## Description

Mathematics: Applications and Interpretation HL 2

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.

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 1010 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 22 and 33 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?<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 yy-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 dy/d}=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 tt-test 797

E Paired tt-tests 800

F The two-sample tt-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?? HYPOTHESIS TESTS 829

A The ?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 ?2 test for independence 844

Review set 31A 851

Review set 31B 853

ANSWERS 855

INDEX 967

Michael Haese, Mark Humphries, Chris Sangwin, Ngoc Vo – Haese Mathematics

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