Cambridge International AS & A Level Mathematics: Pure Mathematics 1

Contents Series introduction vi How to use this book viii Acknowledgements x 1 Quadratics 1 1.1 Solving quadratic equations by factorisation 3 1.2 Completing the square 6 1.3 The quadratic formula 10 1.4 Solving ...

Contents Series introduction vi How to use this book viii Acknowledgements x 1 Quadratics 1 1.1 Solving quadratic equations by factorisation 3 1.2 Completing the square 6 1.3 The quadratic formula 10 1.4 Solving simultaneous equations (one linear and one quadratic) 11 1.5 Solving more complex quadratic equations 15 1.6 Maximum and minimum values of a quadratic function 17 1.7 Solving quadratic inequalities 21 1.8 The number of roots of a quadratic equation 24 1.9 Intersection of a line and a quadratic curve 27 End-of-chapter review exercise 1 31 2 Functions 33 2.1 Definition of a function 34 2.2 Composite functions 39 2.3 Inverse functions 43 2.4 The graph of a function and its inverse 48 2.5 Transformations of functions 51 2.6 Reflections 55 2.7 Stretches 57 2.8 Combined transformations 59 End-of-chapter review exercise 2 67 3 Coordinate geometry 70 3.1 Length of a line segment and midpoint 72 3.2 Parallel and perpendicular lines 75 3.3 Equations of straight lines 78 3.4 The equation of a circle 82 3.5 Problems involving intersections of lines and circles 88 End-of-chapter review exercise 3 92 Contents iii Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Copyright Material - Review Only - Not for Redistribution Cross-topic review exercise 1 95 4 Circular measure 99 4.1 Radians 101 4.2 Length of an arc 104 4.3 Area of a sector 107 End-of-chapter review exercise 4 112 5 Trigonometry 116 5.1 Angles between 0° and 90° 118 5.2 The general definition of an angle 121 5.3 Trigonometric ratios of general angles 123 5.4 Graphs of trigonometric functions 127 5.5 Inverse trigonometric functions 136 5.6 Trigonometric equations 140 5.7 Trigonometric identities 145 5.8 Further trigonometric equations 149 End-of-chapter review exercise 5 153 6 Series 155 6.1 Binomial expansion of a b + n ( ) 156 6.2 Binomial coefficients 160 6.3 Arithmetic progressions 166 6.4 Geometric progressions 171 6.5 Infinite geometric series 175 6.6 Further arithmetic and geometric series 180 End-of-chapter review exercise 6 183 Cross-topic review exercise 2 186 7 Differentiation 190 7.1 Derivatives and gradient functions 191 7.2 The chain rule 198 7.3 Tangents and normals 201 7.4 Second derivatives 205 End-of-chapter review exercise 7 209 Cambridge International AS & A Level Mathematics: Pure Mathematics 1 iv Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Copyright Material - Review Only - Not for Redistribution 8 Further differentiation 211 8.1 Increasing and decreasing functions 213 8.2 Stationary points 216 8.3 Practical maximum and minimum problems 221 8.4 Rates of change 227 8.5 Practical applications of connected rates of change 230 End-of-chapter review exercise 8 235 9 Integration 238 9.1 Integration as the reverse of differentiation 239 9.2 Finding the constant of integration 244 9.3 Integration of expressions of the form ax b n ( ) + 247 9.4 Further indefinite integration 249 9.5 Definite integration 250 9.6 Area under a curve 253 9.7 Area bounded by a curve and a line or by two curves 260 9.8 Improper integrals 264 9.9 Volumes of revolution 268 End-of-chapter review exercise 9 276 Cross-topic review exercise 3 280 Practice exam-style paper 284 Answers 286 Glossary 317 Index 319 Contents v Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Revie