Mathematics Analysis and Approaches for the IB Diploma Standard Level

609 Pages·2019·144.331 MB·English

by  Tim Garry, Ibrahim Wazir

STANDARD LEVEL
Mathematics
Analysis and Approaches
For the IB Diploma
IBRAHIM WAZIR
TIM GARRY
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Mathematics
Analysis and Approaches
for the IB Diploma
IBRAHIM WAZIR
TIM GARRY
Published by Pearson Education Limited, 80 Strand, London, WC2R ORL We are grateful to the following for permission to reproduce copyright material:
www.pearsonglobalschools.com Text
pages 536-537, Edge Foundation Inc.: What Kind of Thing s a Number?
Text © Pearson Education Limited 2019
Theory of Knowledge chapter authored by Ric Sims A Talk with Reuben Hersh, Wed, Oct 24, 2018, Used with permission of Edge
Edited by Jim Newall and Sam Hartburn Foundation Inc.
Proofread by Penny Nicholson and Martin Payne Text extracts relating to the IB syllabus and assessment have been reproduced
Indexed by Georgic Bowden from IBO documents. Our thanks go to the International Baccalaureate for
Designed by © Pearson Education Limited 2019 permission to reproduce its copyright.
Typeset by © Tech-Ser Ltd, Gateshead, UK
This work has been developed independently from and is not endorsed by the
Original illustrations © Pearson Education Limited 2019
International Baccalaureate (IB). International Baccalaureate® is a registered
Hlustrated by © Tech-Set Ltd, Gateshead, UK
trademark of the International Baccalaureate Organization.
Cover design by © Pearson Education Limited 2019
This work s produced by Pearson Education and is not endorsed by any
Cover images: Front: © Getty Images: Busa Photography
trademark owner referenced in this publication.
Inside front cover: Shutterstock.com: Dmitry Lobanov
The rights of Ibrahim Wazir and Tim Garry to be identified as the authors of this Dedications
work have been asserted by them in accordance with the Copyright, Designs and
First and foremost, [ want to thank my wife, friend and devotee, Lody, for all the support she
Patents Act 1988. fas given through all of these years of my work and career. Most of that work occurred on
First published 2019
weekends, nights, while on vacation, and other times inconvenient to my family. I could not
have completed this effort without her assistance, tolerance and enthusiasm.
24 23 22 21 20 19
IMP10987 654321 Most importantly, I dedicate this book to my four grandchildren, Marco, Roberto, Lukas and
Sophia, who lived through my frequent absences from their events.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Iwould also like to extend my thanks to Catherine Barber, our Commissioning Editor at
Pearson, for all her support, flexibility and help.
ISBN 978129226741 8
Ibrahim Wazir
Copyright notice
Allrights reserved. No part of this publication may be reproduced in any form or In loving memory of my parents.
by any means (including photocopying or storing it in any medium by electronic Iwish to express my deepest thanks and love to my wife, Val, for her unflappable good nature
means and whether or not transiently or incidentally to some other use of this and support —and for smiling and laughing with me each day. I am infinitely thankful for
publication) without the written permission of the copyright owner, except in our wonderful and kind-hearted children — Bethany, Neil and Rhona. My love for you all is
accordance with the provisions of the Copyright, Designs and Patents Act 1988 or immeasurable.
under the terms of a licence issued by the Copyright Licensing Agency, Barnard’s Tim Garry
Inn, 86 Fetter Lane, London, EC4A 1EN (www.cla.co.uk). Applications for the
copyright owner’s written permission should be addressed to the publisher.
Printed in Slovakia by Neografia
Acknowledgements
The authors and publisher would like to thank the following individuals and
organisations for their kind permission to reproduce copyright material.
Photographs
(Key: b-bottom; c-centre; I-left; r-right; t-top)
Getty Images: JPL/Moment/Getty Images 1, baxsyl/Moment/Getty Images 45,
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BTT R
Contents
Introduction v
n Algebra and function basics 1
E Functions, equations, and inequalities 45
E Sequences and series 71
n Exponential and logarithmic functions 113
E Trigonometric functions and equations 149
n Geometry and trigonometry 191
Statistics 235
B Probabiiy 307
| E Differential calculus 1 349
| m Differential calculus 2 399
; Integral calculus 431
‘ Probability distributions 475
| Internal assessment 519
’ Theory of knowledge 526
‘[ Answers 550
|
Index 594
|
|
INntroduction
This textbook comprehensively covers all of the material in the syllabus for the
IB Mathematics: . . )
Analysis and two-year Mathematics: Analysis and Approaches Standard Level course of the
Approaches Standard International Baccalaureate (IB) Diploma Programme (DP). First teaching of this course
Level syllabus topics starts in the autumn of 2019 with first exams occurring in May 2021. We, the authors,
1. Number and Algebra have strived to thoroughly explain and demonstrate the mathematical concepts and
2. Functions . ; |
methods listed in the course syllabus.
3. Geometry and
Trigonometry
4. Statistics and
Probability Content
5. Calculus
As you will see when you look at the table of contents, the five syllabus topics (see
margin note) are fully covered, though some are split over different chapters in order
to group the information as logically as possible. This textbook has been designed so
that the chapters proceed in a manner that supports effective learning of the course
content. Thus — although not essential — it is recommended that you read and study
the chapters in numerical order. It is particularly important that you thoroughly
review and understand all of the content in the first chapter, Algebra and function
basics, before studying any of the other chapters.
Other than the final two chapters (Theory of knowledge and Internal assessment),
each chapter has a set of exercises at the end of every section. Also, at the end of each
chapter there is a set of practice questions, which are designed to expose you to
questions that are more ‘exam-like’. Many of the end-of-chapter practice questions are
taken from past IB exam papers. Near the end of the book, you will find answers to all
of the exercises and practice questions. There are also numerous worked examples
throughout the book, showing you how to apply the concepts and skills you are
studying.
The Internal assessment chapter provides thorough information and advice on the
required mathematical exploration component. Your teacher will advise you on
the timeline for completing your exploration and will provide critical support during
the process of choosing your topic and writing the draft and final versions of your
exploration.
The final chapter in the book will support your involvement in the Theory of
knowledge course. It is a thought-provoking chapter that will stimulate you to think
more deeply and critically about the nature of knowledge in mathematics and the
relationship between mathematics and other areas of knowledge.
eBook
Included with this textbook is an eBook that contains a digital copy of the textbook and
additional high-quality enrichment materials to promote your understanding of a wide
range of concepts and skills encountered throughout the course. These materials include:
« Interactive GeoGebra applets demonstrating key concepts
* Worked solutions for all exercises and practice questions
* Graphical display calculator (GDC) support
To access the eBook, please follow the instructions located on the inside cover.
Information boxes
As you read this textbook, you will encounter numerous boxes of different colours
containing a wide range of helpful information. |
Learning objectives
You will find learning objectives at the start of each chapter. They set out the content
and aspects of learning covered in the chapter.
Learning objectives
By the end of this chapter, you should be familiar with...
» different forms of equations of lines and their gradients and intercepts
« parallel and perpendicular lines
« different methods to solve a system of linear equations (maximum of
three equations in three unknowns)
Key facts A function is one-to-one
Key facts are drawn from the main text and if each element y in the
highlighted for quick reference to help you "angel is the hlmge of
; : i S exactly one elementx in
identify clear learning points. thedomain.
Hints If you use a graph to
Specific hints can be found alongside answer a question on an
explanations, questions, exercises, and worked 1B mathe@al]ic;exacr?,
examples, provid1i ng 1enk? 1ght i3 nto howAto . y2nodu mwuelslt- laibnecllluedde sa ketcclhea r
analyse/answer a question. They also identify in your working,
common errors and pitfalls.
Notes Quadratic equations will
Notes include general information or advice. be covered in detail in
Chapter 2.
Examples
Worked examples show you how to tackle questions and apply the concepts and skills
you are studying.
Find x such that the distance between points (1, 2) and (x, —10) is 13 units.

Solution
d=11+3 (=1=0 — 2y> = 1/3> = (xx — 1)— + (— 12)?
=2169=x>=2x+1+144=>x>—2x—24=0
> (x—6)(x+4)=0=>x—6=0o0orx+4=0
=x=6o0r x=—4
How to use this book
This book is designed to be read by you — the student. It is very important that you
read this book carefully. We have strived to write a readable book —and we hope that
your teacher will routinely give you reading assignments from this textbook, thus
giving you valuable time for productive explanations and discussions in the classroom.
Developing your ability to read and understand mathematical explanations will prove
to be valuable to your long-term intellectual development, while also helping you to
comprehend mathematical ideas and acquire vital skills to be successful in the Analysis
and Approaches SL course. Your goal should be understanding, not just remembering.
You should always read a chapter section thoroughly before attempting any of the
exercises at the end of the section.
Our aim is to support genuine inquiry into mathematical concepts while maintaining
a coherent and engaging approach. We have included material to help you gain insight
into appropriate and wise use of your GDC and an appreciation of the importance
of proof as an essential skill in mathematics. We endeavoured to write clear and
thorough explanations supported by suitable worked examples, with the overall goal
of presenting sound mathematics with sufficient rigour and detail at a level appropriate
for a student of SL mathematics.
For over 10 years, we have been writing successful textbooks for IB mathematics
courses. During that time, we have received many useful comments from both
teachers and students. If you have suggestions for improving this textbook, please feel
free to write to us at [email protected]. We wish you all the best in your
mathematical endeavours.
Ibrahim Wazir and Tim Garry

vi
Algebra and R
function basics \
Learning objectives
By the end of this chapter, you should be familiar with...
» different forms of equations of lines and their gradients and intercepts
o parallel and perpendicular lines
o the concept of a function and its domain, range and graph
mathematical notation for functions
¢ composite functions
» characteristics of an inverse function and finding the inverse function f~(x)
» transformations of graphs and composite transformations of graphs.
Equations and formulae

Equations, identities and formulae
You will encounter a wide variety of equations in this course. Essentially, an
equation is a statement equating two algebraic expressions that may be true or
false depending upon the value(s) substituted for the variable(s). Values of the
variables that make the equation true are called solutions or roots of the equation.
All of the solutions to an equation comprise the solution set of the equation.
An equation that is true for all possible values of the variable is called an identity.
Many equations are often referred to as a formula (plural: formulae) and
typically contain more than one variable and, often, other symbols that
represent specific constants or parameters (constants that may change in value
but do not alter the properties of the expression). Formulae with which you are
familiar include: A = 7r2, d = rt,d = \[(x;, — x)> + 01— y2)?and V= %m-‘.
Whereas most equations that we encounter will have numerical solutions, we
can solve a formula for one variable in terms of other variables - often referred
to as changing the subject of a formula.
(a) Solve for b in the formula a> + b* = ¢
(b) Solve for I in the formula T = 217‘/% ;
(c) Solve for R in the formula M = nR
Rtr
T o o i S R R N S e A
Solution
@ @+b=c2=b=c2—a’=b=*+/—a?
If b is a length thenb = Vc2 — a2