**Mathematics for Economics and Business, Ninth Edition**

by Ian Jacques

**Contents**

INTRODUCTION: Getting Started 1

Notes for students: how to use this text 1

Chapter 1 Linear Equations 5

1.1 Introduction to algebra 6

1.1.1 Negative numbers 7

1.1.2 Expressions 9

1.1.3 Brackets 12

Key Terms 17

Exercise 1.1 18

Exercise 1.1* 20

1.2 Further algebra 22

1.2.1 Fractions 22

1.2.2 Equations 29

1.2.3 Inequalities 33

Key Terms 36

Exercise 1.2 36

Exercise 1.2* 38

1.3 Graphs of linear equations 40

Key Terms 51

Exercise 1.3 52

Exercise 1.3* 53

1.4 Algebraic solution of simultaneous linear equations 55

Key Term 65

Exercise 1.4 65

Exercise 1.4* 66

1.5 Supply and demand analysis 67

Key Terms 80

Exercise 1.5 80

Exercise 1.5* 82

1.6 Transposition of formulae 84

Key Terms 91

Exercise 1.6 91

Exercise 1.6* 92

1.7 National income determination 93

Key Terms 105

Exercise 1.7 105

Exercise 1.7* 106

Formal mathematics 109

Multiple choice questions 112

Examination questions 116

Chapter 2 Non-linear Equations 121

2.1 Quadratic functions 122

Key Terms 136

Exercise 2.1 137

Exercise 2.1* 138

2.2 Revenue, cost and profit 140

Key Terms 148

Exercise 2.2 148

Exercise 2.2* 150

2.3 Indices and logarithms 151

2.3.1 Index notation 151

2.3.2 Rules of indices 155

2.3.3 Logarithms 161

2.3.4 Summary 167

Key Terms 168

Exercise 2.3 168

Exercise 2.3* 170

2.4 The exponential and natural logarithm functions 172

Key Terms 182

Exercise 2.4 182

Exercise 2.4* 183

Formal mathematics 186

Multiple choice questions 189

Examination questions 193

Chapter 3

**Mathematics of Finance**197

3.1 Percentages 198

3.1.1 Index numbers 204

3.1.2 Inflation 208

Key Terms 210

Exercise 3.1 210

Exercise 3.1* 213

3.2 Compound interest 216

Key Terms 226

Exercise 3.2 226

Exercise 3.2* 228

3.3 Geometric series 230

Key Terms 238

Exercise 3.3 238

Exercise 3.3* 239

3.4 Investment appraisal 241

Key Terms 253

Exercise 3.4 253

Exercise 3.4* 255

Formal mathematics 257

Multiple choice questions 259

Examination questions 263

Chapter 4 Differentiation 267

4.1 The derivative of a function 268

Key Terms 277

Exercise 4.1 277

Exercise 4.1* 278

4.2 Rules of differentiation 279

Rule 1 The constant rule 279

Rule 2 The sum rule 280

Rule 3 The difference rule 281

Key Terms 286

Exercise 4.2 286

Exercise 4.2* 288

4.3 Marginal functions 290

4.3.1 Revenue and cost 290

4.3.2 Production 297

4.3.3 Consumption and savings 299

Key Terms 301

Exercise 4.3 301

Exercise 4.3* 302

4.4 Further rules of differentiation 304

Rule 4 The chain rule 305

Rule 5 The product rule 307

Rule 6 The quotient rule 310

Exercise 4.4 312

Exercise 4.4* 313

4.5 Elasticity 314

Key Terms 326

Exercise 4.5 326

Exercise 4.5* 327

4.6 Optimisation of economic functions 329

Key Terms 345

Exercise 4.6 345

Exercise 4.6* 347

4.7 Further optimisation of economic functions 348

Key Term 359

Exercise 4.7* 359

4.8 The derivative of the exponential and natural logarithm functions 361

Exercise 4.8 370

Exercise 4.8* 371

Formal mathematics 373

Multiple choice questions 376

Examination questions 382

Chapter 5 Partial Differentiation 389

5.1 Functions of several variables 390

Key Terms 400

Exercise 5.1 401

Exercise 5.1* 402

5.2 Partial elasticity and marginal functions 404

5.2.1 Elasticity of demand 404

5.2.2 Utility 407

5.2.3 Production 413

Key Terms 415

Exercise 5.2 416

Exercise 5.2* 418

5.3 Comparative statics 420

Key Terms 429

Exercise 5.3* 429

5.4 Unconstrained optimisation 433

Key Terms 444

Exercise 5.4 444

Exercise 5.4* 445

5.5 Constrained optimisation 447

Key Terms 456

Exercise 5.5 457

Exercise 5.5* 458

5.6 Lagrange multipliers 460

Key Terms 468

Exercise 5.6 469

Exercise 5.6* 470

Formal mathematics 472

Multiple choice questions 474

Examination questions 477

Chapter 6 Integration 483

6.1 Indefinite integration 484

Key Terms 495

Exercise 6.1 496

Exercise 6.1* 497

6.2 Definite integration 499

6.2.1 Consumer’s surplus 503

6.2.2 Producer’s surplus 504

6.2.3 Investment flow 506

6.2.4 Discounting 508

Key Terms 509

Exercise 6.2 509

Exercise 6.2* 510

Formal mathematics 513

Multiple choice questions 515

Examination questions 518

Chapter 7 Matrices 523

7.1 Basic matrix operations 524

7.1.1 Transposition 526

7.1.2 A ddition and subtraction 527

7.1.3 Scalar multiplication 530

7.1.4 Matrix multiplication 531

7.1.5 Summary 539

Key Terms 539

Exercise 7.1 540

Exercise 7.1* 542

7.2 Matrix inversion 545

Key Terms 560

Exercise 7.2 560

Exercise 7.2* 561

7.3 Cramer’s rule 564

Key Term 572

Exercise 7.3 572

Exercise 7.3* 573

Formal mathematics 576

Multiple choice questions 577

Examination questions 581

Chapter 8 Linear Programming 585

8.1 Graphical solution of linear programming problems 586

Key Terms 600

Exercise 8.1 601

Exercise 8.1* 602

8.2 Applications of linear programming 604

Key Terms 612

Exercise 8.2 612

Exercise 8.2* 614

Formal mathematics 617

Multiple choice questions 618

Examination questions 623

Chapter 9 Dynamics 627

9.1 Difference equations 628

9.1.1 National income determination 634

9.1.2 Supply and demand analysis 636

Key Terms 639

Exercise 9.1 639

Exercise 9.1* 640

9.2 Differential equations 643

9.2.1 National income determination 649

9.2.2 Supply and demand analysis 651

Key Terms 653

Exercise 9.2 654

Exercise 9.2* 655

Formal mathematics 658

Multiple choice questions 659

Examination questions 662

Answers to Problems 664

Chapter 1 664

Chapter 2 674

Chapter 3 683

Chapter 4 687

Chapter 5 698

Chapter 6 705

Chapter 7 709

Chapter 8 715

Chapter 9 719

Glossary 723

Index 730

**Preface**

This text is intended primarily for students on economics, business studies and management courses. It assumes very little prerequisite knowledge, so it can be read by students who have not undertaken a mathematics course for some time. The style is informal, and the text contains a large number of worked examples. Students are encouraged to tackle problems for themselves as they read through each section. Detailed solutions are provided so that all answers can be checked. Consequently, it should be possible to work through this text on a self-study basis. The material is wide ranging and varies from elementary topics such as percentages and linear equations to more sophisticated topics such as constrained optimisation of multivariate functions. The text should therefore be suitable for use on both low- and high-level quantitative methods courses.

This text was first published in 1991. The prime motivation for writing it then was to try to produce a text that students could actually read and understand for themselves. This remains the guiding principle when writing this ninth edition. One of the main improvements is the inclusion of over 200 additional questions. Each chapter now ends with both multiple choice questions and a selection of longer examination-style questions.

Students usually enjoy tackling multiple choice questions since they provide a quick way of testing recall of the material covered in each chapter. Several universities include multiple choice as part of their assessment. The final section in each chapter entitled “Examination Questions” contains longer problems which require knowledge and understanding of more than one topic. Although these have been conveniently placed at the end of each chapter it may be best to leave these until the end of the academic year so that they can be used during the revision period just before the examinations.