Introduction� vii

1�Fundamentals� 1 1.1 Sets, inequalities, absolute value and properties of real numbers 1 1.2 Roots and radicals (surds) 14 1.3 Exponents (indices) 20 1.4 Scientific notation (standard form) 24 1.5 Algebraic expressions� 26 1.6 Equations and formulae 35

2�Functions�� 46 2.1 Definition of a function 46 2.2 Composite functions 57 2.3 Inverse functions 61 2.4 Transformations of functions 70

3�Algebraic�Functions,�Equations�and�Inequalities�� 90 3.1 Polynomial functions�� 91 3.2 Quadratic functions�� 99 3.3 Zeros, factors and remainders�� 112 3.4 Rational functions�� 126 3.5 Other equations and inequalities �� 132 3.6 Partial fractions �� 144

4�Sequences�and�Series��� 151 4.1 Sequences�� 151 4.2 Arithmetic sequences�� 155 4.3 Geometric sequences�� 158 4.4 Series�� 164 4.5 Counting principles�� 174 4.6 The binomial theorem�� 183 4.7 Mathematical induction�� 190

5�Exponential�and�Logarithmic�Functions��� 206 5.1 Exponential functions�� 206 5.2 Exponential growth and decay�� 211 5.3 The number e�� 216 5.4 Logarithmic functions�� 224 5.5 Exponential and logarithmic equations�� 234

6�Matrix�Algebra��� 246 6.1 Basic definitions�� 247 6.2 Matrix operations�� 249 6.3 Applications to systems�� 256 6.4 Further properties and applications�� 267

Contents

A01_MATHS_SB_4968_PREL.indd 3 01/02/2012 09:03

Contents

7�Trigonometric�Functions�and�Equations�� 279 7.1 Angles, circles, arcs and sectors�� 280 7.2 The unit circle and trigonometric functions �� 288 7.3 Graphs of trigonometric functions�� 301 7.4 Trigonometric equations�� 314 7.5 Trigonometric identities�� 322 7.6 Inverse trigonometric functions�� 335

8�Triangle�Trigonometry��� 350 8.1 Right triangles and trigonometric functions of acute angles�� 350 8.2 Trigonometric functions of any angle�� 361 8.3 The law of sines�� 369 8.4 The law of cosines�� 376 8.5 Applications�� 383

9�Vectors�� 398 9.1 Vectors as displacements in the plane �� 399 9.2 Vector operations �� 402 9.3 Unit vectors and direction angles �� 409 9.4 Scalar product of two vectors�� 419

10�Complex�Numbers�� 42810.1 Complex numbers, sums, products and quotients�� 42910.2 The complex plane �� 44010.3 Powers and roots of complex numbers�� 449

11�Statistics��� 46311.1 Graphical tools�� 46511.2 Measures of central tendency�� 48011.3 Measures of variability�� 486

12�Probability��� 51612.1 Randomness�� 51612.2 Basic definitions �� 51912.3 Probability assignments�� 52512.4 Operations with events�� 53712.5 Bayes’ theorem�� 552

13�Differential�Calculus�I:�Fundamentals��� 57113.1 Limits of functions �� 57213.2 The derivative of a function: definition and basic rules �� 58013.3 Maxima and minima – first and second derivatives�� 59913.4 Tangents and normals�� 615

14�Vectors,�Lines�and�Planes��� 626

14.1 Vectors from a geometric viewpoint �� 627

14.2 Scalar (dot) product�� 637

14.3 Vector (cross) product�� 644

A01_MATHS_SB_4968_PREL.indd 4 01/02/2012 09:03

v

14.4 Lines in space�� 653

14.5 Planes�� 670

15�Differential�Calculus�II:�Further�Techniques�and�Applications�� �700

15.1 Derivatives of composite functions, products and quotients � 701

15.2 Derivatives of trigonometric and exponential functions� 71615.3 Implicit differentiation, logarithmic functions and inverse

trigonometric functions � 729

15.4 Related rates� 745

15.5 Optimization � 753

16�Integral�Calculus�� 771

16.1 Anti-derivative � 771

16.2 Methods of integration: integration by parts� 781

16.3 More methods of integration� 787

16.4 Area and definite integral� 795

16.5 Integration by method of partial fractions (Optional)� 809

16.6 Areas � 812

16.7 Volumes with integrals� 819

16.8 Modelling linear motion� 826

16.9 Differential equations (Optional)� 836

17�Probability�Distributions�� 854

17.1 Random variables� 854

17.2 The binomial distribution � 870

17.3 Poisson distribution� 881

17.4 Continuous distributions� 889

17.5 The normal distribution 902�

18�Mathematical�Exploration�–�Internal�Assessment� ���922

19�Sample�Examination�Papers� 932

20�Theory�of�Knowledge� 952

Answers� 970

Index 1035

Options��Topic 8 – Statistics and probabilityTopic 9 – Sets, relations and groupsTopic 10 – CalculusTopic 11 – Discrete mathematicsAll accessed through the online e-book (see page ix)

A01_MATHS_SB_4968_PREL.indd 5 01/02/2012 09:03