About the Book

For an online preview, please email IB@haesemathematics.com.

This book has been written for the IB Diploma Programme course Mathematics: Analysis and Approaches HL, for first assessment in May 2021.

This book is designed to complete the course in conjunction with the Mathematics: Core Topics HL textbook. It is expected that students will start using this book approximately 6-7 months into the two-year course, upon the completion of the Mathematics: Core Topics HL textbook.

This product has been developed independently from and is not endorsed by the International Baccalaureate Organization.  International Baccalaureate, Baccalaureát International, Bachillerato Internacional and IB are registered trademarks owned by the International Baccalaureate Organization.

Year Published: 2019
Page Count: 912
ISBN: 978-1-925489-59-0 (9781925489590)
Online ISBN: 978-1-925489-71-2 (9781925489712)

Table of Contents

Mathematics: Analysis and Approaches HL

1 FURTHER TRIGONOMETRY 17
A Reciprocal trigonometric functions 18
B Inverse trigonometric functions 20
C Algebra with trigonometric functions 23
D Double angle identities 27
E Compound angle identities 31
Review set 1A 38
Review set 1B 39
2 EXPONENTIAL FUNCTIONS 41
A Rational exponents 42
B Algebraic expansion and factorisation 44
C Exponential equations 47
D Exponential functions 49
E Growth and decay 54
F The natural exponential 60
Review set 2A 64
Review set 2B 65
3 LOGARITHMS 67
A Logarithms in base $10$ 68
B Logarithms in base $a$ 71
C Laws of logarithms 73
D Natural logarithms 76
E Logarithmic equations 80
F The change of base rule 81
G Solving exponential equations using logarithms 82
H Logarithmic functions 87
Review set 3A 92
Review set 3B 94
4 INTRODUCTION TO COMPLEX NUMBERS 97
A Complex numbers 99
B The sum of two squares factorisation 101
C Operations with complex numbers 102
D Equality of complex numbers 104
E Properties of complex conjugates 106
Review set 4A 107
Review set 4B 108
5 REAL POLYNOMIALS 109
A Polynomials 110
B Operations with polynomials 111
C Zeros, roots, and factors 114
D Polynomial equality 117
E Polynomial division 120
F The Remainder theorem 124
G The Factor theorem 127
H The Fundamental Theorem of Algebra 128
I Sum and product of roots theorem 131
J Graphing cubic functions 133
K Graphing quartic functions 139
L Polynomial equations 143
M Cubic inequalities 145
Review set 5A 146
Review set 5B 148
6 FURTHER FUNCTIONS 151
A Even and odd functions 152
B The graph of $y=[f(x)]^2$ 154
C Absolute value functions 156
D Rational functions 164
E Partial fractions 169
Review set 6A 171
Review set 6B 172
7 COUNTING 175
A The product principle 176
B The sum principle 178
C Factorial notation 179
D Permutations 181
E Combinations 186
Review set 7A 190
Review set 7B 191
8 THE BINOMIAL THEOREM 193
A Binomial expansions 194
B The binomial theorem for $n \in \, \mathbb{Z}^+$ 198
C The binomial theorem for $n \in \, \mathbb{Q}$ 202
Review set 8A 206
Review set 8B 207
9 REASONING AND PROOF 209
A Logical connectives 212
B Proof by deduction 213
C Proof by equivalence 217
D Definitions 219
E Proof by exhaustion 222
F Disproof by counter example 223
G Proof by contrapositive 225
H Proof by contradiction: reductio ad absurdum 227
Review set 9A 230
Review set 9B 231
10 PROOF BY MATHEMATICAL INDUCTION 233
A The process of induction 234
B The principle of mathematical induction 237
Review set 10A 251
Review set 10B 252
11 LINEAR ALGEBRA 253
A Systems of linear equations 255
B Row operations 257
C Solving $2 \times 2$ systems of linear equations 259
D Solving $3 \times 3$ systems of linear equations 261
Review set 11A 266
Review set 11B 267
12 VECTORS 269
A Vectors and scalars 270
B Geometric operations with vectors 273
C Vectors in the plane 279
D The magnitude of a vector 281
E Operations with plane vectors 282
F Vectors in space 285
G Operations with vectors in space 287
H Vector algebra 289
I The vector between two points 290
J Parallelism 296
K The scalar product of two vectors 299
L The angle between two vectors 301
M Proof using vector geometry 307
N The vector product of two vectors 309
Review set 12A 318
Review set 12B 320
13 VECTOR APPLICATIONS 323
A Lines in $2$ and $3$ dimensions 324
B The angle between two lines 328
C Constant velocity problems 330
D The shortest distance from a point to a line 333
E Intersecting lines 336
F Relationships between lines 338
G Planes 345
H Angles in space 353
I Intersecting planes 355
Review set 13A 360
Review set 13B 363
14 COMPLEX NUMBERS 367
A The complex plane 368
B Modulus and argument 371
C Geometry in the complex plane 375
D Polar form 379
E Euler's form 386
F De Moivre's theorem 388
G Roots of complex numbers 392
Review set 14A 395
Review set 14B 396
15 LIMITS 399
A Limits 401
B The existence of limits 404
C Limits at infinity 406
D Trigonometric limits 409
E Continuity 410
Review set 15A 413
Review set 15B 413
16 INTRODUCTION TO DIFFERENTIAL CALCULUS 415
A Rates of change 417
B Instantaneous rates of change 420
C The gradient of a tangent 423
D The derivative function 424
E Differentiation from first principles 426
F Differentiability and continuity 430
Review set 16A 432
Review set 16B 433
17 RULES OF DIFFERENTIATION 435
A Simple rules of differentiation 436
B The chain rule 441
C The product rule 444
D The quotient rule 446
E Derivatives of exponential functions 449
F Derivatives of logarithmic functions 454
G Derivatives of trigonometric functions 457
H Derivatives of inverse trigonometric functions 461
I Second and higher derivatives 463
J Implicit differentiation 466
Review set 17A 469
Review set 17B 471
18 PROPERTIES OF CURVES 475
A Tangents 476
B Normals 483
C Increasing and decreasing 485
D Stationary points 489
E Shape 494
F Inflection points 497
G Understanding functions and their derivatives 502
H L'Hôpital's rule 504
Review set 18A 508
Review set 18B 512
19 APPLICATIONS OF DIFFERENTIATION 517
A Rates of change 518
B Optimisation 524
C Related rates 533
Review set 19A 538
Review set 19B 540
20 INTRODUCTION TO INTEGRATION 543
A Approximating the area under a curve 544
B The Riemann integral 547
C Antidifferentiation 551
D The Fundamental Theorem of Calculus 553
Review set 20A 558
Review set 20B 559
21 TECHNIQUES FOR INTEGRATION 561
A Discovering integrals 562
B Rules for integration 565
C Particular values 570
D Integrating $f(ax + b)$ 571
E Partial fractions 576
F Integration by substitution 577
G Integration by parts 583
Review set 21A 585
Review set 21B 587
22 DEFINITE INTEGRALS 589
A Definite integrals 590
B Definite integrals involving substitution 594
C The area under a curve 596
D The area above a curve 601
E The area between two functions 603
F The area between a curve and the $y$-axis 608
G Solids of revolution 610
H Problem solving by integration 616
I Improper integrals 620
Review set 22A 623
Review set 22B 626
23 KINEMATICS 629
A Displacement 631
B Velocity 633
C Acceleration 640
D Speed 644
Review set 23A 649
Review set 23B 651
24 MACLAURIN SERIES 653
A Maclaurin series 656
B Convergence 659
C Composite functions 661
D Addition and subtraction 663
E Differentiation and integration 664
F Multiplication 668
G Division 669
Review set 24A 670
Review set 24B 671
25 DIFFERENTIAL EQUATIONS 673
A Differential equations 674
B Euler's method for numerical integration 677
C Differential equations of the form $\frac{dx}{dy}=f(x)$ 680
D Separable differential equations 684
E Logistic growth 690
F Homogeneous differential equations $\frac{dx}{dy}=f(\frac{y}{x})$ 694
G The integrating factor method 696
H Maclaurin series developed from a differential equation 697
Review set 25A 702
Review set 25B 704
26 BIVARIATE STATISTICS 707
A Association between numerical variables 708
B Pearson's product-moment correlation coefficient 713
C Line of best fit by eye 718
D The least squares regression line 722
E The regression line of $x$ against $y$ 729
Review set 26A 732
Review set 26B 734
27 DISCRETE RANDOM VARIABLES 737
A Random variables 738
B Discrete probability distributions 740
C Expectation 745
D Variance and standard deviation 750
E Properties of $aX + b$ 753
F The binomial distribution 756
G Using technology to find binomial probabilities 760
H The mean and standard deviation of a binomial distribution 763
Review set 27A 765
Review set 27B 766
28 CONTINUOUS RANDOM VARIABLES 769
A Probability density functions 771
B Measures of centre and spread 774
C The normal distribution 778
D Calculating normal probabilities 782
E The standard normal distribution 789
F Normal quantiles 793
Review set 28A 799
Review set 28B 800
ANSWERS 803
INDEX 910

Authors

  • Michael Haese
  • Mark Humphries
  • Chris Sangwin
  • Ngoc Vo

Author

Michaelhaese

Michael Haese

Michael completed a Bachelor of Science at the University of Adelaide, majoring in Infection and Immunity, and Applied Mathematics. He studied laminar heat flow as part of his Honours in Applied Mathematics, and finished a PhD in high speed fluid flows in 2001. He has been the principal editor for Haese Mathematics since 2008.

What motivates you to write mathematics books?

My passion is for education as a whole, rather than just mathematics. In Australia I think it is too easy to take education for granted, because it is seen as a right but with too little appreciation for the responsibility that goes with it. But the more I travel to places where access to education is limited, the more I see children who treat it as a privilege, and the greater the difference it makes in their lives. But as far as mathematics goes, I grew up with mathematics textbooks in pieces on the kitchen table, and so I guess it continues a tradition.

What do you aim to achieve in writing?

I think a few things:

  • I want to write to the student directly, so they can learn as much as possible from the text directly. Their book is there even when their teacher isn't.
  • I therefore want to write using language which is easy to understand. Sure, mathematics has its big words, and these are important and we always use them. But the words around them should be as simple as possible, so the meaning of the terms can be properly explained to ESL (English as a Second Language) students.
  • I want to make the mathematics more alive and real, not by putting it in contrived "real-world" contexts which are actually over-simplified and fake, but rather through its history and its relationship with other subjects.

What interests you outside mathematics?

Lots of things! Horses, show jumping and course design, alpacas, badminton, running, art, history, faith, reading, hiking, photography ....

Author

Markhumphries

Mark Humphries

Mark has a Bachelor of Science (Honours), majoring in Pure Mathematics, and a Bachelor of Economics, both of which were completed at the University of Adelaide. He studied public key cryptography for his Honours in Pure Mathematics. He started with the company in 2006, and is currently the writing manager for Haese Mathematics.

What got you interested in mathematics? How did that lead to working at Haese Mathematics?

I have always enjoyed the structure and style of mathematics. It has a precision that I enjoy. I spend an inordinate amount of my leisure time reading about mathematics, in fact! To be fair, I tend to do more reading about the history of mathematics and how various mathematical and logic puzzles work, so it is somewhat different from what I do at work.

How did I end up at Haese Mathematics?

I was undertaking a PhD, and I realised that what I really wanted to do was put my knowledge to use. I wanted to pass on to others all this interesting stuff about mathematics. I emailed Haese Mathematics (Haese and Harris Publications as they were known back then), stating that I was interested in working for them. As it happened, their success with the first series of International Baccalaureate books meant that they were looking to hire more people at the time. I consider myself quite lucky!

What are some interesting things that you get to do at work?

On an everyday basis, it's a challenge (but a fun one!) to devise interesting questions for the books. I want students to have questions that pique their curiosity and get them thinking about mathematics in a different way. I prefer to write questions that require students to demonstrate that they understand a concept, rather than relying on rote memorisation.

When a new or revised syllabus is released for a curriculum that we write for, a lot of work goes into devising a structure for the book that addresses the syllabus. The process of identifying what concepts need to be taught, organising those concepts into an order that makes sense from a teaching standpoint, and finally sourcing and writing the material that addresses those concepts is very involved – but so rewarding when you hold the finished product in your hands, straight from the printer.

What interests you outside mathematics?

Apart from the aforementioned recreational mathematics activities, I play a little guitar, and I enjoy playing badminton and basketball on a social level.

Author

Chrissangwin

Chris Sangwin

Chris completed a BA in Mathematics at the University of Oxford, and an MSc and PhD in Mathematics at the University of Bath. He spent thirteen years in the Mathematics Department at the University of Birmingham, and from 2000-2011 was seconded half time to the UK Higher Education Academy "Maths Stats and OR Network" to promote learning and teaching of university mathematics. He was awarded a National Teaching Fellowship in 2006. Chris Sangwin joined the University of Edinburgh in 2015 as Professor of Technology Enhanced Science Education.

What are your learning and teaching interests in mathematics?

I teach mathematics at university but am particularly interested in core pure mathematics which starts in school and continues to be taught at university. Solving mathematical problems is at the heart of mathematics, and I enjoy teaching problem solving at university.

What interests you outside mathematics?

I really enjoy hill walking and mountaineering, particularly spending time with friends in the hills.

Why do you choose to collaborate with a small publisher on the other side of the world?

There is a unique team spirit in Haese which other publishers don't have. This makes authorship much more collaborative than my previous experiences, which is really enjoyable and I'm sure leads to much better quality books for students which are, after all, the whole point.

Author

Ngocvo

Ngoc Vo

Ngoc Vo completed a Bachelor of Mathematical Sciences at the University of Adelaide, majoring in Statistics and Applied Mathematics. Her Mathematical interests include regression analysis, Bayesian statistics, and statistical computing. Ngoc has been working at Haese Mathematics as a proof reader and writer since 2016.

What drew you to the field of mathematics?

Originally, I planned to study engineering at university, but after a few weeks I quickly realised that it wasn't for me. So I switched to a mathematics degree at the first available opportunity. I didn't really have a plan to major in statistics, but as I continued my studies I found myself growing more fond of the discipline. The mathematical rigor in proving distributional results and how they link to real-world data -- it all just seemed to click.

What are some interesting things that you get to do at work?

As the resident statistician here at Haese Mathematics, I get the pleasure of writing new statistics chapters and related material. Statistics has always been a challenging subject to both teach and learn, however it doesn't always have to be that way. To bridge that gap, I like to try and include as many historical notes, activities, and investigations as I can to make it as engaging as possible. The reasons why we do things, and the people behind them are often important things we forget to talk about. Statistics, and of course mathematics, doesn't just exist within the pages of your textbook or even the syllabus. There's so much breadth and depth to these disciplines, most of the time we just barely scratch the surface.

What interests you outside mathematics?

In my free time I like studying good typography and brushing up on my TeX skills to become the next TeXpert. On the less technical side of things, I also enjoy scrapbooking, painting, and making the occasional card.

Features

  • Snowflake (24 months)

    A complete electronic copy of the textbook, with interactive, animated, and/or printable extras.

  • Self Tutor

    Animated worked examples with step-by-step, voiced explanations.

  • Theory of Knowledge

    Activities to guide Theory of Knowledge projects.

  • Graphics Calculator Instructions

    For Casio fx-CG50, TI-84 Plus CE, TI-nspire, and HP Prime

Icon selftutor ib

This book offers SELF TUTOR for every worked example. On the electronic copy of the textbook, access SELF TUTOR by clicking anywhere on a worked example to hear a step-by-step explanation by a teacher. This is ideal for catch-up and revision, or for motivated students who want to do some independent study outside school hours.

Icon graphics calculator instructions%20%282%29

Graphics calculator instructions for Casio fx-CG50, TI-84 Plus CE, TI-nspire, and HP Prime are included with this textbook. The textbook will either have comprehensive instructions at the start of the book, specific instructions available from icons located throughout, or both. The extensive use of graphics calculators and computer packages throughout the book enables students to realise the importance, application, and appropriate use of technology.

Icon theory of knowledge

Theory of Knowledge is a core requirement in the International Baccalaureate Diploma Programme.

Students are encouraged to think critically and challenge the assumptions of knowledge. Students should be able to analyse different ways of knowing and kinds of knowledge, while considering different cultural and emotional perceptions, fostering an international understanding.

Snowflake

This book is available on electronic devices through our Snowflake learning platform. This book includes 24 months of Snowflake access, featuring a complete electronic copy of the textbook.

Where relevant, Snowflake features include interactive geometry, graphing, and statistics software, demonstrations, games, spreadsheets, and a range of printable worksheets, tables, and diagrams. Teachers are provided with a quick and easy way to demonstrate concepts, and students can discover for themselves and re-visit when necessary.

Support material

  • Errata

    Last updated - 12 Nov 2020