Mathematics for the international student: international baccalaureat mathematics HL (options), international baccalaureate diploma programme

416 Pages·2005·4.168 MB·English

by  Blythe, Peter

HAESE HARRIS PUBLICATIONS
&
Specialists in mathematics publishing
Mathematics
for the international student
Mathematics HL (Options)
Including coverage on CD of the
Geometry option for Further Mathematics SL
Peter Blythe
Peter Joseph
Paul Urban
David Martin
Robert Haese
Michael Haese
International
Baccalaureate
Diploma
Programme
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MATHEMATICSFORTHEINTERNATIONALSTUDENT
InternationalBaccalaureateMathematicsHL(Options)
PeterBlythe B.Sc.
PeterJoseph M.A.(Hons.),Grad.Cert.Ed.
PaulUrban B.Sc.(Hons.),B.Ec.
DavidMartin B.A.,B.Sc.,M.A.,M.Ed.Admin.
RobertHaese B.Sc.
MichaelHaese B.Sc.(Hons.),Ph.D.
Haese&HarrisPublications
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haeseandharris.com.au
NationalLibraryofAustraliaCardNumber&ISBN 1876543337
©Haese&HarrisPublications2005
PublishedbyRaksarNomineesPtyLtd
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2005 Reprinted 2006(twice)
CartoonartworkbyJohnMartin.ArtworkbyPiotrPoturajandDavidPurton.
CoverdesignbyPiotrPoturaj.
ComputersoftwarebyDavidPurton.
TypesetinAustraliabySusanHaeseandCharlotteSabel(RaksarNominees).
TypesetinTimesRoman10\Qw_/11\Qw_
ThetextbookanditsaccompanyingCDhavebeendevelopedindependentlyoftheInternational
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,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,
mechanical,photocopying,recordingorotherwise,withoutthepriorpermissionofthepublisher.
EnquiriestobemadetoHaese&HarrisPublications.
Copying for educational purposes: Where copies of part or the whole of the book are made
underPartVBoftheCopyrightAct,thelawrequiresthattheeducationalinstitutionorthebody
that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For
information,contacttheCopyrightAgencyLimited.
Acknowledgements: The publishers acknowledge the cooperation of many teachers in the
preparationofthisbook.Afulllistappearsonpage4.
Whileeveryattempthasbeenmadetotraceandacknowledgecopyright,theauthorsandpublishers
apologiseforanyaccidentalinfringementwherecopyrighthasproveduntraceable. Theywouldbe
pleasedtocometoasuitableagreementwiththerightfulowner.
Disclaimer: All the internet addresses (URL’s) given in this book were valid at the time of
printing.Whiletheauthorsandpublisherregretanyinconveniencethatchangesofaddressmay
causereaders,noresponsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthe
publisher.
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FOREWORD
Mathematics forthe International Student: Mathematics HL(Options) has been written
asacompanionbooktotheMathematicsHL(Core)textbook.Together,theyaimtoprovide
students and teachers with appropriate coverage of the two-year Mathematics HL Course
(first examinations 2006), which is one of the courses of study in the International
BaccalaureateDiplomaProgramme.
Itisnotourintentiontodefinethecourse.Teachersareencouragedtouseotherresources.We
have developed the book independently of the International Baccalaureate Organization
(IBO) in consultation with many experienced teachers of IB Mathematics. The text is not
endorsedbytheIBO.
On the accompanying CD, we offer coverage of the Euclidean Geometry Option for students
undertaking the IB Diploma course FurtherMathematics SL.This Option (with answers)
canbeprintedfromtheCD.
The interactive features of the CD allow immediate access to our own specially designed
geometry packages, graphing packages and more. Teachers are provided with a quick and
easywaytodemonstrateconcepts,andstudentscandiscoverforthemselvesandre-visitwhen
necessary.
InstructionsappropriatetoeachgraphicscalculatorproblemareontheCDandcanbeprinted
forstudents.TheseinstructionsarewrittenforTexasInstrumentsandCasiocalculators.
In this changing world of mathematics education, we believe that the contextual approach
shown in this book, with associated use of technology, will enhance the students
understanding,knowledgeandappreciationofmathematicsanditsuniversalapplication.
Wewelcomeyourfeedback Email: [email protected]
Web: www.haeseandharris.com.au
PJB PJ PMU
DCM RCH PMH
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ACKNOWLEDGEMENTS
Theauthorsandpublisherswouldliketothankallthoseteacherswhohavereadtheproofsof
thisbookandofferedadviceandencouragement.
Special thanks to MarkWillis for permission to include some of his questions in HLTopic 8
‘Statistics and probability’. Others who offered to read and comment on the proofs include:
Mark William Bannar-Martin, Nick Vonthethoff, Hans-Jørn Grann Bentzen, IsaacYoussef,
Sarah Locke, Ian Fitton, Paola San Martini, Nigel Wheeler, Jeanne-Mari Neefs, Winnie
Auyeungrusk, Martin McMulkin, Janet Huntley, Stephanie DeGuzman, Simon Meredith,
RupertdeSmidt,ColinJeavons,DaveLoveland,JanDijkstra,ClareByrne,PeterDuggan,Jill
Robinson, Sophia Anastasiadou, Carol A. Murphy, Janet Wareham, Robert Hall, Susan
Palombi, Gail A. Chmura, Chuck Hoag, Ulla Dellien, Richard Alexander, Monty
Winningham, Martin Breen, Leo Boissy, Peter Morris, Ian Hilditch, Susan Sinclair, Ray
Chaudhuri,GrahamCramp.Toanyonewemayhavemissed,weofferourapologies.
Thepublisherswishtomakeitclearthatacknowledgingtheseindividualsdoesnotimplyany
endorsement of this book by any of them, and all responsibility for the content rests with the
authorsandpublishers.
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TABLEOFCONTENTS 5
TABLE OF CONTENTS
FURTHER MATHEMATICS SL TOPIC 1
TOPIC1
GEOMETRY
Availableonlybyclickingontheiconalongside.
Thischapterplusanswersisfullyprintable.
HL TOPIC 8
(FurthermathematicsSLTopic2)
STATISTICS AND PROBABILITY 9
A Expectationalgebra 10
B Cumulativedistributionfunctions 19
C Distributionsofthesamplemean 45
D Confidenceintervalsformeansandproportions 60
E Significanceandhypothesistesting 73
F TheChi-squareddistribution 88
Reviewset8A 101
Reviewset8B 104
HL TOPIC 9
(FurthermathematicsSLTopic3)
SETS, RELATIONS AND GROUPS 109
A Sets 110
B Orderedpairs 119
C Functions 131
D Binaryoperations 136
E Groups 145
F Furthergroups 159
Reviewset9A 166
Reviewset9B 169
HL TOPIC 10
(FurthermathematicsSLTopic4)
SERIES AND DIFFERENTIAL EQUATIONS 171
A Somepropertiesoffunctions 174
B Sequences 190
C Infiniteseries 199
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6 TABLEOFCONTENTS
D TaylorandMaclaurinseries 223
E Firstorderdifferentialequations 229
Reviewset10A 242
Reviewset10B 242
Reviewset10C 243
Reviewset10D 244
Reviewset10E 245
HL TOPIC 11
(FurthermathematicsSLTopic5)
DISCRETE MATHEMATICS 247
A NUMBERTHEORY 248
A.1 Numbertheoryintroduction 248
A.2 Orderpropertiesandaxioms 249
A.3 Divisibility,primalityandthedivisionalgorithm 256
A.4 Gcd,lcmandtheEuclideanalgorithmgreatestcommondivisor(gcd) 263
A.5 Thelineardiophantineequation ax(cid:2)(cid:3)(cid:2)by(cid:2)(cid:4)(cid:2)c 270
A.6 Primenumbers 274
A.7 Linearcongruence 278
A.8 TheChineseremaindertheorem 286
A.9 Divisibilitytests 289
A.10 Fermat’slittletheorem 292
B GRAPHTHEORY 296
B.1 Preliminaryproblemsinvolvinggraphtheory 296
B.2 Terminology 297
B.3 Fundamentalresultsofgraphtheory 301
B.4 Journeysongraphsandtheirimplication 310
B.5 Planargraphs 316
B.6 Treesandalgorithms 319
B.7 TheChinesepostmanproblem 332
B.8 Thetravellingsalesmanproblem(TSP) 336
Reviewset11A 339
Reviewset11B 340
Reviewset11C 341
Reviewset11D 342
Reviewset11E 343
APPENDIX (Methods of proof) 345
ANSWERS 351
INDEX 411
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SYMBOLS AND NOTATION f .......... g the set of all elements..........
2 is an element of
E(X) the expected value of X,
2= is not an elementof
which is ¹
fxj....... the set of all x such that ......
Var(X) the varianceof X, N the set of all natural numbers
which is ¾ 2
X Z the set of integers
X¡¹
Z = the standardisedvariable Q the set of rational numbers
¾
R the set of real numbers
P(.......) the probability
of ........ occurring C the set of all complex
numbers
» is distributedas Z+ the set of positive integers
¼ is approximatelyequal to P the set of all prime numbers
x the sample mean U the universal set
sn2 the sample variance ; or f g the empty (null) set
sn2¡1 the unbiassedestimate of ¾2 µ is a subset of
¹ the mean of random ½ is a proper subset of
X
variableX P(A) the power of set A
¾ the standarddeviation A\B the intersectionof sets
X
of random variable X A and B
DU(n) the discrete uniform A[B the union of sets A and B
distribution ) implies that
B(n, p) the binomialdistribution )Á does not imply that
A0 the complementof the set A
B(1, p) the Bernoulli distribution
n(A) the number of elements
Hyp(n, M, N) the hypergeometric
in the set A
distribution
AnB the difference of sets
Geo(p) the geometricdistribution
A and B
NB(r, p) the negativebinomial
A¢B the symmetricdifference
distribution
of sets A and B
Po(m) the Poisson distribution
A£B the Cartesianproduct of
U(a, b) the continuousuniform
sets A and B
distribution
R a relation of ordered pairs
Exp(¸) the exponentialdistribution
xRy x is related to y
N(¹, ¾2) the normal distribution x´y(modn) x is equivalentto y, modulon
pb the random variable Z the set of residue classes,
n
of sample proportions modulo n
X the random variable £n multiplication,modulo n
of sample means 2Z the set of even integers
f : A!B f is a function under which
T the random variable
each element of set A has
of the t-distribution
an image in set B
º the number of degrees f : x7!y f is a function under which
of freedom x is mapped to y
f(x) the image of x under
H0 the null hypothesis
the functionf
H1 the alternativehypothesis f¡1 the inverse function of
Â2 the chi-squaredstatistic
calc the function f
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f ±g or f(g(x)) the compositefunction of f and g
jxj the modulusor absolutevalue of x
[a, b] the closed interval, a6x6b
]a, b[ the open interval a<x<b
u the nth term of a sequenceor series
n
fu g the sequencewith nth term u
n n
S the sum of the first n terms of a sequence
n
S1 the sum to infinity of a series
Xn
ui u1+u2+u3+:::::+un
i=1
Qn
ui u1£u2£u3£:::::£un
i=1
lim f(x) the limit of f(x) as x tends to a
x!a
lim f(x) the limit of f(x) as x tends to a from the positive side of a
x!a+
maxfa, bg the maximumvalue of a or b
X1
c xn the power series whose terms have form c xn
n n
n=0
ajb a divides b, or a is a factor of b
aÁj b a does not divide b, or a is a not a factor of b
gcd(a, b) the greatest commondivisor of a and b
lcm(a, b) the least commonmultiple of a and b
»= is isomorphicto
G is the complementof G
A matrix A
An matrix A to the power of n
A(G) the adjacencymatrix of G
A(x, y) the point A in the plane with Cartesiancoordinatesx and y
[AB] the line segment with end points A and B
AB the length of [AB]
(AB) the line containingpoints A and B
b
A the angle at A
C[AB or ]CAB the angle between [CA] and [AB]
¢ABC the triangle whose vertices are A, B and C
or the area of triangle ABC
k is parallel to
Ák is not parallel to
? is perpendicularto
AB.CD length AB £ length CD
PT2 PT £ PT
Power M the power of point M relative to circle C
C
¡!
AB the vector from A to B
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88
HL Topic
(Further Mathematics SL Topic 2)
Before beginning any work on this option, it is recommended that a careful revision of
thecorerequirementsforstatisticsandprobabilityismade.
Thisisidentifiedby“Topic6–Core:StatisticsandProbability”asexpressedinthesyl-
labus guide on pages 26–29 of IBO document on the Diploma Programme Mathe-
maticsHLforthefirstexamination2006.
Throughout this booklet, there will be many references to the core requirements,
taken from “Mathematics for the International Student Mathematics HL (Core)” Paul
Urban et al, published by Haese and Harris, especially chapters 18, 19, and 30. This
willbereferredtoas“fromthetext”.
Statistics and probability
Contents: A Expectation algebra
B Cumulative distribution functions
(for discrete and continuous
variables)
C Distribution of the sample mean
and the Central Limit Theorem
D Confidence intervals for means and
proportions
E Significance and hypothesis testing
and errors
F The Chi-squared distribution, the
“goodness of fit” test, the test for
the independence of two variables.
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10 STATISTICSANDPROBABILITY (Topic8)
A EXPECTATION ALGEBRA
E(X), THE EXPECTED VALUE OF X
Recallthat if a randomvariableX has mean ¹ then ¹ is knownas the expectedvalueof X,
or simply E(X).
P
( xP(x), for discrete X
¹ = E(X) = R
xf(x) dx, for continuous X
From section 30E.1 of the text (Investigation1) we noticed that
E(aX +b) = aE(X)+b
P
Proof: (discrete case only) E(aX+b) = (ax+b)P(x)
P
= [axP(x)+bP(x)]
P P
=a xP(x)+b P(x)
P
=aE(X)+b(1) fas P(x)=1g
=aE(X)+b
Var(X(cid:2)), THE VARIANCE OF X
A random variable X, has variance ¾2, also known as Var(X)
where ¾2 = Var(X) = E((X ¡¹)2)
P
Notice that for discrete X ² Var(X) = (x¡¹)2p(x)
P
² Var(X) = x2p(x)¡¹2
² Var(X) = E(X2)¡fE(X)g2
Again, from Investigation1 of Section 30E.1, Var(aX +b) = a2 Var(X)
Proof: (discrete case only)
Var(aX +b) = E((aX +b)2)¡fE(aX +b)g2
¡ ¢
= E a2X2+2abX+b2 ¡faE(X)+bg2
=a2 E(X2)+2ab E(X)+b2¡a2fE(X)g2¡2ab E(X)¡b2
=a2E(X2)¡a2fE(X)g2
=a2[E(X2)¡fE(X)g2]
=a2Var (X)
THE STANDARDISED VARIABLE, Z
2
If a random variable X is normally distributed with mean ¹ and variance ¾ we write
X » N(¹, ¾2), where » reads is distributed as.
X ¡¹
The standardised variable Z is defined as Z = and has mean 0 and variance 1.
¾
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