Mathematics for the international student : mathematics SL, for use with IB diploma programme

760 Pages·2012·18.199 MB·English

by  Sandra Haese, Michael Haese, Robert Haese, Mark Humphries, Marjut Mäenpää

HAESE MATHEMATICS
Specialists in mathematics publishing
Mathematics
for the international student
Mathematics SL
third edition
Robert Haese
Sandra Haese
Michael Haese
Marjut Mäenpää
Mark Humphries
for use with
IB Diploma
Programme
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MATHEMATICSFORTHEINTERNATIONALSTUDENT
MathematicsSLthirdedition
RobertHaese B.Sc.
SandraHaese B.Sc.
MichaelHaese B.Sc.(Hons.),Ph.D.
MarjutMäenpää B.Sc.,Dip.Ed.
MarkHumphries B.Sc.(Hons.)
HaeseMathematics
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-921972-08-9
©HaeseMathematics2012
PublishedbyHaeseMathematics.
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2004
Reprinted 2005threetimes(withminorcorrections),2006,2007,2008twice
SecondEdition 2009
Reprinted 2010twice(withminorcorrections),2011
ThirdEdition 2012
TypesetinTimesRoman10\Qw_.
The textbook and its accompanying CD have been developed independently of the International
Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by,
theIBO.
This book is copyright. Except as permitted by the CopyrightAct (any fair dealing for the purposes of
private study, research, criticism or review), no part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese
Mathematics.
Copying foreducational purposes:Where copies of part or the whole of the book are made under Part
VB of the CopyrightAct, the law requires that the educational institution or the body that administers it
has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the
CopyrightAgencyLimited.
Acknowledgements: While every attempt has been made to trace and acknowledge copyright, the authors
and publishers apologise for any accidental infringement where copyright has proved untraceable. They
wouldbepleasedtocometoasuitableagreementwiththerightfulowner.
Disclaimer:Alltheinternetaddresses(URLs)giveninthisbookwerevalidatthetimeofprinting.While
the authors and publisher regret any inconvenience that changes of address may cause readers, no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
Mathematics for the International Student: Mathematics SL has been written to embrace the
syllabus for the two-year Mathematics SL Course, to be first examined in 2014. It is not our
intention to define the course. Teachers are encouraged to use other resources. We have developed
this book independently of the International Baccalaureate Organization (IBO) in consultation with
manyexperiencedteachersofIBMathematics.ThetextisnotendorsedbytheIBO.
In the third edition, chapters are arranged to follow the same order as the chapters in our
MathematicsHL(Core)thirdedition,makingiteasierforteacherswhohavecombinedclassesofSL
andHLstudents.
Syllabus references are given at the beginning of each chapter. The new edition reflects the new
Mathematics SLsyllabus. More lower level questions have been added at the start of the exercises,
to make them more suitable for a range of ability levels. Discussion topics for the Theory of
Knowledgehavebeenincludedinthisedition.Seepage12forasummary.
In response to the introduction of a calculator-free examination paper, a large number of questions
have been added and categorised as ‘calculator’ or ‘non calculator’. In particular, the final chapter
containsover160examination-stylequestions.
ComprehensivegraphicscalculatorinstructionsforCasiofx-9860GPlus,Casiofx-CG20,TI-84Plus
and TI-nspire are accessible as printable pages on the CD (see page 16) and, occasionally, where
additional help may be needed, more detailed instructions are available from icons located
throughout the book. The extensive use of graphics calculators and computer packages throughout
the book enables students to realise the importance, application, and appropriate use of technology.
No single aspect of technology has been favoured. It is as important that students work with a pen
andpaperasitisthattheyusetheirgraphicscalculator,oruseaspreadsheetorgraphingpackageon
computer.
This package is language rich and technology rich. The combination of textbook and interactive
StudentCDwillfosterthemathematicaldevelopmentofstudentsinastimulatingway.Frequentuse
of the interactive features on the CD is certain to nurture a much deeper understanding and
appreciation of mathematical concepts. The CD also offers Self Tutor for every worked example.
Self Tutor is accessed via the CD – click anywhere on any worked example to hear a teacher’s
voice explain each step in that worked example. This is ideal for catch-up and revision, or for
motivatedstudentswhowanttodosomeindependentstudyoutsideschoolhours.
For students who may not have a good understanding of the necessary background knowledge for
this course, we have provided printable pages of information, examples, exercises, and answers on
theStudentCD–see‘Backgroundknowledge’(page16).
The interactive features of the CD allow immediate access to our own specially designed geometry
software, graphing software and more. Teachers are provided with a quick and easy way to
demonstrateconcepts,andstudentscandiscoverforthemselvesandre-visitwhennecessary.
continuednextpage
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Itisnotourintentionthateachchapterbeworkedthroughinfull.Timeconstraintsmaynotallowfor
this.Teachers must select exercises carefully, according to the abilities and prior knowledge of their
students, to make the most efficient use of time and give as thorough coverage of work as possible.
Investigations throughout the book will add to the discovery aspect of the course and enhance
studentunderstandingandlearning.
In this changing world of mathematics education, we believe that the contextual approach shown in
this book, with the associated use of technology, will enhance the students’ understanding,
knowledgeandappreciationofmathematics,anditsuniversalapplication.
Wewelcomeyourfeedback.
Email: [email protected]
Web: www.haesemathematics.com.au
RCH SHH PMH
EMM MAH
ACKNOWLEDGEMENTS
CartoonartworkbyJohnMartin.ArtworkbyPiotrPoturajandBenjaminFitzgerald.
CoverdesignbyPiotrPoturaj.
Computer software by Thomas Jansson, Troy Cruickshank, Ashvin Narayanan, Adrian Blackburn,
EdwardRossandTimLee.
TypesetinAustraliabyCharlotteFrost.
EditorialreviewbyCatherineQuinnandDavidMartin.
The authors and publishers would like to thank all those teachers who offered advice and
encouragementonthisbook.Manyofthemreadthepageproofsandofferedconstructivecomments
and suggestions. These teachers include: Dr. Andrzej Cichy, Paul Thompson, Chris Carter, Glen
Whiffen, Leslie Miller, Annette MacDougall, Fatima Remtulla, Sandra Tweedy, Vivienne
Verschuren, Dhruv Prajapati, Matt Kaun and Pamela Vollmar. To anyone we may have missed, we
offerourapologies.
The publishers wish to make it clear that acknowledging these individuals does not imply any
endorsementofthisbookbyanyofthem,andallresponsibilityforthecontentrestswiththeauthors
andpublishers.
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USING THE INTERACTIVE STUDENT CD
TheinteractiveCDisidealforindependentstudy.
MMaaIINNttTTEEhhRRAACCTTeeIIVVEEmmSSTTUUDDEEaaNNTTCCDDttiiccss
Students can revisit concepts taught in class and undertake their own revision fortheinternationalstudent
SSLL
tthhiirrddeeddiittiioonn
andpractice.TheCDalsohasthetextofthebook,allowingstudentstoleave includeSself Tutor ©2012
MMaatthheemmaattiiccssSSLL
thetextbookatschoolandkeeptheCDathome. ffoorruusseewwiitttthhhhIIiiBBrrddDDeeiippddllooiimmttiiaaoonnPPrrooggrraammmmee
By clicking on the relevant icon, a range of interactive features can be
accessed:
INTERACTIVE
(cid:2) Self Tutor LINK
(cid:2) Graphicscalculatorinstructions
(cid:2) Backgroundknowledge(asprintablepages)
(cid:2) Interactive links to spreadsheets, graphing and geometry software,
computerdemonstrationsandsimulations
Graphics calculator instructions: Detailed instructions are available on the
CD, as printable pages (see page 16). Click on the icon for Casio fx-9860G GRAPHICS
CALCULATOR
Plus,Casiofx-CG20,TI-84Plus,orTI-nspireinstructions.
INSTRUCTIONS
SELF TUTOR is an exciting feature of this book.
The Self Tutor icon on each worked example denotes an active link on the CD.
Simply ‘click’on the Self Tutor (or anywhere in the example box) to access the worked
example, with a teacher’s voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using
movement and colour on the screen.
Ideal for students who have missed lessons or need extra help.
Example2 Self Tutor
µ ¶
3
Alinepassesthroughthepoint A(1,5) andhasdirectionvector . Describethelineusing:
2
a a vector equation b parametricequations c a Cartesianequation.
a The vector eµqua¶tionis r = aµ+ t¶b where µ ¶
¡! 1 3 b= 3 R
a = OA= and b = 2
5 2
µ ¶ µ ¶ µ ¶
A
x 1 3
) = +t , t2R
y 5 2
b x=1+3t and y =5+2t, t2R r
a
x¡1 y¡5
c Now t= =
3 2
) 2x¡2=3y¡15
O
) 2x¡3y =¡13
SeeChapter13,Vectorapplications,p.323
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6 TABLEOFCONTENTS
TABLE OF CONTENTS
SYMBOLSANDNOTATION 2 FUNCTIONS 53
USEDINTHISBOOK 10 A Relationsandfunctions 54
B Functionnotation 57
BACKGROUNDKNOWLEDGE 16 C Domainandrange 59
D Compositefunctions 64
A Surdsandradicals CD
E Signdiagrams 66
B Scientificnotation(standardform) CD
F Rationalfunctions 69
C Numbersystemsandsetnotation CD
G Inversefunctions 73
D Algebraicsimplification CD
Reviewset2A 77
E Linearequationsandinequalities CD
Reviewset2B 78
F Modulusorabsolutevalue CD
Reviewset2C 80
G Productexpansion CD
H Factorisation CD
3 EXPONENTIALS 81
I Formularearrangement CD
J Addingandsubtractingalgebraicfractions CD A Exponents 82
K Congruenceandsimilarity CD B Lawsofexponents 84
L Pythagoras’theorem CD C Rationalexponents 87
M Coordinategeometry CD D Algebraicexpansionandfactorisation 90
N Rightangledtriangletrigonometry CD E Exponentialequations 92
F Exponentialfunctions 94
Factsaboutnumbersets CD G Growthanddecay 98
H Thenaturalexponentialex 101
Summaryofcircleproperties CD Reviewset3A 105
Reviewset3B 106
Summaryofmeasurementfacts CD Reviewset3C 107
GRAPHICSCALCULATOR 4 LOGARITHMS 109
INSTRUCTIONS 16 A Logarithmsinbase10 110
B Logarithmsinbasea 113
Casiofx-9860GPLUS CD
C Lawsoflogarithms 116
Casiofx-CG20 CD
D Naturallogarithms 120
TexasInstrumentsTI-84Plus CD
TexasInstrumentsTI-nspire CD E Exponentialequationsusinglogarithms 123
F Thechangeofbaserule 125
1 QUADRATICS 17 G Graphsoflogarithmicfunctions 126
H Growthanddecay 130
A Quadraticequations 19
Reviewset4A 132
B Thediscriminantofaquadratic 25
Reviewset4B 133
C Quadraticfunctions 28
Reviewset4C 134
D Findingaquadraticfromitsgraph 38
E Wherefunctionsmeet 42
5 TRANSFORMINGFUNCTIONS 135
F Problemsolvingwithquadratics 44
A Graphingfunctions 136
G Quadraticoptimisation 47
B Transformationofgraphs 140
Reviewset1A 49
C Translationsy=f(x)+bandy=f(x-a) 141
Reviewset1B 50
D Stretchesy=pf(x),p>0and
Reviewset1C 51
y=f(qx),q>0 143
E Reflectionsy=-f(x)andy=f(-x) 144
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TABLEOFCONTENTS 7
F Miscellaneoustransformations 146 10 TRIGONOMETRIC
Reviewset5A 148 FUNCTIONS 231
Reviewset5B 149
A Periodicbehaviour 232
Reviewset5C 150
B Thesinefunction 236
C Modellingusingsinefunctions 243
6 SEQUENCESANDSERIES 151
D Thecosinefunction 246
A Numbersequences 152 E Thetangentfunction 248
B Thegeneraltermofanumbersequence 153 F Generaltrigonometricfunctions 251
C Arithmeticsequences 155 Reviewset10A 253
D Geometricsequences 159 Reviewset10B 253
E Series 166 Reviewset10C 254
F Arithmeticseries 167
G Geometricseries 170 11 TRIGONOMETRICEQUATIONS
Reviewset6A 176 ANDIDENTITIES 255
Reviewset6B 176
A Trigonometricequations 256
Reviewset6C 177
B Usingtrigonometricmodels 263
C Trigonometricrelationships 265
7 THEBINOMIALEXPANSION 179
D Doubleangleformulae 268
A Binomialexpansions 180
E Trigonometricequationsinquadraticform 271
B Thebinomialcoefficient(nr) 183
Reviewset11A 272
C Thebinomialtheorem 185
Reviewset11B 273
Reviewset7A 187
Reviewset11C 274
Reviewset7B 188
12 VECTORS 275
8 THEUNITCIRCLEAND
A Vectorsandscalars 276
RADIANMEASURE 189
B Geometricoperationswithvectors 279
A Radianmeasure 190 C Vectorsintheplane 286
B Arclengthandsectorarea 193 D Themagnitudeofavector 288
C Theunitcircleandthetrigonometricratios 196 E Operationswithplanevectors 290
D Applicationsoftheunitcircle 201 F Thevectorbetweentwopoints 293
E Multiplesof ?h_ and ?f_ 205 G Vectorsinspace 296
F Theequationofastraightline 209 H Operationswithvectorsinspace 300
Reviewset8A 210 I Parallelism 304
Reviewset8B 211 J Thescalarproductoftwovectors 307
Reviewset8C 212 Reviewset12A 314
Reviewset12B 315
9 NON-RIGHTANGLED Reviewset12C 317
TRIANGLETRIGONOMETRY 213
13 VECTORAPPLICATIONS 319
A Areasoftriangles 214
B Thecosinerule 217 A Problemsinvolvingvectoroperations 320
C Thesinerule 220 B Linesin2-Dand3-D 322
D Usingthesineandcosinerules 224 C Theanglebetweentwolines 326
Reviewset9A 228 D Constantvelocityproblems 328
Reviewset9B 229 E Theshortestdistancefromalinetoapoint 331
Reviewset9C 230 F Intersectinglines 335
G Relationshipsbetweenlines 337
Reviewset13A 340
Reviewset13B 341
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8 TABLEOFCONTENTS
Reviewset13C 342 D Integration 454
E Rulesforintegration 457
14 INTRODUCTIONTO F Integratingf(ax+b) 462
DIFFERENTIALCALCULUS 343 G Integrationbysubstitution 465
H Definiteintegrals 468
A Limits 345
Reviewset18A 472
B Limitsatinfinity 347
Reviewset18B 473
C Ratesofchange 350
Reviewset18C 473
D Thederivativefunction 353
E Differentiationfromfirstprinciples 355
19 APPLICATIONSOF
Reviewset14A 357
Reviewset14B 358 INTEGRATION 475
Reviewset14C 358 A Theareaunderacurve 476
B Theareabetweentwofunctions 479
15 RULESOFDIFFERENTIATION 359 C Kinematics 483
A Simplerulesofdifferentiation 360 D Solidsofrevolution 489
B Thechainrule 364 Reviewset19A 494
C Theproductrule 367 Reviewset19B 496
D Thequotientrule 369 Reviewset19C 497
E Derivativesofexponentialfunctions 371
F Derivativesoflogarithmicfunctions 375 20 DESCRIPTIVESTATISTICS 499
G Derivativesoftrigonometricfunctions 378 A Keystatisticalconcepts 500
H Secondandhigherderivatives 381 B Measuringthecentreofdata 505
Reviewset15A 383 C Measuringthespreadofdata 517
Reviewset15B 383 D Boxplots 521
Reviewset15C 384 E Cumulativefrequencygraphs 526
F Varianceandstandarddeviation 531
16 PROPERTIESOFCURVES 385 Reviewset20A 539
A Tangentsandnormals 386 Reviewset20B 541
B Increasinganddecreasingfunctions 392 Reviewset20C 542
C Stationarypoints 397
D Inflectionsandshape 401 21 LINEARMODELLING 545
Reviewset16A 409 A Correlation 546
Reviewset16B 410 B Pearson’scorrelationcoefficient 550
Reviewset16C 411 C Lineofbestfit 554
D Theleastsquaresregressionline 557
17 APPLICATIONSOF E Interpolationandextrapolation 558
DIFFERENTIALCALCULUS 413 Reviewset21A 562
A Kinematics 414 Reviewset21B 563
B Ratesofchange 423 Reviewset21C 565
C Optimisation 428
Reviewset17A 437 22 PROBABILITY 567
Reviewset17B 438 A Experimentalprobability 569
Reviewset17C 439 B Samplespace 574
C Theoreticalprobability 575
18 INTEGRATION 441 D Tablesofoutcomes 579
A Theareaunderacurve 442 E Compoundevents 581
B Antidifferentiation 448 F Treediagrams 585
C Thefundamentaltheoremofcalculus 449
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TABLEOFCONTENTS 9
G Samplingwithandwithoutreplacement 588
H SetsandVenndiagrams 591
I Lawsofprobability 597
J Independentevents 601
Reviewset22A 604
Reviewset22B 604
Reviewset22C 606
23 DISCRETERANDOM
VARIABLES 607
A Discreterandomvariables 608
B Discreteprobabilitydistributions 610
C Expectation 614
D Thebinomialdistribution 618
Reviewset23A 626
Reviewset23B 627
Reviewset23C 628
24 THENORMALDISTRIBUTION 629
A Thenormaldistribution 631
B Probabilitiesusingacalculator 636
C Thestandardnormaldistribution
(Z-distribution) 639
D Quantilesork-values 644
Reviewset24A 648
Reviewset24B 649
Reviewset24C 650
25 MISCELLANEOUSQUESTIONS 651
A Non-calculatorquestions 652
B Calculatorquestions 665
ANSWERS 679
INDEX 755
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10
SYMBOLS AND NOTATION USED IN THIS BOOK
N thesetofpositiveintegersandzero, ajb adividesb
f0,1,2,3,....g u thenthtermofasequenceorseries
n
Z thesetofintegers,f0,§1,§2,§3,....g d thecommondifferenceofanarithmetic
Z+ thesetofpositiveintegers,f1,2,3,....g sequence
Q thesetofrationalnumbers r thecommonratioofageometricsequence
Q+ thesetofpositiverationalnumbers, Sn thesumofthefirstntermsofasequence,
fxjx>0, x2Q g u1+u2+::::+un
R thesetofrealnumbers S1 or S thesumtoinfinityofasequence,
u +u +::::
R+ thesetofpositiverealnumbers, 1 2
Xn
fxjx>0, x2Rg u u +u +::::+u
i 1 2 n
fx1,x2,....g thesetwithelements x1,x2,.... µi=1 ¶
n(A) thenumberofelementsinsetA n
therth binomialcoefficient,r=0,1,2,::::
r
fxj.... thesetofallxsuchthat intheexpansionof (a+b)n
2 isanelementof f : A!B f isafunctionunderwhicheachelement
2= isnotanelementof ofsetAhasanimageinsetB
? theempty(null)set f : x7!y f isafunctionwhichmapsxontoy
U theuniversalset f(x) theimageofxunderthefunctionf
[ union f¡1 theinversefunctionofthefunctionf
\ intersection
f±g thecompositefunctionoff andg
½ isapropersubsetof
lim f(x) thelimitoff(x)asxtendstoa
µ isasubsetof x!a
A0 thecomplementofthesetA dy
thederivativeofy withrespecttox
1 p 1 dx
an, na atothepowerof n, nthrootofa f0(x) thederivativeoff(x)withrespecttox
p
(if a>0 then na>0) d2y
thesecondderivativeofy withrespecttox
1 p 1 dx2
a2, a atothepower 2, squarerootofa f00(x) thesecondderivativeoff(x)withrespect
(if a>0 then pa>0) tox
dny
jxj themo½dulusorabsolutevalueofx dxn thenthderivativeofy withrespecttox
xforx>0 x2R
jxj= ¡xforx<0 x2R fR(n)(x) thenthderivativeoff(x)withrespecttox
ydx theindefiniteintegralofy withrespecttox
´ identity or isequivalentto
Z
¼ isapproximatelyequalto b
ydx thedefiniteintegralofy withrespecttox
> isgreaterthan
a betweenthelimits x=a and x=b
¸or> isgreaterthanorequalto ex exponentialfunctionofx
< islessthan log x logarithmtothebaseaofx
a
·or6 islessthanorequalto lnx thenaturallogarithmofx, log x
e
isnotgreaterthan
sin,cos,tan thecircularfunctions
¥ isnotlessthan
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