About this resource ix
About eBookPLUS and studyON xii
Acknowledgements xiii
Unit 3 Further Calculus 1
Topic 1 the Logarithmic Function 2
1 The logarithmic function 2 1
1.1 Overview 1
1.2 Review of the index laws 2
1.3 Logarithmic laws and equations 7
1.4 Logarithmic scales 17
1.5 Indicial equations 21
1.6 Logarithmic graphs 25
1.7 Applications 32
1.8 Review: exam practice 37
Answers 41
Revision Unit 3 Topic 1
Chapter 1 47
Topic 2 Further Differentiation and Applications 2
2 Calculus of exponential functions 48
2.1 Overview 48
2.2 Review of limits and differentiation 49
2.3 The exponential function 54
2.4 Differentiation of exponential functions 61
2.5 Applications of exponential functions 64
2.6 Review: exam practice 74
Answers 78
3 Calculus of logarithmic functions 83
3.1 Overview 83
3.2 The natural logarithm and the function y = loge (x) 84
3.3 The derivative of y = loge (x) 91
3.4 Applications of logarithmic functions 94
3.5 Review: exam practice 97
Answers 100
4 Calculus of trigonometric functions 107
4.1 Overview 107
4.2 Review of the unit circle, symmetry and exact values 108
4.3 Review of solving trigonometric equations with and without the use of technology 117
4.4 Review of graphs of trigonometric functions of the form y = A sin(B (x + C)) + D and
y = A cos(B (x + C)) + D 122
4.5 Derivatives of the sine and cosine functions 130
4.6 Applications of trigonometric functions 136
4.7 Review: exam practice 145
Answers 148
Contents v
5 Further differentiation and applications 156
5.1 Overview 156
5.2 The chain rule 157
5.3 The product rule 162
5.4 The quotient rule 165
5.5 Applications of differentiation 170
5.6 Review: exam practice 188
Answers 191
Revision Unit 3 Topic 2
Chapters 2, 3, 4 and 5 196
Topic 3 Integrals
6 Antidifferentiation 197
6.1 Overview 197
6.2 Antidifferentiation of rational functions 198
6.3 Antidifferentiation of exponential functions 202
6.4 Antidifferentiation of logarithmic functions 205
6.5 Antidifferentiation of sine and cosine functions 208
6.6 Further integration 211
6.7 Review: exam practice 220
Answers 224
7 Integration 228
7.1 Overview 228
7.2 Estimating the area under a curve 229
7.3 The fundamental theorem of calculus and definite integrals 240
7.4 Areas under curves 247
7.5 Areas between curves 257
7.6 Applications of integration 265
7.7 Review: exam practice 275
Answers 280
Revision Unit 3 Topic 3
Chapters 6 and 7 285
Practice Assessment 1
Mathematical Methods: Problem solving and modelling task 286
Practice Assessment 2
Mathematical Methods: Unit 3 examination 290
Unit 4 Further Functions and Statistics 302
Topic 1 Further Differentiation and Applications 3
8 The second derivative and applications of differentiation 302
8.1 Overview 302
8.2 Second derivatives 303
8.3 Concavity and points of inflection 306
8.4 Curve sketching 311
8.5 Applications of the second derivative 321
8.6 Review: exam practice 327
Answers 332
Revision Unit 4 Topic 1
Chapter 8 337
Topic 2 Trigonometric Functions 2
9 Cosine and sine rules 338
9.1 Overview 338
9.2 Review of trigonometric ratios and the unit circle 339
9.3 The sine rule 347
9.4 The cosine rule 355
9.5 Area of a triangle 362
9.6 Applications of the sine and cosine rules 367
9.7 Review: exam practice 372
Answers 375
Revision Unit 4 Topic 2
Chapter 9 377
Topic 3 Discrete Random Variables 2
10 Discrete random variables 378
10.1 Overview 378
10.2 Bernoulli distributions 379
10.3 Bernoulli random variables 380
10.4 Binomial distributions 385
10.5 The mean and variance of a binomial distribution 393
10.6 Applications 398
10.7 Review: exam practice 401
Answers 404
Revision Unit 4 Topic 3
Chapter 10 406
Topic 4 Continuous Random Variables and the Normal Distribution
11 General continuous random variables 407
11.1 Overview 407
11.2 Continuous random variables and the probability density function 408
11.3 Cumulative distribution functions 419
11.4 Measures of centre and spread 429
11.5 Review: exam practice 441
Answers 446
12 The normal distribution 449
12.1 Overview 449
12.2 The normal distribution 450
12.3 Standardised normal variables 457
12.4 The inverse normal distribution 468
12.5 Applications of the normal distribution 473
12.6 Review: exam practice 479
Answers 482
Revision Unit 4 Topic 4
Chapters 11 and 12 484
13 Sampling and confidence intervals 485
13.1 Overview 485
13.2 Sample statistics 486
13.3 The distribution of p? 491
13.4 Confidence intervals 499
13.5 Review: exam practice 504
Answers 508
Revision Unit 4 Topic 5
Chapter 13 510
Practice Assessment 3
Mathematical Methods: Unit 4 examination 511
Practice Assessment 4
Mathematical Methods: Units 3 & 4 examination 522
Glossary 532
Index 535
