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
