Bridging the Gap to University Mathematics - M. Gould, E. Hurst (Springer, 2009) WW.pdf

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Bridging the Gap to University Mathematics
Martin Gould
• Edward Hurst
Bridging the Gap
to University Mathematics
123
ISBN: 978-1-84800-289-0
e-ISBN: 978-1-84800-290-6
DOI: 10.1007/978-1-84800-290-6
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2008942154
Mathematics Subject Classification (2000): 00–01, 00A05, 00A35, 00A99, 03-00, 03-01, 97-00, 97-01
Springer-Verlag London Limited 2009
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as per-
mitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publish-
ers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the
Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to
the publishers.
The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a
specific statement, that such names are exempt from the relevant laws and regulations and therefore free
for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the information
contained in this book and cannot accept any legal responsibility or liability for any errors or omissions
that may be made.
Printed on acid-free paper
Springer Science+Business Media
springer.com
c
Preface
Mathematics has always been an exciting challenge for both of us. Even before
university, we thoroughly enjoyed getting to grips with calculus and meeting
stufflikevectorsandmechanicsforthefirsttime.Theonlyproblemthatwe
both found when we first started our degrees was that amongst our new peers,
different people had very different mathematical backgrounds. Some had done
“single maths,” some “double,” and some an entirely different syllabus alto-
gether. We quickly learned that a mathematics degree requires mastery of a
whole range of ideas: including those that you haven’t studied before.
The aim of this book is in no way to teach you everything that you would
learn in the first year of a mathematics degree. Instead, our aim was to write
a book that you could read before going to university that would give you a
solid foundation on which to build all of the new skills that you will acquire.
That way, when you actually arrive at university you will have much more
time for all of the other amazing things that being a student offers, rather than
having to spend hours looking up something that you could easily learn in a few
minutes from a straightforward book like this. To this end, we have included an
appendix of loads of formulae and identities so that you can spend your nights
partying rather than searching in the library for the integral of tan x .
I suppose being young means being radical, so we’ve written this book
backwards. Not crazy, “read in the mirror” backwards, but the chapters are
set out in the reverse of what you are probably used to seeing. Each chapter
is designed to be a completely stand-alone entity, and chapters always start
with some questions. Our reason for doing this is so that if you see a chapter
title about something with which you are familiar, you can dive straight into
some questions then head off to the next chapter without having to read any
explanation of the topic. If you really want to fly through the material, the
vi
Preface
first ten “Test Yourself” questions of each chapter are designed to cover the
key points. If you can score close to full marks on these, you’re doing pretty
well. If what the questions are asking looks foreign to you, work through the
chapter and the rest of the exercises and come back to these last. All being well,
you’ll be able to do them within a reasonably short time. If you have already
studied a lot of “pre-university” maths, you may well have a good knowledge of
quite a few of the earlier chapters’ contents. If you haven’t, there will obviously
be more chapters where you’ll be starting afresh. For those readers taking
the International Baccalaureate (or any other equivalent qualification) the key
to success is exactly the same – study what you need to, pass over what you
don’t. In any case, the latter chapters move further away from school and college
mathematics and towards degree-style thinking, so as the book progresses there
is certainly something for everyone.
Please don’t ever be disheartened if you’re finding some things dicult. We
wrotethisbookinthehopethatitcouldhelpprepareyouasfullyaspossible
for your studies, so that you can have the best time at university, both in terms
of academic achievement and your student life as a whole. There may be times
when you would like more practice with a new skill, and so we thoroughly
recommend that in these instances you search out some more questions to do,
either from the Internet or from other books.
On the whole, we hope that regardless of whether you are a student who
has just decided to study maths further, or you are someone just about to set
off to university, this book will serve you well as a single, cohesive guide that
draws on your knowledge thus far and helps shape it so that you are ready to
tackle the challenges of the mathematics ahead.
Martin Gould and Edward Hurst

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