Cambridge Senior Mathematics: Mathematical Methods VCE Units 1 & 2

862 Pages·2022·11.826 MB·English

by  Michael Evans, Kay Lipson, Douglas Wallace, David Greenwood

MATHEMATICAL
METHODS
VCE UNITS 1 & 2
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
MICHAEL EVANS | KAY LIPSON
DOUGLAS WALLACE | DAVID GREENWOOD
MATHEMATICAL
METHODS
VCE UNITS 1 & 2
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
MICHAEL EVANS | KAY LIPSON
DOUGLAS WALLACE | DAVID GREENWOOD
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Contents
Introductionandoverview ix
Acknowledgements xiv
1
Reviewinglinearequations 1
1A Linearequations . . . . . . . . . . . . . . . . . . . . . . . 2
1B Constructinglinearequations . . . . . . . . . . . . . . . . . 8
1C Simultaneousequations . . . . . . . . . . . . . . . . . . . 11
1D Constructingsimultaneouslinearequations . . . . . . . . . 16
1E Solvinglinearinequalities . . . . . . . . . . . . . . . . . . . 19
1F Usingandtransposingformulas . . . . . . . . . . . . . . . 22
ReviewofChapter1 . . . . . . . . . . . . . . . . . . . . . . 27
2
Reviewingcoordinategeometry 32
2A Distanceandmidpoints . . . . . . . . . . . . . . . . . . . . 33
2B Thegradientofastraightline . . . . . . . . . . . . . . . . . 36
2C Theequationofastraightline . . . . . . . . . . . . . . . . . 42
2D Graphingstraightlines . . . . . . . . . . . . . . . . . . . . 50
2E Parallelandperpendicularlines . . . . . . . . . . . . . . . 54
2F Familiesofstraightlines . . . . . . . . . . . . . . . . . . . 58
2G Linearmodels . . . . . . . . . . . . . . . . . . . . . . . . . 61
2H Simultaneouslinearequations . . . . . . . . . . . . . . . . 64
ReviewofChapter2 . . . . . . . . . . . . . . . . . . . . . . 70
iv Contents
3
Quadratics 76
3A Expandingandcollectingliketerms . . . . . . . . . . . . . . 77
3B Factorising . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3C Quadraticequations . . . . . . . . . . . . . . . . . . . . . 87
3D Graphingquadratics . . . . . . . . . . . . . . . . . . . . . 91
3E Completingthesquareandturningpoints . . . . . . . . . . . 96
3F Graphingquadraticsinpolynomialform. . . . . . . . . . . . 101
3G Solvingquadraticinequalities . . . . . . . . . . . . . . . . . 104
3H Thegeneralquadraticformula . . . . . . . . . . . . . . . . 106
3I Thediscriminant . . . . . . . . . . . . . . . . . . . . . . . 110
3J Solvingsimultaneouslinearandquadraticequations . . . . . 114
3K Familiesofquadraticpolynomialfunctions . . . . . . . . . . 117
3L Quadraticmodels . . . . . . . . . . . . . . . . . . . . . . . 126
ReviewofChapter3 . . . . . . . . . . . . . . . . . . . . . . 129
4
Agalleryofgraphs 137
4A Rectangularhyperbolas . . . . . . . . . . . . . . . . . . . 138
4B Thetruncus . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4C Thegraphofy2 =x . . . . . . . . . . . . . . . . . . . . . . 144
4D Thegraphofy=√x . . . . . . . . . . . . . . . . . . . . . . 145
4E Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4F Determiningrules . . . . . . . . . . . . . . . . . . . . . . . 154
ReviewofChapter4 . . . . . . . . . . . . . . . . . . . . . . 158
5
Functionsandrelations 164
5A Setnotationandsetsofnumbers . . . . . . . . . . . . . . . 165
5B Relations,domainandrange . . . . . . . . . . . . . . . . . 169
5C Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5D One-to-onefunctionsandimplieddomains . . . . . . . . . . 183
5E Piecewise-definedfunctions . . . . . . . . . . . . . . . . . 187
5F Applyingfunctionnotation . . . . . . . . . . . . . . . . . . 189
5G Inversefunctions . . . . . . . . . . . . . . . . . . . . . . . 191
5H Functionsandmodellingexercises . . . . . . . . . . . . . . 194
ReviewofChapter5 . . . . . . . . . . . . . . . . . . . . . . 198
6
Polynomials 205
6A Thelanguageofpolynomials . . . . . . . . . . . . . . . . . 206
6B Divisionofpolynomials . . . . . . . . . . . . . . . . . . . . 210
6C Factorisationofpolynomials . . . . . . . . . . . . . . . . . 216
6D Solvingcubicequations . . . . . . . . . . . . . . . . . . . . 225
6E Cubicfunctionsoftheformf(x)=a(x h)3+k . . . . . . . . . 228
−
Contents v
6F Graphsoffactorisedcubicfunctions . . . . . . . . . . . . . 232
6G Solvingcubicinequalities . . . . . . . . . . . . . . . . . . . 238
6H Familiesofcubicpolynomialfunctions . . . . . . . . . . . . 239
6I Quarticandotherpolynomialfunctions . . . . . . . . . . . . 243
6J Applicationsofpolynomialfunctions . . . . . . . . . . . . . 248
6K Thebisectionmethod . . . . . . . . . . . . . . . . . . . . . 253
ReviewofChapter6 . . . . . . . . . . . . . . . . . . . . . . 257
7
Transformations 265
7A Translationsoffunctions . . . . . . . . . . . . . . . . . . . 266
7B Dilationsandreflections . . . . . . . . . . . . . . . . . . . 270
7C Combinationsoftransformations . . . . . . . . . . . . . . . 274
7D Determiningtransformations . . . . . . . . . . . . . . . . . 277
7E Transformationsofgraphsoffunctions . . . . . . . . . . . . 280
ReviewofChapter7 . . . . . . . . . . . . . . . . . . . . . . 286
8
RevisionofChapters2–7 292
8A Technology-freequestions . . . . . . . . . . . . . . . . . . 292
8B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 295
8C Extended-responsequestions . . . . . . . . . . . . . . . . . 298
8D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 302
9
Probability 304
9A Samplespacesandprobability . . . . . . . . . . . . . . . . 305
9B Estimatingprobabilities . . . . . . . . . . . . . . . . . . . . 312
9C Multi-stageexperiments . . . . . . . . . . . . . . . . . . . 317
9D Combiningevents . . . . . . . . . . . . . . . . . . . . . . . 322
9E Probabilitytables . . . . . . . . . . . . . . . . . . . . . . . 327
9F Conditionalprobability . . . . . . . . . . . . . . . . . . . . 331
9G Independentevents . . . . . . . . . . . . . . . . . . . . . . 339
9H Solvingprobabilityproblemsusingsimulation . . . . . . . . . 346
9I Pseudocodeforprobabilityandsimulation . . . . . . . . . . 350
ReviewofChapter9 . . . . . . . . . . . . . . . . . . . . . . 356
10
Countingmethods 363
10A Additionandmultiplicationprinciples . . . . . . . . . . . . . 364
10B Arrangements . . . . . . . . . . . . . . . . . . . . . . . . 367
10C Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
10D Applicationstoprobability. . . . . . . . . . . . . . . . . . . 379
ReviewofChapter10 . . . . . . . . . . . . . . . . . . . . . 382
vi Contents
11
Discreteprobabilitydistributions 386
11A Discreterandomvariables . . . . . . . . . . . . . . . . . . 387
11B Samplingwithoutreplacement . . . . . . . . . . . . . . . . 393
11C Samplingwithreplacement:thebinomialdistribution . . . . . 396
ReviewofChapter11 . . . . . . . . . . . . . . . . . . . . . 407
12
RevisionofChapters9–11 413
12A Technology-freequestions . . . . . . . . . . . . . . . . . . 413
12B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 415
12C Extended-responsequestions . . . . . . . . . . . . . . . . . 418
12D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 422
13
Exponentialfunctionsandlogarithms 424
13A Theindexlaws . . . . . . . . . . . . . . . . . . . . . . . . 425
13B Rationalindices . . . . . . . . . . . . . . . . . . . . . . . . 432
13C Graphsofexponentialfunctions . . . . . . . . . . . . . . . . 435
13D Solvingexponentialequationsandinequalities . . . . . . . . 441
13E Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 445
13F Usinglogarithmstosolveexponentialequations
andinequalities . . . . . . . . . . . . . . . . . . . . . . . . 449
13G Graphsoflogarithmfunctions . . . . . . . . . . . . . . . . . 452
13H Exponentialmodelsandapplications . . . . . . . . . . . . . 458
ReviewofChapter13 . . . . . . . . . . . . . . . . . . . . . 468
14
Circularfunctions 474
14A Measuringanglesindegreesandradians . . . . . . . . . . . 475
14B Definingcircularfunctions:sineandcosine . . . . . . . . . . 478
14C Anothercircularfunction:tangent . . . . . . . . . . . . . . 480
14D Symmetrypropertiesofcircularfunctions . . . . . . . . . . 481
14E Valuesofcircularfunctions . . . . . . . . . . . . . . . . . . 484
14F Graphsofsineandcosine . . . . . . . . . . . . . . . . . . . 487
14G Solutionoftrigonometricequations . . . . . . . . . . . . . . 494
14H Sketchgraphsofy=asinn(t ε)andy=acosn(t ε) . . . . 499
± ±
14I Sketchgraphsofy=asinn(t ε) bandy=acosn(t ε) b. 501
± ± ± ±
14J FurthersymmetrypropertiesandthePythagoreanidentity . . 503
14K Thetangentfunction . . . . . . . . . . . . . . . . . . . . . 506
14L NumericalmethodswithaCAScalculator . . . . . . . . . . . 509
14M Generalsolutionoftrigonometricequations . . . . . . . . . 512
14N Applicationsofcircularfunctions . . . . . . . . . . . . . . . 515
ReviewofChapter14 . . . . . . . . . . . . . . . . . . . . . 518
Contents vii
15
RevisionofChapters13–14 524
15A Technology-freequestions . . . . . . . . . . . . . . . . . . 524
15B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 526
15C Extended-responsequestions . . . . . . . . . . . . . . . . . 529
15D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 533
16
Ratesofchange 535
16A Recognisingrelationships . . . . . . . . . . . . . . . . . . . 536
16B Constantrateofchange . . . . . . . . . . . . . . . . . . . . 540
16C Averagerateofchange . . . . . . . . . . . . . . . . . . . . 543
16D Instantaneousrateofchange . . . . . . . . . . . . . . . . . 548
16E Positionandaveragevelocity . . . . . . . . . . . . . . . . . 554
ReviewofChapter16 . . . . . . . . . . . . . . . . . . . . . 561
17
Differentiationandantidifferentiationofpolynomials 566
17A Thederivative . . . . . . . . . . . . . . . . . . . . . . . . . 568
17B Rulesfordifferentiation . . . . . . . . . . . . . . . . . . . . 576
17C Differentiatingxnwherenisanegativeinteger . . . . . . . . 584
17D Graphsofthederivativefunction . . . . . . . . . . . . . . . 587
17E Antidifferentiationofpolynomialfunctions . . . . . . . . . . 595
17F Limitsandcontinuity . . . . . . . . . . . . . . . . . . . . . 601
17G Whenisafunctiondifferentiable? . . . . . . . . . . . . . . . 607
ReviewofChapter17 . . . . . . . . . . . . . . . . . . . . . 610
18
Applicationsofdifferentiationofpolynomials 615
18A Tangentsandnormals . . . . . . . . . . . . . . . . . . . . 616
18B Ratesofchange . . . . . . . . . . . . . . . . . . . . . . . . 619
18C Stationarypoints . . . . . . . . . . . . . . . . . . . . . . . 624
18D Typesofstationarypoints . . . . . . . . . . . . . . . . . . . 627
18E Applicationstomaximumandminimumproblems . . . . . . . 632
18F Applicationstomotioninastraightline . . . . . . . . . . . . 639
18G Familiesoffunctionsandtransformations . . . . . . . . . . 646
18H Newton’smethodforfindingsolutionstoequations . . . . . . 649
ReviewofChapter18 . . . . . . . . . . . . . . . . . . . . . 654
19
RevisionofChapters16–18 663
19A Technology-freequestions . . . . . . . . . . . . . . . . . . 663
19B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 664
19C Extended-responsequestions . . . . . . . . . . . . . . . . . 670
19D Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 674
viii Contents
20
Furtherdifferentiationandantidifferentiation 677
20A Thechainrule. . . . . . . . . . . . . . . . . . . . . . . . . 678
20B Differentiatingrationalpowers . . . . . . . . . . . . . . . . 683
20C Antidifferentiatingrationalpowers . . . . . . . . . . . . . . 686
20D Thesecondderivative . . . . . . . . . . . . . . . . . . . . . 689
20E Sketchgraphs . . . . . . . . . . . . . . . . . . . . . . . . 690
ReviewofChapter20 . . . . . . . . . . . . . . . . . . . . . 693
21
Integration 697
21A Estimatingtheareaunderagraph . . . . . . . . . . . . . . 698
21B Findingtheexactarea:thedefiniteintegral . . . . . . . . . . 703
21C Signedarea . . . . . . . . . . . . . . . . . . . . . . . . . . 708
ReviewofChapter21 . . . . . . . . . . . . . . . . . . . . . 714
22
RevisionofChapters20–21 721
22A Technology-freequestions . . . . . . . . . . . . . . . . . . 721
22B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 722
23
RevisionofChapters1–22 724
23A Technology-freequestions . . . . . . . . . . . . . . . . . . 724
23B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . 727
23C Extended-responsequestions . . . . . . . . . . . . . . . . . 731
A
Algorithmsandpseudocode 735
A1 Algorithmsandflowcharts . . . . . . . . . . . . . . . . . . 736
A2 Iterationandselection . . . . . . . . . . . . . . . . . . . . 741
A3 Introductiontopseudocode . . . . . . . . . . . . . . . . . . 747
Glossary 756
Answers 764
OnlineappendicesaccessedthroughtheInteractiveTextbookorPDFTextbook
AppendixB Furtherpolynomialsandsystemsoflinearequations
AppendixC GuidetotheTI-NspireCAScalculatorinVCEmathematics
AppendixD GuidetotheCasioClassPadIICAScalculatorinVCEmathematics
AppendixE IntroductiontocodingusingPython
AppendixF IntroductiontocodingusingtheTI-Nspire
AppendixG IntroductiontocodingusingtheCasioClassPad
Introduction and overview
CambridgeMathematicalMethodsVCEUnits1&2SecondEditionprovidesacomplete
teachingandlearningresourcefortheVCEStudyDesigntobefirstimplementedin2023.
Ithasbeenwrittenwithunderstandingasitschiefaim,andwithamplepracticeoffered
throughtheworkedexamplesandexercises. Theworkhasbeentrialledintheclassroom,and
theapproachesofferedarebasedonclassroomexperienceandtheresponsesofteachersto
earliereditionsofthisbookandtherequirementsofthenewStudyDesign.
MathematicalMethodsUnits1and2provideanintroductorystudyofsimpleelementary
functions,algebra,calculus,andprobabilityandstatisticsandtheirapplicationsinavariety
ofpracticalandtheoreticalcontexts. ThecourseisdesignedaspreparationforMathematical
MethodsUnits3and4,andtogetherwithCambridgeSpecialistMathematicsVCEUnits1&2
SecondEdition,aspreparationforSpecialistMathematicsUnits3and4. Itcontainsassumed
knowledgeandskillsfortheseunits.
ThefirsttwochaptersproviderevisionofmaterialfromYears9and10mathematics. This
materialisessentialtotheintroductionoffunctionsandcalculus.
Thebookhasbeencarefullypreparedtoreflecttheprescribedcourse. IntheStudyDesignwe
havetheKeyKnowledgedotpointinOutcome2:
(cid:4) keyelementsofalgorithmdesign: sequencing,decision-making,repetitionand
representationincludingtheuseofpseudocode
andinOutcome3:
(cid:4) thepurposeandeffectofsequencing,decision-makingandrepetitionstatementson
relevantfunctionalitiesoftechnology,andtheirroleinthedesignofalgorithmsand
simulations.
Theseareaddressedgenerallyin‘AppendixA:Algorithmsandpseudocode’andalso
specificallyforsimulationandcountinginprobabilityandthenumericalsolutionof
equationswiththebisectionmethodandNewton’smethod. Inadditiontotheonline
appendicesonthegeneraluseofcalculators,therearethreeonlineappendicesforusing
boththeprogramminglanguagePythonandtheinbuiltcapabilitiesofstudents’CAS
calculators.
Thebookcontainssixrevisionchapters. Theseprovidetechnology-free,multiple-choiceand
extended-responsequestions.
Thefirstfourrevisionchapterscontainmaterialsuitableforstudentinvestigations,afeature
ofthenewcourse. TheStudyDesignsuggeststhat‘[a]nInvestigationcomprisesonetotwo
weeksofinvestigationintooneortwopracticalortheoreticalcontextsorscenariosbased
oncontentfromareasofstudyandapplicationofkeyknowledgeandkeyskillsforthe
outcomes’. Wehaveaimedtoprovidestrongsupportforteachersinthedevelopmentofthese
investigations.
TheTI-NspirecalculatorexamplesandinstructionshavebeencompletedbyPeterFlynn,
andthosefortheCasioClassPadbyMarkJelinek,andwethankthemfortheirhelpful
contributions.