Some other Pergamon Press titles of interest C. PLUMPTON & W. A. TOMKYS Theoretical Mechanics for Sixth Forms 2nd SI Edition Volumes 1 & 2 D. T. E. MARJORAM Exercises in Modern Mathematics Further Exercises in Modern Mathematics Modern Mathematics in Secondary Schools D. G. H. B. LLOYD Modern Syllabus Algebra Sixth Form Pure Mathematics VOLUME ONE C. PIUMPTON Queen Mary College, London W. A. TOMKYS Belle Vue Boys Grammar School Bradford SECOND EDITION PERGAMON PRESS OXFORD NEW YORK TORONTO SYDNEY PARIS KRANKKURT U. K. Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 OBW, England U. S. A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York, 10523, U.S.A. CANADA Pergamon of Canada Ltd., 75 The East Mall, Toronto, Ontario, Canada AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia FRANCE Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France FEDERAL RE PUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus, OF GERMANY Pferdstrasse 1, Federal Republic of Germany Copyright © 1968 Pergamon Press Ltd. 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, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers First edition 1962 Reprinted with corrections 1965 Second edition 1968 Reprinted 1972, 1973, 1975, 1978 Library of Congress Catalog Card No. 67-30688 Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter ISBN 0 08 009374 4 Flexi-cover PREFACE TO THE SECOND EDITION THIS book is the first of a series of volumes on Pure Mathematics and Theoretical Mechanics for Sixth Form students whose aim is entrance into British and Commonwealth Universities or Technical Colleges. A knowledge of Pure Mathematics up to G.C.E. O-level is assumed and the subject is developed by a concentric treatment in which each new topic is used to illustrate ideas already treated. The major topics of Algebra, Calculus, Coordinate Geometry and Trigonometry are developed together. This volume covers most of the Pure Mathematics required for the single subject Mathematics at Advanced Level. Volume Two covers the remainder, with the exception of Pure Geometry, of the Pure Mathe­ matics required for a double subject at Advanced Level. Early and rapid progress in calculus is made at the beginning of this volume in order to facilitate the student's progress along the most satisfactory lines in Pure Mathematics, in Theoretical Mechanics and in Physics. The worked examples are an essential feature of this book and they are followed by routine exercises within the text of each chapter, asso­ ciated closely with the work on which they are dependent. The exercises at the end of each chapter collectively embody all the topics of that chapter and, where possible, the preceding chapters also. They are grad­ ed in difficulty and in all cases the last few of these exercises might well be deferred, by most students, until a second reading. Most of these miscellaneous exercises are taken from the examination papers of the University of London (L.), the Northern Universities Joint Matricula­ tion Board (N.), the Oxford and Cambridge Schools Examination Board (O.C.) and the Cambridge Local Examinations Syndicate (C). We are grateful to the authorities concerned for permission to reprint and use these questions. ix X PREFACE TO THE SECOND EDITION The sections, exercises and equations are numbered according to the chapters; e.g. § 5.4 is the fourth section of Chapter V; Ex. 9.5 is the set of exercises at the end of § 9.5; equation (3.7) is the seventh (number­ ed) equation of Chapter m. Only those equations to which subsequent references are made are numbered. In this second edition we have made some minor changes and cor­ rections. We are grateful to all those who have made suggestions for improvements and corrections. In particular we thank Mr. J. A. Croft and Mr. G. Hawkes who read the proofs. C. PLUMPTON W. A. TOMKYS CHAPTER I INTRODUCTION TO THE CALCULUS 1.1 Coordinates and Loci Coordinates. The position of a point in a plane may be defined by reference to a pair of rectangular axes Ox, Oy with which the student will be familiar from drawing the graphs of functions of x. Thus in Fig. 1 (#1, yi) denotes the point whose x-value or x-coordinate is x\ and whose j-coordinate is y±. Negative jc-coordinates are by convention to the left of the origin and negative ^-coordinates below the origin. Fig. 1 shows the points A(3, 2), B(-l 2), C(-2, -1) and Z)(l, -2). 9 4 •(-1,2) *(3.2) * *, y. y 0 C»(-2,-l) D.(..-21 Fig. 1. vi 2 SIXTH FORM PURE MATHEMATICS The Equation of a Curve. The relation which is satisfied by the coordinates of any point on a particular curve is called the equation of the curve. Thus if the equation of the curve is y = f(x) [y is a function of x] and if (xi, y{) is a point on the curve, then y± =f(x{). The curve is the locus of a point which moves so that its coordinates always satisfy the equation. Example. If f(x) = ax2+bx+c and if the points (4, —9), ( — 1, —4), 9 (0, —1) lie on the curve y =/(*), calculate the values of a, b and c. Since (4, —9), ( — 1, —4), and (0, —1) each lie on y = /(*), then -9=/(4), -4=/(-l) and -l=/(0); .-. -9=16a + 46 + c, -4 = a-6-fc, -l = c from which a= - 1, b = 2, c == - 1. The Gradient of a Straight Line. The gradient of the straight line joining the points P(a, b) and Q(p q) is defined as (q—b)/(p—a). It re­ 9 presents the rate of change of y compared with x along PQ. Exercises 1.1 1. Show on a diagram the positions of the following points whose coordinates are given: (2,4), (5, -1), (-3, 2), (0,4), (5,0). 2. In each of the following, the equation of a curve is given together with one coordinate of a point on the curve. Calculate, in each case, the possible values of the other coordinate: (i) y — JC!+5X+4, (* = 3) (ii) x*+y* = 25, (y = 3) (iii) y = logi„ x, 0- = 2) (iv) y = x3-x, (y = 0) (v) y = 5x+4, (x = a^- h). 3. (i) If /■<*) = 3^1 > calculate/(-l); (ii) If f(x) = (xt+5x+l3)1", calculate/(2); INTRODUCTION TO THE CALCULUS 3 (iii) If /(*) = — +4, calculate /( - 2); (iv) If /(*) = log x, write down/(—j ; 10 (v) If /(*) = sin (2x+10)°, write down /(40). 4. Write down the gradients of the lines joining each of the following pairs of points: (i) (2,4), (-1,5); (ii) (-2,3), (4, -2); (iii) (-3, -2), (-5, -3); (iv) (a, -*), (-«,«; (v) (2/, 2//), (-2/f, -20. b c 5. If f(x) = a+—+— and if f(x) = 3-5 when x = -2, /(*) = 7 when t JC = — 1, and f(x) = 16 when x = 0*5, calculate the values of a, b, and c. / 1 1\ 6. Show that the gradient of the line joining (h, k) to I - —, —r-l is equal to the gradient of the line joining (/i, k) to the origin and hence show that the three points are collinear. State which of the following sets of points are collinear: (i) (3,5), (-2,1), (4,0); (ii) (1,3), (-5,1), (7,5); (iii) (2m, -/), (0,0), (/, m); (iv) (2//, -1/m), (l//,0), (0, 1/m). 7. Calculate the gradient of the line joining the points whose x-coordinates are 1 and -2 respectively on the curve y — x+4/x. Show that this chord is parallel to the chord joining the two points on the curve whose ^-coordinates are 4 and — \ respectively. 8. The curve y = f(x) cuts the jc-axis three times at x = 2, x = 3, and x = — 2. If/(x) is a polynomial function of x of degree three of the form x3 + ax*+bx+c, find/(x). 4 SIXTH FORM PURE MATHEMATICS 1.2 The Idea of a Limit Consider the sum of the series of numbers 1+Y + J+-|-+ We have: i . X _ A i -r 2 — 2 » 1 . JL . JL - .7. 1 T" 2 T- 4 — 4 » 1 4--1-4.i4.JL - 15 1 + 2 T 4 T8 — 8 > The addition of each number halves the gap between the sum that has been arrived at and 2. By taking sufficient numbers the sum can be made as near to 2 as we please. In these circumstances 2 is defined as the limit of the sum of the series as the number of terms increases indefinitely (or as the number of terms tends to infinity) and this statement is written, n 1 + + = 2 l L( 4 T -4) - n A similar result to this is illustrated by a recurring decimal. Thus by division -|- = 0-1 and this is equivalent to the statement . . / 1 11 1 \ 1 h?14To+Io-2+ios+ •• +io*) = T- n The tangent to a curve can be considered as the limiting position of a chord as the two points of intersection approach one another. Thus in Fig. 2 the tangent to the curve at P is the limiting position of the chord PQ as Q approaches P. x2—a2 Algebraic Limits. Consider lim . Direct substitution of x2—a2 x = a in the expression gives an indeterminate result, but x—a x2 — a2 lim = lim (x-\-a)= 2a. x-*-a ^ ^ x-*- a INTRODUCTION TO THE CALCULUS This result can be extended: xn-an lim if n is a positive integer ^ x-a x a = lim (xn-1+xn-2a+xn-*a2+. . , + a71-1) by division; -. xn-an , n .'. lim = na71"1. (1.1) -+ x-a x a Fig. 2. It can be proved, but it will not be proved here, that result (1.1) is true also when n is fractional or negative. The result is illustrated by two particular cases thus: = l i m ^ - ^= (V^^)(V^+Va) lim lim ^ x-a -+ (x-a)(y/x+y/a) ^ x-a x a x a x a i -\ lim _1 1_ . x~x—a" x a .. 1 1 flim = lim = lim =—=. _* x-a ^ ax a2 x a x a