Mathematics Applications and Interpretation for the IB Diploma Higher Level

1016 Pages·2019·151.516 MB·English

by  Tim Garry, Ibrahim Wazir, Kevin Frederick, Jim Nakamoto, Stephen Lumb

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Mathematics
Applications and Interpretation
for the IB Dlploma IBRAHIM WAZIR
GAI 3338
JIM NAKAMOTO
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etyR (UYL EEEEEEEEEEEEEE
Weare grateful to the following for permission to reproduce copyright material:
Published by Pearson Education Limited, 80 Strand, London, WC2R ORL
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pages 904-905, Edge Foundation Inc.: What Kind of Thing Is a Number?
Text © Pearson Education Limited 2019
ATalk with Reuben Hersh, Wed, Oct 24, 2018, Used with permission of Edge
Theory of Knowledge chapter authored by Ric Sims
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Development edited by Jim Newall
Copy edited by Linnet Bruce Text extracts relating to the 1B syllabus and assessment have been reproduced
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Dedications
‘The rights of Ibrahim Wazir, Tim Garry, Jim Nakamoto, Kevin Frederick and
dedicate this work 0 the memory of my pareisand my brother, Saced, who passed avay
Stephen Lumb to be identified as the authors of this work have been asserted by
during the caly stages of work on this editon.
them in accordance with the Copyright, Designs and Patents Act 1988.
My specal thanks ga a my wife, Lody, for standing besde me throughout witing his book.
First published 2019 She has ben my inspiration and motivation for continuing to improve my knowledge and
move my career forward. She i my rock, and | dedicate this book 10 hr:
242322212019
IMP 10987654321 My thanks go 10 allthe students and teachers who used the earlcr editions and sent us their
British Library Cataloguing in Publication Data comments
A catalogue record for this book is available from the British Library Ibrahim Wazir
ISBN 9780435193447
Inloving memory of my parents,
Copyright notice
Iyvish o express my despest thanks and ove to my wif, Val, for her unflappable good nature
Al rights reserved. No part of this publication may be reproduced in any form or
and support — and for smiling and laughing with me cach day. | am: infinitely thankfi for
by any means (including photocopying or storing it in any medium by electronic
ourwonderfuland kind-hearted children ~Bethany, Neil and Rhona. My lov foryou ll s
means and whether or no transiently or incidentally to some other se of this
immeasurabe.
publication) without the written permission of the copyright owner, exceptin
accordance with the provisions of the Copyright, Designs and Patents Act 1988 or Tim Garry
under the terms of licence issued by the Copyright Licensing Agency, 5th Floor,
To Penny
Shackleton House, 4 Battlebridge Lane, London, SE1 2HX (www.cla.co.uk).
Applications for the copyright owner's written permission should be addressed Jim Nakamoto
o the publisher.
Ihave inexpresibl gratitude and apprecation for my i uli who has suppormte eadnd our
Printed in Slovakia by Neografia
Jaily as ook o this second job. The only e raf hee s lve.
Acknowledgements
Lena and Willam, my heart ached every time had to turn you away 1o get work done. Thank
‘The authors and publisher would like to thank the following individuals and
you foryour acceptance andI hape you forgive me.
organisations for their kind permission to reproduce copyright material.
rahim, thank you foryour confidence in me and thinkingI am worthy of this ask.
Photographs
Thankyout 0 my grade 12 students who worked through some early chaper drafis and
(Key: b-bottom; c-centre; Left; r-right; 1op)
cornected plnty oftypos. Slackerz unie!
Getty Images: Chris eberhardt/Getty Images1, ] Broughton Photography/
Finally,t everyone lse who picked up looseends while I was otherwise oceupied: thank you for
Moment/Getty Images 15, Howard Pugh (Marais)/Moment/Getty Images 57,
casing the bunden o his tas.
aaaaimages/Moment/Getty Images 99, Ryan Trussler/Getty Images 123, Johner
Images/Getty Images 173, Gabriel Perez/Moment/Getty Images 201, Witthaya Kevin Frederick
Prasongsin/Moment/Getty Images 273, David Crunelle(EyeEm|Getty Images 297,
Iyvould like t thank my familyfor their fove and invaluable support. 1 would also lke to thank
Kuninobu Sato|EyeEm/Getty Images 361, Wonsup Im EyeEm.Getty Images 407,
all om y students and collagues who make teaching Mathemaics such a joy.
Daniela White Images|Getty Images 449, JPL/Moment/Getty Images 525, baxsyl/
Moment/Getty Images 577, d 3sign/Moment/Getty Images 633, Image Source/ Stephen Lumb
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Allother images © Pearson Education
Contents
Introduction

Number and algebra basics
E 15
Functions
R
57
Sequences and series
E
99
Geometry and trigonometry 1
O
123
O Geometry and trigonometry 2
O 178
Complex numbers
E
201
Matrix algebra
E
273
Vectors
O
297
Modeliing real-life phenomena
E
361
E
Descriptive statistics
E 407
Probability of events
E
449
Graph theory
E
525
Introduction to differential calculus
E
577
Further differential calculus
B
E 633
Probability distributions
E
665
Integral calculus
H
733
Inferential statistics
E
751
Statistical tests and analyses
E
781
E Bivariate analysis
B 833
Integral calculus 2
887
Internal assessment
894
Theory of knowledge
918
Answers
1004
Index
Introduction
This textbook comprehensively covers all of the material in the syllabus for the two-
IB Mathematics:
Applications and year Mathematics: Applications and Interpretation Higher Level course of the
Interpretation Higher 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
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
if each clement y in the
Key facts are drawn from the main text and
range is the image of
highlighted for quick reference to help you
exactly one element xin
identify clear learning points.
the domain.
Hints
Ifyou use a graph to
Specific hints can be found alongside answer a question on an
explanations, questions, exercises, and worked IB mathematics exam,
you mustinclude a clear
examples, providing insight into how to
and well-labelled sketch
analyse[answer a question. They also identify in your working.
common errors and pitfalls.
Notes
Quadratic equations will
be covered in detail in
Notes include general information or advice.
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=13=(10J— 2x7 = —13" =1 (x —2 1)> F+ (- 12)
=169=x>—2x+1+144=x>—2x—24=0
=>(x—6)(x+4)=0=>x—-6=0o0rx+4=0
=X =G orx——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
Applications and Interpretation HL 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 HL mathematics.
Our thanks go to Jim Nakamoto, Kevin Frederick and Stephen Lumb who joined our
team for this edition, helping us to add richness and variety to the series.
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
Number and algebra

basics
Number and algebra basics
Learning objectives
By the end of this chapter, you should be familiar with...
+ making reasonable estimations and better approximations
« demonstrating an understanding of the rules of exponents
+ using correct scientific notation
+ demonstrating an understanding of the rules of logarithms.
This chapter revises and consolidates previous knowledge of scientific notation,
exponential expressions, logarithms and estimation skills.
Estimation and approximation
‘While the terms estimation and approximation are often used to mean a
guess, their inferences are different. Although both terms suggest a lack of
precision, estimation infers a lack of precision in the process of measurement,
while approximation lacks precision in the statement of the measurement.
Both estimation and approximation skills are important in mathematics,
but they are skills that are practiced every day in many contexts.
Here are some examples of estimation and approximation.
Estimation
1. You are cycling to a campsite that is 100 km away. Estimate your arrival time
if you depart at 08:00.
2. Estimate the number of olives that would fill a litre jar.
3. Estimate the number of pages in this textbook.
Approximation
1. Approximate the diameter of a circle that has a circumference of 109.2 cm.
2. Your bank has a digital device that scans the waiting line of customers and
suggests the approximate waiting time in line is 6 minutes.
3. According to data published by our local airport, “Approximately 2 million
passengers used the airport in December”.
In giving an estimation or approximation, measurements are often rounded
to some level of accuracy, with the rule simply being that digits less than 5 are
rounded to 0; and digits that are 5 or greater increase the preceding digit by 1.
(a) Round each value to the nearest unit.
(i) 256.4 (i) 1.49 (iii) 63.5 (v) 700.9
(b) Round each value to the nearest one-hundredth.
(i) 1.006 (i) 7.295 (iii) 67.085 (iv) 34.113
_———————
Solution
(a) (i) 256.4 4isless than 5, so round down to 0
256.4 rounded to the nearest unit is 256
(i) 1.49 4isless than 5, so round down to 0
1.49 rounded to the nearest unit is 1
(i) 63.5 5 is rounded up, adding I to the units digit
63.5 rounded to the nearest unit is 64
(iv) 700.9 9is rounded up, adding 1 to the units digit
700.9 rounded to the nearest unit is 701
(b) (i) 1.006 6 isrounded up, adding I to the hundredths digit
1.006 rounded to the nearest one-hundredth is 1.01
(ii) 7.295 5 isrounded up, adding I to the hundredths digit,
which in turn adds 1 to the tenths digits in this case
7.295 rounded to the nearest one-hundredth is 7.30
(iii) 67.085 5 is rounded up, adding 1 to the hundredths digit
67.085 rounded to the nearest one-hundredth is 67.09
(iv) 34.113 3 s less than 5, so round down to 0
34.113 rounded to the nearest one-hundredth is 34.11
The approximate answer produced as a result of rounding depends on the digit
to which it is rounded, and may or may not be appropriate.
The rounded values in part (a) of Example 1.1 produced differences of
(i) 256.4 —256 =0.4 (i) 1.49 —1=10.49
(iii) 63.5 — 64 = —0.5 (iv) 700.9 — 701 = —0.1
What are the percentage errors in rounding if the original values are
assumed to be precise measurements?
Number and algebra basics
]
Solution
Dividing the differences by the original values we obtain:
0.4 0.49
(i) mzo.l%% (ii) mz329%
Lo =05 e =il
(iii) a5~ 0.787% (iv) 7009~ 0.0143%

The errors are all quite small except for the second one. (Note that this
percentage is rounded, too!) Choosing an arbitrary decimal place to which a
measurement is rounded produces inaccuracies that may not be acceptable.
In IB mathematics, where an exact final answer is not required, an approximate
answer, to the required accuracy, is important. To achieve this, a thorough
understanding of the notion of significant figures (s.f.) is critical. We will
revisit percentage errors after studying significant figures.
Significant figures (s.f.)
Rule Example

All non-zero digits are significant 74818226 has 8 s.f.
123.45 has 5 s.f.

All zeros between non-zero digits are 103.05 has 5 s.f.
significant 780002 has 6 s.f.

Zeros to the left of an implied decimal point | 23000 has 2 s.f., while 23000.0 has 6 s.f.
are not significant, whereas zeros to the
right of an explicit decimal are significant.

To the right ofa decimal point, all leading 0.0043 has 2 s.f, while 0.0043000 has 5 s.f.
zeros are not significant, whereas all zeros
that follow non-zero digits are significant.
Table 1.1 Significant figures rules and examples
Indicate the number of significant figures in each value.
(a) 30020 (b) 30020.0 (c) 0.008 (d) 1000.0
(e) 1.09 (f) 7.00101 (g) 0.02 (h) 0.020
ey
Solution
(a) 4: The non-zero digits and the zeros in between are significant.
(b) 6: All digits between the leading non-zero digit and the decimal point
are significant. The zero after the decimal point is also significant.
(c) 1:Only the 8’ is significant.
(d) 5: All digits between the leading non-zero digit and the decimal point
are significant. The zero after the decimal point is also significant.