About the Book

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

This book has been written for the IB Diploma Programme course Mathematics: Applications and Interpretation 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 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: 968
ISBN: 978-1-925489-60-6 (9781925489606)
Online ISBN: 978-1-925489-72-9 (9781925489729)

Table of Contents

Mathematics: Applications and Interpretation HL

1 EXPONENTIALS 13
A Rational exponents 14
B Algebraic expansion and factorisation 16
C Exponential functions 19
D Graphing exponential functions from a table of values 20
E Graphs of exponential functions 21
F Exponential equations 25
G Growth and decay 27
H The natural exponential 34
I The logistic model 39
Review set 1A 40
Review set 1B 42
2 LOGARITHMS 45
A Logarithms in base $10$ 46
B Laws of logarithms 49
C Natural logarithms 52
D Logarithmic functions 56
E Logarithmic scales 59
Review set 2A 64
Review set 2B 65
3 APPROXIMATIONS AND ERROR 67
A Errors in measurement 68
B Absolute and percentage error 71
Review set 3A 75
Review set 3B 75
4 LOANS AND ANNUITIES 77
A Loans 78
B Annuities 84
Review set 4A 88
Review set 4B 89
5 MODELLING 91
A The modelling cycle 92
B Linear models 98
C Piecewise models 101
D Systems of equations 108
Review set 5A 111
Review set 5B 114
6 DIRECT AND INVERSE VARIATION 117
A Direct variation 118
B Powers in direct variation 122
C Inverse variation 123
D Powers in inverse variation 126
E Determining the variation model 127
F Using technology to find variation models 129
Review set 6A 131
Review set 6B 133
7 BIVARIATE STATISTICS 135
A Association between numerical variables 136
B Pearson's product-moment correlation coefficient 141
C The coefficient of determination 146
D Line of best fit by eye 148
E The least squares regression line 152
F Statistical reliability and validity 160
G Spearman's rank correlation coefficient 164
Review set 7A 169
Review set 7B 172
8 NON-LINEAR MODELLING 175
A Logarithmic models 176
B Exponential models 178
C Power models 181
D Problem solving 184
E Non-linear regression 186
Review set 8A 189
Review set 8B 191
9 VECTORS 193
A Vectors and scalars 194
B Geometric operations with vectors 197
C Vectors in the plane 202
D The magnitude of a vector 204
E Operations with plane vectors 205
F Vectors in space 209
G Operations with vectors in space 211
H The vector between two points 212
I Parallelism 217
J The scalar product of two vectors 220
K The angle between two vectors 222
L The vector product of two vectors 227
M Vector components 234
Review set 9A 237
Review set 9B 239
10 VECTOR APPLICATIONS 241
A Lines in $2$ and $3$ dimensions 242
B The angle between two lines 246
C Constant velocity problems 248
D The shortest distance from a point to a line 251
E The shortest distance between two objects 254
F Intersecting lines 256
Review set 10A 261
Review set 10B 263
11 COMPLEX NUMBERS 265
A Real quadratics with $\Delta < 0$ 266
B Complex numbers 268
C Operations with complex numbers 269
D Equality of complex numbers 272
E The complex plane 273
F Modulus and argument 276
G Geometry in the complex plane 279
H Polar form 282
I Exponential form 290
J Frequency and phase 292
Review set 11A 293
Review set 11B 295
12 MATRICES 297
A Matrix structure 298
B Matrix equality 300
C Addition and subtraction 301
D Scalar multiplication 303
E Matrix algebra 305
F Matrix multiplication 307
G The inverse of a matrix 314
H Simultaneous linear equations 318
Review set 12A 323
Review set 12B 325
13 EIGENVALUES AND EIGENVECTORS 327
A Eigenvalues and eigenvectors 328
B Matrix diagonalisation 333
C Matrix powers 336
D Markov chains 338
Review set 13A 349
Review set 13B 351
14 AFFINE TRANSFORMATIONS 353
A Translations 355
B Rotations about the origin 356
C Reflections 359
D Stretches 361
E Enlargements 362
F Composite transformations 363
G Area 367
Review set 14A 372
Review set 14B 373
15 GRAPH THEORY 375
A Graphs 376
B Properties of graphs 380
C Routes on graphs 383
D Adjacency matrices 385
E Transition matrices for graphs 392
F Trees 394
G Minimum spanning trees 395
H Eulerian graphs 401
I The Chinese Postman Problem 404
J Hamiltonian graphs 406
K The Travelling Salesman Problem 409
Review set 15A 416
Review set 15B 419
16 VORONOI DIAGRAMS 423
A Voronoi diagrams 424
B Constructing Voronoi diagrams 428
C Adding a site to a Voronoi diagram 433
D Nearest neighbour interpolation 437
E The Largest Empty Circle problem 439
Review set 16A 443
Review set 16B 445
17 INTRODUCTION TO DIFFERENTIAL CALCULUS 447
A Rates of change 449
B Instantaneous rates of change 452
C Limits 455
D The gradient of a tangent 460
E The derivative function 462
F Differentiation from first principles 464
Review set 17A 466
Review set 17B 467
18 RULES OF DIFFERENTIATION 469
A Simple rules of differentiation 470
B The chain rule 474
C The product rule 477
D The quotient rule 479
E Derivatives of exponential functions 482
F Derivatives of logarithmic functions 485
G Derivatives of trigonometric functions 487
H Second derivatives 490
Review set 18A 492
Review set 18B 493
19 PROPERTIES OF CURVES 495
A Tangents 496
B Normals 502
C Increasing and decreasing 503
D Stationary points 508
E Shape 513
F Inflection points 515
Review set 19A 519
Review set 19B 521
20 APPLICATIONS OF DIFFERENTIATION 523
A Rates of change 524
B Optimisation 529
C Modelling with calculus 537
D Related rates 540
Review set 20A 543
Review set 20B 545
21 INTRODUCTION TO INTEGRATION 547
A Approximating the area under a curve 548
B The Riemann integral 552
C Antidifferentiation 556
D The Fundamental Theorem of Calculus 558
Review set 21A 563
Review set 21B 564
22 TECHNIQUES FOR INTEGRATION 565
A Discovering integrals 566
B Rules for integration 568
C Particular values 572
D Integrating $f(ax+b)$ 574
E Integration by substitution 577
Review set 22A 580
Review set 22B 581
23 DEFINITE INTEGRALS 583
A Definite integrals 584
B Definite integrals involving substitution 587
C The area under a curve 588
D The area above a curve 592
E The area between a curve and the $y$-axis 596
F Solids of revolution 598
G Problem solving by integration 602
Review set 23A 604
Review set 23B 606
24 KINEMATICS 609
A Displacement 611
B Velocity 613
C Acceleration 619
D Speed 623
E Velocity and acceleration in terms of displacement 626
F Motion with variable velocity 628
G Projectile motion 634
Review set 24A 638
Review set 24B 640
25 DIFFERENTIAL EQUATIONS 643
A Differential equations 644
B Solutions of differential equations 646
C Differential equations of the form $\frac{dy}{dx}=f(x)$ 649
D Separable differential equations 652
E Slope fields 658
F Euler's method for numerical integration 661
Review set 25A 666
Review set 25B 667
26 COUPLED DIFFERENTIAL EQUATIONS 669
A Phase portraits 671
B Coupled linear differential equations 678
C Second order differential equations 686
D Euler's method for coupled equations 688
Review set 26A 693
Review set 26B 694
27 DISCRETE RANDOM VARIABLES 697
A Random variables 698
B Discrete probability distributions 700
C Expectation 703
D Variance and standard deviation 709
E Properties of $aX + b$ 711
F The binomial distribution 714
G Using technology to find binomial probabilities 718
H The mean and standard deviation of a binomial distribution 721
I The Poisson distribution 722
Review set 27A 726
Review set 27B 728
28 THE NORMAL DISTRIBUTION 731
A Introduction to the normal distribution 733
B Calculating probabilities 736
C The standard normal distribution 743
D Quantiles 748
Review set 28A 751
Review set 28B 753
29 ESTIMATION AND CONFIDENCE INTERVALS 755
A Linear combinations of random variables 756
B The sum of two independent Poisson random variables 759
C Linear combinations of normal random variables 760
D The Central Limit Theorem 762
E Confidence intervals for a population mean with known variance 768
F Confidence intervals for a population mean with unknown variance 776
Review set 29A 779
Review set 29B 781
30 HYPOTHESIS TESTING 783
A Statistical hypotheses 785
B The $Z$-test 787
C Critical values and critical regions 794
D Student's $t$-test 797
E Paired $t$-tests 800
F The two-sample $t$-test for comparing population means 803
G Hypothesis tests for the mean of a Poisson population 805
H Hypothesis tests for a population proportion 809
I Hypothesis tests for a population correlation coefficient 813
J Error probabilities and statistical power 817
Review set 30A 824
Review set 30B 826
31 $\chi^2$ HYPOTHESIS TESTS 829
A The $\chi^2$ goodness of fit test 830
B Estimating distribution parameters in a goodness of fit test 838
C Critical regions and critical values 842
D The $\chi^2$ test for independence 844
Review set 31A 851
Review set 31B 853
ANSWERS 855
INDEX 967

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 - 18 Aug 2021