Mathematics - Analysis and Approaches SL

2019·151.22 MB·english

by  Paul Fannon

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ISBN: 9781510462359
eISBN: 9781510461888
© Paul Fannon, Vesna Kadelburg, Ben Woolley, Stephen Ward 2019
First published in 2019 by
Hodder Education,
An Hachette UK Company
Carmelite House
50 Victoria Embankment
London EC4Y 0DZ
www.hoddereducation.com
Impression number 10 9 8 7 6 5 4 3 2 1
Year 2023 2022 2021 2020 2019
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Illustrations by Integra Software Services Pvt. Ltd., Pondicherry, India and also the authors
Typeset in Integra Software Services Pvt. Ltd., Pondicherry, India
Printed in Italy
A catalogue record for this title is available from the British Library.
Contents
Introduction
viii
The toolkit and the mathematical exploration
xi
Core SL content
Chapter 1 Core: Exponents and logarithms 2
■ 1A Laws of exponents 4
■ 1B Operations with numbers in the form a × 10k, where 1 # a < 10 and
k is an integer 11
■ 1C Logarithms 14
Chapter 2 Core: Sequences 22
■ 2A Arithmetic sequences and series 24
■ 2B Geometric sequences and series 33
■ 2C Financial applications of geometric sequences and series 39
Chapter 3 Core: Functions 48
■ 3A Concept of a function 50
■ 3B Sketching graphs 61
Chapter 4 Core: Coordinate geometry 74
■ 4A Equations of straight lines in two dimensions 76
■ 4B Three-dimensional coordinate geometry 86
Chapter 5 Core: Geometry and trigonometry 92
■ 5A Volumes and surface areas of three-dimensional solids 94
■ 5B Rules of trigonometry 101
■ 5C Applications of trigonometry 113
Chapter 6 Core: Statistics 130
■ 6A Sampling 132
■ 6B Summarizing data 139
■ 6C Presenting data 150
■ 6D Correlation and regression 159
Chapter 7 Core: Probability 178
■ 7A Introduction to probability 180
■ 7B Probability techniques 184
Chapter 8 Core: Probability distributions 200
■ 8A Discrete random variables 202
■ 8B Binomial distribution 207
■ 8C The normal distribution 212
vi Contents
Chapter 9 Core: Differentiation 222
■ 9A Limits and derivatives 224
■ 9B Graphical interpretation of derivatives 230
■ 9C Finding an expression for the derivative 240
■ 9D Tangents and normals at a given point and their equations 245
Chapter 10 Core: Integration 256
■ 10A Anti-differentiation 258
■ 10B Definite integration and the area under a curve 262
Core SL content: Review Exercise
272
Additional Analysis and approaches SL content
Chapter 11 Analysis and approaches: Proof 278
■ 11A The structure of mathematical proof 280
Chapter 12 Analysis and approaches: Exponents and logarithm28s6
■ 12A Laws of exponents with rational exponents 288
■ 12B Logarithms 290
Chapter 13 Analysis and approaches: Sequences and series 300
■ 13A The sum of infinite convergent geometric sequences 302
■ 13B The binomial expansion 306
Chapter 14 Analysis and approaches: Functions 314
■ 14A Composite functions 316
■ 14B Inverse functions 319
Chapter 15 Analysis and approaches: Quadratics 326
■ 15A Graphs of quadratic functions 328
■ 15B Solving quadratic equations and inequalities 342
■ 15C The discriminant 346
Chapter 16 Analysis and approaches: Graphs 354
■ 16A Transformations of graphs 356
■ 16B Rational functions 366
■ 16C Exponential and logarithmic functions 370
Chapter 17 Analysis and approaches: Equations 382
■ 17A Solving equations analytically 384
■ 17B Solving equations graphically 387
■ 17C Applications of equations 389
Contents vii
Chapter 18 Analysis and approaches: Trigonometry 394
■ 18A Radian measure of angles 396
■ 18B Trigonometric functions 405
■ 18C Trigonometric identities 413
■ 18D Graphs of trigonometric functions 418
■ 18E Trigonometric equations 426
Chapter 19 Analysis and approaches: Statistics and probabilit4y40
■ 19A Linear regression 442
■ 19B Conditional probability 446
■ 19C Normal distribution 449
Chapter 20 Analysis and approaches: Differentiation 458
■ 20A Extending differentiation 460
■ 20B The chain rule for composite functions 464
■ 20C The product and quotient rules 466
■ 20D The second derivative 470
■ 20E Local maximum and minimum points 476
■ 20F Points of inflection with zero and non-zero gradients 481
Chapter 21 Analysis and approaches: Integration 490
■ 21A Further indefinite integration 492
■ 21B Further links between area and integrals 497
■ 21C Kinematics 507
Analysis and approaches SL: Practice Paper 1
516
Analysis and approaches SL: Practice Paper 2
520
Answers
523
Glossary
614
Acknowledgements
616
Index 617
Introduction
Welcome to your coursebook for Mathematics for the IB Diploma: Analysis and
approaches SL. The structure and content of this coursebook follow the structure and
content of the 2019 IB Mathematics: Analysis and approaches guide, with headings that
correspond directly with the content areas listed in the guide.
This is also the first book required by students taking the higher level course. Students
should be familiar with the content of this book before moving on to Mathematics for
the IB Diploma: Analysis and approaches HL.
Using this book
The book begins with an introductory chapter on the ‘toolkit’, a set of mathematical
thinking skills that will help you to apply the content in the rest of the book to any
type of mathematical problem. This chapter also contains advice on how to complete
your mathematical exploration.
The remainder of the book is divided into two sections. Chapters 1 to 10 cover the
core content that is common to both Mathematics: Analysis and approaches and
Mathematics: Applications and interpretation. Chapters 11 to 21 cover the remaining
SL content required for Mathematics: Analysis and approaches.
Special features of the chapters include:
ESSENTIAL UNDERSTANDINGS
Each chapter begins with a summary of the key ideas to be explored and a list
of the knowledge and skills you will learn. These are revisited in a checklist at
the end of each chapter.
CONCEPTS
The IB guide identifies 12 concepts central to the study of mathematics that will help you
make connections between topics, as well as with the other subjects you are studying.
These are highlighted and illustrated with examples at relevant points throughout the
book.
KEY POINTS
Important mathematical rules and formulae are presented as Key Points, making them easy
to locate and refer back to when necessary.
WORKED EXAMPLES
There are many Worked Examples in each chapter, demonstrating how the Key Points and
mathematical content described can be put into practice. Each Worked Example comprises
two columns:
On the left, how to think about the On the right, what to write, prompted by the
problem and what tools or methods left column, to produce a formal solution to
will be needed at each step. the question.
Using this book ix
Exercises
Each section of each chapter concludes with a comprehensive exercise so that students can test their knowledge of the
content described and practise the skills demonstrated in the Worked Examples. Each exercise contains the following
types of questions:
n Drill questions: These are clearly linked to particular Worked Examples and gradually increase in difficulty. Each
of them has two parts – a and b – desgined such that if students get a wrong, b is an opportunity to have another go
at a very similar question. If students get a right, there is no need to do b as well.
n Problem-solving questions: These questions require students to apply the skills they have mastered in the drill
questions to more complex, exam-style questions. They are colour-coded for difficulty.
1 Green questions are closely related to standard techniques and require a small number of processes. They
should be approachable for all candidates.
2 Blue questions require students to make a small number of tactical decisions about how to apply the standard
methods and they will often require multiple procedures. They should be achievable for SL students aiming
for higher grades.
3 Red questions often require a creative problem-solving approach and extended technical procedures. They
will stretch even the advanced SL students and be challenging for HL students aiming for the top grades.
4 Black questions go beyond what is expected in IB examinations, but provide an enrichment opportunity for
the most advanced students.
The questions in the Mixed Practice section at the end of each chapter are similarly
colour-coded, and contain questions taken directly from past IB Diploma Mathematics
exam papers. There is also a review exercise halfway through the book covering all of
the core content, and two practice examination papers at the end of the book.
Answers to all exercises can be found at the back of the book.
Throughout the first ten chapters it is assumed that you have access to a calculator for
all questions. A calculator symbol is used where we want to remind you that there is a
particularly important calculator trick required in the question.
A non-calculator icon suggests a question is testing a particular skill for the non-
calculator paper.
The guide places great emphasis on the importance of technology in mathematics and
expects you to have a high level of fluency with the use of your calculator and other
relevant forms of hardware and software. Therefore, we have included plenty of
screenshots and questions aimed at raising awareness and developing confidence in
these skills, within the contexts in which they are likely to occur. This icon is used to
indicate topics for which technology is particularly useful or necessary.
Making connections: Mathematics is all about making links. You might be
interested to see how something you have just learned will be used elsewhere in
the course and in different topics, or you may need to go back and remind yourself
of a previous topic.
Be the Examiner
These are activities that present you with three different worked solutions to a particular
question or problem. Your task is to determine which one is correct and to work out where
the other two went wrong.