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Advanced Pure Mathematics

Macmillan Master Series Accounting Advanced English Language Advanced Pure Mathematics Arabic Banking Basic Management Biology British Politics Business Administration Business Communication Business Law C Programming Chemistry COBOL Programming Commerce Communication Computers Databases Economic and Social History Economics Electrical Engineering Electronic and Electrical Calculations Electronics English as a Foreign Language English Grammar English Language English Literature English Spelling French

French 2 German German 2 Human Biology Italian Italian 2 Japanese Manufacturing Marketing Mathematics Mathematics for Electrical and Electronic Engineering Modern British History Modern European History Modern World History Pascal Programming Philosophy Photography Physics Psychology Science Social Welfare Sociology Spanish Spanish 2 Spreadsheets Statistics Study Skills Word Processing

Mastering

Advanced Pure

Mathematics Geoff Buckwell

MACMILLAN

© Geoff Buckwell 1996 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 9HE. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 1996 by MACMILLAN PRESS LTD Houndmills, Basingstoke, Hampshire RG21 6XS and London Companies and representatives throughout the world ISBN 978-0-333-62049-6 ISBN 978-1-349-13551-6 (eBook) DOI 10.1007/978-1-349-13551-6 A catalogue record for this book is available from the British Library. 10 9 05 04

8 7 6 5 4 03 02 01 00 99

3 98

2 1 97 96

Copy-edited and typeset by Povey-Edmondson Okehampton and Rochdale, England

Qcontents

ix

Introduction Acknowledgements

X

1 Number Negative and fractional indices 1.1

1.2 1.3 1.4

1

2 7 9 17

Surds Logarithms The number e and other bases

2 Polynomials 2.1 Polynomials (manipulation) 2.2 Finding linear factors of a polynomial 2.3 Solving equations 2.4 The quadratic polynomial 2.5 Non-linear simultaneous equations

22 22 26 28 31 36

3 Series 3.1 The summation notation (~] 3.2 Arithmetic Series 3.3 Geometric series 3.4 Infinite geometric series 3.5 The binomial expansion 3.6 The binomial expansion when n is not a positive integer

40

4 Coordinates 4.1 Distance between two points 4.2 Gradient of a line 4.3 Other formulae 4.4 The equation of a straight line 4.5 Perpendicular lines 4.6 Conversion of data to fit a straight line graph

65

5 The differential calculus 5.1 The concept of differentiation 5.2 Differentiating a function of a function (the chain rule) 5.3 The product rule 5.4 The quotient rule

89 89 93 96 97

41 42 47 50 53 59

66 68 69 74 78 81

CONTENTS

v

Small increm