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CALCULUSSETFREE

CalculusSetFree: InfinitesimalstotheRescue

CharlesBryanDawson

UniversityProfessorofMathematics,UnionUniversity,USA

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries

©C.BryanDawson2022

Themoralrightsoftheauthorhavebeenasserted Impression:1

Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove

Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer

PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica

BritishLibraryCataloguinginPublicationData Dataavailable

LibraryofCongressControlNumber:2021937201

ISBN978–0–19–289559–2(hbk.)

ISBN978–0–19–289560–8(pbk.)

DOI:10.1093/oso/9780192895592.001.0001

Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY

LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork.

3.3LocalExtrema

3.9Newton’sMethod

IVIntegration

4.1Antiderivatives

4.2FiniteSums

4.3AreasandSums

4.4DefiniteIntegral

4.5FundamentalTheoremofCalculus

4.6SubstitutionforIndefiniteIntegrals

4.7SubstitutionforDefiniteIntegrals

4.8NumericalIntegration,PartI

4.9NumericalIntegration,PartII

4.10InitialValueProblemsandNetChange

VTranscendentalFunctions

5.1Logarithms,PartI

5.2Logarithms,PartII

5.3InverseFunctions

5.4Exponentials

5.5GeneralExponentials

5.6GeneralLogarithms

5.7ExponentialGrowthandDecay

5.8InverseTrigonometricFunctions

5.9HyperbolicandInverseHyperbolicFunctions

5.10ComparingRatesofGrowth

5.11LimitswithTranscendentalFunctions:L’Hospital’sRule,PartI 827

5.12L’Hospital’sRule,PartII:MoreIndeterminateForms

5.13FunctionswithoutEnd

VIApplicationsofIntegration

6.1AreabetweenCurves

6.2Volumes,PartI

6.3Volumes,PartII 907

6.4ShellMethodforVolumes 919

6.5Work,PartI 935

6.6Work,PartII

7.4TrigonometricSubstitution

7.5PartialFractions,PartI

7.6PartialFractions,PartII

7.7OtherTechniquesofIntegration

7.8StrategyforIntegration

7.9TablesofIntegralsandUseofTechnology

VIIIAlternateRepresentations:ParametricandPolarCurves

8.1ParametricEquations

8.2TangentstoParametricCurves

8.3PolarCoordinates

8.4TangentstoPolarCurves

8.5ConicSections

8.6ConicSectionsinPolarCoordinates

IXAdditionalApplicationsofIntegration

9.1ArcLength

9.4LengthsandSurfaceAreaswithParametricCurves

9.5HydrostaticPressureandForce

9.6CentersofMass

9.7ApplicationstoEconomics

9.8LogisticGrowth

XSequencesandSeries

PrefacefortheStudent

Formany,thestudyofcalculusisseenasariteofpassage—toconquercalculusistopassthroughthegatewaytothesciences,engineering,mathematics,business,economics,technology,andmanyotherfields. Forsome,thestudyofcalculusisindicativeofachievement,ahallmarkofaqualityeducation.Afewcan’t waittostudycalculus,theircuriosityoverflowingwithenthusiasm.Yetothersseecalculusasanannoyance,somethingtotolerateinpursuitofmoreimportantormoreinterestingsubjects.Thisbookisforall ofyou.

Whateveryourreasonforstudyingcalculus,itismyhopethatthistextfacilitatesnotjustthemastering oftechnicalskillsandtheunderstandingofmathematicalconcepts,butalsotraininginthinkinginapatient, systematic,disciplined,andlogicalmanner.Althoughtechnicalskillscanbeusefulforsomestudentsin theircareers,andtheunderstandingofmathematicalconceptscanbeofusetoevenmore,thehabitsof mindcreatedbycarefulthinkingcanbeofusetoeveryone,atanytime,inanyplace.

Preparationforsuccess

Ifyouhavelearnedtodriveacar,thenyoumayrecallhowdrivingtookmuchconsciousthoughtatfirst; butlater,withpractice,drivingbecamemuchmoreofabackgroundtask.Thesameistrueofaddition andmultiplicationfacts;thetask2+4takesverylittlementalenergy.Thisisthehallmarkofdeeplearning: whenataskhasbeenlearnedthoroughly,thenitcanbeperformedaccuratelywithlittleeffort.

Successincalculusismucheasierifbasicalgebraicandtrigonometricskillshavebeenlearnedthis deeply.Ifthequadraticformulaandlawsofexponentscanbeappliedaccuratelyupondemand,thenthe mindisfreetoconcentrateontheconceptsathand.Ifnot,theninsteadofjugglingthreenewconcepts consciously,adozenormoredistractingitemsthatmustberelearnedcompetewiththenewconceptsfor mentalenergy,hamperingone’slearningofthenewmaterial.

Mentalskill-buildingismuchthesameasphysicalskill-building;ittakestimeandconsistentefforton thepartofthelearner.Liftingweightsseveraltimesperweekforamonthisamuchmoreeffectivestrategy thanwaitinguntilthenightbeforetheskillstesttotrytocramtheentiremonth’srepsintooneevening’s workout.Body-buildingsimplydoesnotworkthatway,andneitherdoeslearningmathematics.

Onefinalbitofadvice:learnfromfailure.Everyonemakeserrors,eventextbookauthorswithdecades ofexperienceinthesubject.Nooneisperfect.Butwhenyoumakeanerror,makesureyouunderstand whyitwasanerror,whyadifferentapproachmustbeused,andhowtoavoidmakingthesameerrorin thefuture.Thereisoftensomethingtobelearnedfromyourerrors.Mistakesarenottobefeared,butto beusedtoyouradvantage!

Featuresofthistextbook

Whatmakesthistextbookdifferent?Themostobviousansweristhatitusesinfinitesimals,whichare infinitelysmallnumbersthatyoumightnothaveencounteredinyourpreviouscourses.

x PrefacefortheStudent

Althoughinfinitesimalswereanessentialpartofthedevelopmentofcalculus,theyhavebeenabsent fromnearlyallcalculustextbooksformorethanacentury.Thelargestfactorintheswitchawayfrom infinitesimalswasthefactthat,atthetime,noonehadbeenabletodeveloprigorouslytherequirednumber system.Thisstateofaffairschangedduringthe1960s,andnowtheuseofinfinitesimalsisonceagain seenasmathematicallylegitimate.UsingnotationandproceduresthatIhavedevelopedandpublished, thestudyofcertainportionsofcalculusinthistextisbothmoreintuitiveandsimpleralgebraicallythan inothercalculustextbooks.

Additionalfeaturesinclude:

• Areadableandstudent-friendlynarrative. Thenarrativeiswrittentohelpyouthinkthroughthe developmentofconceptsandthinkthroughsolutionstoexamples.Followingthethinkingprocess helpsyoucreatemeaningandretainideasmoreeasily.

• Readingexercises. Readingexercisesaremeanttobeworkedwhenencounteredduringreading. Thesolutionisplacedinthemarginonetothreepageslater.

• Hundredsofdiagrams. Consistencyofcolorusethroughoutthetext’sdiagramshelpswith interpretation.

• Marginnotes. Marginnotesareusedtoaddexplanations,tips,cautionsagainstmakingcommon errors,andhistoricalnotes.

• Examples. Hundredsofexampleswithcompletesolutionsareincluded.Somesolutionsarewritten compactly,demonstratingthelevelofdetailexpectedofstudentwork.Othersincludemoredetails ofhowtothinkthroughthesolution.

• Thousandsofexercises. Exercisesrangefromtheroutinetothechallenging.Manysectionsinclude “rapid-response”exercisesmeanttohelpyoudistinguishbetweenobjectsoralgebraicforms.Some exercisesareverysimilartoexamplesinthenarrative.Otherexercisesrequireyoutothinkcreatively orexploretheideasmoredeeply.Sometimesexercisesfrommucholdertextbooksareincluded,such asthoselabeled“(GSL)”fromtheclassicearly-20th-centurytextofGranville,Smith,andLongley.

• Answerstoodd-numberedexercises. Theanswerstoodd-numberedexercisessometimesinclude hints,briefexplanationsofwhysomeattemptedanswersareincorrect,alternateformsofanswers, orbothsimplifiedandnonsimplifiedanswerstohelpyoudeterminethesourceofanerror.

• Anextensiveindex.

Itismyprayerthatthistextbookisablessingtoyou,thatithelpsyouunderstandtheconceptsand developtheskillsofcalculusasyoucontinueyoureducationaljourney.Enjoy! BryanDawson UniversityProfessorofMathematics UnionUniversity

July2021

PrefacefortheInstructor

Thistextbookcoverssingle-variablecalculusthroughsequencesandseries,andcorrespondstothefirst twosemestersofcollege-levelcalculusatmostuniversitiesintheUnitedStates.Theorganizationissimilar tothatofotherpopularcalculustexts.

Whatmakesthistextbookdifferentisitsuseofinfinitesimals(andhyperrealnumbersingeneral)for alllimitingprocesses,includingthedefinitionsofderivativeandintegral.Thenotationandprocedures usedwithhyperrealnumbersinthistextbook(whichdifferfromthoseusedinothernonstandardanalysis sources)weredevelopedbyme1 andwereintroducedinarticlesin TheAmericanMathematicalMonthly (February2018)and TheCollegeMathematicsJournal (November2019),withadditionalarticlesplanned. Theutilityofthehyperrealnumbersreachesintootherareasofthecalculusaswell,suchascomparing ratesofgrowthoffunctionsandarelatedprocedurefortestingseries.

Inadditiontomakingcalculusconceptsmoreintuitive,theuseofinfinitesimalscorrespondsmore closelytothewayourcolleaguesinotherdisciplinesteachstudentstoanalyzetheirideas.Theprocedures usedinthistextbookforlimitsarealsoconnectedmoredirectlytothedefinitions,arealgebraicallysimpler, andaremetwithamuchgreaterdegreeofstudentsuccess.

Additionaldifferencesfromothertextbooksincludethesplittingofsomematerialintotwosectionsto facilitatemoreeasilythemultiple-daycoveragetypicalofthosetopics,aswellassomeminorreorganization oftopicscomparedtootherbooks.

Featuresofthistextbook

Featuresofthistextbookinclude:

• Areadableandstudent-friendlynarrative. Thenarrativeiswrittentohelpstudentsthinkthrough thedevelopmentofconceptsandthinkthroughsolutionstoexamples.

• Readingexercises. Readingexercisesaremeanttobeworkedwhenencounteredduringreading. Thesolutionisplacedinthemarginonetothreepageslater.

• Hundredsofdiagrams. Consistencyofcolorusethroughoutthetext’sdiagramshelpswith interpretation.Forinstance,graphsoffunctionsarebluewhereastangentlinesareorange.

• Marginnotes. Marginnotesareusedtoaddexplanations,tips,cautionsagainstmakingcommon errors,andlinkstobiographiesontheMacTutorHistoryofMathematicswebsite.

• Thousandsofexercises. Exercisesrangefromthesimpleandtheroutinetothechallenging.Many sectionsinclude“rapid-response”exercisesmeanttohelpstudentsdistinguishbetweenobjectsor

1 SeetheAcknowledgmentssectionforoneexception.

algebraicforms;theseexercisescouldbeconsideredforin-classroomuse.Someexercisesarevery similartoexamplesinthenarrative.Otherexercisesrequirestudentstothinkcreativelyorexplore theideasmoredeeply.Sometimesexercisesfrommucholdertextbooksareincluded(oradapted forinclusion),suchasthoselabeled“(GSL)”fromtheclassicearly-20th-centurytextofGranville, Smith,andLongley.

Enoughexercisesareincludedtoallowyoutohavechoicesofwhichodd-numberedexercisesto includetocraftanappropriatehomeworkset.Even-numberedexercisescorrespondroughlytooddnumberedexercisesforadditionalstudentpractice.Althoughitisnowcommontofindsolutionsto textbookexercisesontheinternet,Istillfollowthecustomofonlyprovidingtheanswerstooddnumberedexercisesinthetextbook.

• Areviewsectiononavoidingcommonerrors. Eachsubsectionofsection0.6focusesonone particulartypeoferrorsothatyoucanreferstudentstohelpasneeded.Forinstance,studentswho arepronetocancellationerrorscanbereferredtosection0.6“Cancellation.”Theexercisesineach subsectionaredesignedtohelpstudentsrecognizewhethersuchanerrorhasbeenmade,inthe hopesofhelpingthemavoidsucherrorsinthefuture.

• Anextensiveindex.

Another,perhapsunusual,featureofthistextbookisthatitdoesnotcontainfictitiousnamesinword problemsanditdoesnotcontainanygender-specificwordsaftertheprefaces.

Teachinginfinitesimals

Sections1.1–1.3containbasicideas,notation,concepts,andproceduresformanipulatinghyperrealnumbers.JustasweallowcalculusstudentstouserealnumberswithoutfirstsubjectingthemtoDedekindcuts, studentsshouldbeallowedtousehyperrealnumberswithoutreferencetoultrafilters.Aswithastudent’s introductiontoanyothertypeofnumber,thesesectionshelpastudentlearnwhatinfinitesimalsandother hyperrealsare,howtomanipulatethemalgebraically,andwheretheyfitonnumberlines.Notethatthe symbols ε and δ arenotusedforinfinitesimalsinthisbook,becausethesesymbolsmaybeusedforreal numbersinlatercourses.

Sections1.1–1.3arefundamentalforworkingwithhyperrealsandthereforeshouldbecoveredthoroughlyandmasteredbythestudent.Icoveronesectionper50-minuteclassperiod,spendingthreeclass daystotalonthismaterial.

Section1.3givesstudentstheopportunitytopracticethecalculationsinvolvedinfindinglimits;then, insection1.4,studentscanconcentrateontheconceptoflimitshavingalreadylearnedthemanipulations. Thisseparationalsoallowstheflexibilitytoskiplimitsandcoversections1.8and2.1onthederivative immediatelyaftersection1.3,orcoversections4.2–4.4onsumsandthedefiniteintegralimmediatelyafter section1.3.However,itisassumedthatatsomepointthestudentslearnthematerialinsections1.4–1.7. Whatissometimesknownasthe directsubstitutionproperty oflimits,whichiscalled evaluatinglimits usingcontinuity inthistextbook,isnotcovereduntilsection1.7.Thisallowstheverificationofcontinuity

insection1.6tobebothnaturalandmeaningful.Whatothertextbookscall limitlaws arenotnecessaryin thiscurriculum;theequivalentisimplicitinthemanipulationsofsections1.1–1.3.

Moreaboutsectiondependencies

Chapter0isentirelyoptional.Iusuallycoversections0.4and0.5priortobeginningchapter1.Section 1.0isalsooptionalandmaybeskipped;muchofthismaterialisrepeatedinsection1.8.

Section3.6,limitsatinfinity,canbecoveredimmediatelyaftersection1.5.Althoughsections3.5and 3.7oncurvesketchingarenotstrictlynecessaryforlatermaterial,itismyopinionthatmoststudentslearn propertiesofgraphsbestiftheysketchafewbyhand.

Section4.1canbedelayeduntiljustbeforesection4.5,onthefundamentaltheorem.Sections4.8and 4.9canbedelayedaslongasyoudesire.

Theorderofchapters5and6canbeswapped,andtheorderofsectionsinchapter6canbevaried. Sections5.13,7.7,and7.9areeasilyomitted.Althoughtheuseofl’Hospital’sruleinsections5.11and 5.12istraditional(andIstillcoverit),manyofthelimitsofthesesectionscanbecalculatedinotherways, suchasusingtechniquesfromsection5.10andfromseriesinsection10.10.

Sections9.2and9.4dependonportionsofchapter8,butothersectionsinchapter9canbecovered afterchapter7or,withajudiciousselectionofexercises,alongwithchapter6.Sections8.1–8.4canbe coveredasearlyasimmediatelyfollowingchapter5,ifdesired.

Finally,asisusualincalculustextbooks,numeroussectionscontainsubsectionsuponwhichnolater materialdepends.Thisallowsyoutocoverfavorite(sub)topics.However,coverageofeveryproblemtype demonstratedinthenarrativemaynotalwaysbepractical.

Acknowledgments

Ithankmywifeforherpatiencewithmewhileembarkingonthelongjourneyofwritingthisbook.Without hersupport,thisprojectwouldnotexist.

ThefirstcolleaguetojoinmeinteachingcalculususingtheseinfinitesimalmethodswasTroyRiggs (UnionUniversity).Troysuggestedtheuseofthesymbol =forrenderingarealresult(seesection1.3)and wasthefirsttousethesymbolintheclassroom.HeandIspentmorehoursthaneitherofuswishtoadmit discussingtheuseofinfinitesimalideas,andthesediscussionswereessentialtothisproject’ssuccess. Otherswhoclassroom-testedapreliminaryversionofthistextareGeorgeMoss(UnionUniversity), NicholasZoller(SouthernNazareneUniversity),andMoNiazi(SouthernNazareneUniversity).

Thanksalsogotothehundredsofstudentswhoprovidedencouragementandfeedback,bothexplicitly andimplicitly.Knowinghowstudentsinteractwiththematerialhasshapedmanyaspectsofthisbook.

Additionalthankstocolleaguesincludeeditorialboardsandrefereesofarticlesandbookproposals;administratorswhograntedaresearchleaveandotherreleasetimetowrite;andmanyotherswhoparticipated inhallwaydiscussions,attendedmyworkshops,orprovidedencouragement.

Theseedforinvestigatinginfinitesimalmethodsincalculuswasplantedbyastudentinacalculusclass in2004,RobertMichael,whoaskedmanyquestionsaboutinfinitesimalstowhichIdidnothaveadequate answers.Thatseedsatdormantforseveralyears,butwhenitsprouted,itgrewlargerthanIcouldhave imagined.

Somejourneystakegenerations.Mypaternalgrandfather’sformaleducationendedaftertheeighth grade(asitdidforallbutoneofmygrandparents).Hewasasharecropper,leasingthesamelandyear byyearfornearlyfourdecades.Someoneonceaskedhimwhyheneverpurchasedlandofhisown.His replywasthatifhepurchasedland,hecouldleavealegacyforoneofhischildren;ifheinsteadspent thatmoneyforcollegeeducations,hecouldleavealegacyforallfiveofhischildren.Myfathermajored inmathematicsandthenenrichedmymathematicaleducationenoughasachildtoinstillacuriositythat ledtoanacademiccareer.Inadditiontopassingthatlegacyontomyownchildren,itismyprivilegeasa professortohelpotherfamiliesbuildtheirlegacies.Whatablessing!

Last,IthankmyCreator,notonlyforgivingmelife,butalsoforgivingmetheinsightskeytothe developmentoftheinfinitesimalmethodsandfornotlettingmequitwhenIweariedofthejourney.

IwillblesstheLordwhohasgivenmecounsel; Myheartalsoinstructsmeinthenightseasons.

Psalm16:7NKJV

Ihaveoftenprayedthatthistextbookwillbeablessingtobothstudentsandinstructors.Mayitalways beso.

BryanDawson UniversityProfessorofMathematics UnionUniversity

July2021

Chapter0 Review

AlgebraReview,PartI

Someofyouhavealreadylearnedalgebra

andtrigonometryatadeep levelandarereadytojumpintochapter1withconfidence.Others needtospendtimeinthischapter,perhapsmuchtime.Themore accuratelyandreadilyonecanperformalgebra,themoreeasilyone canlearncalculus.Ifitisneeded,aninvestmentintimeandeffort nowwillpaygreatdividendslater.

Becausethematerialofchapter0isareview,thenarrativeisrelativelysparse.Motivationanddevelopmentofformulasarenotalways presented.

Realnumberline

Theword algebra derivesfromthename ofthefirstbookaboutalgebra, Hisab al-jabrw’al-muqabala,writtenbyAbuJa’far MuhammadibnMusaAl-Khwarizmi.

AbuJa’farMuhammadibnMusaAl-Khwarizmi,780–850(approximately) http://www-history.mcs.st-andrews.ac.uk/ Biographies/Al-Khwarizmi.html.TheMac TutorHistoryofMathematicsarchive, hostedbytheUniversityofSt.Andrews inScotland,isoneofthemosttrusted sourcesforthehistoryofmathematics.The websitehasbeeninoperationsincebefore theturnofthecentury.Linkstobiographies ofmathematiciansinthisbookaretothe MacTutorarchive.

Thefirstnumbersachildlearnsarethecountingnumbers,1,2,3, , andsoon,alsoknownasthe naturalnumbers N.Thesenumbersare picturedonahorizontalline,equallyspaced,withlargernumbersto therightandsmallernumberstotheleft(figure 1). 1234

Nextonelearnsthe integers Z,whichincludethenaturalnumbers, theirnegatives,andzero.Thesearealsoplacedonthenumberline (figure 2).

Thencamenumbersoftheform a b ,where a and b areintegers. Thesearethe rationalnumbers Q.Rationalnumbers

Figure1 Countingnumbersonthenumber line

−4−3−2−11234 0

Figure2 Integersonthenumberline

Theterm rational isderivedfromtheword ratio.Rationalnumbersareratiosofintegers. arealsoplaced proportionallyonthenumberline,asalwayswithlargernumbersto therightandsmallernumberstotheleft.Itiscommontorepresent numbersaspointsonaline,asshowninfigure 3.Rationalnumbers havedecimalexpansionsthateitherterminateorrepeat,suchas

or 4 3 =1.3=1.33333

Irrationalnumbers havedecimalexpansionsthatneitherterminate norrepeat,suchas √2=1.4142 ... and π =3.14159 ... .They alsofindtheirplaceonthenumberline.Thecollectionofallofthese

−4−3−2−11234 0 1 2 7 3 2 1

Figure3 Rationalnumbers(inorange)as pointsonthenumberline

Figure4 Realnumbers(inorange)aspoints onthenumberline

numbersiscalledthe realnumbers R (figure 4).Therealnumbersfill outthenumberline;theycanbeplacedinone-to-onecorrespondence withthepointsontheline.Evenso,staytunedformorenumbersin chapter1!

Inequalities

Thestatement a < b meansthenumber a islessthanthenumber b, and a istotheleftof b onthenumberline.Thestatement a > b means thenumber a isgreaterthanthenumber b,and a istotherightof b on thenumberline.

Anyinequalitycanbewrittenintwodifferentways: 4<9 meansthesamethingas 9>4 −4−3−2−11234 0

Figure5 Anumberline.Because 1 istotheleftof 3, 1<3.Multiplyboth numbersby 1 andtheorderreverses: 1> 3

CAUTION:Theword smaller canbeambiguous.Onemightusesmallerassynonymouswith“lessthan,”inwhichcase 3 is smallerthan 1.Onemightusesmalleras synonymouswith“closertozero”(smaller inmagnitude),inwhichcase 1 issmaller than 3

Noticeinfigure 5 thatalthough1istotheleftof3onthenumber line, 1istotherightof 3.Thus,1<3,but 1> 3.Whenmultiplying(ordividing)aninequalitybyanegativenumber,thedirection oftheinequalitymustbereversed.

INEQUALITIES:MULTIPLYINGORDIVIDINGBY NEGATIVES

Whenmultiplyingordividingbothsidesofaninequalitybya negativenumber,thedirectionoftheinequalitymustbereversed.

The reciprocal ofanumberis 1 dividedby thatnumber;thereciprocalof 35 is 1 35 .

Thesameistruewhentakingreciprocals.

Although2<7,notice infigure 6 that 1 2 > 1 7 .Thelargerthedenominator,thesmallerthe fraction.

−11234567 1 2 1 7

Figure6 Anumberline.Because 2 istotheleftof 7, 2<7.Takereciprocals ofbothnumbersandtheorderreverses: 1 2 > 1 7

INEQUALITIES:RECIPROCALS

Whentakingreciprocalsofbothsidesofaninequality,thedirectionoftheinequalitymustbereversed.

Solvinglinearinequalitiesissimilartosolvinglinearequations.Care mustbetaken,however,tochangethedirectionoftheinequalityunder thecircumstancesjustdescribed.

Example1 Solvetheinequality 4x +2< x 7.

Solution Firstwesubtract x frombothsides:

Thegoalistoisolatethevariable x onone sideoftheequationbyitself. 3x +2< 7.

Nextwesubtract2frombothsides: 3x < 9.

Finally,wedividebothsidesby3:

Becausewearedividingbyapositivenumber,thedirectionoftheinequalitydoesnot change. x < 3.

Thesolutiontotheinequalityis x < 3.

Solutionstoinequalitiesmayalsobepresentedingraphicalform. Theideaistoindicatewhichnumbersonthenumberlinesatisfythe inequality.For x < 3,thevariable x satisfiestheinequalityaslongas x istotheleftof 3(figure 7):

−3

Figure7 Theinequality x < 3,shadedingreen.Theopencircleindicates that 3 isnotincluded

Drawinganopencircleindicatesthepointisnotincluded.An alternateversionthatissometimesusedistodrawaparenthesis instead(figure 8).

Recallthat a ≤ b meansthateither a < b or a = b

Example2 Solvetheinequality 7x +1≤4.

−3 )

Figure8 Theinequality x < 3,shadedin green.Theparenthesisindicatesthat 3 isnot included

3 7 [

Figure10 Theinequality x ≥ 3 7 ,shadedin green.Thebracketindicatesthat 3 7 isincluded

Solution Firstwesubtract1frombothsides: 7x ≤3.

Nextwedividebothsidesby 7,whichrequiresswitchingthedirectionoftheinequality: x ≥ 3 7

Thesolutiontotheinequalityis x ≥ 3 7 .

Becausethevalue x = 3 7 isincludedinthesolutiontoexample 2, whenpresentingthesolutioningraphicalform,thecircleisfilledin (figure 9):

Figure9 Theinequality x ≥ 3 7 ,shadedingreen.Thefilledcircleindicates that 3 7 isincluded

Analternateversionofthediagramistouseabracketinsteadofa filledcircle(figure 10).

Intervals

Intervalnotation,andthevarioustypesofintervals,aresummarized intable1.Parenthesesindicatetheendpointisnotincluded;brackets indicatetheendpointisincluded.Thesymbols ∞ and −∞ arenot numbers,butmerelyindicatorsthattheintervalhasnoendpointon therightortheleft,respectively.Intervalsthatdonotcontainanyof theirendpointsare openintervals;intervalsthatcontainalloftheirendpointsare closedintervals.Becauseneither ∞ nor −∞ areendpoints, then [a, ∞) isaclosedinterval.Boundedintervalshavetwoendpoints; unboundedintervalsrangeto ∞ or −∞.Thetwonumbersorsymbols intheintervalarealwayswrittenwiththesmallervalueor −∞ onthe leftandthelargervalueor ∞ ontheright.Intable1,thegreenshading onthegraphsindicatesthesolutionset.

Example3 Fortheinequality 3< x ≤5,(a)writetheinequalityin intervalnotationand(b)classifytheinterval(open/closed,bounded/ unbounded).

Table1 Notation,visualization,andclassificationofintervals

inequality interval graph open/closed bounded?

a < x < b (a, b) ab open bounded

a ≤ x ≤ b [a, b] ab closed bounded

a < x ≤ b (a, b] ab neither bounded

a ≤ x < b [a, b) ab neither bounded

a < x (a, ∞) a open unbounded

a ≤ x [a, ∞) a closed unbounded

x < b (−∞, b) b open unbounded

x ≤ b (−∞, b] b closed unbounded

all x ∈ R (−∞, ∞) both unbounded

Solution (a)Because3isnotincluded,weuseaparenthesisfor thatendpoint,andbecause5isincluded,weuseabracketforthat endpoint.Theintervalis (3,5].

(b)Theintervalisneitheropennorclosedbecauseitcontainsone, butnottheother,endpoint.Theintervalisboundedbecauseitdoes notrangeto ∞ orto −∞.

Example4 Graphtheinterval:(a) [4, ∞),(b) ( 5, 3).

Solution (a)Theinterval [4, ∞) includes4,sowedrawafilledcircle at4toindicateitsinclusion.Theshading(green)hasnoboundonthe right.

(b)Fortheinterval ( 5, 3)

wedonotincludeeitherendpoint,so wedrawopencirclesat 5andat 3.Theshading(green)isbetween thosetwonumbers.

−5−3

Intervalsarecharacterizedbythepropertythatallnumbersbetweentwooftheinterval’snumbersarealsointheinterval.Ifthereis any“gap”intheset,thenitisnotaninterval.Theset ( 4,0) ∪ (3,5) isnotaninterval;seefigure 11

Boundedopenintervalssuchas ( 5, 3) shareanotationwithpointsinthe xy-plane. Contextisnearlyalwaysenoughtodeterminewhichismeant.

−4035

Figure11 Theset ( 4,0) ∪ (3,5) isnotan intervalbecausethereisa“gap”from 0 to 3

CAUTION:MISTAKETOAVOID

Theheuristic“absolutevaluechanges to +”isthesourceofmanyalgebraicerrorsand shouldbeavoided.Forinstance,if x = 5, then |− x| = |− ( 5)| = |5| =5= x, soforthisvalueof x, |− x|̸= x

“Removingthenegative”doesnotwork forabsolutevaluesofvariableexpressions, because x isnotnecessarilyanegative number.

Absolutevalue

Theideaofabsolutevaluecanbevisualizedasthedistancefromthe numbertozeroonthenumberline(figure 12).

−203 |3| |=3 −2| =2

Figure12 Absolutevalueasthedistanceonthenumberlinebetweenthe numberandzero

Becausethenumber3islocated3unitsawayfromzeroonthe numberline, |3| =3.Becausethenumber 2islocated2unitsaway fromzeroonthenumberline, |− 2| =2.

Alternately,wecanthinkoffoldingthenumberlineatthe number0,foldingtheleftsideofthenumberlineontotherightside. Theabsolutevalueofanumberiswhereitislocatedafterthefolding.Theformaldefinitionofabsolutevaluesaysthatweleavepositive numbersalonebut“reflect”or“fold”thenegativenumbersbytaking theirnegatives.

Definition1 ABSOLUTEVALUE

Foranyrealnumber a, |a| = a, if a≥0 a, if a<0. Because 2<0,thedefinition(using a = 2)saysthat |−2| = ( 2) =2.

PROPERTIESOFABSOLUTEVALUE

Foranyrealnumbers a and b, |a b| = |a|·|b| and a b = |a| |b| (assuming b =0).

CAUTION:MISTAKETOAVOID |a + b|̸= |a| + |b| |a b|̸= |a|−|b|

Thepropertiesofabsolutevaluegiveusruleswecanusetosimplify expressions.Forinstance,

Howcanwesimplify √x2?Manyassumetheansweris x,butthis isnotcorrect,becausethesquarerootoperationalwaysgivesusthe nonnegativenumberwiththesquarethatisinside.Forinstance, √16= 4andnot 4andnot ±4.

Therefore, √42 = √16=4,

whereas

Thesolutionsto x 2 =16 are x = ±√16= ±4,butthesymbol √16 meansonlythe positivevalue, 4

CAUTION:MISTAKETOAVOID

x2 = x ( 4)2 = √16=4(not 4)

Thisillustratesthefollowingfact:

x2 = |x|.

Absolutevalueequations andinequalities

Whichnumberssatisfy |x| =3?Thetwonumbers x =3and x = 3 aretheonlysolutionstothisequation.Seefigure 13.

ABSOLUTEVALUEEQUATIONS

If a ≥0,then |x| = a ifandonlyif x = ±a.

Example5 Solve x2 =16

Solution Webeginbytakingthesquarerootofbothsidesofthe equationandsimplifying:

x2 = √16

Nextwesolvetheabsolutevalueequation: x = ±4.

SQUAREROOTSOFSQUARES

x2 = |x|

|3| |=3 −3|=3

Figure13 Thetwonumberslocated 3 units fromzero

Thephrase“x = ±a”hasthesamemeaning as“x = a or x = a.”

Rememberthat √x2 = |x|