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Trang 2MATHEMATICS FOR THE INTERNATIONAL STUDENT Mathematics HL (Core) second edition
Paul Urban B.Sc.(Hons.),B.Ec
David Martin B.A.,B.Sc.,M.A.,M.Ed.Admin
Robert Haese B.Sc
Sandra Haese B.Sc
Michael Haese B.Sc.(Hons.),Ph.D
Mark Humphries B.Sc.(Hons.)
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
Web: www.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
First Edition 2004
Reprinted 2005 three times (with minor corrections), 2006, 2007 Second Edition 2008
Reprinted 2009 (with minor corrections), 2010
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
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
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
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, 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
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
a range of interactive features:
¢ spreadsheets
¢ videoclips INTERACTIVE ¢ graphing and geometry software LINK
¢ graphics calculator instructions T
¢ 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
that occurs, click on the relevant icon to access the instructions for the other type Examples in the textbook are not always given for both types of calculator Where tì
of calculator € By
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
This is a companion to the Mathematics HL (Core) textbook It offers coverage |
of each of the following options: Bist
¢ Topic 8—Statistics and probability Ry
¢ 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
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 MTA TT TODTMOADS FUNCTIONS Relations and functions
Function notation, domain and range Composite functions, fo g
Sign diagrams
Inequalities (inequations)
The modulus function 1
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 —'*T QUMOADS = 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?>, N EXPONENTIALS Index notation Evaluating powers Index laws Algebraic expansion and factorisation Exponential equations tHØƠ(œW>»> 0 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 6O HnmjmUOOĐPè + mo 7 ul œ ØCtw> ỔẰ@Ồ1Hđđ1mĐOỢODjØ> OOœakwW>
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
Trang 7TABLE OF CONTENTS 7 QAmMUaAWS AWPY © 10 QW p> 11 QW > | 2 AmMmMoawP Graphing polynomials 201 Theorems for real polynomials 208 Review set 7A 210 Review set 7B 211 Review set 7C 212 COUNTING AND THE BINOMIAL EXPANSION 213 The product principle 214 Counting paths 216 Factorial notation 217 Permutations 219 Combinations 223 Binomial expansions 226 The general binomial expansion 229 Review set 8A 231 Review set 8B 232 MATHEMATICAL INDUCTION 233
The process of induction 234
The principle of mathematical induction 236
Indirect proof (extension) 244 Review set 9A 245 Review set 9B 245 Review set 9C 246 THE UNIT CIRCLE AND RADIAN MEASURE 247 BACKGROUND KNOWLEDGE — TRIGONOMETRY WITH RIGHT
ANGLED TRIANGLES -— Printable pages CD
Radian measure 248
Arc length and sector area 250
The unit circle and the basic trigonometric ratios 253 Areas of triangles 263 Review set 10A 266 Review set 10B 267 Review set 10C 268 NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY 269
The cosine rule 270
The sine rule 272
Using the sine and cosine rules 277
Review set 11A 280
Review set 11B 281
ADVANCED TRIGONOMETRY 283
Observing periodic behaviour 285
The sine function 288
Modelling using sine functions 293
The cosine function 296
The tangent function 297
Trigonometric equations 299
G Using trigonometric models 305
H Reciprocal trigonometric functions 307
I Trigonometric relationships 309
J Compound angle formulae 310
K Double angle formulae 314
L _ Trigonometric equations in quadratic form 318
M Trigonometric series and products 318
Review set 12A 319 Review set 12B 320 Review set 12C 321 Review set 12D 322 13 MATRICES 323 A Matrix structure 324
B- Matrix operations and definitions 326
C The inverse of a 2 x 2 matrix 342
D 3x3 and larger matrices 348
E Solving systems of linear equations 350
F Solving systems using row operations 354
G Induction with matrices 364
Review set 13A 366 Review set 13B 367 Review set 13C 368 Review set 13D 369 Review set 13E 370 14 VECTORS IN 2 AND 3 DIMENSIONS 371 A Vectors 372
B Operations with vectors 375
C 2-D vectors in component form 383
D 3-D coordinate geometry 388
E 3-D vectors in component form 390
F Algebraic operations with vectors 393
G Parallelism 398
H Unit vectors 400
I The scalar product of two vectors 402
J The vector product of two vectors 407
Review set 14A 416 Review set 14B 417 Review set 14C 418 Review set 14D 419 Review set 14E 420 15 COMPLEX NUMBERS 421
A Complex numbers as 2-D vectors 422
B Modulus, argument, polar form 425
C De Moivre's Theorem 438
D Roots of complex numbers 441
E Further complex number problems 445
Trang 8TABLE OF CONTENTS Tmo ~“ QAAMUAWPS = " ATTTAAMUANDSYS = VAP = Ne 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 pS 23 DERIVATIVES OF CIRCULAR FUNCTIONS AND RELATED RATES
A Derivatives of circular functions
B The derivatives of reciprocal circular functions C The derivatives of inverse circular functions D = Maxima and minima with trigonometry E Related rates 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 trị w N Q ØO > 25 APPLICATIONS OF INTEGRATION Finding areas between curves Motion problems Ww >
Trang 9TABLE OF CONTENTS 9 26 mMmoOaQw »> »> ØOOœw> ™ ty 29 QW > 30
Problem solving by integration 748
Review set 25A 752 Review set 25B 753 Review set 25C 755 VOLUMES OF REVOLUTION 757 Solids of revolution 758 Volumes for two defining functions 762 Review set 26 765 FURTHER INTEGRATION AND DIFFERENTIAL EQUATIONS 767
The integrals of = and —— 768
Further integration by substitution 769
Integration by parts 771
Miscellaneous integration 773
Separable differential equations 774
Review set 27A 783
Review set 27B 784
STATISTICAL DISTRIBUTIONS
OF DISCRETE RANDOM
VARIABLES 785
Discrete random variables 786
Discrete probability distributions 788
Expectation 791
The measures of a discrete random
variable 794
The binomial distribution 801
The Poisson distribution 807
Review set 28A 810 Review set 28B 812 STATISTICAL DISTRIBUTIONS OF CONTINUOUS RANDOM VARIABLES 813 Continuous probability density functions 814 Normal distributions 817 The standard normal distribution (Z-distribution) 821
Applications of the normal distribution 828
Review set 29A 830
Review set 29B 831
MISCELLANEOUS QUESTIONS 833
ANSWERS 857
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, }
oo <or< is less than or equal to
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, -
Q (nl > 0 te QO} [a,b] the closed interval ø < z < b
R the set of real numbers ] a, b[ the open interval a<a<b
Rt the set of positive real numbers, Un the nth term of a sequence or series
{z|z>0,z€R} d the common difference of an arithmetic
C the set of complex numbers, sequence
{a + bila, b € R} r the common ratio of a geometric sequence
a v-1 Sn the sum of the first n terms of a 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, p jus 21 + úa + |z| the modulus of z n arg Zz the argument of z » tú UL TUR Pv Un ¿=1 Re z the real part of z
Im z the imaginary part of z r ri(n—r)!
{x1, £2, } the set with elements #1, #2, f: A—B f isa function under which each element of
n(A) the number of elements in the finite set A set A has an image in set B
{z| the set of all z such that ƒ: zr+U f is a function under which x is mapped to
c is an element of f(x) the image of x under the function f
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 C is a subset of Ƒ) the derivative of f(x) with respect to x d2 A’ the complement of the set A Ta the second derivative of y with respect to x x 1 n, 2 to th f+, nth root of oo
an, Wa “ oune Power ohne Mm Towore f" (2) the second derivative of f(x) with respect to x
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
Gf a>0 then a > 0) f™ (x) the nth deriviative of f(x) with respect to x
Iz| the modulus ` absolute value of z, that is f y dz the indefinite integral of y with respect to x
x forx > x
{ tere co œ€R b
= identity or is equivalent to / y dx the definite integral of y with respect to x