Foundation Maths

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Foundation Maths fourth edition Anthony Croft and Robert Davison

Features of the book • Mathematical processes are described in everyday language – mathematical ideas are usually developed by example rather than formal proof, thereby encouraging students’ learning.

• Key points highlight important results that need to be referred to easily or remembered. • Worked examples are included throughout the book to reinforce learning. • Self-assessment questions are provided at the end of most sections to test understanding of important parts of the section. Answers are given at the back of the book.

• Exercises provide a key opportunity to develop competence and understanding through practice. Answers are given at the back of the book.

• Test and assignment exercises (with answers provided in a separate Lecturers’ Manual on the website) allow lecturers and tutors to set regular assignments or tests throughout the course. New to this edition • Six new chapters: Chapter 4 Sets, Chapter 8 Number Bases, Chapter 9 Elementary Logic, Chapter 31 Integration by parts, Chapter 36 Correlation and Chapter 37 Regression.

• Extra end-of-chapter questions for students (with answers) on the website at www.pearsoned.co.uk/croft

Foundation Maths

Foundation Maths has been written for students taking higher and further education courses who have not specialised in mathematics on post-16 qualifications and need to use mathematical tools in their courses. It is ideally suited to those studying marketing, business studies, management, science, engineering, social science, geography, combined studies and design. It will be useful for those who lack confidence and who need careful, steady guidance in mathematical methods. For those whose mathematical expertise is already established, the book will be a helpful revision and reference guide. The style of the book also makes it suitable for self-study and distance learning.

fourth edition

• PowerPoint slides for lecturers on the website featuring Key Points from the book with their related Worked Examples.

Robert Davison has twenty five years experience teaching mathematics in both further and higher education. He is currently Head of Quality in the Faculty of Computing Sciences and Engineering at De Montfort University, where he also teaches mathematics. He has authored many very successful mathematics textbooks including several for engineering students.

ISBN 0-131-97921-3

9 780131 979215 www.pearson-books.com

Croft and Davison

Anthony Croft has taught mathematics in further and higher education institutions for twenty four years. He is currently Director of the Mathematics Education Centre at Loughborough university, which has been designated a Centre for Excellence in Teaching and Learning by the Higher Education Funding Council for England. He teaches mathematics and engineering undergraduates, and has championed mathematics support for students who find the transition from school to university difficult and for students with learning difficulties. He has authored many very successful mathematics textbooks including several for engineering students.

Foundation Maths Anthony Croft and Robert Davison fourth edition

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Foundation Maths Visit the Foundation Maths, fourth edition Companion Website at www.pearsoned.co.uk/croft to find valuable student learning material including:  Student Support Pack. Contains one-page summaries of key topics complete with examples, exercises and answers  Extra end-of-chapter questions with answers

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We work with leading authors to develop the strongest educational materials in mathematics, bringing cutting-edge thinking and best learning practice to a global market. Under a range of well-known imprints, including Prentice Hall, we craft high quality print and electronic publications which help readers to understand and apply their content, whether studying or at work. To find out more about the complete range of our publishing, please visit us on the World Wide Web at: www.pearsoned.co.uk

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Foundation Maths Fourth edition Anthony Croft Loughborough University

Robert Davison De Montfort University

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Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk

First published 1995 Third edition 2003 Fourth edition published 2006 # Pearson Education Limited 1995, 2006 The rights of Anthony Croft and Robert Davison to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP. ISBN: 978-0-13-197921-5

British Library Cataloging-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Croft, Tony, 1957Foundation maths / Anthony Croft, Robert Davison. 4th ed. p. cm. Includes index. ISBN-13: 978-0-13-197921-5 (alk. paper) ISBN-10: 0-13-197921-3 (alk. paper) 1. Mathematics. I. Davison, Robert. II. Title. QA37.3.C76 2006 510 dc22 2006007747 10 9 8 7 6 5 4 3 2 10 09 08 07 Typeset in 10=12.5 Times by 72 Printed and bound by Ashford Colour Press, Gosport The Publisher’s policy is to use paper manufactured from sustainable forests.

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Contents Preface Guided tour Mathematical symbols 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Arithmetic of whole numbers Fractions Decimal fractions Sets Percentage and ratio Algebra Indices Number bases Elementary logic Simplifying algebraic expressions Factorisation Algebraic fractions Transposing formulae Solving equations Sequences and series Functions Graphs of functions The straight line The exponential function The logarithm function Angles Introduction to trigonometry The trigonometrical functions and their graphs Trigonometrical identities and equations Solution of triangles Matrices Measurement Gradients of curves The product and quotient rules of differentiation Integration and areas under curves Integration by parts Functions of more than one variable and partial differentiation Tables and charts Statistics

vii ix xi 1 14 26 34 46 54 63 78 89 100 108 115 129 135 146 161 174 194 207 216 234 244 252 265 277 294 300 316 333 340 357 365 382 399

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Contents

35 36 37

Probability Correlation Regression

413 422 437

Solutions Index

444 520

Supporting resources Visit www.pearsoned.co.uk/croft to find valuable online resources Companion Website for students  Student Support Pack. Contains one-page summaries of key topics complete with examples, exercises and answers  Extra end-of-chapter questions with answers For instructors  Solutions Manual containing solutions to the test and assignment exercises at the end of each chapter  PowerPoint slides featuring figures from the book and key points from the book with their related worked examples For more information please contact your local Pearson Education sales representative or visit www.pearsoned.co.uk/croft

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Preface

Today, a huge variety of disciplines require their students to have knowledge of certain mathematical tools in order to appreciate the quantitative aspects of their subjects. At the same time, higher education institutions have widened access so that there is much greater variety in the pre-university mathematical experiences of the student body. Some students are returning to education after many years in the workplace or at home bringing up families. Foundation Maths has been written for those students in higher education who have not specialised in mathematics at A or AS level. It is intended for non-specialists who need some but not a great deal of mathematics as they embark upon their courses of higher education. It is likely to be especially useful to those students embarking upon a Foundation Degree with mathematical content. It takes students from around the lower levels of GCSE to a standard which will enable them to participate fully in a degree or diploma course. It is ideally suited for those studying marketing, business studies, management, science, engineering, social science, geography, combined studies and design. It will be useful for those who lack confidence and need careful, steady guidance in mathematical methods. Even for those whose mathematical expertise is already established, the book will be a helpful revision and reference guide. The style of the book also makes it suitable for those who wish to engage in self-study or distance learning. We have tried throughout to adopt an informal, user-friendly approach and have described mathematical processes in everyday language. Mathematical ideas are usually developed by example rather than by formal proof. This reflects our experience that students learn better from examples than from abstract development. Where appropriate, the examples contain a great deal of detail so that the student is not left wondering how one stage of a calculation leads to the next. In Foundation Maths, objectives are clearly stated at the beginning of each chapter, and key points and formulae are highlighted throughout the book. Selfassessment questions are provided at the end of most sections. These test understanding of important features in the section and answers are given at the back of the book. These are followed by exercises; it is essential that these are attempted as the only way to develop competence and

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Preface

understanding is through practice. Solutions to these exercises are given at the back of the book and should be consulted only after the exercises have been attempted. A further set of test and assignment exercises is given at the end of each chapter. These are provided so that the tutor can set regular assignments or tests throughout the course. Solutions to these are not provided. Feedback from students who have used earlier editions of this book indicates that they have found the style and pace of the book helpful in their study of mathematics at university. In order to keep the size of the book reasonable we have endeavoured to include topics which we think are most important, cause the most problems for students, and which have the widest applicability. We have started the book with materials on arithmetic including whole numbers, fractions and decimals. This is followed by several chapters which gradually introduce important and commonly used topics in algebra. The remaining chapters introduce functions, trigonometry, calculus, matrices, statistics and probability. These will be found useful in the courses previously listed. The best strategy for those using the book would be to read through each section, carefully studying all of the worked examples and solutions. Many of these solutions develop important results needed later in the book. It is then a good idea to cover up the solution and try to work the example again independently. It is only by doing the calculation that the necessary techniques will be mastered. At the end of each section the self-assessment questions should be attempted. If these cannot be answered then the previous few pages should be worked through again in order to find the answers in the text, before checking with answers given at the back of the book. Finally, the exercises should be attempted, and again, answers should be checked regularly with those given at the back of the book. In conclusion, remember that learning mathematics takes time and effort. Carrying out a large number of exercises allows the student to experience a greater variety of problems, thus building up expertise and confidence. Armed with these the student will be able to tackle more unfamiliar and demanding problems that arise in other aspects of their course. We hope that you find Foundation Maths useful and wish you the very best of luck. Anthony Croft, Robert Davison 2006

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Guided tour

Key points highlight important results that need to be referred to easily or remembered.

11111112

011

Worked examples are included throughout the book to reinforce student learning and to illustrate a stepby-step approach to solving mathematical problems.

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x

Guided tour

Self-assessment questions are provided at the end of most sections to test understanding of important parts of the section. Answers are given at the back of the book.

11111112

Exercises provide a key opportunity to develop competence and understanding through practice. Answers are given at the back of the book.

11111112

011

Test and assignment exercises (with answers provided in a separate Lecturers’ Manual) allow lecturers and tutors to set regular assignments or tests throughout the course.

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Mathematical symbols

þ

plus



minus



plus or minus



multiply by



multiply by



divide by

¼

is equal to



is identically equal to



is approximately equal to

6¼

is not equal to

>

is greater than

Ç

is greater than or equal to