Cambridge Senior Mathematics VCE: Specialist Mathematics VCE Units 3 & 4

865 Pages·2023·14.007 MB·English

by  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
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
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(cid:13)c MichaelEvans,NeilCracknell,JosianAstruc,KayLipsonandPeterJones2015
(cid:13)c MichaelEvans,DavidTreeby,KayLipson,JosianAstruc,NeilCracknell,GarethAinsworth
andDanielMathews2023
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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.