Ophthalmology

Cover......Page 1
Contents......Page 3
Overarching themes......Page 5
Extra online content......Page 7
Chapter 1: Algebraic methods......Page 10
 1.1: Proof by contradiction......Page 11
 1.2: Algebraic fractions......Page 14
 1.3: Partial fractions......Page 18
 1.4: Repeated factors......Page 21
 1.5: Algebraic division......Page 23
 Mixed exercise: 1......Page 28
Chapter 2: Functions and graphs......Page 31
 2.1: The modulus function......Page 32
 2.2: Functions and mappings......Page 36
 2.3: Composite functions......Page 41
 2.4: Inverse functions......Page 45
 2.5: y = |f(x)| and y = f(|x|)......Page 49
 2.6: Combining transformations......Page 53
 2.7: Solving modulus problems......Page 57
 Mixed exercise: 2......Page 62
Chapter 3: Sequences and series......Page 68
 3.1: Arithmetic sequences......Page 69
 3.2: Arithmetic series......Page 72
 3.3: Geometric sequences......Page 75
 3.4: Geometric series......Page 79
 3.5: Sum to infinity......Page 82
 3.6: Sigma notation......Page 85
 3.7: Recurrence relations......Page 88
 3.8: Modelling with series......Page 92
 Mixed exercise: 3......Page 95
Chapter 4: Binomial expansion......Page 100
 4.1: Expanding (1 + x)n......Page 101
 4.2: Expanding (a + bx)n......Page 106
 4.3 Using partial fractions......Page 110
 Mixed exercise: 4......Page 113
Review exercise: 1......Page 116
Chapter 5: Radians......Page 122
 5.1: Radian measure......Page 123
 5.2: Arc length......Page 127
 5.3: Areas of sectors and segments......Page 131
 5.4: Solving trigonometric equations......Page 137
 5.5: Small angle approximations......Page 142
 Mixed exercise: 5......Page 144
Chapter 6: Trigonometric functions......Page 151
 6.1: Secant, cosecant and cotangent......Page 152
 6.2: Graphs of sec x, cosec x and cot x......Page 154
 6.3: Using sec x, cosec x and cot x......Page 158
 6.4: Trigonometric identities......Page 162
 6.5: Inverse trigonometric functions......Page 167
 Mixed exercise: 6......Page 171
Chapter 7: Trigonometry and modelling......Page 175
 7.1: Addition formulae......Page 176
 7.2: Using the angle addition formulae......Page 180
 7.3: Double-angle formulae......Page 183
 7.4: Solving trigonometric equations......Page 186
 7.5: Simplifying α cos x ± b sin x......Page 190
 7.6: Proving trigonometric identities......Page 195
 7.7: Modelling with trigonometric functions......Page 198
 Mixed exercise: 7......Page 201
Chapter 8: Parametric equations......Page 206
 8.1: Parametric equations......Page 207
 8.2: Using trigonometric identities......Page 211
 8.3: Curve sketching......Page 215
 8.4: Points of intersection......Page 218
 8.5: Modelling with parametric equations......Page 222
 Mixed exercise: 8......Page 229
Review exercise: 2......Page 234
Chapter 9: Differentiation......Page 240
 9.1: Differentiating sin x and cos x......Page 241
 9.2: Differentiating exponentials and logarithms......Page 244
 9.3: The chain rule......Page 246
 9.4: The product rule......Page 250
 9.5: The quotient rule......Page 252
 9.6: Differentiating trigonometric functions......Page 255
 9.7: Parametric differentiation......Page 260
 9.8: Implicit differentiation......Page 263
 9.9: Using second derivatives......Page 266
 9.10: Rates of change......Page 270
 Mixed exercise: 9......Page 274
Chapter 10: Numerical methods......Page 282
 10.1: Locating roots......Page 283
 10.2: Iteration......Page 287
 10.3: The Newton–Raphson method......Page 291
 10.4: Applications to modelling......Page 295
 Mixed exercise: 10......Page 298
Chapter 11: Integration......Page 302
 11.1: Integrating standard functions......Page 303
 11.2: Integrating f(αx + b)......Page 305
 11.3: Using trigonometric identities......Page 307
 11.4: Reverse chain rule......Page 309
 11.5: Integration by substitution......Page 312
 11.6: Integration by parts......Page 316
 11.7: Partial fractions......Page 319
 11.8: Finding areas......Page 322
 11.9: The trapezium rule......Page 326
 11.10: Solving differential equations......Page 331
 11.11: Modelling with differential equations......Page 335
 Mixed exercise: 11......Page 338
Chapter 12: Vectors......Page 345
 12.1: 3D coordinates......Page 346
 12.2: Vectors in 3D......Page 348
 12.3: Solving geometric problems......Page 353
 12.4: Application to mechanics......Page 356
 Mixed exercise: 12......Page 358
Review exercise: 3......Page 361
Exam-style practice: Paper 1: Pure Mathematics......Page 367
Exam-style practice: Paper 2: Pure Mathematics......Page 370
Answers......Page 374
Index......Page 432