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INTRODUCTORY MATHEMATICAL ANALYSIS FOURTEENTH EDITION ERNEST F. HAEUSSLER JR.
RICHARD S. PAUL
RICHARD J. WOOD
FOR BUSINESS, ECONOMICS, AND THE LIFE AND SOCIAL SCIENCES This page intentionally left blank
AlgebraicRulesfor Realnumbers
a C b D b C a
ab D ba
a C .b C c/ D .a C b/ C c
a.bc/ D .ab/c
a.b C c/ D ab C ac
a.b c/ D ab ac
.a C b/c D ac C bc
.a b/c D ac bc
a C 0 D a
a 0 D 0
a 1 D a
a C . a/ D 0 . a/ D a
. 1/a D a
a b D a C . b/
a . b/ D a C b
a 1 a D 1
a b D a 1 b
. a/b D .ab/ D a. b/
. a/. b/ D ab
a b D a b
a b D a b D a b a c C b c D a C b c
a c b c D a b c
a b c d D ac bd
a=b
c=d D ad bc
a b D ac bc .c ¤ 0/
SummationFormulas
ALGEBRA Exponents
FactoringFormulas SpecialProducts
QuadraticFormula
If ax2 C bx C c D 0,where a ¤ 0,then x D b ˙pb2 4ac 2a
Inequalities
If a < b,then a C c < b C c. If a < b and c > 0,then ac < bc
SpecialSums
StraightLines
m D y2 y1 x2 x1 (slopeformula)
y y1 D m.x x1/ (point-slopeform)
y D mx C b (slope-interceptform)
x D constant(verticalline)
y D constant(horizontalline)
AbsoluteValue
jabjDjaj jbj
ˇ ˇ ˇ a b ˇ ˇ ˇ D jaj jbj ja bjDjb aj
jaj a jaj ja C bj jajCjbj (triangleinequality)
Logarithms
logb x D y ifandonlyif x D by
logb.mn/ D logb m C logb n
logb m n D logb m logb n
logb mr D r logb m
logb 1 D 0
logb b D 1
logb br D r blogb p D p .p > 0/
logb m D loga m loga b
BusinessRelations
FINITEMATHEMATICS Interest D (principal)(rate)(time)
Totalcost D variablecost C fixedcost
Averagecostperunit D totalcost quantity
Totalrevenue D (priceperunit)(numberofunitssold) Profit D totalrevenue totalcost
OrdinaryAnnuityFormulas A D R 1 .1 C r/ n r D Ran r (presentvalue)
S D R .1 C r/n 1 r D Rsn r (futurevalue)
Counting
nPr D nŠ .n r/Š
nCr D nŠ rŠ.n r/Š
nC0 C nC1 C C nCn 1 C nCn D 2n
nC0 D 1 D nCn
nC1CrC1 D nCr C nCrC1
PropertiesofEvents
For E and F eventsforanexperimentwithsamplespace S
E [ E D E
E \ E D E
.E0/0 D E
E [ E0 D S
E \ E0 D;
E [ S D S
E \ S D E
E [;D E
E \;D;
E [ F D F [ E
E \ F D F \ E
.E [ F/0 D E0 \ F0
.E \ F/0 D E0 [ F0
E [ .F [ G/ D .E [ F/ [ G
E \ .F \ G/ D .E \ F/ \ G
E \ .F [ G/ D .E \ F/ [ .E \ G/
E [ .F \ G/ D .E [ F/ \ .E [ G/
CompoundInterestFormulas
S D P.1 C r/n
P D S.1 C r/ n
re D 1 C r n n 1
S D Pert
P D Se rt
re D er 1
MatrixMultiplication
.AB/ik D n X jD1 AijBjk D Ai1B1k C Ai2B2k C C Ainbnk
.AB/T D BTAT A 1A D I D AA 1 .AB/ 1 D B 1A 1
Probability
P.E/ D #.E/ #.S/
P.EjF/ D #.E \ F/ #.F/
P.E [ F/ D P.E/ C P.F/ P.E \ F/
P.E0/ D 1 P.E/
P.E \ F/ D P.E/P.FjE/ D P.F/P.EjF/
For X adiscreterandomvariablewithdistribution f X x f.x/ D 1
D .X/ D E.X/ D X x xf.x/
Var.X/ D E..X /2/ D X x .x /2f.x/
D .X/ D pVar.X/
Binomialdistribution
f.x/ D P.X D x/ D nCxpxqn x
D np
Dpnpq
(quotientrule) dy dx D
DefinitionofDerivativeof f.x/
DifferentiationFormulas
CALCULUS GraphsofElementaryFunctions
IntegrationFormulas
Producers’SurplusforSupply
INTRODUCTORY MATHEMATICAL ANALYSIS ERNEST F. HAEUSSLER JR.
The Pennsylvania State University
RICHARD S. PAUL
The Pennsylvania State University
RICHARD J. WOOD
Dalhousie University
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Introductorymathematicalanalysisforbusiness,economics,andthelife andsocialsciences/ErnestF.Haeussler,Jr.(ThePennsylvaniaState University),RichardS.Paul(ThePennsylvaniaStateUniversity),Richard J.Wood(DalhousieUniversity).—Fourteenthedition.
Includesbibliographicalreferencesandindex. ISBN978-0-13-414110-7(hardcover)
1.Mathematicalanalysis.2.Economics,Mathematical.3.Business mathematics.I.Paul,RichardS.,authorII.Wood,RichardJames,author III.Title.
QA300.H322017 515 C2017-903584-3
ForBronwen This page intentionally left blank
PARTI COLLEGEALGEBRA CHAPTER0 ReviewofAlgebra 1
0.1 SetsofRealNumbers 2
0.2 SomePropertiesofRealNumbers 3
0.3 ExponentsandRadicals 10
0.4 OperationswithAlgebraicExpressions 15
0.5 Factoring 20
0.6 Fractions 22
0.7 Equations,inParticularLinearEquations 28
0.8 QuadraticEquations 39 Chapter0Review 45
CHAPTER1 ApplicationsandMoreAlgebra 47
1.1 ApplicationsofEquations 48
1.2 LinearInequalities 55
1.3 ApplicationsofInequalities 59
1.4 AbsoluteValue 62
1.5 SummationNotation 66
1.6 Sequences 70 Chapter1Review 80
CHAPTER2 FunctionsandGraphs 83
2.1 Functions 84
2.2 SpecialFunctions 91
2.3 CombinationsofFunctions 96
2.4 InverseFunctions 101
2.5 GraphsinRectangularCoordinates 104
2.6 Symmetry 113
2.7 TranslationsandReflections 118
2.8 FunctionsofSeveralVariables 120 Chapter2Review 128
CHAPTER3 Lines,Parabolas,andSystems 131
3.1 Lines 132
3.2 ApplicationsandLinearFunctions 139
3.3 QuadraticFunctions 145
3.4 SystemsofLinearEquations 152
3.5 NonlinearSystems 162
3.6 ApplicationsofSystemsofEquations 164 Chapter3Review 172
CHAPTER4 ExponentialandLogarithmicFunctions 175
4.1 ExponentialFunctions 176
4.2 LogarithmicFunctions 188
4.3 PropertiesofLogarithms 194
4.4 LogarithmicandExponentialEquations 200 Chapter4Review 204
PARTII FINITEMATHEMATICS CHAPTER5 MathematicsofFinance 208
5.1 CompoundInterest 209
5.2 PresentValue 214
5.3 InterestCompoundedContinuously 218
5.4 Annuities 222
5.5 AmortizationofLoans 230
5.6 Perpetuities 234 Chapter5Review 237
CHAPTER6 MatrixAlgebra 240
6.1 Matrices 241
6.2 MatrixAdditionandScalarMultiplication 246
6.3 MatrixMultiplication 253
6.4 SolvingSystemsbyReducingMatrices 264
6.5 SolvingSystemsbyReducingMatrices(Continued) 274
6.6 Inverses 279
6.7 Leontief’sInput--OutputAnalysis 286 Chapter6Review 292
CHAPTER7 LinearProgramming 294
7.1 LinearInequalitiesinTwoVariables 295
7.2 LinearProgramming 299
7.3 TheSimplexMethod 306
7.4 ArtificialVariables 320
7.5 Minimization 330
7.6 TheDual 335 Chapter7Review 344
CHAPTER8 IntroductiontoProbabilityandStatistics 348
8.1 BasicCountingPrincipleandPermutations 349
8.2 CombinationsandOtherCountingPrinciples 355
8.3 SampleSpacesandEvents 367
8.4 Probability 374
8.5 ConditionalProbabilityandStochasticProcesses 388
8.6 IndependentEvents 401
8.7 Bayes’Formula 411 Chapter8Review 419
CHAPTER9 AdditionalTopicsinProbability 424
9.1 DiscreteRandomVariablesandExpectedValue 425
9.2 TheBinomialDistribution 432
9.3 MarkovChains 437 Chapter9Review 447
PARTIII CALCULUS CHAPTER10
LimitsandContinuity 450
10.1 Limits 451
10.2 Limits(Continued) 461
10.3 Continuity 469
10.4 ContinuityAppliedtoInequalities 474 Chapter10Review 479
CHAPTER11 Differentiation 482
11.1 TheDerivative 483
11.2 RulesforDifferentiation 491
11.3 TheDerivativeasaRateofChange 499
11.4 TheProductRuleandtheQuotientRule 509
11.5 TheChainRule 519 Chapter11Review 527
CHAPTER12
AdditionalDifferentiationTopics 531
12.1 DerivativesofLogarithmicFunctions 532
12.2 DerivativesofExponentialFunctions 537
12.3 ElasticityofDemand 543
12.4 ImplicitDifferentiation 548
12.5 LogarithmicDifferentiation 554
12.6 Newton’sMethod 558
12.7 Higher-OrderDerivatives 562 Chapter12Review 566
CHAPTER13
CurveSketching 569
13.1 RelativeExtrema 570
13.2 AbsoluteExtremaonaClosedInterval 581
13.3 Concavity 583
13.4 TheSecond-DerivativeTest 591
13.5 Asymptotes 593
13.6 AppliedMaximaandMinima 603 Chapter13Review 614
CHAPTER14 Integration 619
14.1 Differentials 620
14.2 TheIndefiniteIntegral 625
14.3 IntegrationwithInitialConditions 631
14.4 MoreIntegrationFormulas 635
14.5 TechniquesofIntegration 642
14.6 TheDefiniteIntegral 647
14.7 TheFundamentalTheoremofCalculus 653 Chapter14Review 661
CHAPTER15 ApplicationsofIntegration 665
15.1 IntegrationbyTables 666
15.2 ApproximateIntegration 672
15.3 AreaBetweenCurves 678
15.4 Consumers’andProducers’Surplus 687
15.5 AverageValueofaFunction 690
15.6 DifferentialEquations 692
15.7 MoreApplicationsofDifferentialEquations 699
15.8 ImproperIntegrals 706 Chapter15Review 709
CHAPTER16 ContinuousRandomVariables 713
16.1 ContinuousRandomVariables 714
16.2 TheNormalDistribution 721
16.3 TheNormalApproximationtotheBinomialDistribution 726 Chapter16Review 730
CHAPTER17 MultivariableCalculus 732
17.1 PartialDerivatives 733
17.2 ApplicationsofPartialDerivatives 738
17.3 Higher-OrderPartialDerivatives 744
17.4 MaximaandMinimaforFunctionsofTwoVariables 746
17.5 LagrangeMultipliers 754
17.6 MultipleIntegrals 761 Chapter17Review 765
APPENDIXA CompoundInterestTables 769
APPENDIXB TableofSelectedIntegrals 777
APPENDIXC AreasUndertheStandardNormalCurve 780
AnswerstoOdd-NumberedProblems AN-1
Index I-1
Preface Thefourteentheditionof IntroductoryMathematicalAnalysisforBusiness,Economics,andtheLifeandSocialSciences(IMA) continuestoprovideamathematical foundationforstudentsinavarietyoffieldsandmajors,assuggestedbythetitle. Asbeguninthethirteenthedition,thebookhasthreeparts:CollegeAlgebra,Chapters0–4; FiniteMathematics,Chapters5–9;andCalculus,Chapters10–17.
SchoolsthathavetwoacademictermsperyeartendtogiveBusinessstudentsaterm devotedtoFiniteMathematicsandatermdevotedtoCalculus.FortheseschoolswerecommendChapters0through9forthefirstcourse,startingwhereverthepreparationofthe studentsallows,andChapters10through17forthesecond,includingasmuchasthestudents’backgroundallowsandtheirneedsdictate.
Forschoolswiththreequarterorthreesemestercoursesperyearthereareanumber ofpossibleusesforthisbook.IftheirprogramallowsthreequartersofMathematics,wellpreparedBusinessstudentscanstartafirstcourseonFiniteMathematicswithChapter1 andproceedthroughtopicsofinterestuptoandincludingChapter9.Inthisscenario,a secondcourseonDifferentialCalculuscouldstartwithChapter10onLimitsandContinuity,followedbythethree“differentiationchapters”,11through13inclusive.Here,Section 12.6onNewton’sMethodcanbeomittedwithoutlossofcontinuity,whilesomeinstructors mayprefertoreviewChapter4onExponentialandLogarithmicFunctionspriortostudyingthemasdifferentiablefunctions.Finally,athirdcoursecouldcompriseChapters14 through17onIntegralCalculuswithanintroductiontoMultivariableCalculus.Notethat Chapter16iscertainlynotneededforChapter17andSection15.8onImproperIntegrals canbesafelyomittedifChapter16isnotcovered.
Approach IntroductoryMathematicalAnalysisforBusiness,Economics,andtheLifeandSocial Sciences(IMA) takesauniqueapproachtoproblemsolving.Ashasbeenthecaseinearliereditionsofthisbook,weestablishanemphasisonalgebraiccalculationsthatsetsthis textapartfromotherintroductory,appliedmathematicsbooks.Theprocessofcalculating withvariablesbuildsskillinmathematicalmodelingandpavesthewayforstudentstouse calculus.Thereaderwillnotfinda“definition-theorem-proof”treatment,butthereisasustainedefforttoimpartagenuinemathematicaltreatmentofappliedproblems.Inparticular, ourguidingphilosophyleadsustoincludeinformalproofsandgeneralcalculationsthat shedlightonhowthecorrespondingcalculationsaredoneinappliedproblems.Emphasis ondevelopingalgebraicskillsisextendedtotheexercises,ofwhichmany,eventhoseof thedrilltype,aregivenwithgeneralratherthannumericalcoefficients.
Wehaverefinedtheorganizationofourbookovermanyeditionstopresentthecontent inverymanageableportionsforoptimalteachingandlearning.Inevitably,thatprocess tendstoput“weight”onabook,andthepresenteditionmakesaveryconcertedeffortto parethebookbacksomewhat,bothwithrespecttodesignfeatures—makingforacleaner approach—andcontent—recognizingchangingpedagogicalneeds.
ChangesfortheFourteenthEdition Wecontinuetomaketheelementarynotionsintheearlychapterspavethewayfortheir useinmoreadvancedtopics.Forexample,whilediscussingfactoring,atopicmanystudentsfindsomewhatarcane,wepointoutthattheprinciple“ab D 0implies a D 0or b D 0”,togetherwithfactoring,enablesthesplittingofsomecomplicatedequationsinto severalsimplerequations.Wepointoutthatpercentagesarejustrescalednumbersviathe “equation” p% D p 100 sothat,incalculus,“relativerateofchange”and“percentagerate ofchange”arerelatedbythe“equation” r D r 100%.Wethinkthatatthistime,when negativeinterestratesareoftendiscussed,evenifseldomimplemented,itiswisetobe absolutelypreciseaboutsimplenotionsthatareoftentakenforgranted.Infact,inthe
Finance,Chapter5,weexplicitlydiscussnegativeinterestratesandask,somewhatrhetorically,whybanksdonotusecontinuouscompounding(giventhatforalongtimenow continuouscompoundinghasbeenabletosimplifycalculations inpractice aswellasin theory).
Wheneverpossible,wehavetriedtoincorporatetheextraideasthatwereinthe“Explore andExtend”chapter-closersintothebodyofthetext.Forexample,thefunctionstaxrate t.i/ andtaxpaid T.i/ ofincome i,areseenforwhattheyare:everydayexamplesofcase-defined functions.Wethinkthatintheprocessoflearningaboutpolynomialsitishelpfultoinclude Horner’sMethodfortheirevaluation,sincewithevenasimplecalculatorathandthismakes thecalculationmuchfaster.Whiledoinglinearprogramming,itsometimeshelpstothink oflinesandplanes,etcetera,intermsofinterceptsalone,soweincludeanexercisetoshow thatifalinehas(nonzero)intercepts x0 and y0 thenitsequationisgivenby
and,moreover,(forpositive x0 and y0)weaskforageometricinterpretationoftheequivalent equation y0x C x0y D x0y0
But,turningtoour“paring”oftheprevious IMA,letusbeginwithLinearProgramming.ThisissurelyoneofthemostimportanttopicsinthebookforBusinessstudents.We nowfeelthat,whilestudentsshouldknowaboutthepossibilityof MultipleOptimumSolutions and DegeneracyandUnboundedSolutions,theydonothaveenoughtimetodevote anentire,albeitshort,sectiontoeachofthese.TheremainingsectionsofChapter7are alreadydemandingandwenowcontentourselveswithprovidingsimplealertstothese possibilitiesthatareeasilyseengeometrically.(Thedeletedsectionswerealwaystagged as“omittable”.)
Wethinkfurtherthat,inIntegralCalculus,itisfarmoreimportantforAppliedMathematicsstudentstobeadeptatusingtablestoevaluateintegralsthantoknowabout IntegrationbyParts and PartialFractions.Infact,thesetopics,ofendlessjoytosomeasrecreationalproblems,donotseemtofitwellintothegeneralschemeofseriousproblemsolving. Itisafactoflifethatanelementaryfunction(inthetechnicalsense)caneasilyfailtohave anelementaryantiderivative,anditseemstousthat Parts doesnotgofarenoughtorescue thisdifficultytowarranttheconsiderabletimeittakestomasterthetechnique.Since PartialFractions ultimatelyleadtoelementaryantiderivativesforall rational functions,they are partofseriousproblemsolvingandabettercasecanbemadefortheirinclusioninan appliedtextbook.However,itisvainglorioustodosowithouttheinversetangentfunction athandand,bylongstandingtacitagreement,appliedcalculusbooksdonotventureinto trigonometry.
Afterdeletingthesectionsmentionedabove,wereorganizedtheremainingmaterialof the“integrationchapters”,14and15,torebalancethem.ThefirstconcludeswiththeFundamentalTheoremofCalculuswhilethesecondismoreproperly“applied”.Wethinkthatthe formerlydauntingChapter17hasbenefitedfromdeletionof ImplicitPartialDifferentiation,the ChainRule forpartialdifferentiation,and LinesofRegression.SinceMultivariable CalculusisextremelyimportantforAppliedMathematics,wehopethatthismoremanageablechapterwillencourageinstructorstoincludeitintheirsyllabi.
ExamplesandExercises Mostinstructorsandstudentswillagreethatthekeytoaneffectivetextbookisinthe qualityandquantityoftheexamplesandexercisesets.Tothatend,morethan850examplesareworkedoutindetail.Someoftheseexamplesincludea strategy boxdesigned toguidestudentsthroughthegeneralstepsofthesolutionbeforethespecificsolution isobtained.(See,forexample,Section14.3Example4.)Inaddition,anabundantnumberofdiagrams(almost500)andexercises(morethan5000)areincluded.Oftheexercises,approximately20percenthavebeeneitherupdatedorwrittencompletelyanew.In eachexerciseset,groupedproblemsareusuallygiveninincreasingorderofdifficulty. Inmostexercisesetstheproblemsprogressfromthebasicmechanicaldrill-typetomore
interestingthought-provokingproblems.Theexerciseslabeledwithacolouredexercise numbercorrelatetoa“NowWorkProblemN”statementandexampleinthesection. Basedonthefeedbackwehavereceivedfromusersofthistext,thediversityofthe applicationsprovidedinboththeexercisesetsandexamplesistrulyanassetofthisbook. Manyrealappliedproblemswithaccuratedataareincluded.Studentsdonotneedtolook hardtoseehowthemathematicstheyarelearningisappliedtoeverydayorwork-related situations.Agreatdealofefforthasbeenputintoproducingaproperbalancebetween drill-typeexercisesandproblemsrequiringtheintegrationandapplicationoftheconcepts learned.
PedagogyandHallmarkFeatures Applications: Anabundanceandvarietyofapplicationsfortheintendedaudienceappear throughoutthebooksothatstudentsseefrequentlyhowthemathematicstheyarelearningcanbeused.Theseapplicationscoversuchdiverseareasasbusiness,economics, biology,medicine,sociology,psychology,ecology,statistics,earthscience,andarchaeology.Manyoftheseapplicationsaredrawnfromliteratureandaredocumentedby references,sometimesfromtheWeb.Insome,thebackgroundandcontextaregiven inordertostimulateinterest.However,thetextisself-contained,inthesensethatit assumesnopriorexposuretotheconceptsonwhichtheapplicationsarebased.(See,for example,Chapter15,Section7,Example2.)
NowWorkProblemN: Throughoutthetextwehaveretainedthepopular NowWork ProblemN feature.Theideaisthatafteraworkedexample,studentsaredirectedto anend-of-sectionproblem(labeledwithacoloredexercisenumber)thatreinforcesthe ideasoftheworkedexample.Thisgivesstudentsanopportunitytopracticewhatthey havejustlearned.Becausethemajorityofthesekeyedexercisesareodd-numbered,studentscanimmediatelychecktheiranswerinthebackofthebooktoassesstheirlevelof understanding.Thecompletesolutionstotheodd-numberedexercisescanbefoundin theStudentSolutionsManual.
Cautions: Cautionarywarningsarepresentedinverymuchthesamewayaninstructor wouldwarnstudentsinclassofcommonlymadeerrors.Theseappearinthemargin, alongwithotherexplanatorynotesandemphases.
Definitions,keyconcepts,andimportantrulesandformulas: Theseareclearlystated anddisplayedasawaytomakethenavigationofthebookthatmucheasierforthe student.(See,forexample,theDefinitionofDerivativeinSection11.1.)
Reviewmaterial: Eachchapterhasareviewsectionthatcontainsalistofimportant termsandsymbols,achaptersummary,andnumerousreviewproblems.Inaddition, keyexamplesarereferencedalongwitheachgroupofimportanttermsandsymbols.
Inequalitiesandslackvariables: InSection1.2,wheninequalitiesareintroducedwe pointoutthat a b isequivalentto“thereexistsanon-negativenumber, s,suchthat a C s D b”.Theideaisnotdeepbutthepedagogicalpointisthat slackvariables,key toimplementingthesimplexalgorithminChapter7,shouldbefamiliarandnotdistract fromtherathertechnicalmaterialinlinearprogramming.
Absolutevalue: Itiscommontonotethat ja bj providesthedistancefrom a to b.In Example4eofSection1.4wepointoutthat“x islessthan unitsfrom ”translatesas jx j < .InSection1.4thisisbutanexercisewiththenotation,asitshouldbe,but thepointhereisthatlater(inChapter9) willbethemeanand thestandarddeviation ofarandomvariable.Againwehaveseparated,inadvance,asimpleideafromamore advancedone.Ofcourse,Problem12ofProblems1.4,whichasksthestudenttosetup jf.x/ Lj < ,hasasimilaragendatoChapter10onlimits.
Earlytreatmentofsummationnotation: ThistopicisnecessaryforstudyofthedefiniteintegralinChapter14,butitis useful longbeforethat.Sinceitisanotationthatis newtomoststudentsatthislevel,butnomorethananotation,wegetitoutoftheway inChapter1.Byusingitwhenconvenient, beforecoverageofthedefiniteintegral,itis notadistractionfromthatchallengingconcept.
Section1.6onsequences: Thissectionprovidesseveralpedagogicaladvantages. Theverydefinitionisstatedinafashionthatpavesthewayforthemoreimportantand morebasicdefinitionoffunctioninChapter2.Insummingthetermsofasequencewe areabletopracticetheuseofsummationnotationintroducedintheprecedingsection. Themostobviousbenefitthoughisthat“sequences”allowsusabetterorganization intheannuitiessectionofChapter5.Boththepresentandthefuturevaluesofanannuityareobtainedbysumming(finite)geometricsequences.Laterinthetext,sequences ariseinthedefinitionofthenumber e inChapter4,inMarkovchainsinChapter9,and inNewton’smethodinChapter12,sothatahelpfulunifyingreferenceisobtained.
Sumofaninfinitesequence: Inthecourseofsummingthetermsofafinitesequence, itisnaturaltoraisethepossibilityofsummingthetermsofaninfinitesequence.Thisis anonthreateningenvironmentinwhichtoprovideafirstforayintotheworldoflimits. Wesimplyexplainhowcertaininfinitegeometricsequenceshavewell-definedsumsand phrasetheresultsinawaythatcreatesatoeholdfortheintroductionoflimitsinChapter 10.Theseparticularinfinitesumsenableustointroducetheideaofaperpetuity,first informallyinthesequencesection,andthenagaininmoredetailinaseparatesectionin Chapter5.
Section2.8,FunctionsofSeveralVariables: Theintroductiontofunctionsofseveral variablesappearsinChapter2becauseitisatopicthatshouldappearlongbeforeCalculus.Oncewehavedonesomecalculusthereareparticularwaystousecalculusinthe studyoffunctionsofseveralvariables,buttheseaspectsshouldnotbeconfusedwiththe basicsthatweusethroughoutthebook.Forexample,“a-sub-n-angle-r”and“s-sub-nangle-r”studiedintheMathematicsofFinance,Chapter5,areperfectlygoodfunctions oftwovariables,andLinearProgrammingseekstooptimizelinearfunctionsofseveral variablessubjecttolinearconstraints.
Leontief’sinput-outputanalysisinSection6.7: Inthissectionwehaveseparatedvariousaspectsofthetotalproblem.WebeginbydescribingwhatwecalltheLeontiefmatrix A asanencodingoftheinputandoutputrelationshipsbetweensectorsofaneconomy. Sincethismatrixcanoftenbeassumedtobeconstantforasubstantialperiodoftime, webeginbyassumingthat A isagiven.Thesimplerproblemisthentodeterminethe production, X,whichisrequiredtomeetanexternaldemand, D,foraneconomywhose Leontiefmatrixis A.Weprovideacarefulaccountofthisasthesolutionof .I A/X D D Since A canbeassumedtobefixedwhilevariousdemands, D,areinvestigated,thereis some justificationtocompute .I A/ 1 sothatwehave X D .I A/ 1D.However,use ofamatrixinverseshouldnotbeconsideredanessentialpartofthesolution.Finally,we explainhowtheLeontiefmatrixcanbefoundfromatableofdatathatmightbeavailable toaplanner.
BirthdayprobabilityinSection8.4: Thisisatreatmentoftheclassicproblemofdeterminingtheprobabilitythatatleast2of n peoplehavetheirbirthdayonthesameday. Whilethisproblemisgivenasanexampleinmanytexts,therecursiveformulathatwe giveforcalculatingtheprobabilityasafunctionof n isnotacommonfeature.Itisreasonabletoincludeitinthisbookbecauserecursivelydefinedsequencesappearexplicitlyin Section1.6.
MarkovChains: Wenoticedthatconsiderablesimplificationoftheproblemoffinding steadystatevectorsisobtainedbywritingstatevectorsascolumnsratherthanrows. Thisdoesnecessitatethatatransitionmatrix T D Œtij have tij D“probabilitythatnext stateis i giventhatcurrentstateis j”butavoidsseveralartificialtranspositions.
SignChartsforafunctioninChapter10: Thesignchartsthatweintroducedinthe 12theditionnowmaketheirappearanceinChapter10.Ourpointisthatthesecharts canbemadeforanyreal-valuedfunctionofarealvariableandtheirhelpingraphingafunctionbeginspriortotheintroductionofderivatives.Ofcoursewecontinueto exploittheiruseinChapter13“CurveSketching”where,foreachfunction f,weadvocatemakingasignchartforeachof f, f0,and f00,interpretedfor f itself.Whenthisis possible,thegraphofthefunctionbecomesalmostself-evident.Wefreelyacknowledge thatthisisablackboardtechniqueusedbymanyinstructors,butitappearstoorarelyin textbooks.
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E.Adibi, ChapmanUniversity
Acknowledgments Weexpressourappreciationtothefollowingcolleagueswhocontributedcommentsand suggestionsthatwerevaluabletousintheevolutionofthistext.(Professorsmarkedwith anasteriskreviewedthefourteenthedition.)
R.M.Alliston, PennsylvaniaStateUniversity
R.A.Alo, UniversityofHouston
K.T.Andrews, OaklandUniversity
M.N.deArce, UniversityofPuertoRico
E.Barbut, UniversityofIdaho
G.R.Bates, WesternIllinoisUniversity
*S.Beck, NavarroCollege
D.E.Bennett, MurrayStateUniversity
C.Bernett, HarperCollege
A.Bishop, WesternIllinoisUniversity
P.Blau, ShawneeStateUniversity
R.Blute, UniversityofOttawa
S.A.Book, CaliforniaStateUniversity
A.Brink, St.CloudStateUniversity
R.Brown, YorkUniversity
R.W.Brown, UniversityofAlaska
S.D.Bulman-Fleming, WilfridLaurierUniversity
D.Calvetti, NationalCollege
D.Cameron, UniversityofAkron
K.S.Chung, KapiolaniCommunityCollege
D.N.Clark, UniversityofGeorgia
E.L.Cohen, UniversityofOttawa
J.Dawson, PennsylvaniaStateUniversity
A.Dollins, PennsylvaniaStateUniversity
T.J.Duda, ColumbusStateCommunityCollege
G.A.Earles, St.CloudStateUniversity
B.H.Edwards, UniversityofFlorida
J.R.Elliott, WilfridLaurierUniversity
J.Fitzpatrick, UniversityofTexasatElPaso
M.J.Flynn, RhodeIslandJuniorCollege
G.J.Fuentes, UniversityofMaine
L.Gerber, St.John’sUniversity
T.G.Goedde, TheUniversityofFindlay
S.K.Goel, ValdostaStateUniversity
G.Goff, OklahomaStateUniversity
J.Goldman, DePaulUniversity
E.Greenwood, TarrantCountyCollege,Northwest Campus
J.T.Gresser, BowlingGreenStateUniversity
L.Griff, PennsylvaniaStateUniversity
R.Grinnell, UniversityofTorontoatScarborough
F.H.Hall, PennsylvaniaStateUniversity
V.E.Hanks, WesternKentuckyUniversity
*T.Harriott, MountSaintVincentUniversity
R.C.Heitmann, TheUniversityofTexasatAustin
J.N.Henry, CaliforniaStateUniversity
W.U.Hodgson, WestChesterStateCollege
*J.Hooper, AcadiaUniversity
B.C.Horne,Jr., VirginiaPolytechnicInstituteandState University
J.Hradnansky, PennsylvaniaStateUniversity
P.Huneke, TheOhioStateUniversity
C.Hurd, PennsylvaniaStateUniversity
J.A.Jiminez, PennsylvaniaStateUniversity
*T.H.Jones, Bishop’sUniversity
W.C.Jones, WesternKentuckyUniversity
R.M.King, GettysburgCollege
M.M.Kostreva, UniversityofMaine
G.A.Kraus, GannonUniversity
J.Kucera, WashingtonStateUniversity
M.R.Latina, RhodeIslandJuniorCollege
L.N.Laughlin, UniversityofAlaska,Fairbanks
P.Lockwood-Cooke, WestTexasA&MUniversity
J.F.Longman, VillanovaUniversity
*F.MacWilliam, AlgomaUniversity
I.Marshak, LoyolaUniversityofChicago
D.Mason, ElmhurstCollege
*B.Matheson, UniversityofWaterloo
F.B.Mayer, Mt.SanAntonioCollege
P.McDougle, UniversityofMiami
F.Miles, CaliforniaStateUniversity
E.Mohnike, Mt.SanAntonioCollege
C.Monk, UniversityofRichmond
R.A.Moreland, TexasTechUniversity
J.G.Morris, UniversityofWisconsin-Madison
J.C.Moss, PaducahCommunityCollege
D.Mullin, PennsylvaniaStateUniversity
E.Nelson, PennsylvaniaStateUniversity
S.A.Nett, WesternIllinoisUniversity
R.H.Oehmke, UniversityofIowa
Y.Y.Oh, PennsylvaniaStateUniversity
J.U.Overall, UniversityofLaVerne
*K.Pace, TarrantCountyCollege
A.Panayides, WilliamPattersonUniversity
D.Parker, UniversityofPacific
N.B.Patterson, PennsylvaniaStateUniversity
V.Pedwaydon, LawrenceTechnicalUniversity
E.Pemberton, WilfridLaurierUniversity
M.Perkel, WrightStateUniversity
D.B.Priest, HardingCollege
J.R.Provencio, UniversityofTexas
L.R.Pulsinelli, WesternKentuckyUniversity
M.Racine, UniversityofOttawa
*B.Reed, NavarroCollege
N.M.Rice, Queen’sUniversity
A.Santiago, UniversityofPuertoRico
J.R.Schaefer, UniversityofWisconsin–Milwaukee
S.Sehgal, TheOhioStateUniversity
W.H.Seybold,Jr., WestChesterStateCollege
*Y.Shibuya, SanFranciscoStateUniversity
G.Shilling, TheUniversityofTexasatArlington
S.Singh, PennsylvaniaStateUniversity
L.Small, LosAngelesPierceCollege
E.Smet, HuronCollege
J.Stein, CaliforniaStateUniversity,LongBeach
M.Stoll, UniversityofSouthCarolina
T.S.Sullivan, SouthernIllinoisUniversityEdwardsville
E.A.Terry, St.Joseph’sUniversity
A.Tierman, SaginawValleyStateUniversity
B.Toole, UniversityofMaine
J.W.Toole, UniversityofMaine
*M.Torres, AthabascaUniversity
D.H.Trahan, NavalPostgraduateSchool
J.P.Tull, TheOhioStateUniversity
L.O.Vaughan,Jr., UniversityofAlabamain Birmingham
L.A.Vercoe, PennsylvaniaStateUniversity
M.Vuilleumier, TheOhioStateUniversity
B.K.Waits, TheOhioStateUniversity
A.Walton, VirginiaPolytechnicInstituteandState University
H.Walum, TheOhioStateUniversity
E.T.H.Wang, WilfridLaurierUniversity
A.J.Weidner, PennsylvaniaStateUniversity
L.Weiss, PennsylvaniaStateUniversity
N.A.Weigmann, CaliforniaStateUniversity
S.K.Wong, OhioStateUniversity
G.Woods, TheOhioStateUniversity
C.R.B.Wright, UniversityofOregon
C.Wu, UniversityofWisconsin–Milwaukee
B.F.Wyman, TheOhioStateUniversity
D.Zhang, WashingtonStateUniversity
SomeexercisesaretakenfromproblemsupplementsusedbystudentsatWilfridLaurier University.WewishtoextendspecialthankstotheDepartmentofMathematicsofWilfrid LaurierUniversityforgrantingPrenticeHallpermissiontouseandpublishthismaterial, andalsotoPrenticeHall,whointurnallowedustomakeuseofthismaterial.
WeagainexpressoursinceregratitudetothefacultyandcoursecoordinatorsofThe OhioStateUniversityandColumbusStateUniversitywhotookakeeninterestinthisand othereditions,offeringanumberofinvaluablesuggestions.
SpecialthanksareduetoMPSNorthAmerica,LLC.fortheircarefulworkonthesolutionsmanuals.Theirworkwasextraordinarilydetailedandhelpfultous.Wealsoappreciate thecarethattheytookincheckingthetextandexercisesforaccuracy.
ErnestF.Haeussler,Jr. RichardS.Paul RichardJ.Wood
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0 0.1 SetsofRealNumbers
0.2 SomePropertiesofReal Numbers
0.3 ExponentsandRadicals
0.4 Operationswith AlgebraicExpressions
0.5 Factoring
0.6 Fractions
0.7 Equations,inParticular LinearEquations
0.8 QuadraticEquations Chapter0 Review
ReviewofAlgebra LesleyGriffithworkedforayachtsupplycompanyinAntibes,France.Often, sheneededtoexaminereceiptsinwhichonlythetotalpaidwasreportedand thendeterminetheamountofthetotalwhichwasFrench“value-addedtax”. ItisknownasTVAfor“TaxeàlaValueAjouté”.TheFrenchTVAratewas 19.6%(butinJanuaryof2014itincreasedto20%).AlotofLesley’sbusinesscame fromItaliansuppliersandpurchasers,soshealsohadtodealwiththesimilarproblem ofreceiptscontainingItaliansalestaxat18%(now22%).
Aproblemofthiskinddemandsaformula,sothattheusercanjustpluginatax ratelike19.6%or22%tosuitaparticularplaceandtime,butmanypeopleareable toworkthroughaparticularcaseoftheproblem,usingspecifiednumbers,without knowingtheformula.Thus,ifLesleyhada200-EuroFrenchreceipt,shemighthave reasonedasfollows:Iftheitemcost100Eurosbeforetax,thenthereceipttotalwould befor119.6Euroswithtaxof19.6,so taxinareceipttotalof200isto200as19.6is to119.6.Statedmathematically,
4%
IfherreasoningiscorrectthentheamountofTVAina200-Euroreceiptisabout16.4% of200Euros,whichis32.8Euros.Infact,manypeoplewillnowguessthat
taxin R D R p 100 C p
givesthetaxinareceipt R,whenthetaxrateis p%.Thus,ifLesleyfeltconfidentabout herdeduction,shecouldhavemultipliedherItalianreceiptsby 18 118 todeterminethetax theycontained.
Ofcourse,mostpeopledonotrememberformulasforverylongandareuncomfortablebasingamonetarycalculationonanassumptionsuchastheoneweitalicized above.Therearelotsofrelationshipsthataremorecomplicatedthansimpleproportionality!Thepurposeofthischapteristoreviewthealgebranecessaryforyoutoconstruct yourownformulas, withconfidence, asneeded.Inparticular,wewillderiveLesley’s formulafromprincipleswithwhicheverybodyisfamiliar.Thisusageofalgebrawill appearthroughoutthebook,inthecourseofmaking generalcalculationswithvariable quantities
Inthischapterwewillreviewrealnumbersandalgebraicexpressionsandthebasic operationsonthem.Thechapterisdesignedtoprovideabriefreviewofsometermsand methodsofsymboliccalculation.Probably,youhaveseenmostofthismaterialbefore. However,becausethesetopicsareimportantinhandlingthemathematicsthatcomes later,animmediatesecondexposuretothemmaybebeneficial.Devotewhatevertime isnecessarytothesectionsinwhichyouneedreview.
Objective Tobecomefamiliarwithsets,in particularsetsofrealnumbers,and thereal-numberline.
0.1SetsofRealNumbers A set isacollectionofobjects.Forexample,wecanspeakofthesetofevennumbers between5and11,namely,6,8,and10.Anobjectinasetiscalledan element of thatset.Ifthissoundsalittlecircular,don’tworry.Thewords set and element arelike line and point ingeometry.Wecannotdefinetheminmoreprimitiveterms.Itisonly withpracticeinusingthemthatwecometounderstandtheirmeaning.Thesituationis alsoratherlikethewayinwhichachildlearnsafirstlanguage.Withoutknowing any words,achildinfersthemeaningofafewverysimplewordsbywatchingandlistening toaparentandultimatelyusestheseveryfewwordstobuildaworkingvocabulary. Noneofusneedstounderstandthemechanicsofthisprocessinordertolearnhowto speak.Inthesameway,itispossibletolearnpracticalmathematicswithoutbecoming embroiledintheissueofundefinedprimitiveterms.
Onewaytospecifyasetisbylistingitselements,inanyorder,insidebraces.For example,theprevioussetis f6; 8; 10g,whichwecoulddenotebyalettersuchas A, allowingustowrite A Df6; 8; 10g.Notethat f8; 10; 6g alsodenotesthesameset,as does f6; 8; 10; 10g. Asetisdeterminedbyitselements,andneitherrearrangementsnor repetitionsinalistingaffecttheset.Aset A issaidtobeasubsetofaset B ifand onlyifeveryelementof A isalsoanelementof B.Forexample,if A Df6; 8; 10g and B Df6; 8; 10; 12g,then A isasubsetof B but B isnotasubsetof A.Thereisexactly onesetwhichcontains no elements.Itiscalled theemptyset andisdenotedby ;. Certainsetsofnumbershavespecialnames.Thenumbers1,2,3,andsoonform thesetof positiveintegers:
setofpositiveintegers Df1; 2; 3;:::g
Thethreedotsareaninformalwayofsayingthatthelistingofelementsisunending andthereaderisexpectedtogenerateasmanyelementsasneededfromthepattern.
Thepositiveintegerstogetherwith0andthe negativeintegers 1; 2; 3;:::; formthesetof integers:
setofintegers Df:::; 3; 2; 1; 0; 1; 2; 3;:::g
Thereasonfor q ¤ 0isthatwecannot dividebyzero.
Everyintegerisarationalnumber.
Thesetof rationalnumbers consistsofnumbers,suchas 1 2 and 5 3 ,thatcanbe writtenasaquotientoftwointegers.Thatis,arationalnumberisanumberthatcan bewrittenas p q ,where p and q areintegersand q ¤ 0.(Thesymbol“¤”isread“isnot equalto.”)Forexample,thenumbers 19 20 , 2 7 ,and 6 2 arerational.Weremarkthat 2 4 , 1 2 , 3 6 , 4 8 ,0 5,and50%allrepresentthesamerationalnumber.Theinteger2isrational, since2 D 2 1 .Infact,everyintegerisrational.
Allrationalnumberscanberepresentedbydecimalnumbersthat terminate, such as 3 4 D 0 75and 3 2 D 1 5,orby nonterminating,repeatingdecimalnumbers (composed ofagroupofdigitsthatrepeatswithoutend),suchas 2 3 D 0:666 :::; 4 11 D 0:3636 :::; and 2 15 D 0 1333 Numbersrepresentedby nonterminating,nonrepeating decimals
Everyrationalnumberisarealnumber. arecalled irrationalnumbers.Anirrationalnumbercannotbewrittenasaninteger dividedbyaninteger.Thenumbers (pi)and p2areexamplesofirrationalnumbers. Together,therationalnumbersandtheirrationalnumbersformthesetof realnumbers
Thesetofrealnumbersconsistsofall decimalnumbers.
Realnumberscanberepresentedbypointsonaline.Firstwechooseapointonthe linetorepresentzero.Thispointiscalledthe origin.(SeeFigure0.1.)Thenastandard measureofdistance,calleda unitdistance, ischosenandissuccessivelymarkedoff bothtotherightandtotheleftoftheorigin.Witheachpointonthelineweassociatea directeddistance,whichdependsonthepositionofthepointwithrespecttotheorigin.
Positionstotherightoftheoriginareconsideredpositive .C/ andpositionstotheleft arenegative . /.Forexample,withthepoint 1 2 unittotherightoftheoriginthere correspondsthenumber 1 2 ,whichiscalledthe coordinate ofthatpoint.Similarly,the coordinateofthepoint1.5unitstotheleftoftheoriginis 1:5.InFigure0.1,the coordinatesofsomepointsaremarked.Thearrowheadindicatesthatthedirectionto therightalongthelineisconsideredthepositivedirection.
Toeachpointonthelinetherecorrespondsauniquerealnumber,andtoeach realnumbertherecorrespondsauniquepointontheline.Thereisa one-to-onecorrespondence betweenpointsonthelineandrealnumbers.Wecallsuchaline,with coordinatesmarked,a real-numberline.Wefeelfreetotreatrealnumbersaspoints onareal-numberlineandviceversa.
EXAMPLE1IdentifyingKindsofRealNumbers Isittruethat0:151515 ::: isanirrationalnumber?
Solution: Thedotsin0:151515 ::: areunderstoodtoconveyrepetitionofthedigit string“15”.Irrationalnumbersweredefinedtoberealnumbersthatarerepresentedbya nonterminating,nonrepeating decimal,so0 151515 isnotirrational.Itisthereforea rationalnumber.Itisnotimmediatelyclearhowtorepresent0 151515 asaquotient ofintegers.InChapter1wewilllearnhowtoshowthat0 151515 D 5 33 .Youcan checkthatthisis plausible byentering5 33onacalculator,butyoushouldalsothink aboutwhythecalculatorexercisedoesnot prove that0 151515 D 5 33
NowWorkProblem7 G
PROBLEMS0.1 InProblems1–12,determinethetruthofeachstatement.Ifthe statementisfalse,giveareasonwhythatisso.
1. p 13isaninteger.
2. 2 7 isrational.
3. 3isapositiveinteger.
4. 0isnotrational.
5. p3isrational.
6. 1 0 isarationalnumber.
Objective Toname,illustrate,andrelate propertiesoftherealnumbersand theiroperations.
7. p25isnotapositiveinteger.
8. p2isarealnumber.
9. 0 0 isrational.
10. isapositiveinteger.
11. 0istotherightof p2onthereal-numberline.
12. Everyintegerispositiveornegative.
13. Everyterminatingdecimalnumbercanberegardedasa repeatingdecimalnumber.
14. p 1isarealnumber.
0.2SomePropertiesofRealNumbers Wenowstateafewimportantpropertiesoftherealnumbers.Let a, b,and c bereal numbers.
1. TheTransitivePropertyofEquality
If a D b and b D c; then a D c
Thus,twonumbersthatarebothequaltoathirdnumberareequaltoeachother. Forexample,if x D y and y D 7,then x D 7.
Zerodoesnothaveareciprocalbecause thereisnonumberthatwhenmultiplied by0gives1.Thisisaconsequenceof 0 a D 0in7.TheDistributiveProperties.
2. TheClosurePropertiesofAdditionandMultiplication
Forallrealnumbers a and b,thereareuniquerealnumbers a C b and ab.
Thismeansthatanytwonumberscanbeaddedandmultiplied,andtheresultin eachcaseisarealnumber.
3. TheCommutativePropertiesofAdditionandMultiplication a C b D b C a and ab D ba
Thismeansthattwonumberscanbeaddedormultipliedinanyorder.Forexample, 3
4. TheAssociativePropertiesofAdditionandMultiplication
C .b C c/ D .a C b/ C c and a.bc/ D .ab/c
Thismeansthat,forbothadditionandmultiplication,numberscanbegroupedin anyorder.Forexample,2 C .3 C 4/ D .2 C 3/ C 4;inbothcases,thesumis9.Similarly,2x C .x C y/ D .2x C x/ C y,andobservethattherightsidemoreobviouslysimplifiesto3x C y thandoestheleftside.Also, .6 1 3 / 5 D 6. 1 3 5/,andheretheleftside obviouslyreducesto10,sotherightsidedoestoo.
5. TheIdentityProperties
Thereareuniquerealnumbersdenoted0and1suchthat,foreachrealnumber a, 0 C a D a and1a D a
6. TheInverseProperties
Foreachrealnumber a,thereisauniquerealnumberdenoted a suchthat a C . a/ D 0
Thenumber a iscalledthe negative of a
Forexample,since6 C . 6/ D 0,thenegativeof6is 6.Thenegativeofanumberisnotnecessarilyanegativenumber.Forexample,thenegativeof 6is6,since . 6/ C .6/ D 0.Thatis,thenegativeof 6is6,sowecanwrite . 6/ D 6.
Foreachrealnumber a, except 0,thereisauniquerealnumberdenoted a 1 such that a a 1 D 1
Thenumber a 1 iscalledthe reciprocal of a
Thus,allnumbers except 0haveareciprocal.Recallthat a 1 canbewritten 1 a .For example,thereciprocalof3is 1 3 ,since3. 1 3 / D 1.Hence, 1 3 isthereciprocalof3.The reciprocalof 1 3 is3,since . 1 3 /.3/ D 1. Thereciprocalof0isnotdefined
7. TheDistributiveProperties
a.b C c/ D ab C ac and .b C c/a D ba C ca
0 a D 0 D a 0
Similarly,
and
Thedistributivepropertycanbeextendedtotheform
Infact,itcanbeextendedtosumsinvolvinganynumberofterms. Subtraction isdefinedintermsofaddition:
a b means a C . b/ where b isthenegativeof b.Thus,6 8means6 C . 8/ Inasimilarway,wedefine division intermsofmultiplication.If b ¤ 0,then a b means a.b 1/
Usually,wewriteeither
Thus, 3 5 means3times 1 5 ,where 1 5 isthereciprocalof5.Sometimeswereferto a b as the ratio of a to b.Weremarkthatsince0doesnothaveareciprocal, divisionby0is notdefined. a b means a timesthereciprocalof b. Thefollowingexamplesshowsomemanipulationsinvolvingthepreceding properties.
EXAMPLE1ApplyingPropertiesofRealNumbers a. x.y 3z C 2w/ D .y 3z C 2w/x,bythecommutativepropertyofmultiplication.
b. Bytheassociativepropertyofmultiplication,3.4 5/ D .3 4/5.Thus,theresultof multiplying3bytheproductof4and5isthesameastheresultofmultiplyingthe productof3and4by5.Ineithercase,theresultis60.
c. Showthat a.b c/ ¤ .ab/ .ac/
Solution: Toshowthenegationofageneralstatement,itsufficestoprovidea counterexample.Here,taking a D 2and b D 1 D c,weseethatthat a.b c/ D 2while .ab/ .ac/ D 4.
NowWorkProblem9 G
EXAMPLE2ApplyingPropertiesofRealNumbers a. Showthat2 p2 D p2 C 2.
Solution: Bythedefinitionofsubtraction,2 p2 D 2 C . p2/.However,bythe commutativepropertyofaddition,2 C . p2/ D p2 C 2.Hence,bythetransitive propertyofequality,2 p2 D p2 C 2.Similarly,itisclearthat,forany a and b, wehave a b D b C a
b. Showthat .8 C x/ y D 8 C .x y/
