Mathematics: Analysis and Approaches HL-Textbook (Paperback)

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