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Trang 1%* HAESE s 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
Trang 2MATHEMATICS FOR THE INTERNATIONAL STUDENT
Mathematics HL (Core) second edition
Paul Urban B.Sc.(Hons.),B.Ec
Michael Haese B.Sc.(Hons.),Ph.D
Haese & Harris Publications
3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA
Telephone: +61 8 8355 9444, Fax: +618 8355 9471
Email: info@haeseandharris.com.au
National Library of Australia Card Number & ISBN 978-1-876543-11-2
© Haese & Harris Publications 2008
Published by Raksar Nominees Pty Ltd
3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA
Reprinted 2005 three times (with minor corrections), 2006, 2007
Second Edition 2008
Cartoon artwork by John Martin Artwork by Piotr Poturaj and David Purton
Cover design by Piotr Poturaj
Computer software by David Purton, Thomas Jansson and Troy Cruickshank
Typeset in Australia by Susan Haese (Raksar Nominees) Typeset in Times Roman 105 1 lệ
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, the IBO
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 Enquiries to be made to Haese & Harris Publications
Copying for educational purposes: Where copies of part or the whole of the book are made
under Part VB of the Copyright Act, 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 Copyright Agency Limited
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 would be pleased to come to a suitable agreement with the nghtful owner
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
Trang 3FOREWORD
Mathematics for the International Student: Mathematics HL has been written to reflect the
syllabus for the two-year IB Diploma Mathematics HL course It is not our intention to define the
course Teachers are encouraged to use other resources 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 endorsed by the IBO
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 A large number of questions,
including some to challenge even the best students, have been added In particular, the final chapter
contains over 200 miscellaneous questions, some of which require the use of a graphics calculator 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
The combination of textbook and interactive Student CD will foster the mathematical development
of students in a stimulating way Frequent use of the interactive features on the CD is certain to nurture a much deeper understanding and appreciation of mathematical concepts
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
Emphasis is placed on the gradual development of concepts with appropriate worked examples, 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
running the CD
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
acceptable portfolio pieces for their students that meet the requirement criteria for the portfolios
Review sets appear at the end of each chapter and a suggested order for teaching the two-year course is given at the end of this Foreword
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 and paper as it is
that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on computer
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 easy way to
demonstrate concepts, and students can discover for themselves and re-visit when necessary
Trang 4Instructions appropriate to each graphic calculator problem are on the CD and can be printed for stu- dents These instructions are written for Texas Instruments and Casio calculators
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, knowledge and
appreciation of mathematics, and its universal application
We welcome your feedback
Email: info@haeseandharris.com.au
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, John Poole and Marjut Maenpaéa4 We acknowledge the contributions of John Owen and Mark Bruce in 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, Rupert de Smidt, Terry Swain, Marie-Therese Filippi, Nigel Wheeler, Sarah Locke, Rema George 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
Teachers are encouraged to carefully check the BACKGROUND KNOWLEDGE sections supplied
on the accompanying CD to ensure that basics have been mastered relatively early in the two-year
HL course Some of these topics naturally occur at the beginning of a specific chapter, as indicated
in the table of contents Click on the BACKGROUND KNOWLEDGE active icons to access the printable pages on the CD
Teachers will have their personal preferences for the order in which the chapters are tackled A suggestion is to work progressively from Chapter 1 through to Chapter 20, but leave Chapters 9, 15 and, possibly, 16 for the second year The remaining chapters can be worked through in order
Alternatively, for the first year, students could work progressively from Chapter 1 to Chapter 23 but 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 the final year of school
Another approach could be to teach just those topics that are included in the Mathematics SL syllabus in the first year and leave the remaining topics for completion in the second year
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 requirements and preferences that they may have
Trang 5USING THE INTERACTIVE STUDENT CD
The CD 1s 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,
allowing students to leave the textbook at school and keep the CD at home
The icon denotes an Interactive Link on the CD Simply ‘click’ the icon to access
¢ spreadsheets
¢ videoclips INTERACTIVE
¢ graphing and geometry software LINK
¢ computer demonstrations and simulations
¢ background knowledge (as printable pages)
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 Click the ‘Background knowledge’ icon on pages 12 and 248
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 (TI or C) to access printable instructions
NOTE ON ACCURACY
Students are reminded that in assessment tasks, including examination papers, unless otherwise stated 1n the question, all numerical answers must be given exactly or to three significant figures
HL & SL COMBINED CLASSES
Refer to our website www.haeseandharris.com.au for guidance in using this textbook in HL and SL combined classes
HL OPTIONS
¢ Topic 9—Sets, relations and groups
¢ Topic 10—Series and differential equations
¢ Topic 11 —Discrete mathematics
In addition, coverage of the Geometry option for students undertaking the
IB Diploma course Further Mathematics is presented on the CD that accompanies the HL Options 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 The HL (CORE) EXAMINATION PREPARATION & PRACTICE GUIDE offers additional questions and practice exams to help students prepare for the Mathematics HL examination For more
information email info@haeseandharris.com.au
Trang 66 TABLE OF CONTENTS
TABLE OF CONTENTS
SYMBOLS AND NOTATION
USED IN THIS BOOK
BACKGROUND KNOWLEDGE
to access, ‘click’ active icon on CD
Surds and radicals
Scientific notation (Standard form)
Number systems and set notation
Algebraic simplification
Linear equations and inequalities
Modulus or absolute value
Product expansion
Factorisation
Formula rearrangement
Adding and subtracting algebraic fractions
Congruence and similarity
Coordinate geometry
ANSWERS
FUNCTIONS
Relations and functions
Function notation, domain and range
Composite functions, fo g
Sign diagrams
Inequalities (inequations)
The reciprocal function x > z
Asymptotes of other rational functions
Inverse functions
Functions which have inverses
Review set 1A
Review set 1B
Review set 1C
SEQUENCES AND SERIES
Number patterns
Sequences of numbers
Arithmetic sequences
Geometric sequences
Series
Miscellaneous problems
Review set 2A
Review set 2B
Review set 2C
7MUOUaQAW?>,
EXPONENTIALS
Index notation
Evaluating powers
Index laws
Algebraic expansion and factorisation
Exponential equations
10
12
CD
CD
CD
CD
CD
CD
CD
CD
CD
CD
CD
CD
CD
17
18
21
27
28
32
35
41
42
44
46
49
50
51
53
54
54
56
59
65
72
74
75
76
77
78
79
80
84
87
ØCtw>
Graphs of exponential functions Growth and decay
The natural exponential ‘e’
Review set 3A Review set 3B Review set 3C
LOGARITHMS
Logarithms Logarithms in base 10 Laws of logarithms Natural logarithms Exponential equations using logarithms The change of base rule
Graphs of logarithmic functions Growth and decay
Review set 4A Review set 4B Review set 4C Review set 4D
88
91
95
98
99
99
101
102
104
106
110
112
114
115
120
122
123
123
124 GRAPHING AND TRANSFORMING FUNCTIONS
Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A
Review set 5B QUADRATIC EQUATIONS AND FUNCTIONS
Solving quadratic equations (Review) The discriminant of a quadratic The sum and product of the roots Graphing quadratic functions Finding a quadratic from its graph Where functions meet
Problem solving with quadratics Quadratic optimisation
Review set 6A Review set 6B Review set 6C Review set 6D Review set 6E COMPLEX NUMBERS AND
POLYNOMIALS
Solutions of real quadratics with A < 0 Complex numbers
Real polynomials Roots, zeros and factors
125
126
128
133
137
140
141
143
145
149
152
153
161
165
167
170
173
174
175
175
176
177
178
180
188 193
Trang 7TABLE OF CONTENTS 7
10
QW
11
QW
2
AmMmMoawP
COUNTING AND THE
MATHEMATICAL INDUCTION 233
THE UNIT CIRCLE AND RADIAN
MEASURE 247
BACKGROUND KNOWLEDGE —
TRIGONOMETRY WITH RIGHT
ANGLED TRIANGLES -— Printable pages CD
The unit circle and the basic
NON-RIGHT ANGLED TRIANGLE
TRIGONOMETRY 269
ADVANCED TRIGONOMETRY 283
14 VECTORS IN 2 AND
3 DIMENSIONS 371
Trang 8TABLE OF CONTENTS
Tmo
VAP
QAAMUAWSY
Relationship between lines
Planes and distances
Angles in space
The intersection of two or more planes
Review set 16A
Review set 16B
Review set 16C
Review set 16D
DESCRIPTIVE STATISTICS
461
466
471
473
477
478
479
481
483 Continuous numerical data and histograms 485
Measuring the centre of data
Cumulative data
Measuring the spread of data
Statistics using technology
Variance and standard deviation
The significance of standard deviation
Review set 17A
Review set 17B
PROBABILITY
Experimental probability
Sample space
Theoretical probability
Compound events
Using tree diagrams
Sampling with and without replacement
Binomial probabilities
Sets and Venn diagrams
Laws of probability
Independent events
Probabilities using permutations and
combinations
Bayes’ theorem
Review set 18A
Review set 18B
Review set 18C
Review set 18D
489
500
502
510
512
518
520
522
525
528
532
533
537
541
543
546
549
554
558
560
562
564
565
566
568
INTRODUCTION TO CALCULUS 569
Limits
Finding asymptotes using limits
Trigonometric limits
Calculation of areas under curves
Review set 19
DIFFERENTIAL CALCULUS
The derivative function
Derivatives at a given #z-value
Simple rules of differentiation
The chain rule
Product and quotient rules
Tangents and normals
Higher derivatives
Review set 20A
Review set 20B
Review set 20C
570
574
577
579
586
589
592
595
600
604
607
611
616
618
619
620
21 APPLICATIONS OF DIFFERENTIAL CALCULUS
Time rate of change General rates of change Motion in a straight line Some curve properties Rational functions Inflections and shape Optimisation Implicit differentiation Review set 21A Review set 21B Review set 21C ZATMAMOAWSY
22 DERIVATIVES OF EXPONENTIAL
AND LOGARITHMIC
FUNCTIONS
Exponential e Natural logarithms Derivatives of logarithmic functions Applications
Some special exponential functions Review set 22A
Review set 22B
tH©Ợ(@œ\wW
23 DERIVATIVES OF CIRCULAR FUNCTIONS AND RELATED RATES
functions
functions
D = Maxima and minima with trigonometry
Review set 23A Review set 23B INTEGRATION
Antidifferentiation
Integration Integrating e®**° and (ax +b)”
Integrating circular functions Definite integrals
Review set 24A Review set 24B Review set 24C
25 APPLICATIONS OF INTEGRATION
Finding areas between curves Motion problems
The fundamental theorem of calculus Integrating f(u)u’(x) by substitution
621
622
623
627
634
642
647
652
661
664
665
666
667
668
673
677
679
683
684
685
687
688
693
694
697
699
704
705
707
708
710
715
720
722
724
730
734
735
736
737
738 744
Trang 9TABLE OF CONTENTS 9
26
ty
29
30
VOLUMES OF REVOLUTION 757
FURTHER INTEGRATION AND
STATISTICAL DISTRIBUTIONS
OF DISCRETE RANDOM
VARIABLES 785
The measures of a discrete random
STATISTICAL DISTRIBUTIONS
OF CONTINUOUS RANDOM
Continuous probability density functions 814
The standard normal distribution
MISCELLANEOUS QUESTIONS 833
Trang 10SYMBOLS AND NOTATION USED IN THIS BOOK
N the set of positive integers and zero, > ores is greater than or equal to
{0, 1, 2, 3, .}
< is less than
Z the set of integers, {0, +1, +2, +3, }
Zt the set of positive integers, {1, 2, 3, }
is not greater than
Q the set of rational numbers
+ rd is not less than
the set of positive rational numbers, -
Rt the set of positive real numbers, Un the nth term of a sequence or series
{a + bila, b € R} r the common ratio of a geometric sequence
Zz a complex number 1 T 2 TT so + Un
x* the complex conjugate of z Soo or S the sum to infinity of a sequence,
¿=1
Re z the real part of z
{z| the set of all z such that ƒ: zr+U f is a function under which x is mapped to
ớ is not an element of f-} the inverse function of the function f
Ø the empty (null) set fog the composite function of f and g
; lim f(z the limit of f(z) as a tends to a
U the universal set oa f(a) f(z)
U union dy ¬-
— the derivative of y with respect to x
Nn intersection dz
d2
x
1
1 (ha 0 ten 6> 0 đ”ụ the nth derivative of y with tt
a2, Ja a to the power 4, square root of a dx” © TMD CETIVERIVG OF 9 WIN TEBPCC! NO 7
x forx > x { tere co œ€R b
; ; a between the limits «=a and rc=b
is approximately equal to
; e* exponential function of x
> is greater than
log, x logarithm to the base a of x