Mathematics for the IB Diploma Standard Level with CD-ROM

616 Pages·2012·39.085 MB·English

by  Paul Fannon, Vesna Kadelburg, Ben Woolley, Stephen Ward

Mathematics
Standard Level
for the IB Diploma
Paul Fannon, Vesna Kadelburg,
Ben Woolley and Stephen Ward
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town,
Singapore, São Paulo, Delhi, Mexico City
Cambridge University Press
Th e Edinburgh Building, Cambridge CB2 8RU, UK
www.cambridge.org
Information on this title: www.cambridge.org/9781107613065
© Cambridge University Press 2012
Th is publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2012
Printed and bound in Great Britain by the MPG Books Group
A catalogue record for this publication is available from the British Library
ISBN 978-1-107-61306-5 Paperback with CD-ROM for Windows® and Mac®
Cover image: David Robertson/Alamy
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Worksheets and copies of them remain in the copyright of
Cambridge University Press and such copies may not be
distributed or used in any way outside the purchasing institution.
Contents
Introduction vi
Algebra, functions and equations
(Topic 1: Algebra and Topic 2: Functions & equations)
1 Quadratic functions 1 5 Transformations of graphs 113
1A The quadratic form y = ax2 + bx + c 1 5A Translations 113
1B The completed square form y = a(x − h)2 + k 8 5B Stretches 116
1C The factorised form y = a (x − p)(x − q) 13 5C Refl ections 119
1D The quadratic formula and the discriminant 15 5D Consecutive transformations 121
1E Intersections of graphs and simultaneous
6 Sequences and series 132
equations 21
1F Using quadratic functions to solve problems 24 6A General sequences 132
6B General series and sigma notation 136
2 Exponents and logarithms 34
6C Arithmetic sequences 138
2A Laws of exponents 34
6D Arithmetic series 141
2B Exponential functions 43
6E Geometric sequences 144
2C The number e 49
6F Geometric series 148
2D Introduction to logarithms 51
6G Infi nite geometric series 150
2E Laws of logarithms 56
6H M ixed questions on sequences
2F Graphs of logarithms 61 and series 155
2G Solving exponential equations 62
7 Binomial expansion 161
3 Algebraic structures 71 7A I ntroduction to the binomial theorem 161
3A Solving equations by factorising 72 7B Binomial coeffi cients 164
3B Solving equations by substitution 73 7C Applying the binomial theorem 166
3C Features of graphs 76
3D Using a graphical calculator to solve equations 80
3E Working with identities 83
4 The theory of functions 86
4A Function notation 86
4B Domain and range 88
4C Composite functions 92
4D Inverse functions 95
4E Rational functions 101
© Cambridge University Press 2012. Contents iii
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Geometry
(Topic 3: Circular functions & trigonometry & Topic 4: Vectors)
8 Circular measure and 10 Geometry of triangles
trigonometric functions 176 and circles 250
8A Measuring angles 177 10A Right-angled triangles 251
8B Defi nitions and graphs of the sine and 10B The sine rule 254
cosine functions 184
10C The cosine rule 261
8C Defi nition and graph of the tangent
10D Area of a triangle 267
function 194
10E Trigonometry in three dimensions 270
8D Exact values of trigonometric functions 198
10F Length of an arc 275
8E Transformations of trigonometric graphs 201
10G Area of a sector 279
8F Modelling using trigonometric functions 208
10H Triangles and circles 282
9 Trigonometric equations
and identities 216 11 Vectors 295
9A Introducing trigonometric equations 216 11A Positions and displacements 295
9B Harder trigonometric equations 224 11B Vector algebra 304
9C Trigonometric identities 233 11C Distances 309
9D Using identities to solve equations 236 11D Angles 313
9E Double angle identities 239 11E Properties of the scalar product 318
11F Vector equation of a line 322
11G Solving problems involving lines 330
Calculus (Topic 6: Calculus)
12 Basic differentiation and its 13 Basic integration and its
applications 348 applications 393
12A Sketching derivatives 348 13A Reversing differentiation 393
12B Differentiation from fi rst principles 356 13B Constant of integration 395
12C Rules of differentiation 358 13C Rules of integration 396
12D Interpreting derivatives and second 13D Integrating x−1 and ex 399
derivatives 361
13E Integrating trigonometric functions 400
12E Differentiating trigonometric functions 368
13F Finding the equation of a curve 401
12F Differentiating exponential and natural
13G Defi nite integration 404
logarithm functions 370
13H G eometrical signifi cance of defi nite
12G Tangents and normals 372
integration 406
12H Stationary points 376
13I The area between two curves 412
12I General points of infl exion 382
12J Optimisation 384
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14 Further differentiation 420 15 Further integration 442
14A Differentiating composite functions 15A Reversing standard derivatives 442
using the chain rule 420
15B Integration by substitution 446
14B Differentiating products using the
15C Kinematics 453
product rule 425
15D Volumes of revolution 457
14C Differentiating quotients using the
quotient rule 429
14D Optimisation with constraints 433
Statistics and probability
(Topic 5: Statistics & probability)
16 Summarising data 466 17D Tree diagrams and fi nding intersections 520
16A Measures of the centre of data 466 17E Independent events 525
16B Measures of spread 470 17F Conditional probability 528
16C Frequency tables and grouped data 472 17G Further Venn diagrams 533
16D Cumulative frequency 478 17H Selections with and without replacement 538
16E Histograms 485
18 Probability distributions 546
16F Constant changes to data 489
18A Random variables 546
16G Correlation 491
18B Expectation of a discrete random variable 550
16H Linear regression 497
18C The binomial distribution 553
17 Probability 507 18D The normal distribution 562
17A Empirical probability 507 18E The inverse normal distribution 569
17B Theoretical probability 510
17C Combined events and Venn diagrams 515
19 Questions crossing chapters 578
Answers 588
Glossary 649
Index 663
Acknowledgements 669
Terms and conditions for use of the CD-ROM 670
© Cambridge University Press 2012. Contents v
Not for printing, sharing or distribution.
Introduction
Structure of the book
Th e book is split roughly into four blocks. Chapters 1 to 7 cover algebra and
functions, chapters 8 to 11 cover geometry, chapters 12 to 15 cover calculus, and
chapters 16 to 18 cover probability and statistics. Chapter 19 contains questions
that mix together diff erent parts of the course – a favourite trick in International
Baccalaureate® (IB) examinations.
You do not have to work through the book in the order presented, but given
how much the IB likes to mix up topics, you will fi nd that some questions refer
to material in previous chapters. In such cases, a ‘rewind panel’ will tell you that
the material has been covered earlier, so that you can decide whether to remind
yourself or move on.
In the book we have tried to include only material that will be examinable. Th ere
are many proofs and ideas that are useful and interesting but which are not
included in the main text; these can be found on the CD-ROM should you wish
to explore them.
Each chapter starts with a list of learning objectives which give you an idea
about what the chapter contains. Th ere is also an introductory problem that
illustrates what you will be able to do aft er you have completed the chapter.
Some introductory problems relate to ‘real life’ situations, while others are purely
mathematical. You should not expect to be able to solve the problem at the
start, but you may want to think about possible strategies and what sort of new
facts and methods would help you. Th e solution to the introductory problem is
provided at the end of the chapter, aft er the summary of the chapter contents.
Key point boxes
Th e most important ideas and formulae are emphasised in the Key point boxes.
Th ey also highlight which formulae are given in the Formula booklet.
Worked examples
Each worked example is split into two columns. On the right is what you should
write down in your solution. Sometimes examples may go into more detail than
you strictly need, but they are designed to give you an idea of what is required to
score full method marks in examinations. Mathematics, however, is about much
more than remembering methods and preparing for examinations. So, on the left
of each worked example are notes that describe the thought processes and suggest
which approach you could use to tackle the question. We hope that these will
vi Introduction © Cambridge University Press 2012.
Not for printing, sharing or distribution.
help you learn how to solve problems that diff er from the worked examples. It is
very deliberate that some of the exercise questions require you to do more than
just repeat the methods in the worked examples – mathematics is about thinking!
Signposts
Th ere are several kinds of boxes that appear throughout the book.
Theory of knowledge issues
Every lesson is really a ‘theory of knowledge’ lesson, but sometimes the
connections may not be obvious. Although mathematics is frequently cited as an
example of certainty and truth, things are oft en not so clear-cut. In these boxes
we will try to highlight some of the weaknesses and ambiguities in mathematics,
as well as showing how mathematics links to other areas of knowledge.
From another perspective
Mathematics is oft en described as a unifi ed international language, but the
International Baccalaureate encourages looking at things in various ways. As well
as highlighting some diff erences between mathematicians from diff erent parts of
the world, these boxes also discuss other perspectives on the mathematics we are
covering – historical, pragmatic and cultural.
Research explorer
As part of your course, you will be asked to write a report on an area of
mathematics beyond the syllabus, related to a topic that changes from year to
year. It is sometimes diffi cult to know which topics are suitable as a basis for such
reports, so we have tried to show where a topic can act as a jumping-off point for
further work. Th ese can also give you ideas for the extended essay. Th ere is a lot
of great mathematics out there!
Exam hints
Although we encourage you to think of mathematics as more than just a subject
to be studied in order to pass an examination, it is useful to be aware of some exam hint
common errors so that you can try to avoid making them yourself. In these boxes
we highlight common pitfalls; we also point out where graphical calculators can
be used eff ectively to simplify a question or speed up your work, oft en referring
to the relevant calculator skills sheet on the CD-ROM.
Fast forward / Rewind
Mathematics is all about making links. You might be interested in seeing how
something you have just learned will be used elsewhere in the course, or you may
need to go back and remind yourself of a previous topic. Th ese boxes indicate
connections with other sections of the book to help you fi nd your way around.
© Cambridge University Press 2012. Introduction vii
Not for printing, sharing or distribution.
How to use the questions
The colour coding
Th e questions are colour-coded to distinguish between diff erent levels.
Black questions are drill questions. Th ey are meant to help you practise the methods described in
the book, but they are usually not structured like typical questions that appear in the examination.
Th is does not mean they are easy – in fact, some of them are quite tough – but they are generally
similar in style to the worked examples.
Each diff erently numbered drill question tests a diff erent skill. Lettered subparts (a), (b), (c), …
of a question are of increasing diffi culty. Within each lettered part there may be multiple roman-
numeral parts (i), (ii), (iii), …, which are all of similar diffi culty. Unless you want to get lots of
practice, we recommend that you do only one roman-numeral part and then check your answer. If
you have made a mistake, you may want to think about what went wrong and then attempt another
of the roman-numeral parts.
Green questions are examination-style questions which should be accessible to students on
the way to achieving a grade 3 or 4.
Blue questions are harder examination-style questions. If you are aiming for a grade 5 or 6,
you should be able to make signifi cant progress through most of these.
Red questions are at the very top end of diffi culty among examination-style questions. If
you can do these, then you are likely to be on course for a grade 7.
Gold questions are those that are not typically set in the examination but which
are designed to provoke thinking and discussion, in order to help you gain a better
understanding of a particular concept.
At the end of each chapter you will see longer questions typical of the second section of the IB
examination. Th e parts (a), (b), (c), … of these follow the same colour-coding scheme.
Of course, these are just guidelines. If you are aiming for a grade 6, do not be surprised if
occasionally you fi nd a green question you cannot do; people are rarely equally good at all areas
of the syllabus. Similarly, even if you are able to do all the red questions, that does not guarantee
you will get a grade 7 – aft er all, in the examination you will have to deal with time pressure and
examination stress! It is also worth remembering that these questions are graded according to
our experience of the fi nal examination. When you fi rst start the course, you may well fi nd the
questions harder than you would do by the end of the course, so try not to get discouraged.
Calculator versus non-calculator questions
In the fi nal examination there will be one paper in which calculators are not allowed. Some questions
require a calculator, but most could appear in either the calculator or the non-calculator paper.
Certain types of question are particularly common in the non-calculator paper, and you
need to know how to deal with them. Th ey are indicated by the non-calculator symbol.
On the other hand, some questions can be done in a clever way using a calculator, or cannot
realistically be done without using a calculator. Th ese are marked with a calculator symbol.
viii Introduction © Cambridge University Press 2012.
Not for printing, sharing or distribution.
Note, however, that in the fi nal examination you will not get any calculator/non-
calculator indications, so you must make sure to learn which types of questions
have an easy calculator method. Th e calculator skills sheets on the CD-ROM can
help with this.
With questions that are not labelled with either the calculator or the non-
calculator symbol, you could mix up practising with and without a calculator.
Be careful not to become too reliant on your calculator – half of the core
examination needs to be done without one!
On the CD-ROM
On the CD-ROM there are various materials that you might fi nd useful.
Prior learning
Th e International Baccalaureate syllabus lists what candidates are expected to
know before taking the examination. Not all the topics on the list are explicitly
covered in the course, but knowledge of them may be needed to answer
examination questions. Don’t worry, you do not have to be familiar with all the
‘prior learning’ topics before starting the course: we have indicated in the rewind
panels where a particular concept or skill is required, and on the CD-ROM you
can fi nd a self-assessment test for checking your knowledge, as well as some
worksheets to help you learn any skills that you might be missing.
Coursebook support
Supporting worksheets include:
• calculator skills sheets that give instructions for making optimal use of some
of the recommended graphical calculators
• fi ll-in proof sheets to allow you to re-create proofs that are not required in the
examination
• self-discovery sheets to encourage you to investigate new results for yourself
in the examination
• supplementary sheets exploring some applications, international and
historical perspectives of the mathematics covered in the syllabus.
e-version
A fl at pdf of the whole coursebook (for days when you don’t want to carry the
paperback!)
We hope that you will fi nd Standard Level Mathematics for the IB diploma an
interesting and enriching course. You may also fi nd it quite challenging, but
do not get intimidated – frequently, topics start to make sense only aft er lots of
revision and practice. Persevere and you will succeed!
Th e author team.
© Cambridge University Press 2012. Introduction ix
Not for printing, sharing or distribution.