SPECIALIST MATHEMATICS VCE UNITS 3 & 4 CAMBRIDGE SENIOR MATHEMATICS VCE SECOND EDITION MICHAEL EVANS | DAVID TREEBY | KAY LIPSON | JOSIAN ASTRUC NEIL CRACKNELL | GARETH AINSWORTH | DANIEL MATHEWS SPECIALIST MATHEMATICS VCE UNITS 3 & 4 CAMBRIDGE SENIOR MATHEMATICS VCE SECOND EDITION MICHAEL EVANS | DAVID TREEBY | KAY LIPSON | JOSIAN ASTRUC NEIL CRACKNELL | GARETH AINSWORTH | DANIEL MATHEWS ShaftesburyRoad,CambridgeCB28EA,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi-110025,India 103PenangRoad,#05-06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartofCambridgeUniversityPress&Assessment,adepartment oftheUniversityofCambridge. WesharetheUniversity’smissiontocontributetosocietythroughthepursuitofeducation, learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org (cid:13)c MichaelEvans,NeilCracknell,JosianAstruc,KayLipsonandPeterJones2015 (cid:13)c MichaelEvans,DavidTreeby,KayLipson,JosianAstruc,NeilCracknell,GarethAinsworth andDanielMathews2023 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionsof relevantcollectivelicensingagreements,noreproductionofanypartmaytake placewithoutthewrittenpermissionofCambridgeUniversityPress&Assessment. Firstpublished2015 SecondEdition2023 2019181716151413121110987654321 CoverdesignbyDeniseLane(Sardine) TypesetbyJanePitkethly PrintedinChinabyC&COffsetPrintingCo.Ltd. AcataloguerecordforthisbookisavailablefromtheNationalLibraryof Australiaatwww.nla.gov.au ISBN978-1-009-11057-0Paperback Additionalresourcesforthispublicationatwww.cambridge.edu.au/GO ReproductionandCommunicationforeducationalpurposes TheAustralianCopyrightAct1968(theAct)allowsamaximumofonechapteror10%ofthe pagesofthispublication,whicheveristhegreater,tobereproducedand/orcommunicatedbyany educationalinstitutionforitseducationalpurposesprovidedthattheeducationalinstitution(or thebodythatadministersit)hasgivenaremunerationnoticetoCopyrightAgencyLimited (CAL)undertheAct. FordetailsoftheCALlicenceforeducationalinstitutionscontact: CopyrightAgencyLimited Level12,66GoulburnStreet SydneyNSW2000 Telephone:(02)93947600 Facsimile:(02)93947601 Email:[email protected] ReproductionandCommunicationforotherpurposes ExceptaspermittedundertheAct(forexampleafairdealingforthepurposesofstudy,research, criticismorreview)nopartofthispublicationmaybereproduced,storedinaretrievalsystem, communicatedortransmittedinanyformorbyanymeanswithoutpriorwrittenpermission.All inquiriesshouldbemadetothepublisherattheaddressabove. CambridgeUniversityPress&Assessmenthasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublicationanddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Informationregardingprices,traveltimetablesandotherfactualinformationgiveninthisworkis correctatthetimeoffirstprintingbutCambridgeUniversityPress&Assessmentdoesnot guaranteetheaccuracyofsuchinformationthereafter. CambridgeUniversityPressacknowledgestheAustralianAboriginalandTorresStraitIslander peoplesofthisnation.Weacknowledgethetraditionalcustodiansofthelandsonwhichour companyislocatedandwhereweconductourbusiness.Wepayourrespectstoancestorsand Elders,pastandpresent.CambridgeUniversityPressiscommittedtohonouringAustralian AboriginalandTorresStraitIslanderpeoples’uniqueculturalandspiritualrelationshipstothe land,watersandseasandtheirrichcontributiontosociety. Contents Introductionandoverview viii Acknowledgements xiii 1 Preliminarytopics 1 1A Circularfunctions. . . . . . . . . . . . . . . . . . . . . . . 2 1B Thesineandcosinerules . . . . . . . . . . . . . . . . . . . 14 1C Sequencesandseries . . . . . . . . . . . . . . . . . . . . . 19 1D Themodulusfunction . . . . . . . . . . . . . . . . . . . . . 29 1E Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1F Ellipsesandhyperbolas . . . . . . . . . . . . . . . . . . . . 36 1G Parametricequations . . . . . . . . . . . . . . . . . . . . . 43 1H Algorithmsandpseudocode . . . . . . . . . . . . . . . . . . 51 ReviewofChapter1 . . . . . . . . . . . . . . . . . . . . . . 57 2 Logicandproof 65 2A Revisionofprooftechniques . . . . . . . . . . . . . . . . . 66 2B Quantifiersandcounterexamples . . . . . . . . . . . . . . . 75 2C Provinginequalities . . . . . . . . . . . . . . . . . . . . . . 79 2D Telescopingseries . . . . . . . . . . . . . . . . . . . . . . 82 2E Mathematicalinduction . . . . . . . . . . . . . . . . . . . . 84 ReviewofChapter2 . . . . . . . . . . . . . . . . . . . . . . 96 iv Contents 3 Circularfunctions 101 3A Thereciprocalcircularfunctions . . . . . . . . . . . . . . . 102 3B Compoundanddoubleangleformulas . . . . . . . . . . . . 109 3C Theinversecircularfunctions . . . . . . . . . . . . . . . . 115 3D Solutionofequations . . . . . . . . . . . . . . . . . . . . . 122 3E Sumsandproductsofsinesandcosines . . . . . . . . . . . 129 ReviewofChapter3 . . . . . . . . . . . . . . . . . . . . . . 133 4 Vectors 143 4A Introductiontovectors . . . . . . . . . . . . . . . . . . . . 144 4B Resolutionofavectorintorectangularcomponents . . . . . . 155 4C Scalarproductofvectors . . . . . . . . . . . . . . . . . . . 167 4D Vectorprojections . . . . . . . . . . . . . . . . . . . . . . 172 4E Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4F Geometricproofs . . . . . . . . . . . . . . . . . . . . . . . 179 ReviewofChapter4 . . . . . . . . . . . . . . . . . . . . . . 187 5 Vectorequationsoflinesandplanes 197 5A Vectorequationsoflines . . . . . . . . . . . . . . . . . . . 198 5B Intersectionoflinesandskewlines . . . . . . . . . . . . . . 206 5C Vectorproduct . . . . . . . . . . . . . . . . . . . . . . . . 212 5D Vectorequationsofplanes . . . . . . . . . . . . . . . . . . 217 5E Distances,anglesandintersections . . . . . . . . . . . . . . 223 ReviewofChapter5 . . . . . . . . . . . . . . . . . . . . . . 230 6 Complexnumbers 237 6A Startingtobuildthecomplexnumbers . . . . . . . . . . . . 238 6B Modulus,conjugateanddivision . . . . . . . . . . . . . . . 246 6C Polarformofacomplexnumber . . . . . . . . . . . . . . . 251 6D Basicoperationsoncomplexnumbersinpolarform . . . . . 255 6E Solvingquadraticequationsoverthecomplexnumbers . . . . 262 6F Solvingpolynomialequationsoverthecomplexnumbers . . . 266 6G UsingDeMoivre’stheoremtosolveequations. . . . . . . . . 273 6H Sketchingsubsetsofthecomplexplane . . . . . . . . . . . . 277 ReviewofChapter6 . . . . . . . . . . . . . . . . . . . . . . 281 7 RevisionofChapters1–6 289 7A Technology-freequestions . . . . . . . . . . . . . . . . . . 289 7B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 295 7C Extended-responsequestions . . . . . . . . . . . . . . . . . 305 7D Algorithmsandpseudocode . . . . . . . . . . . . . . . . . . 314 Contents v 8 Differentiationandrationalfunctions 316 8A Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . 317 8B Derivativesofx=f(y) . . . . . . . . . . . . . . . . . . . . . 322 8C Derivativesofinversecircularfunctions . . . . . . . . . . . 326 8D Secondderivatives . . . . . . . . . . . . . . . . . . . . . . 331 8E Pointsofinflection . . . . . . . . . . . . . . . . . . . . . . 333 8F Relatedrates . . . . . . . . . . . . . . . . . . . . . . . . . 346 8G Rationalfunctions . . . . . . . . . . . . . . . . . . . . . . . 354 8H Asummaryofdifferentiation . . . . . . . . . . . . . . . . . 363 8I Implicitdifferentiation . . . . . . . . . . . . . . . . . . . . . 365 ReviewofChapter8 . . . . . . . . . . . . . . . . . . . . . . 371 9 Techniquesofintegration 381 9A Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . 382 9B Antiderivativesinvolvinginversecircularfunctions . . . . . . 390 9C Integrationbysubstitution . . . . . . . . . . . . . . . . . . 392 9D Definiteintegralsbysubstitution . . . . . . . . . . . . . . . 398 9E Useoftrigonometricidentitiesforintegration . . . . . . . . . 400 9F Furthersubstitution. . . . . . . . . . . . . . . . . . . . . . 404 9G Partialfractions . . . . . . . . . . . . . . . . . . . . . . . 407 9H Integrationbyparts . . . . . . . . . . . . . . . . . . . . . . 415 9I Furthertechniquesandmiscellaneousexercises . . . . . . . 420 ReviewofChapter9 . . . . . . . . . . . . . . . . . . . . . . 424 10 Applicationsofintegration 429 10A Thefundamentaltheoremofcalculus . . . . . . . . . . . . . 430 10B Areaofaregionbetweentwocurves . . . . . . . . . . . . . 436 10C IntegrationusingaCAScalculator . . . . . . . . . . . . . . 443 10D Volumesofsolidsofrevolution . . . . . . . . . . . . . . . . 449 10E Lengthsofcurvesintheplane . . . . . . . . . . . . . . . . . 458 10F Areasofsurfacesofrevolution . . . . . . . . . . . . . . . . 462 ReviewofChapter10 . . . . . . . . . . . . . . . . . . . . . 467 11 Differentialequations 477 11A Anintroductiontodifferentialequations . . . . . . . . . . . 478 11B Differentialequationsinvolvingafunctionofthe independentvariable . . . . . . . . . . . . . . . . . . . . . 482 11C Differentialequationsinvolvingafunctionofthe dependentvariable . . . . . . . . . . . . . . . . . . . . . . 490 11D Applicationsofdifferentialequations . . . . . . . . . . . . . 493 11E Thelogisticdifferentialequation . . . . . . . . . . . . . . . 504 11F Separationofvariables . . . . . . . . . . . . . . . . . . . . 507 vi Contents 11G Differentialequationswithrelatedrates . . . . . . . . . . . 511 11H Usingadefiniteintegraltosolveadifferentialequation . . . . 516 11I UsingEuler’smethodtosolveadifferentialequation . . . . . 518 11J Slopefieldforadifferentialequation . . . . . . . . . . . . . 525 ReviewofChapter11 . . . . . . . . . . . . . . . . . . . . . 528 12 Kinematics 537 12A Position,velocityandacceleration . . . . . . . . . . . . . . 538 12B Constantacceleration . . . . . . . . . . . . . . . . . . . . . 553 12C Velocity–timegraphs . . . . . . . . . . . . . . . . . . . . . 558 12D Differentialequationsoftheformv=f(x)anda=f(v) . . . . . 565 12E Otherexpressionsforacceleration . . . . . . . . . . . . . . 569 ReviewofChapter12 . . . . . . . . . . . . . . . . . . . . . 574 13 Vectorfunctionsandvectorcalculus 583 13A Vectorfunctions . . . . . . . . . . . . . . . . . . . . . . . 584 13B Positionvectorsasafunctionoftime . . . . . . . . . . . . . 588 13C Vectorcalculus . . . . . . . . . . . . . . . . . . . . . . . . 594 13D Velocityandaccelerationformotionalongacurve . . . . . . 600 ReviewofChapter13 . . . . . . . . . . . . . . . . . . . . . 608 14 RevisionofChapters8–13 615 14A Technology-freequestions . . . . . . . . . . . . . . . . . . 615 14B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 619 14C Extended-responsequestions . . . . . . . . . . . . . . . . . 630 14D Algorithmsandpseudocode . . . . . . . . . . . . . . . . . . 642 15 Linearcombinationsofrandomvariablesandthesamplemean 645 15A Linearfunctionsofarandomvariable . . . . . . . . . . . . . 646 15B Linearcombinationsofrandomvariables . . . . . . . . . . . 651 15C Linearcombinationsofnormalrandomvariables . . . . . . . 661 15D Thesamplemeanofanormalrandomvariable . . . . . . . . 663 15E Investigatingthedistributionofthesamplemean usingsimulation . . . . . . . . . . . . . . . . . . . . . . . 666 15F Thedistributionofthesamplemean . . . . . . . . . . . . . . 671 ReviewofChapter15 . . . . . . . . . . . . . . . . . . . . . 677 Contents vii 16 Confidenceintervalsandhypothesistestingforthemean 682 16A Confidenceintervalsforthepopulationmean . . . . . . . . . 683 16B Hypothesistestingforthemean . . . . . . . . . . . . . . . . 692 16C One-tailandtwo-tailtests . . . . . . . . . . . . . . . . . . . 702 16D Two-tailtestsrevisited . . . . . . . . . . . . . . . . . . . . 708 16E Errorsinhypothesistesting . . . . . . . . . . . . . . . . . . 712 ReviewofChapter16 . . . . . . . . . . . . . . . . . . . . . 717 17 RevisionofChapters15–16 726 17A Technology-freequestions . . . . . . . . . . . . . . . . . . 726 17B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 728 17C Extended-responsequestions . . . . . . . . . . . . . . . . . 730 17D Algorithmsandpseudocode . . . . . . . . . . . . . . . . . . 732 18 RevisionofChapters1–17 734 18A Technology-freequestions . . . . . . . . . . . . . . . . . . 734 18B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 738 18C Extended-responsequestions . . . . . . . . . . . . . . . . . 743 Glossary 752 Answers 764 OnlineappendicesaccessedthroughtheInteractiveTextbookorPDFTextbook AppendixA GuidetotheTI-NspireCAScalculatorinVCEmathematics AppendixB GuidetotheCasioClassPadIICAScalculatorinVCEmathematics AppendixC IntroductiontocodingusingPython AppendixD IntroductiontocodingusingtheTI-Nspire AppendixE IntroductiontocodingusingtheCasioClassPad Introduction and overview CambridgeSpecialistMathematicsVCEUnits3&4SecondEditionprovidesacomplete teachingandlearningresourcefortheVCEStudyDesigntobefirstimplementedin2023. Ithasbeenwrittenwithunderstandingasitschiefaim,andwithamplepracticeoffered throughtheworkedexamplesandexercises. Theworkhasbeentrialledintheclassroom, andtheapproachesofferedarebasedonclassroomexperienceandthehelpfulfeedbackof teacherstoearliereditions. SpecialistMathematicsUnits3and4provideastudyofelementaryfunctions,algebra, calculus,andprobabilityandstatisticsandtheirapplicationsinavarietyofpracticaland theoreticalcontexts. Thisbookhasbeencarefullypreparedtomeettherequirementsofthe newStudyDesign. ThebookbeginswithareviewofsometopicsfromSpecialistMathematicsUnits1and2, includingalgorithmsandpseudocode,circularfunctionsandproof. Theconceptofproofnowfeaturesmorestronglythroughoutthecourse. Toaccountfor this,wehaveaspeciallywrittenProofchapterthatinvolvestopicssuchasdivisibility; inequalities;graphtheory;combinatorics;sequencesandseries,includingpartialsumsand partialproductsandrelatednotations;complexnumbers;matrices;vectorsandcalculus. Otherchaptersalsofeatureexercisesaimedtofurtherdevelopyourstudents’skillsin mathematicalreasoning. Inadditiontotheonlineappendicesonthegeneraluseofcalculators,therearethreeonline appendicesforusingboththeprogramminglanguagePythonandtheinbuiltcapabilities ofstudents’CAScalculators. Thefourrevisionchaptersprovidetechnology-free,multiple-choiceandextended-response questions. Eachofthefirstthreerevisionchapterscontainasectiononalgorithmsand pseudocode. TheTI-NspirecalculatorexamplesandinstructionshavebeencompletedbyPeterFlynn, andthosefortheCasioClassPadbyMarkJelinek,andwethankthemfortheirhelpful contributions. Overview of the print book 1 Gradedstep-by-stepworkedexampleswithpreciseexplanations(andvideoversions) encourageindependentlearning,andarelinkedtoexercisequestions. 2 Sectionsummariesprovideimportantconceptsinboxesforeasyreference. 3 AdditionallinkedresourcesintheInteractiveTextbookareindicatedbyicons,suchas skillsheetsandvideoversionsofexamples. 4 QuestionsthatsuittheuseofaCAScalculatortosolvethemareidentifiedwithin exercises. 5 Chapterreviewscontainachaptersummaryandtechnology-free,multiple-choice,and extended-responsequestions. 6 Revisionchaptersprovidecomprehensiverevisionandpreparationforassessment, includingnewpracticeInvestigations. 7 Theglossaryincludespagenumbersofthemainexplanationofeachterm. 8 Inadditiontocoveragewithinchapters,printandonlineappendicesprovideadditional supportforlearningandapplyingalgorithmsandpseudocode,includingtheuseofPython andTI-NspireandCasioClassPadforcoding. Numbersrefertodescriptionsabove.