .
2
90 15
33. The text states the following: If the decimal expansions of two real numbers a and b agree to k places, then the
distanceR|eap−resbe|nt≤1/170−ankd.S4h/o2w7athsarteptheeatcinognvdeercsiemisalns.ottrue,thatis,foranyk wecanfindrealnumbersaandbwhose
decimalexpansionsdonotagreeatallbut|a−b|≤10−k.
SOLUTION Leta = 1andb = .9(seethediscussionbeforeExample1).Thedecimalexpansionsofa andbdonot
agree,but|1−.9|<10−k forallk.
35. Determinetheequationofthecirclewithcenter(2,4)andradius3.
Ploteachpairofpointsandcomputethedistancebetweenthem:
SOLU(aT)IO(1N,4)Tahnede(q3u,a2t)ionoftheindicatedcircleis(x−2)2+((by)−(24,)12)=an3d2(=2,49).
37. F(cin)d(0a,ll0p)oainndts(−wi2t,h3i)nteger coordinates located at a distan(cde)5(−fr3o,m−t3h)eaonrdig(i−n.2,T3h)en find all points with integer
coordinaDteestelromcianteedthaetaeqduisattaionnceo5ftfhreomcir(c2l,e3w).ithcenter(2,4)passingthrough(1,−1).
SOLUTION
• Tobelocatedadistance5fromtheorigin,thepointsmustlieonthecirclex2+y2 =25.Thisleadsto12points
withintegercoordinates:
(5,0) (−5,0) (0,5) (0,−5)
(3,4) (−3,4) (3,−4) (−3,−4)
(4,3) (−4,3) (4,−3) (−4,−3)
• Tobelocatedadistance5fromthepoint(2,3),thepointsmustlieonthecircle(x−2)2+(y−3)2 =25,which
impliesthatwemustshiftthepointslistedabovetwounitstotherightandthreeunitsup.Thisgivesthe12points:
(7,3) (−3,3) (2,8) (2,−2)
(5,7) (−1,7) (5,−1) (−1,−1)
(6,6) (−2,6) (6,0) (−2,0)
39. Giveanexampleofafunctionwhosedomain Dhasthreeelementsandrange Rhastwoelements.Doesafunction
Determinethedomainandrangeofthefunction
existwhosedomainDhastwoelementsandrangehasthreeelements?
f :{r,s,t,u}→{A,B,C,D,E}
definedby f(r)= A, f(s)= B, f(t)= B, f(u)= E.
4 CHAPTER 1 PRECALCULUS REVIEW
SOLUTION Define f by f :{a,b,c}→{1,2}where f(a)=1, f(b)=1, f(c)=2.
There is no function whose domain has two elements and range has three elements. If that happened, one of the
domainelementswouldgetassignedtomorethanoneelementoftherange,whichwouldcontradictthedefinitionofa
function.
InExercises40–48,findthedomainandrangeofthefunction.
41. g(t)f=(x)t4=−x
SOLUTION D:allreals;R:{y: y≥0}
√
43. g(t)= 2−t
f(x)=x3
SOLUTION D:{t :t ≤2};R:{y: y≥0}
1
45. h(s)f=(x)=|x|
s
SOLUTION D:{s:s (cid:11)=0};R:{y: y(cid:11)=0}
(cid:5)
47. g(t)= 2+t2
f(x)= 1 √
SOLUTION Dx:2allreals;R:{y: y≥ 2}
InExercises49–52,findtheintervalonwhichthefunctionisincreasing.
1
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 =x4isshownbelow.Fromthegraph,weseethatthefunctionisincreasingon
theinterval(0,∞).
y
12
8
4
x
−2 −1 1 2
In Exercises 53–58, find the zeros of the function and sketch its graph by plotting points. Use symmetry and in-
1
crease/dfe(cxre)a=seinformationwhereappropriate.
x2+1
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.