Table of Contents

About this resource ix About eBookPLUS and studyON xii Acknowledgements xiii Unit 1 Algebra, Statistics and Functions 1 Topic 1 Arithmetic and geometric sequences and series 1 1 Arithmetic sequences 1 1.1 Overview 1 1.2 Arithmetic sequences 2 1.3 The general form of an arithmetic sequence 8 1.4 The sum of an arithmetic sequence 15 1.5 Applications of arithmetic sequences 19 1.6 Review: exam practice 26 Answers 29 Revision Unit 1 Topic 1 Chapter 1 31 Topic 2 Functions and graphs 2 Functions 32 2.1 Overview 32 2.2 Functions and relations 33 2.3 Function notation 44 2.4 Transformations of functions 53 2.5 Piece-wise functions 62 2.6 Review: exam practice 74 Answers 80 3 Quadratic relationships 89 3.1 Overview 89 3.2 Graphs of quadratic functions 90 3.3 Solving quadratic equations with rational roots 103 3.4 Factorising and solving quadratics over R 109 3.5 The discriminant 116 3.6 Modelling with quadratic functions 126 3.7 Review: exam practice 134 Answers 138 4 Inverse proportions and graphs of relations 145 4.1 Overview 145 4.2 The hyperbola 146 4.3 Inverse proportion 157 4.4 The circle 162 4.5 The sideways parabola 170 4.6 Review: exam practice 178 Answers 182 5 Powers and polynomials 191 5.1 Overview 191 5.2 Polynomials 192 5.3 Graphs of cubic polynomials 203 5.4 The factor and remainder theorems 217 5.5 Solving cubic equations 224 5.6 Cubic models and applications 229 5.7 Graphs of quartic polynomials 235 5.8 Solving polynomial equations 242 5.9 Review: exam practice 250 Answers 254 Revision Unit 1 Topic 2 Chapters 2 to 5 270 Practice Assessment 1 Problem solving and modelling task: Functions and graphs 271 Topic 3 Counting and probability 6 Counting and probability 274 6.1 Overview 274 6.2 Fundamentals of probability 275 6.3 Relative frequency 286 6.4 Conditional probability 290 6.5 Independence 299 6.6 Permutations and combinations 305 6.7 Pascal's triangle and binomial expansions 319 6.8 Review: exam practice 329 Answers 332 Revision Unit 1 Topic 3 Chapter 6 336 Topic 4 Exponential functions 1 7 Indices 337 7.1 Overview 337 7.2 Index laws 338 7.3 Negative and rational indices 343 7.4 Indicial equations and scientific notation 348 7.5 Review: exam practice 354 Answers 356 Revision Unit 1 Topic 4 Chapter 7 358 Topic 5 Arithmetic and geometric sequences and series 2 8 Geometric sequences 359 8.1 Overview 359 8.2 Recursive definition and the general term of geometric sequences 360 8.3 The sum of a geometric sequence 371 8.4 Geometric sequences in context 378 8.5 Review: exam practice 384 Answers 387 Revision Unit 1 Topic 5 Chapter 8 389 Practice Assessment 2 Unit 1 internal examination 390 Unit 2 Calculus and Further Functions Topic 1 The logarithmic function 1 Topic 2 Exponential functions 2 9 Exponential and logarithmic functions 399 9.1 Overview 399 9.2 Exponential functions 400 9.3 functions 410 9.4 Modelling with exponential functions 423 9.5 Solving equations with indices 432 9.6 Review: exam practice 438 Answers 442 Revision Unit 2 Topic 1 AND 2 Chapter 9 449 Topic 3 Trigonometric functions 1 10 Trigonometric functions 450 10.1 Overview 450 10.2 Trigonometry review 451 10.3 Radian measure 459 10.4 Unit circle definitions 469 10.5 Exact values and symmetry properties 480 10.6 Graphs of the sine, cosine and tangent functions 490 10.7 Transformations of sine and cosine graphs 499 10.8 Solving trigonometric equations 513 10.9 Modelling with trigonometric functions 524 10.10 Review: exam practice 531 Answers 536 Revision Unit 2 Topic 3 Chapter 10 550 Topic 4 Introduction to differential calculus 11 Rates of change 551 11.1 Overview 551 11.2 Exploring rates of change 552 11.3 The difference quotient 561 11.4 Differentiating simple functions 567 11.5 Interpreting the derivative 570 11.6 Review: exam practice 576 Answers 580 12 Properties and applications of derivatives 584 12.1 Overview 584 12.2 Differentiation by formula 585 12.3 The derivative as a function 589 12.4 Properties of the derivative 595 12.5 Differentiation of power and polynomial functions 599 12.6 Review: exam practice 604 Answers 607 13 Applications of derivatives 611 13.1 Overview 611 13.2 Gradient and equation of a tangent 612 13.3 Displacement–time graphs 620 13.4 Sketching curves using derivatives 629 13.5 Modelling optimisation problems 643 13.6 Review: exam practice 648 Answers 651 Revision Unit 2 Topic 4 Chapters 11 to 13 656 Topic 5 Further differentiation and applications 1 14 Differentiation rules 657 14.1 Overview 657 14.2 The product rule 658 14.3 The quotient rule 660 14.4 The chain rule 666 14.5 Applications of the product, quotient and chain rules 672 14.6 Review: exam practice 674 Answers 677 Revision Unit 2 Topic 5 Chapter 14 682 Topic 6 Discrete random variables 1 15 Discrete random variables 1 683 15.1 Overview 683 15.2 Discrete random variables 684 15.3 Expected values 695 15.4 Variance and standard deviation 704 15.5 Applications of discrete random variables 712 15.6 Review: exam practice 718 Answers 723 Revision Unit 2 Topic 6 Chapter 15 727 Practice Assessment 3 Unit 2 internal examination 728 Practice Assessment 4 Units 1 and 2 internal examination 735 Glossary 743 Index 750

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