Analysis & Approaches HL – Textbook

Analysis & Approaches HL – Textbook

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.

The Mathematics: Analysis and Approaches courses have a focus on algebraic rigour, and this book is written with this focus in mind. The material is presented in a clear, easy-to-follow style, free from unnecessary distractions, while effort has been made to contextualise questions so that students can relate concepts to everyday use.

Each chapter begins with an Opening Problem, offering an insight into the application of the mathematics that will be studied in the chapter. Important information and key notes are highlighted, while worked examples provide step-by-step instructions with concise and relevant explanations. Discussions, Activities, Investigations, and Research exercises are used throughout the chapters to develop understanding, problem solving, and reasoning. A large number of more difficult questions have been included to challenge the HL student.

Discussion topics for Theory of Knowledge are included throughout the book.

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 1010 68
B Logarithms in base aa 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 ∈ Z​+​​ 198
C The binomial theorem for n ∈ 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 × 2 systems of linear equations 259
D Solving 3 × 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 22 and 33 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’Hopital’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 yy-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 ​dx/​dy​​ =f(x) 680
D Separable differential equations 684
E Logistic growth 690
F Homogeneous differential equations ​dx​/dy​​ =f(​y/z​​) 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 xx against yy 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

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