Cambridge senior Specialist Mathematics Units 3&4

789 Pages·2016·33.121 MB·English

by  Michael Evans et al.

“9781107587298pre_pi-x” — 2015/9/15 — 11:22 — page i — #1

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Cambridge Senior Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 3&4 Photocopying is restricted under law and this material must not be transferred to another party
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Cambridge Senior Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 3&4 Photocopying is restricted under law and this material must not be transferred to another party
Contents
Introduction viii
AnoverviewoftheCambridgecompleteteacherandlearningresource x
Acknowledgements xii
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 Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 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 Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 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 Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 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 Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 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
IncludedintheInteractiveTextbookandPDFtextbookonly
AppendixA:GuidetotheTI-NspireCASCalculator(OS4)inVCEMathematics
AppendixB:GuidetotheCasioClassPadIICASCalculatorinVCEMathematics
Cambridge Senior Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 3&4 Photocopying is restricted under law and this material must not be transferred to another party
Introduction
Cambridge Specialist Mathematics Australian Curriculum/VCE Units 3 & 4 provides a
complete teaching and learning resource for the VCE Study Design to be implemented in
2016. It has been written with understanding as its chief aim and with ample practice offered
through the worked examples and exercises. All the work has been trialled in the classroom,
and the approaches offered are based on classroom experience and the responses of teachers
to earlier versions of this book.
Specialist Mathematics Units 3 & 4 offers material on topics from the Specialist Mathematics
Study Design.
The book has been carefully prepared to reflect the prescribed course. New material has been
included for the statistics topics: distribution of sample means and hypothesis testing, and the
calculus topics: arc length and separation of variables technique for differential equations.
The book contains five revision chapters. These provide technology-free questions,
multiple-choice questions and extended-response questions.
The TI-Nspire Calculator examples and instructions have been completed by Russell Brown,
and those for the Casio ClassPad have been completed by M aria Schaffner.
The integration of the features of the textbook and the new digital components of the
package, powered by Cambridge HOTmaths, are illustrated in Overview.
About Cambridge HOTmaths
Cambridge HOTmaths is a comprehensive, award-winning mathematics learning system – an
interactive online maths learning, teaching and assessment resource for students and teachers,
for individuals or whole classes, for school and at home. Its digital engine or platform is used
to host and power the interactive textbook and the Online Teaching Suite, and selected topics
from HOTmaths’ own Years 9 and 10 courses area are available for revision of prior
knowledge. All this is included in the price of the textbook.
Cambridge Senior Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 3&4 Photocopying is restricted under law and this material must not be transferred to another party

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Cambridge Senior Mathematics AC/VCE ISBN 978-1-107-58743-4 © Evans, Cracknell, Astruc, Lipson, Jones 2016 Cambridge University Press
Specialist Mathematics Units 3&4 Photocopying is restricted under law and this material must not be transferred to another party