IB Mathematics HL Textbook - Central High School

936 Pages·2008·12.98 MB·English

HAESE HARRIS PUBLICATIONS
&
Specialists in mathematics publishing
Mathematics
for the international student
Mathematics HL (Core)
second edition
Paul Urban
David Martin
Robert Haese
Sandra Haese
Michael Haese
Mark Humphries
for use with
IB Diploma
Programme
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\001IB_HL-2_00.CDR Friday, 25 January 2008 3:13:17 PM PETERDELL
symbol_pp
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:9) swtimes
MATHEMATICSFORTHEINTERNATIONALSTUDENT
MathematicsHL(Core)secondedition
PaulUrban B.Sc.(Hons.),B.Ec.
DavidMartin B.A.,B.Sc.,M.A.,M.Ed.Admin.
RobertHaese B.Sc.
SandraHaese B.Sc.
MichaelHaese B.Sc.(Hons.),Ph.D.
MarkHumphries B.Sc.(Hons.)
Haese&HarrisPublications
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haeseandharris.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-876543-11-2
©Haese&HarrisPublications2008
PublishedbyRaksarNomineesPtyLtd.
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2004
Reprinted 2005threetimes(withminorcorrections),2006,2007
SecondEdition 2008
Reprinted 2009(withminorcorrections),2010
CartoonartworkbyJohnMartin.ArtworkbyPiotrPoturajandDavidPurton.
CoverdesignbyPiotrPoturaj.
ComputersoftwarebyDavidPurton,ThomasJanssonandTroyCruickshank.
TypesetinAustraliabySusanHaese(RaksarNominees).TypesetinTimesRoman10\Qw_/11\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
endorsedby,theIBO.
This book is copyright. Except as permitted by the Copyright Act (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.
EnquiriestobemadetoHaese&HarrisPublications.
Copying for educational 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,contacttheCopyrightAgencyLimited.
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. Theywouldbepleasedtocometoasuitableagreementwiththerightfulowner.
Disclaimer: All the internet addresses (URL’s) given in this book were valid at the time of
printing. While the authors and publisher regret any inconvenience that changes of address may
cause readers, no responsibility for any such changes can be accepted by either the authors or the
publisher.
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
V:\BOOKS\IB_books\IB_HL-2ed\IB_HL-2ed_00\002IB_HL-2_00.CDR Thursday, 11 March 2010 10:12:20 AM PETER
FFOORREEWWOORRDD
Mathematics for the International Student: Mathematics HL has been written to reflect the
syllabus for the two-year IB Diploma Mathematics HLcourse. It is not our intention to define the
course.Teachersareencouragedtouseotherresources.Wehavedevelopedthebookindependently
of the International Baccalaureate Organization (IBO) in consultation with many experienced
teachersofIBMathematics.ThetextisnotendorsedbytheIBO.
This second edition builds on the strengths of the first edition. Many excellent suggestions were
received from teachers around the world and these are reflected in the changes. In some cases
sections have been consolidated to allow for greater efficiency. Changes have also been made in
response to the introduction of a calculator-free examination paper.Alarge number of questions,
includingsometochallengeeventhebeststudents,havebeenadded.Inparticular,thefinalchapter
containsover200miscellaneousquestions,someofwhichrequiretheuseofagraphicscalculator.
These questions have been included to provide more difficult challenges for students and to give
them experience at working with problems that may or may not require the use of a graphics
calculator.
Thecombinationof textbookandinteractiveStudentCDwillfosterthemathematicaldevelopment
of students in a stimulating way. Frequent use of the interactive features on the CD is certain to
nurtureamuchdeeperunderstandingandappreciationofmathematicalconcepts.
The book contains many problems from the basic to the advanced, to cater for a wide range of
student abilities and interests. While some of the exercises are simply designed to build skills,
every effort has been made to contextualise problems, so that students can see everyday uses and
practical applications of the mathematics they are studying, and appreciate the universality of
mathematics.
Emphasisisplacedonthegradualdevelopmentofconceptswithappropriateworkedexamples,but
we have also provided extension material for those who wish to go beyond the scope of the
syllabus. Some proofs have been included for completeness and interest although they will not be
examined.
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
the Student CD. To access these pages, simply click on the ‘Background knowledge’icons when
runningtheCD.
It is not our intention that each chapter be worked through in full.Time constraints will not allow
for 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
student understanding and learning. Many Investigations could be developed into portfolio
assignments. Teachers should follow the guidelines for portfolio assignments to ensure they set
acceptableportfoliopiecesfortheirstudentsthatmeettherequirementcriteriafortheportfolios.
Review sets appear at the end of each chapter and a suggested order for teaching the two-year
courseisgivenattheendofthisForeword.
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
oftechnologyhasbeenfavoured.Itisasimportantthatstudentsworkwithapenandpaperasitis
that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on
computer.
TheinteractivefeaturesoftheCDallowimmediateaccesstoourownspeciallydesignedgeometry
packages, graphing packages and more. Teachers are provided with a quick and easy way to
demonstrateconcepts,andstudentscandiscoverforthemselvesandre-visitwhennecessary.
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\003IB_HL-2_00.CDR Friday, 25 January 2008 3:56:01 PM PETERDELL
InstructionsappropriatetoeachgraphiccalculatorproblemareontheCDandcanbeprintedforstu-
dents.TheseinstructionsarewrittenforTexasInstrumentsandCasiocalculators.
Inthischangingworldofmathematicseducation,webelievethatthecontextualapproachshowninthis
book,withtheassociateduseoftechnology,willenhancethestudents’understanding,knowledgeand
appreciationofmathematics,anditsuniversalapplication.
Wewelcomeyourfeedback.
Email:[email protected]
Web:www.haeseandharris.com.au PMU DCM RCH
SHH PMH MAH
ACKNOWLEDGEMENTS
The authors and publishers would like to thank all those teachers who have offered advice and
encouragement. Many of them have read page proofs and made constructive comments and
suggestions.
Particular thanks go to Stephen Hobbs who has given generously of his time in reviewing the first
edition and making suggestions for improvement in this second edition. Thanks are also due to
Dr. Andrzej Cichy, Peter Blythe, Brendan Watson, Myrricia Holmann, Jeff Jones, Mark Willis,
JohnPooleandMarjutMäenpää. WeacknowledgethecontributionsofJohnOwenandMarkBrucein
the preparation of the first edition and we also want to thank others who provided assistance – they
include: Cameron Hall, Fran O'Connor, Glenn Smith, Anne Walker, Malcolm Coad, Ian Hilditch,
Phil Moore, Julie Wilson, Kerrie Clements, Margie Karbassioun, Brian Johnson, Carolyn Farr,
RupertdeSmidt,TerrySwain,Marie-ThereseFilippi,NigelWheeler,SarahLocke,RemaGeorge.
The publishers wish to make it clear that acknowledging these individual does not imply any
endorsement of this book by any of them and all responsibility for content rests with the authors and
publishers.
TEACHING THE TWO-YEAR COURSE – A SUGGESTED ORDER
TeachersareencouragedtocarefullychecktheBACKGROUNDKNOWLEDGEsectionssupplied
on the accompanying CD to ensure that basics have been mastered relatively early in the two-year
HLcourse.Someofthesetopicsnaturallyoccuratthebeginningofaspecificchapter,asindicated
in the table of contents. Click on the BACKGROUND KNOWLEDGE active icons to access the
printablepagesontheCD.
Teachers will have their personal preferences for the order in which the chapters are tackled. A
suggestionistoworkprogressivelyfromChapter1throughtoChapter20,butleaveChapters9,15
and,possibly,16forthesecondyear.Theremainingchapterscanbeworkedthroughinorder.
Alternatively,forthefirstyear,studentscouldworkprogressivelyfromChapter1toChapter23but
not necessarily including chapters 7, 15 and 16. Chapter 9 ‘Mathematical Induction’could also be
attempted later, perhaps early in the second year. In some parts of the world, the topics of
Polynomials, Complex Numbers, 3-D Vector Geometry and Calculus are not usually covered until
thefinalyearofschool.
Another approach could be to teach just those topics that are included in the Mathematics SL
syllabusinthefirstyearandleavetheremainingtopicsforcompletioninthesecondyear.
However, it is acknowledged that there is no single best way for all teachers to work through the
syllabus. Individual teachers have to consider particular needs of their students and other
requirementsandpreferencesthattheymayhave.
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\004IB_HL-2_00.CDR Friday, 25 January 2008 3:20:02 PM PETERDELL
USING THE INTERACTIVE STUDENT CD
The CD is ideal for independent study. Frequent use will nurture a deeper
understanding of Mathematics. Students can revisit concepts taught in class and
undertake their own revision and practice.The CD also has the text of the book,
allowingstudentstoleavethetextbookatschoolandkeeptheCDathome.
TheicondenotesanInteractiveLinkontheCD.Simply‘click’theicontoaccess
arangeofinteractivefeatures:
(cid:2) spreadsheets
(cid:2) videoclips INTERACTIVE
(cid:2) graphingandgeometrysoftware LINK
(cid:2) graphicscalculatorinstructions
(cid:2) computerdemonstrationsandsimulations
(cid:2) backgroundknowledge(asprintablepages)
For those who want to make sure they have the prerequisite levels of understanding for this course,
printable pages of background information, examples, exercises and answers are provided on the CD.
Clickthe‘Backgroundknowledge’icononpages12and248.
Graphics calculators: Instructions for using graphics calculators are also given on the CD and can be
printed. Instructions are given for Texas Instruments and Casio calculators. Click on the relevant icon
(TIorC)toaccessprintableinstructions.
Examplesinthetextbookarenotalwaysgivenforbothtypesofcalculator.Where
TI
thatoccurs,clickontherelevanticontoaccesstheinstructionsfortheothertype
ofcalculator. C
NOTE ON ACCURACY
Studentsareremindedthatinassessmenttasks,includingexaminationpapers,unlessotherwisestatedin
thequestion,allnumericalanswersmustbegivenexactlyortothreesignificantfigures.
HL & SL COMBINED CLASSES
Refer to our website www.haeseandharris.com.au for guidance in using this textbook in HLand SL
combinedclasses.
HL OPTIONS
ThisisacompaniontotheMathematicsHL(Core)textbook.Itofferscoverage
ofeachofthefollowingoptions:
(cid:2) Topic8–Statisticsandprobability
(cid:2) Topic9–Sets,relationsandgroups
(cid:2) Topic10–Seriesanddifferentialequations
(cid:2) Topic11–Discretemathematics
In addition, coverage of the Geometry option for students undertaking the
IB Diploma course Further Mathematics is presented on the CD that accompanies the HLOptions
book.
SUPPLEMENTARY BOOKS
A separate book of WORKED SOLUTIONS gives the fully worked solutions for every question
(discussions, investigations and projects excepted) in each chapter of the Mathematics HL (Core)
textbook.TheHL(CORE)EXAMINATIONPREPARATION&PRACTICEGUIDEoffersadditional
questions and practice exams to help students prepare for the Mathematics HLexamination. For more
[email protected]
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\005IB_HL-2_00.CDR Friday, 25 January 2008 12:53:29 PM PETERDELL
6 TABLEOFCONTENTS
TABLE OF CONTENTS
SYMBOLSANDNOTATION F Graphsofexponentialfunctions 88
USEDINTHISBOOK 10 G Growthanddecay 91
H Thenaturalexponential‘e’ 95
Reviewset3A 98
BACKGROUNDKNOWLEDGE 12 Reviewset3B 99
Reviewset3C 99
toaccess,‘click’activeicononCD
A Surdsandradicals CD
4 LOGARITHMS 101
B Scientificnotation(Standardform) CD
C Numbersystemsandsetnotation CD A Logarithms 102
D Algebraicsimplification CD B Logarithmsinbase10 104
E Linearequationsandinequalities CD C Lawsoflogarithms 106
F Modulusorabsolutevalue CD D Naturallogarithms 110
G Productexpansion CD E Exponentialequationsusinglogarithms 112
H Factorisation CD F Thechangeofbaserule 114
I Formularearrangement CD G Graphsoflogarithmicfunctions 115
J Addingandsubtractingalgebraicfractions CD H Growthanddecay 120
K Congruenceandsimilarity CD Reviewset4A 122
L Coordinategeometry CD Reviewset4B 123
ANSWERS CD Reviewset4C 123
Reviewset4D 124
1 FUNCTIONS 17
5 GRAPHINGANDTRANSFORMING
A Relationsandfunctions 18
FUNCTIONS 125
B Functionnotation,domainandrange 21
C Compositefunctions,f±g 27 A Familiesoffunctions 126
D Signdiagrams 28 B Transformationsofgraphs 128
E Inequalities(inequations) 32 C Simplerationalfunctions 133
F Themodulusfunction 35 D Furthergraphicaltransformations 137
1
G Thereciprocalfunction x 41 Reviewset5A 140
x
Reviewset5B 141
H Asymptotesofotherrationalfunctions 42
I Inversefunctions 44 6 QUADRATICEQUATIONSAND
J Functionswhichhaveinverses 46 FUNCTIONS 143
Reviewset1A 49
A Solvingquadraticequations(Review) 145
Reviewset1B 50
B Thediscriminantofaquadratic 149
Reviewset1C 51
C Thesumandproductoftheroots 152
2 SEQUENCESANDSERIES 53 D Graphingquadraticfunctions 153
E Findingaquadraticfromitsgraph 161
A Numberpatterns 54
F Wherefunctionsmeet 165
B Sequencesofnumbers 54
G Problemsolvingwithquadratics 167
C Arithmeticsequences 56
H Quadraticoptimisation 170
D Geometricsequences 59
Reviewset6A 173
E Series 65
Reviewset6B 174
F Miscellaneousproblems 72
Reviewset6C 175
Reviewset2A 74
Reviewset6D 175
Reviewset2B 75
Reviewset6E 176
Reviewset2C 76
7 COMPLEXNUMBERSAND
3 EXPONENTIALS 77
POLYNOMIALS 177
A Indexnotation 78
B Evaluatingpowers 79 A SolutionsofrealquadraticswithΔ<0 178
C Indexlaws 80 B Complexnumbers 180
D Algebraicexpansionandfactorisation 84 C Realpolynomials 188
E Exponentialequations 87 D Roots,zerosandfactors 193
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\006IB_HL-2_00.CDR Friday, 25 January 2008 12:53:50 PM PETERDELL
TABLEOFCONTENTS 7
E Graphingpolynomials 201 G Usingtrigonometricmodels 305
F Theoremsforrealpolynomials 208 H Reciprocaltrigonometricfunctions 307
Reviewset7A 210 I Trigonometricrelationships 309
Reviewset7B 211 J Compoundangleformulae 310
Reviewset7C 212 K Doubleangleformulae 314
L Trigonometricequationsinquadraticform 318
8 COUNTINGANDTHE M Trigonometricseriesandproducts 318
BINOMIALEXPANSION 213 Reviewset12A 319
Reviewset12B 320
A Theproductprinciple 214
Reviewset12C 321
B Countingpaths 216
Reviewset12D 322
C Factorialnotation 217
D Permutations 219 13 MATRICES 323
E Combinations 223
A Matrixstructure 324
F Binomialexpansions 226
B Matrixoperationsanddefinitions 326
G Thegeneralbinomialexpansion 229
C Theinverseofa2×2matrix 342
Reviewset8A 231
D 3×3andlargermatrices 348
Reviewset8B 232
E Solvingsystemsoflinearequations 350
F Solvingsystemsusingrowoperations 354
9 MATHEMATICALINDUCTION 233
G Inductionwithmatrices 364
A Theprocessofinduction 234 Reviewset13A 366
B Theprincipleofmathematicalinduction 236 Reviewset13B 367
C Indirectproof(extension) 244 Reviewset13C 368
Reviewset9A 245 Reviewset13D 369
Reviewset9B 245 Reviewset13E 370
Reviewset9C 246
14 VECTORSIN2AND
10 THEUNITCIRCLEANDRADIAN 3DIMENSIONS 371
MEASURE 247
A Vectors 372
BACKGROUNDKNOWLEDGE– B Operationswithvectors 375
TRIGONOMETRYWITHRIGHT C 2-Dvectorsincomponentform 383
ANGLEDTRIANGLES–Printablepages CD D 3-Dcoordinategeometry 388
A Radianmeasure 248 E 3-Dvectorsincomponentform 390
B Arclengthandsectorarea 250 F Algebraicoperationswithvectors 393
C Theunitcircleandthebasic G Parallelism 398
trigonometricratios 253 H Unitvectors 400
D Areasoftriangles 263 I Thescalarproductoftwovectors 402
Reviewset10A 266 J Thevectorproductoftwovectors 407
Reviewset10B 267 Reviewset14A 416
Reviewset10C 268 Reviewset14B 417
Reviewset14C 418
11 NON-RIGHTANGLEDTRIANGLE Reviewset14D 419
TRIGONOMETRY 269 Reviewset14E 420
A Thecosinerule 270 15 COMPLEXNUMBERS 421
B Thesinerule 272
A Complexnumbersas2-Dvectors 422
C Usingthesineandcosinerules 277
B Modulus,argument,polarform 425
Reviewset11A 280
C DeMoivre'sTheorem 438
Reviewset11B 281
D Rootsofcomplexnumbers 441
E Furthercomplexnumberproblems 445
12 ADVANCEDTRIGONOMETRY 283
Reviewset15A 445
A Observingperiodicbehaviour 285 Reviewset15B 446
B Thesinefunction 288 Reviewset15C 447
C Modellingusingsinefunctions 293
D Thecosinefunction 296 16 LINESANDPLANESINSPACE 449
E Thetangentfunction 297 A Linesin2-Dand3-D 451
F Trigonometricequations 299 B Applicationsofalineinaplane 456
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\007IB_HL-2_00.CDR Friday, 25 January 2008 10:27:06 AM PETERDELL
8 TABLEOFCONTENTS
C Relationshipbetweenlines 461 21 APPLICATIONSOF
D Planesanddistances 466 DIFFERENTIALCALCULUS 621
E Anglesinspace 471
A Timerateofchange 622
F Theintersectionoftwoormoreplanes 473
B Generalratesofchange 623
Reviewset16A 477
C Motioninastraightline 627
Reviewset16B 478
D Somecurveproperties 634
Reviewset16C 479
E Rationalfunctions 642
Reviewset16D 481
F Inflectionsandshape 647
17 DESCRIPTIVESTATISTICS 483 G Optimisation 652
H Implicitdifferentiation 661
A Continuousnumericaldataandhistograms 485
Reviewset21A 664
B Measuringthecentreofdata 489
Reviewset21B 665
C Cumulativedata 500
D Measuringthespreadofdata 502 Reviewset21C 666
E Statisticsusingtechnology 510
22 DERIVATIVESOFEXPONENTIAL
F Varianceandstandarddeviation 512
ANDLOGARITHMIC
G Thesignificanceofstandarddeviation 518
Reviewset17A 520 FUNCTIONS 667
Reviewset17B 522 A Exponentiale 668
B Naturallogarithms 673
18 PROBABILITY 525
C Derivativesoflogarithmicfunctions 677
A Experimentalprobability 528 D Applications 679
B Samplespace 532
E Somespecialexponentialfunctions 683
C Theoreticalprobability 533
Reviewset22A 684
D Compoundevents 537
Reviewset22B 685
E Usingtreediagrams 541
F Samplingwithandwithoutreplacement 543 23 DERIVATIVESOFCIRCULAR
G Binomialprobabilities 546 FUNCTIONSANDRELATED
H SetsandVenndiagrams 549
RATES 687
I Lawsofprobability 554
J Independentevents 558 A Derivativesofcircularfunctions 688
K Probabilitiesusingpermutationsand B Thederivativesofreciprocalcircular
combinations 560 functions 693
L Bayes’theorem 562 C Thederivativesofinversecircular
Reviewset18A 564 functions 694
Reviewset18B 565 D Maximaandminimawithtrigonometry 697
Reviewset18C 566 E Relatedrates 699
Reviewset18D 568 Reviewset23A 704
Reviewset23B 705
19 INTRODUCTIONTOCALCULUS 569
24 INTEGRATION 707
A Limits 570
B Findingasymptotesusinglimits 574 A Antidifferentiation 708
C Trigonometriclimits 577 B Thefundamentaltheoremofcalculus 710
D Calculationofareasundercurves 579 C Integration 715
Reviewset19 586 D Integrating eax+b and (ax+b)n 720
E Integrating f(u)u0(x) bysubstitution 722
20 DIFFERENTIALCALCULUS 589
F Integratingcircularfunctions 724
A Thederivativefunction 592 G Definiteintegrals 730
B Derivativesatagivenx-value 595 Reviewset24A 734
C Simplerulesofdifferentiation 600 Reviewset24B 735
D Thechainrule 604
Reviewset24C 736
E Productandquotientrules 607
F Tangentsandnormals 611 25 APPLICATIONSOF
G Higherderivatives 616 INTEGRATION 737
Reviewset20A 618
A Findingareasbetweencurves 738
Reviewset20B 619
B Motionproblems 744
Reviewset20C 620
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\008IB_HL-2_00.CDR Friday, 25 January 2008 10:28:36 AM PETERDELL
TABLEOFCONTENTS 9
C Problemsolvingbyintegration 748
Reviewset25A 752
Reviewset25B 753
Reviewset25C 755
26 VOLUMESOFREVOLUTION 757
A Solidsofrevolution 758
B Volumesfortwodefiningfunctions 762
Reviewset26 765
27 FURTHERINTEGRATIONAND
DIFFERENTIALEQUATIONS 767
1 1
A Theintegralsof p and 768
a2¡x2 x2+a2
B Furtherintegrationbysubstitution 769
C Integrationbyparts 771
D Miscellaneousintegration 773
E Separabledifferentialequations 774
Reviewset27A 783
Reviewset27B 784
28 STATISTICALDISTRIBUTIONS
OFDISCRETERANDOM
VARIABLES 785
A Discreterandomvariables 786
B Discreteprobabilitydistributions 788
C Expectation 791
D Themeasuresofadiscreterandom
variable 794
E Thebinomialdistribution 801
F ThePoissondistribution 807
Reviewset28A 810
Reviewset28B 812
29 STATISTICALDISTRIBUTIONS
OFCONTINUOUSRANDOM
VARIABLES 813
A Continuousprobabilitydensityfunctions 814
B Normaldistributions 817
C Thestandardnormaldistribution
(Z-distribution) 821
D Applicationsofthenormaldistribution 828
Reviewset29A 830
Reviewset29B 831
30 MISCELLANEOUSQUESTIONS 833
ANSWERS 857
INDEX 933
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
Y:\HAESE\IB_HL-2ed\IB_HL-2ed_00\009IB_HL-2_00.CDR Friday, 25 January 2008 10:28:19 AM PETERDELL
SYMBOLS AND NOTATION USED IN THIS BOOK
N thesetofpositiveintegersandzero, >or> isgreaterthanorequalto
f0,1,2,3,......g
< islessthan
Z thesetofintegers, f0,§1,§2,§3,......g
·or6 islessthanorequalto
Z+ thesetofpositiveintegers, f1,2,3,......g
isnotgreaterthan
Q thesetofrationalnumbers
isnotlessthan
Q+ thesetofpositiverationalnumbers,
fxjx>0, x2Q g [a,b] theclosedinterval a6x6b
R thesetofrealnumbers ]a,b[ theopeninterval a<x<b
R+ thesetofpositiverealnumbers, un thenthtermofasequenceorseries
fxjx>0, x2R g d thecommondifferenceofanarithmetic
C thesetofcomplexnumbers, sequence
fa+bija,b2Rg r thecommonratioofageometricsequence
p
i ¡1 Sn thesumofthefirstntermsofasequence,
z acomplexnumber u1+u2+:::::+un
z¤ thecomplexconjugateofz S1 or S thesumtoinfinityofasequence,
u1+u2+:::::
jzj themodulusofz Xn
ui u1+u2+:::::+un
argz theargumentofz
i=1
Rez therealpartofz ³ ´
n n!
Imz theimaginarypartofz r r!(n¡r)!
fx1,x2,....g thesetwithelements x1,x2,..... f : A!B f isafunctionunderwhicheachelementof
setAhasanimageinsetB
n(A) thenumberofelementsinthefinitesetA
f : x7!y f isafunctionunderwhichxismappedtoy
fxj...... thesetofallxsuchthat
f(x) theimageofxunderthefunctionf
2 isanelementof
2= isnotanelementof f¡1 theinversefunctionofthefunctionf
? theempty(null)set f±g thecompositefunctionoff andg
U theuniversalset lim f(x) thelimitof f(x) asxtendstoa
x!a
[ union dy
thederivativeofywithrespecttox
\ intersection dx
f0(x) thederivativeoff(x)withrespecttox
µ isasubsetof
A0 thecomplementofthesetA d2y thesecondderivativeofywithrespecttox
dx2
an1, pna atothepowerofpn1, nthrootofa f00(x) thesecondderivativeof f(x) withrespecttox
(if a>0 then na>0)
dny
a12,pa atothepower 12, squarerootofa dxn thenthderivativeofywithrespecttox
p
(if a>0 then a>0) f(n)(x) thenthderiviativeof f(x) withrespecttox
R
jxj tnhemodulusorabsolutevalueofx,thatis ydx theindefiniteintegralofywithrespecttox
xforx>0 x2R
¡xforx<0 x2R Z b
´ identity or isequivalentto ydx thedefiniteintegralofywithrespecttox
a betweenthelimits x=a and x=b
¼ isapproximatelyequalto
ex exponentialfunctionofx
> isgreaterthan
logax logarithmtothebaseaofx
0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 IB_HL-2ed
cyan magenta yellow black
V:\BOOKS\IB_books\IB_HL-2ed\IB_HL-2ed_00\010IB_HL-2_00.CDR Thursday, 11 March 2010 10:14:51 AM PETER