Cambridge Seniour Mathematics VCE: Specialist Mathematics Units 3 & 4

789 Pages·2015·30.769 MB·English

by  Michael Evans, Peter Jones, Kay Lipson, Neil Cracknell, Josian Astruc

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Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
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Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathema t i c s U n i t s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party
Contents
Introduction viii
Acknowledgements ix
AnoverviewoftheCambridgecompleteteacherandlearningresource x
1
Preliminarytopics 1
1A Circularfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 2
1B Thesineandcosinerules . . . . . . . . . . . . . . . . . . . . . 14
1C Geometryprerequisites . . . . . . . . . . . . . . . . . . . . . . 19
1D Sequencesandseries . . . . . . . . . . . . . . . . . . . . . . . 23
1E Themodulusfunction . . . . . . . . . . . . . . . . . . . . . . 31
1F Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1G Ellipsesandhyperbolas . . . . . . . . . . . . . . . . . . . . . . 38
1H Parametricequations . . . . . . . . . . . . . . . . . . . . . . . 45
1I Distributionofsamplemeans . . . . . . . . . . . . . . . . . . 53
ReviewofChapter1 . . . . . . . . . . . . . . . . . . . . . . . 57
2
Vectors 67
2A Introductiontovectors . . . . . . . . . . . . . . . . . . . . . . 68
2B Resolutionofavectorintorectangularcomponents. . . . . . . 79
2C Scalarproductofvectors . . . . . . . . . . . . . . . . . . . . . 91
2D Vectorprojections. . . . . . . . . . . . . . . . . . . . . . . . . 96
2E Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2F Geometricproofs . . . . . . . . . . . . . . . . . . . . . . . . . 102
ReviewofChapter2 . . . . . . . . . . . . . . . . . . . . . . . 108
Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathema t i c s U n i t s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party
iv Contents
3
Circularfunctions 118
3A Thereciprocalcircularfunctions . . . . . . . . . . . . . . . . . 119
3B Compoundanddoubleangleformulas . . . . . . . . . . . . . 126
3C Inversesofcircularfunctions . . . . . . . . . . . . . . . . . . . 133
3D Solutionofequations . . . . . . . . . . . . . . . . . . . . . . . 140
ReviewofChapter3 . . . . . . . . . . . . . . . . . . . . . . . 147
4
Complexnumbers 155
4A Startingtobuildthecomplexnumbers. . . . . . . . . . . . . . 156
4B Modulus,conjugateanddivision . . . . . . . . . . . . . . . . . 164
4C Themodulus–argumentformofacomplexnumber. . . . . . . 169
4D Basicoperationsoncomplexnumbersin
modulus–argumentform . . . . . . . . . . . . . . . . . . . . . 174
4E Solvingquadraticequationsoverthecomplexnumbers . . . . . 180
4F Solvingpolynomialequationsoverthecomplexnumbers . . . . 184
4G UsingDeMoivre’stheoremtosolveequations . . . . . . . . . 191
4H Sketchingsubsetsofthecomplexplane . . . . . . . . . . . . . 194
ReviewofChapter4 . . . . . . . . . . . . . . . . . . . . . . . 198
5
RevisionofChapters1–4 206
5A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 206
5B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 208
5C Extended-responsequestions . . . . . . . . . . . . . . . . . . 217
6
Differentiationandrationalfunctions 224
6A Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6B Derivativesofx=f(y) . . . . . . . . . . . . . . . . . . . . . . 230
6C Derivativesofinversecircularfunctions . . . . . . . . . . . . . 234
6D Secondderivatives . . . . . . . . . . . . . . . . . . . . . . . . 239
6E Pointsofinflection . . . . . . . . . . . . . . . . . . . . . . . . 241
6F Relatedrates . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6G Rationalfunctions . . . . . . . . . . . . . . . . . . . . . . . . 262
6H Asummaryofdifferentiation . . . . . . . . . . . . . . . . . . . 271
6I Implicitdifferentiation . . . . . . . . . . . . . . . . . . . . . . 273
ReviewofChapter6 . . . . . . . . . . . . . . . . . . . . . . . 279
Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathema t i c s U n i t s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party
Contents v
7
Techniquesofintegration 288
7A Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 289
7B Antiderivativesinvolvinginversecircularfunctions . . . . . . . 297
7C Integrationbysubstitution . . . . . . . . . . . . . . . . . . . . 299
7D Definiteintegralsbysubstitution . . . . . . . . . . . . . . . . . 305
7E Useoftrigonometricidentitiesforintegration . . . . . . . . . . 307
7F Furthersubstitution . . . . . . . . . . . . . . . . . . . . . . . . 309
7G Partialfractions . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7H Furthertechniquesandmiscellaneousexercises . . . . . . . . . 320
ReviewofChapter7 . . . . . . . . . . . . . . . . . . . . . . . 324
8
Applicationsofintegration 328
8A Thefundamentaltheoremofcalculus . . . . . . . . . . . . . . 329
8B Areaofaregionbetweentwocurves . . . . . . . . . . . . . . 335
8C IntegrationusingaCAScalculator . . . . . . . . . . . . . . . . 342
8D Volumesofsolidsofrevolution . . . . . . . . . . . . . . . . . . 348
8E Lengthsofcurvesintheplane . . . . . . . . . . . . . . . . . . 356
ReviewofChapter8 . . . . . . . . . . . . . . . . . . . . . . . 360
9
Differentialequations 369
9A Anintroductiontodifferentialequations . . . . . . . . . . . . . 370
9B Differentialequationsinvolvingafunctionofthe
independentvariable . . . . . . . . . . . . . . . . . . . . . . . 374
9C Differentialequationsinvolvingafunctionofthe
dependentvariable . . . . . . . . . . . . . . . . . . . . . . . . 382
9D Applicationsofdifferentialequations . . . . . . . . . . . . . . 384
9E Separationofvariables . . . . . . . . . . . . . . . . . . . . . . 395
9F Differentialequationswithrelatedrates . . . . . . . . . . . . . 399
9G Usingadefiniteintegraltosolveadifferentialequation . . . . . 404
9H UsingEuler’smethodtosolveadifferentialequation . . . . . . 406
9I Slopefieldforadifferentialequation. . . . . . . . . . . . . . . 413
ReviewofChapter9 . . . . . . . . . . . . . . . . . . . . . . . 416
Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathema t i c s U n i t s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party
vi Contents
10
Kinematics 425
10A Position,velocityandacceleration . . . . . . . . . . . . . . . . 426
10B Constantacceleration. . . . . . . . . . . . . . . . . . . . . . . 441
10C Velocity–timegraphs . . . . . . . . . . . . . . . . . . . . . . . 446
10D Differentialequationsoftheformv=f(x)anda=f(v) . . . . . 453
10E Otherexpressionsforacceleration . . . . . . . . . . . . . . . . 457
ReviewofChapter10 . . . . . . . . . . . . . . . . . . . . . . 462
11
RevisionofChapters6–10 471
11A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 471
11B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 473
11C Extended-responsequestions . . . . . . . . . . . . . . . . . . 487
12
Vectorfunctions 494
12A Vectorfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 495
12B Positionvectorsasafunctionoftime . . . . . . . . . . . . . . 499
12C Vectorcalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 505
12D Velocityandaccelerationformotionalongacurve . . . . . . . 511
ReviewofChapter12 . . . . . . . . . . . . . . . . . . . . . . 519
13
Dynamics 526
13A Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
13B Newton’slawsofmotion . . . . . . . . . . . . . . . . . . . . . 536
13C Resolutionofforcesandinclinedplanes . . . . . . . . . . . . . 545
13D Connectedparticles . . . . . . . . . . . . . . . . . . . . . . . . 551
13E Variableforces . . . . . . . . . . . . . . . . . . . . . . . . . . 556
13F Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
13G Vectorfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 565
ReviewofChapter13 . . . . . . . . . . . . . . . . . . . . . . 569
14
RevisionofChapters12–13 577
14A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 577
14B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 580
14C Extended-responsequestions . . . . . . . . . . . . . . . . . . 586
Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathema t i c s U n i t s 3 & 4 Photocopying is restricted under law and this material must not be transferred to another party
Contents vii
15
Linearcombinationsofrandomvariablesand
distributionofsamplemeans 592
15A Linearcombinationsofrandomvariables . . . . . . . . . . . . 593
15B Linearcombinationsofindependent
normalrandomvariables . . . . . . . . . . . . . . . . . . . . . 601
15C Simulatingthedistributionofsamplemeans . . . . . . . . . . . 603
15D Thedistributionofthesamplemeanofa
normallydistributedrandomvariable . . . . . . . . . . . . . . 608
15E Thecentrallimittheorem . . . . . . . . . . . . . . . . . . . . . 614
15F Confidenceintervalsforthepopulationmean . . . . . . . . . . 620
ReviewofChapter15 . . . . . . . . . . . . . . . . . . . . . . 628
16
Hypothesistestingforthemean 635
16A Hypothesistestingforthemean . . . . . . . . . . . . . . . . . 636
16B One-tailandtwo-tailtests . . . . . . . . . . . . . . . . . . . . 645
16C Two-tailtestsrevisited . . . . . . . . . . . . . . . . . . . . . . 652
16D Errorsinhypothesistesting . . . . . . . . . . . . . . . . . . . . 656
ReviewofChapter16 . . . . . . . . . . . . . . . . . . . . . . 658
17
RevisionofChapters15–16 664
17A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 664
17B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 666
17C Extended-responsequestions . . . . . . . . . . . . . . . . . . 668
18
RevisionofChapters1–17 670
18A Technology-freequestions . . . . . . . . . . . . . . . . . . . . 670
18B Multiple-choicequestions . . . . . . . . . . . . . . . . . . . . 672
18C Extended-responsequestions . . . . . . . . . . . . . . . . . . 677
Glossary 684
Answers 694
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AppendixB:GuidetotheCasioClassPadIICASCalculatorinVCEMathematics
Cambridge Senior M a t h e m a t ic s A C / V C E ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
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Introduction
CambridgeSpecialistMathematicsAustralianCurriculum/VCEUnits3&4providesa
completeteachingandlearningresourcefortheVCEStudyDesigntobeimplementedin
2016. Ithasbeenwrittenwithunderstandingasitschiefaimandwithamplepracticeoffered
throughtheworkedexamplesandexercises. Alltheworkhasbeentrialledintheclassroom,
andtheapproachesofferedarebasedonclassroomexperienceandtheresponsesofteachers
toearlierversionsofthisbook.
SpecialistMathematicsUnits3&4offersmaterialontopicsfromtheSpecialistMathematics
StudyDesign.
Thebookhasbeencarefullypreparedtoreflecttheprescribedcourse. Newmaterialhasbeen
includedforthestatisticstopics: distributionofsamplemeansandhypothesistesting,andthe
calculustopics: arclengthandseparationofvariablestechniquefordifferentialequations.
Thebookcontainsfiverevisionchapters. Theseprovidetechnology-freequestions,
multiple-choicequestionsandextended-responsequestions.
TheTI-NspirecalculatorexamplesandinstructionshavebeencompletedbyRussellBrown,
andthosefortheCasioClassPadhavebeencompletedbyMariaSchaffner.
Theintegrationofthefeaturesofthetextbookandthenewdigitalcomponentsofthe
package,poweredbyCambridgeHOTmaths,areillustratedintheoverview.
AboutCambridgeHOTmaths
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interactiveonlinemathslearning,teachingandassessmentresourceforstudentsandteachers,
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usedtohostandpowertheinteractivetextbookandtheOnlineTeachingSuite,andselected
topicsfromHOTmaths’ownYears9and10coursesareaareavailableforrevisionofprior
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