Calculus (Third Edition) [3rd ed.] 9781487599973

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CALCULUS

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CALCULUS BY

R. L. JEFFERY Professor of Mathematics Queen's University

UNIVERSITY OF TORONTO PRESS: 1960

Copyright, Canada, 1954, 1958, 1960 Preliminary edition, October, 1954 First regular edition, May, 1955 Reprinted, November, 1956 Second edition, 1958 Reprinted, August, 1959 Third Edition, 1960

No part of the material covered by this copyright may be reproduced in any form without written permission of the publishers.

PREFACE THIS book came to be for two main reasons. First-year university courses in mathematics frequently presuppose a more complete background in algebra, analytic geometry, trigonometry and statics than is assumed by many of the calculus texts available. As a result, present-day texts often leave nothing for the instructors to do and provide little chance for the students to stretch themselves. This tends to frustrate the instructor and bore the student. The other reason is more general. Over the years a huge mass of material has accumulated at the introductory level with the result that some texts are running to over five hundred pages, and others over six hundred. From some quarters a demand for streamlining has arisen. For example, the Engineering Faculty at Queen's has sanctioned a condensing of the calculus in order to make room for some elementary training in statistics. That there is urgent need for a shorter road to the higher levels of mathematics is beyond question. The present book is such a road. However, the author makes no claim that it is the best short road, and would welcome suggestions for improvement. There is, at the beginning, no discussion of the abstract theory of limits or of properties of functions. These ideas are arrived at by a careful examination of particular cases as they arise. Further space is gained by deferring until they are needed such topics as parametric equations, polar coordinates, and curvature. By the time the student reaches these topics he has acquired some maturity and can more easily assimilate what is new in them. The usual chapter on the constant of integration is omitted, but the principles involved are dealt with in the language of simple differential equations, a language which students of Engineering and of Science must learn but which they usually learn far too late. As the book proceeds, first and second order differential equations are dealt with fairly completely. The familiarity with these subjects thus attained will enable students (and some students of Economics can be included here) to get the facts about equations of higher order as needed. There is no attempt to make plausible the underlying theorems about functions, derivatives, and integrals on which the calculus is based. These theorems are stated, without proof, and used to develop the working tools. The proofs are then given in one short chapter, Chapter XI. The student who wishes can study these proofs, and if he has difficulties can take them up privately with his instructor. This gives the forward-looking students a chance to get what they want without interfering with the main job of giving the rank and file what they must have. For many classes it is desirable that the definite integral be introduced early. The usual way is to do a bit about differentiation, then a bit about V

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PREFACE

integration. This jumping back and forth has its disadvantages. Once students have left the topic of differentiation they do not go back to it with the same interest. This procedure also makes it difficult to lay a proper foundation on which to build the ideas of integration. In this book the study of the anti-differential, which is a necessary preliminary to the use of the definite integral, begins on page 72, and there are several sections in these seventy-two pages that could be by-passed without breaking the continuity. The study of the definite integral begins on page 77. For those who find it necessary to go even faster than this, the material on differentials, anti-differentials, and definite integrals is so arranged that it can be taken up after the chapter on the derivatives of algebraic functions which ends on page 25. Much of the material of this text has been used, in one form or another, during the past five years in freshmen classes at Queen's University, but it was not until February of 1954 that the decision was made to revise and publish it in time for the classes of the 1954-1955 session. There are some acknowledgements to make. For many years the author has used the texts by Granville Smith and Longley, and by Middlemiss. We owe much to their influence. The problems in the present book have been taken from class exercises and examinations, and it is more than likely that some of the problems were suggested by these and other texts. The author has also been influenced by a study of the exposition in the book by Randolph and Kac. Among the problems are a few from the Putnam Competition papers (indicated by the letters PC). The author is grateful to the Mathematical Association of America for permission to use them. Acknowledgement is gratefully made to my colleagues—Professor Norman Miller, who pointed out inconsistencies in notation and in the arrangement of topics; Professor F. M. Wood, whose long experience as an engineer and a teacher of engineering students was drawn upon in the selection of topics and problems; and Professor G. L. Edgett, who used some of the material in his Arts classes and thus induced important changes. Dr. G. F. D. Duff of the University of Toronto assisted by reading critically more chapters of the manuscript than any other person. Thanks are also due to Miss Mary Andrews of the Department of Extension for her help in arranging the work for the extramural students; to my former secretary Mrs. Irene Carabine who stencilled much of the material from my scribbled notes; and to my present secretary, Mrs. Valia Krotkov, and to Mr. Harold Still, who shared the responsibility of checking the page proofs and arranging the index. R. L. JEFFERY Queen's University August 16, 1954

PREFACE TO THE THIRD EDITION THE custom of using the definite integral in the first months of the freshman year in science and engineering classes made it desirable to move this topic ahead from chapter vu to chapter v. As in previous editions the notation dy/dx is not introduced until after the differential is defined (chapter IV). Also as in previous editions f(x)dx is called an anti-differential. It would be preferable to call this an anti-derivative or a primitive but then an explanation for the presence of dx would have to be invented. In fact it would be preferable to teach first and second year courses, all mathematics even, without mentioning the differential. But the concept of the differential is so useful that it should not, nor is it likely to be discarded. For these reasons, after many years of resistance, the author has come around to the belief that the differential and the differential notation should be introduced and made familiar by frequent use soon after the derivative has been defined. The term "integral" is first mentioned when the definite integral is introduced. The term "indefinite integral" is reserved for the definite integral with variable upper limit. This practice of introducing and using differentials and anti-differentials before first using the term integral in association with the definite integral has been followed for several years. It is now possible to say with certainty that in this way much of the usual confusion in regard to the role of these various concepts is avoided. Scientists and engineers learn about mathematics in order to apply the knowledge in a formal way to problems that arise in their work. This consideration has been given first place, and a path has been carefully laid out along the lines of learning rules and applying them. Often, owing to pressure of time, or lack of preparation on the part of the students, it is the only path possible. At the same time a second path, which leads through the fundamental concepts of analysis, has been kept in sight. It starts in the Introduction, comes up in §§ 1.3, 2.4, 4.3, 4.4, and in §§ 5.1-5.5. It culminates in chapter XI in which there is a careful development of some of the underlying theorems of analysis with proofs based on the decimal representation of the real numbers, § 0.9. This development was prompted by more than one motive. The show of intellectual curiosity and ability to go deeply into mathematics on the part of the students in science and engineering is second to that of no other group. There has always been at least one student from the Applied Science Faculty on our Putnam Competition teams, and the records show that these teams have won a fair share of distinction. Furthermore, when engineering students, in their third year, are brought face to face for the first time with the necessity of understanding the methods of analysis they often complain if they have been given no leads into these methods in their vii

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PREFACE

first and second years. Again, it is becoming the practice to base high school mathematics on understanding as well as on formal manipulation, with the result that some students enter university ready and willing to take an interest in fundamental concepts. It is hoped that in these several respects the treatment of the fundamental concepts given in this edition will prove to be helpful. The topics that have been added in this third edition are the chain rule for partial derivatives, directional derivatives, l'Hospital's rule, determinants, and linear dependence relative to vectors. More practice problems, especially in determining anti-differentials, have been added and where experience has shown it desirable, more detailed explanations have been given. Some acknowledgments are in order. Professor Hale Trotter read the manuscript of chapter XI and owing to his suggestions the present form of the book is much better than it otherwise would be. Mr. Bruce Kirby and Professor H. W. Ellis have used the previous editions in their classes and have helped with corrections and with suggestions for improvement. The author is deeply indebted to Mr. Ralfe J. Clench whose careful reading of the manuscript and the proofs resulted in many corrections and improvements. My thanks are due to my secretary, Mrs. Dora Wartman, for typing and helping to prepare the manuscript. I also wish to express my thanks to Miss Eleanor Harman and Mrs. Barbara Sutton of the University of Toronto Press for their helpful co-operation. R. L. JEFFERY Queen's University July 25, 1960

CONTENTS PREFACE.

.........

v

PREFACE TO THE THIRD EDITION ..... vii INTRODUCTION ........ 1 0.1 The real number system. 0.2 Decimal representation of rational numbers. 0.3 Decimals which are neither finite nor repeating. 0.4 Definition of real numbers in terms of rational numbers. 0.5 The number scale. 0.6 The rational points are dense on 1. 0.7 Points on the number scale not marked with rational points. 0.8 Real numbers and their properties. 0.9 Assumptions and working rules. 0.10 Functions and functional relations. 0.11 The double use of symbols. 0.12 The Greek alphabet.

I SPEED AND LIMITS ....... 12 1.1 The idea of speed. 1.2 Speed at a point. 1.3 The idea of limit. 1.4 Properties of limits. 1.5 Improvements in notation.

II THE DERIVATIVE OF A FUNCTION ..... 19 2.1 The derivative of a function. 2.2 The derivative as the slope of the tangent line to a curve. 2.3 The four step rule. 2.4 The limit of a ratio when both numerator and denominator tend to zero.

III RULES AND FORMULAS FOR DIFFERENTIATION ... 26 3.1 Rules for differentiation. 3.2 Formulas for differentiation. 3.3 Proofs of formulas for differentiation. 3.4 The derivative of the square root of a function. 3.5 The derivatives of functions which are defined implicitly.

IV DIFFERENTIALS, DIFFERENTIAL EQUATIONS AND ANTIDIFFERENTIALS ........ 36 4.1 Definition and geometrical interpretation of a differential. 4.2 Relations between dy and Ay. 4.3 Functions with vanishing derivatives. 4.4 The fundamental theorem of the differential calculus. 4.5 Two theorems on differentials. 4.6 Some further applications of differentials. 4.7 Differential relations. 4.8 Rules for determining differentials. 4.9 Anti-differentials. 4.10 Formulas for anti-differentials.

V THE DEFINITE INTEGRAL ...... 50 5.1 The definite integral. 5.2 Continuous function. 5.3 Definition of continuity. 5.4 Maximum and minimum values of a function. 5.5 Assumptions regarding the behaviour of continuous functions. 5.6 Sequences of numbers. 5.7 Notations for sums. 5.8 Areas and volumes. 5.9 A problem on area. 5.10 The definition of the definite integral. 5.11 The fundamental theorem of the integral calculus. 5.12 The solution of the area problem of § 5.9. 5.13 The symbol for the definite integral. 5.14 The double use of symbols. 5.15 The existence of the definite integral. 5.16 The definite ix

x

CONTENTS integral of continuous functions. 5.17 Abbreviated methods. 5.18 Area as a function of the variable x and the double meaning of the symbol d A. 5.19 The existence of the definite integral of a continuous function. 5.20 The indefinite integral. 5.21 The fundamental theorem of the integral calculus.

VI THE TRANSCENDENTAL FUNCTIONS . . . . . 71 6.1 Transcendental functions. 6.2 The trigonometric functions. 6.3 The ratio (sin x)/x. 6.4 The derivatives of the trigonometric functions. 6.5 The exponential function f(x) = ax. 6.6 The logarithmic function. 6.7 The function (1+1/x)x. 6.8 The limit of the function (1 + 1 / x ) x as |x| . 6.9 The derivatives of the logarithmic and exponential functions. 6.10 The inverse trigonometric functions. 6.11 The derivatives of the inverse trigonometric functions. 6.12 Principal values. 6.13 Exponential function with complex exponent. 6.14 Anti-differentials involving transcendental functions.

VII RELATED RATES, MAXIMA AND MINIMA, CURVE PLOTTING . 94 7.1 Related rates. 7.2 Increasing and decreasing functions. 7.3 Extreme values of a function. 7.4 Maximum and minimum values of functions relative to an open interval. 7.5 The first test for maxima and minima. 7.6 Derivatives of order higher than the first and points of inflection. 7.7 The second test for extreme values of a function. 7.8 The graphs of transcendental functions.

VIII DISPLACEMENT, VELOCITY AND ACCELERATION. APPLICATIONS OF RATES OF CHANGE. FIRST MOMENTS, WORK, AND PRESSURE ......... 112 8.1 Velocity and acceleration of bodies in motion. 8.2 Relation between force and acceleration. 8.3 Some applications of rates of change. 8.4 Differential equations in which the variables are not separable. 8.5 First moments and centroids. 8.6 Coordinates of centroids. 8.7 Centre of gravity. 8.8 The centroid of a solid body. 8.9 The coordinates of the centroid of a solid body. 8.10 Centroids of symmetrical bodies. 8.11 The moment of a body relative to its centroid. 8.12 Centroids of areas. 8.13 Work. 8.14 Hydrostatic pressure.

IX FURTHER METHODS OF DETERMINING ANTI-DIFFERENTIALS 139 9.1 The anti-differential of udv. 9.2 Anti-differentials containing trigonometric functions. Trigonometric substitution. 9.3 Improper integrals. 9.4 Anti-differentials containing rational functions. 9.5 Approximate integration. 9.6 The trapezoidal rule. 9.7 Simpson's rule. 9.8 The prismoidal formula.

X ROLLE'S THEOREM, MEAN VALUE THEOREM, PARAMETRIC EQUATIONS, ARC LENGTH, SURFACE OF REVOLUTION, CURVATURE, SECOND ORDER DIFFERENTIAL EQUATIONS, L'HOSPITAL'S RULE ....... 157 10.1 Introduction. 10.2 Rolle's theorem. 10.3 The mean value theorem. 10.4 Functions with vanishing derivatives and the fundamental theorem of

xi

CONTENTS the differential calculus. 10.5 Parametric equations. 10.6 The length of a curve. 10.7 Surface of revolution. 10.8 Differential of arc length. 10.9 Curvature. 10.10 Differential equations of the first and second order. 10.11 The catenary. 10.12 Hyperbolic functions. 10.13 Linear differential equations of the second order with constant coefficients. 10.14 A further discussion of limits. 10.15 I'Hospital's rule.

XI FUNDAMENTAL THEOREMS ...... 189 11.1 Introduction. 11.2 Sets and their properties. 11.3 The supremum and infinum of a set. 11.4 The existence of suprema and Ínfima of sets. 11.5 Sequences. 11.6 The limit of a monotone sequence. 11.7 Functions and their properties. 11.8 The suprema and Ínfima of bounded functions. 11.9 Properties of continuous functions. 11.10 The bounds of functions which are continuous on closed intervals. 11.11 The extrema of functions which are continuous on closed intervals. 11.12 The oscillation of a continuous function. 11.13 Functions continuous on a closed interval. 11.14 Values assumed by continuous functions.

X I I

I N F I N I T E

S E R I E S

. . . . . . . .

2 0 1

12.1 The meaning of sequences and series. 12.2 Convergence, divergence and value of infinite series. 12.3 A second method of assigning a value to a series. 12.4 Tests for convergence. 12.5 The ratio test. 12.6 Series of positive and negative terms. Absolute convergence. 12.7 Alternating series test. 12.8 Functions represented by series. 12.9 Series whose terms are functions of x. 12.10 Operations with series. 12.11 Computation by means of series.

XIII PARTIAL DERIVATIVES, MULTIPLE INTEGRATION, MOMENTS OF INERITIA 220 13.1 Functions of two or more variables. 13.2 The geometrical interpretation of a function of two or more variables. 13.3 Partial derivatives. 13.4 Total differentials. 13.5 Partial derivatives of order higher than the first. 13.6 Double integrals. 13.7 Evaluation of a double integral. 13.8 Fundamental theorem on the evaluation of double integrals. 13.9 Polar coordinates. 13.10 Multiple integrals. 13.11 Second moments, moments of inertia. 13.12 Second moments of solid bodies. 13.13 Interpretation of moments of inertia. 13.14 Moments and centroids by multiple integration. 13.15 The transfer theorem. 13.16 Radius of gyration. 13.17 The chain rule for partial derivatives of composite functions. 13.18 Directional derivatives.

XIV VECTORS AND SOLID GEOMETRY ..... 252 14.1 Definitions and notation. 14.2 Definition of a vector in terms of its components. 14.3 Direction angles, equations of lines. 14.4 Multiplication of vectors. 14.5 The equation of a plane. 14.6 Distance from a point to a plane. 14.7 The tangent plane to a surface.

XV DETERMINANTS, MATRICES AND LINEAR EQUATIONS . 265 15.1 Definition of a determinant. 15.2 Properties of determinants. 15.3 Determinants of order higher than three. 15.4 Matrices. 15.5 Cramer's rule. The solution of ai1x+ai2y+ai3z = d¡, i = 1, 2, 3, if the rank of A is

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CONTENTS three. 15.6 Homogeneous linear equations. 15.7 Two homogeneous equations in three unknowns. 15.8 Vectors which are perpendicular to two vectors.

XVI LINEAR RELATIONS AMONG VECTORS .... 275 16.1 Definition of linear dependence. 16.2 Linear relations among four or more vectors. 16.3 Two-dimensional vector spaces. 16.4 Vectors as ordered sets of numbers.

XVII VECTOR PRODUCTS, APPLICATIONS OF VECTORS . . 281 17.1 Vector products. 17.2 Curvilinear motion.

XVIII QUADRIC SURFACES

289

18.1 The general second degree equation. 18.2 The quadric surfaces. 18.3 Ruled surfaces.

INDEX

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CALCULUS

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INTRODUCTION 0.1 The real number system. We shall take for granted the knowledge of positive and negative numbers which is acquired in high school, and a knowledge of the operations of addition, subtraction, multiplication and division on these numbers. We recall that the positive numbers consist of the integers, the rational fractions or rational numbers, and the irrational numbers. These and their negatives, together with the number zero, make up the real number system. A rational number may be written as the quotient of two integers, 3/4, 91/63, for example. An irrational number cannot be so expressed. In a sense this is a negative definition, for it says what an irrational number is not. There are positive definitions of irrational numbers, but we are not, at this stage, in a position to consider them in full detail. It is hoped that the following discussion will give some insight into what is involved. 0.2 Decimal representation of rational numbers. Consider the rational number 1/3. This number is represented in the decimal notation by .333 . . . which is known as a repeating decimal. What is the meaning of the statement that this repeating decimal represents 1/3? Consider the sequence of decimals, .3, .33, .333, . . . , which is the sequence of rational numbers, (1)

The nth term rn of this sequence is the number represented by the first n places of the repeating decimal .333 . . . and it is easily verified that

The greater n becomes the closer 1/10" gets to zero. Hence the greater n becomes, the closer rn gets to 1/3. This is the sense in which the repeating decimal .333 . . . is said to represent 1/3; the greater n becomes the closer the rational number rn gets to 1/3 where rn is the rational number represented by the first n places of the 1

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INTRODUCTION

repeating decimal, .333 . . . . In a similar way any repeating decimal can be shown to represent a rational number in this sense. We note another property of the sequence (1) to which we shall again refer. Consider the difference between two terms of this sequence rn and rm where, for the sake of being definite, we take m > n. Then

where the digit 3 occurs m — n times in the numerator of the second factor on the right. Consequently this factor is less than unity. Hence rm — rn < 1/10™ and the following statement can be made about the sequence (1). The greater n and m become, independently of each other, the closer rm — rn gets to zero. Now let p/q be a rational number. If p is divided by q the remainder is less than q. If the division is continued, a stage may be reached at which the remainder is zero. If this happens p/q is an integer or a finite decimal. If this does not happen then after at most q — 1 steps the remainder is a repetition of a previous remainder and the quotient begins a repetition. Thus every rational number p/q is an integer, a finite decimal or a repeating decimal. Integers and finite decimals are rational numbers, and a repeating decimal represents a rational number in the sense described above. Hence it can be said that Every finite or repeating decimal represents a rational number, and every rational number can be represented by a finite or repeating decimal. 0.3 Decimals which are neither finite nor repeating. Consider the infinite decimal (2)

.121121112...

This is neither a finite nor a repeating decimal and does not, therefore, represent a rational number. In school work it is asserted that (2) represents an irrational number. But what is the basis for this assertion? Corresponding to the decimal (2) is the sequence of rational numbers (3)

of which the nth term rn is the first n places of the decimal (2). The sequence (3) has the following property in common with the sequence (1). Let rm and rn be the mth and nth terms of the sequence. Then the greater m and n become, independently of each other, the closer \rn — rm\ gets to zero. It is this property which the sequence (3) has in common with the

0.4 DEFINITION OF REAL NUMBERS

3

sequence (1) that forms the basis for saying that the decimal (2) represents a number, according to the following definition. 0.4 Definition of real numbers in terms of rational numbers. Let ri, f2 be a sequence of rational numbers. Let rm, rn represent the rath and wth terms of the sequence. If the sequence is such that the greater m and n become, independently of each other, the closer rn — rm\ gets to zero then the sequence is said to represent a real number.

If two sequences r\, r^, . . . , and ri, r-î, . . . , both satisfy this condition and in addition are such that \rn — rn' approaches zero as n increases then the two sequences represent the same real number. Corresponding to any integer a, or any finite decimal, there is a sequence of rational numbers satisfying the condition of Definition 0.4. For an integer such as 961 there is the sequence 961, 9610/10, 96100/102, . . . , and for the finite decimal such as 961.36 there is the sequence 96136/102, 961360/103 For both of these sequences \rm — r n |—»0, as m, n —» °°. For an infinite decimal a . bibz . . .

repeating or otherwise, the corresponding sequence is and, if n>m,

a, ooi/lO, a6i&2/10 2 ,

It follows that rn — rm tends to zero as m, n increase. Consequently everything that has come to be called a number is included in the defintion of this section. 0.5 The number scale. Let / be a line and 0 a point on /, and let u\ be a segment of a line, Figure 0.1.

Fig. 0.1

Take the length of MI as a unit of distance, and on the point whose distance is one unit to the right of 0 mark the integer 1, on the point two units to the right of 0 mark the integer 2, and so on. Bisect u\ and take one of the halves as a unit of length u2. On the point which is the length of M 2 to the right of 0 mark the rational number 1/2; on the point three times the length of «2 to the right of 0 mark the rational number 3/2, and so on. If MI is divided into q equal parts and one of the parts uq used as a measure,

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INTRODUCTION

the numbers p/q, p = 1, 2, 3, . . . , which have not been previously marked, may be marked on points of /. If g = 6, for example, then the number four times We to the right of 0 would have been marked with 2/3 at the third stage. This process marks every positive rational number, expressed in lowest terms, on points of /. The points so marked are called rational points.

0.6 The rational points are dense on /. Let Pi and P2 be any two

distinct points on / to the right of 0. Regardless of how close together are PI and PI. there is an infinite number of rational points between them. If q is such that l/q is less than the distance between PI and PI then for at least one value of p the number p/q, or the same number in lowest terms, is marked on a point between PI and P%. It then follows that between any two points on /, no matter how close together, there is an infinite number of rational points. This is expressed by saying that the rational points are dense on the number scale.

0.7 Points on the number scale not marked with rational points.

If the rational points are dense on / is there room for points other than rational points? Figure 0.2 is a right triangle with the sides about the right angle each equal to unity where by unity we mean the measure of the length of MI. Then the length h of the hypotenuse is now given by h = (P + l 2 )* = V2. Let Q be a point a distance h to the right of 0. Is Q a rational point? If so it is marked with a rational number p/q in lowest terms and pz/q2 = 2, p* = 2g 2 ; 32 = i"2/2. Because q is an integer it follows that p2/2 is an integer. Consequently ¿>2 is divisible by 2 and it follows that p is divisible by 2. For otherwise p is of the form 2r+l, p"1 = 4r 2 +4r+l which Fig. 0.2 is not divisible by 2. Hence p is of the form 2r, r an integer, and

Because r 2 is an integer it follows that q is divisible by 2. Hence p and q have a common factor 2 and it follows that p/q is not in lowest terms, which is a contradiction. The conclusion that Q is not marked with a rational point now follows. Another immediate conclusion is that if the unit u\ is divided into q equal parts then, no matter how great is q, the measure uq cannot be used to measure the length h exactly. For this reason the length h of the hypotenuse of the triangle of Figure 0.2 is said to be incommensurable with unity, or simply incommensurable.

0.8 REAL NUMBERS

5

If the length h is divided into q equal parts and one of the parts hq is used as a measure to locate points to the right of 0 the points so located can, in a similar way, be shown to be other than rational points and the set so obtained, for all values of q, can be shown to be dense on /. Thus we have on I a set of points, other than rational points, which are dense on I. The set of all points on / which are not rational points are called irrational points. 0.8 Real numbers and their properties. Let R be a point on / to the right of 0. Because the rational points are dense on / there is a rational point r\ between 0 and R and nearer to R than half the distance OR, a rational point rz between r\ and R and nearer to R than half the distance riR, and so on. Let ri, r 2 , . . . , be the sequence of rational points determined in this way. Then the greater n becomes the nearer to R is rn. Hence if rm and rn are two points of the sequence the greater m and n become, the closer to zero the distance rmrn becomes, and the rational numbers rm, rn are such that rn — rm gets closer to zero. Hence this sequence of rational numbers defines a real number in the sense of Definition 0.4. Let this number, postulated by Definition 0.4, be associated with the point R. Thus, there is associated with every point R on / a real number from the set postulated by Definition 0.4. The following question now arises. If r\, r-i, . . . , is a sequence of rational numbers satisfying the conditions of Definition 0.4 is there a point R on / such that if rn is the point marked with the number rn of this sequence, the measure of the distance between rn and R tends to zero as n increases? The answer is yes, but at this stage we do not have the means for making this plausible. We leave it for the reader to think about. (See Exercise 11 (a), Problem 20.) The discussions of §§0.2-0.8 take considerable space, as much as can be spared, yet these discussions cannot be considered as an adequate foundation for the real number system. For one thing they are in terms of addition, subtraction, multiplication, and division, and these have not been defined. The most that can be hoped is that they throw a revealing light on the problems involved in defining the real numbers. Although the definition of a real number in terms of rational numbers given in §0.4 is generally accepted, it is not the only definition. 1 There remains the problem of making precise statements of the assumptions on which the work of this book is based. This is done in the following section. 'For another see G. H. Hardy, A Course of Pure Mathematics (Cambridge University Press, 1952). Other references are, Richard Courant and Herbert Robbins, What is Mathematics? (Oxford University Press, 1941); Edward G. Begle, Introductory Calculus With Analytic Geometry (New York: Henry Holt and Company, 1954); R. L. Jeffery, The Theory of Functions of a Real Variable (University of Toronto Press, 1953).

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INTRODUCTION

0.9 Assumptions and working rules. It is assumed that there exists a set of elements a, b, c, . . . x, y, . . . , called real numbers. This set, denoted by R, contains the positive and negative numbers of arithmetic. To each number x in the set R there corresponds a single point x on the x-axis of the Cartesian coordinate system, and to each point on the x-axis there corresponds a single number in the set R. Because of this correspondence the terms point and number are used interchangeably. A positive number x measures the distance of the corresponding point x to the right of the origin, and the points corresponding to the negative numbers are similarly located to the left of the origin. The number zero corresponds to the origin, and conversely. It is assumed that the operations of addition, subtraction, multiplication, and division of arithmetic are valid for the numbers in R. If a set of rules or operations leads to the formation of a number a . bibz . . . bk

in decimal form where a is an integer and each b is one of the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and the set of rules permit the determination of &i for any given integer k then it is assumed that this set of rules or operations determines a number in R denoted by a.bibz... in the decimal notation. If o is a positive real number and q is an integer there is a positive real number \\A\-\B\\. (c) For a number d > 0 the relations A — C\ < d, B—C\^d hold. Show that \A—B = B — ^4| < 2d. Give an example for which the equality holds. 20. Show that the numbers \/3, SXo cannot be expressed in the form p/q where p and q are integers with no common factor except unity. 21. Show that the following repeating decimals represent rational numbers in the sense of §0.2. (a) .3222 . . . , (b) .23575757 22. Determine the repeating decimal which represents the rational number 52/31. 23. Determine all the functions implied by each of the following relations. (a) y2 — xzy+xy — x3 = 0. Ans. f(x) = x1; g(x) = —x. (b) y2 — xy+y — x = 0. Ans. f(x) = — I ; g(x) = x. (c) / —* 2 ;y+2ry 2 -2x 3 = 0.

CHAPTER I

SPEED AND LIMITS 1.1 The idea of speed. If an automobile covers 100 miles in four hours, we say that its average speed is 25 miles per hour. The term average speed suggests that its speed has not been uniform from the instant it started until the instant it stopped. We also speak of the speed of a moving body at a given instant, or as it passes a given point. What do we mean by saying that an automobile was travelling at the rate of 50 miles per hour when it crossed a railway track? If you attempt to write a formal definition of speed at a point you will find it difficult. We shall try to throw some light on this question by a study of the motion of a body falling from rest in a vacuum. It has been established that if a body falls from rest at a point 0 the distance 5 is given by where 5 is the measure of the distance from 0 in feet, t is the measure of the time, in seconds, that the body has been falling from 0, and g is a constant due to the attraction of gravity. g is approximately 32 at sea level, and it is assumed that the problem is under consideration at an altitude for which g is exactly 32. Then (1) 5 = 16i2. Let PI, Figure 1.1, be the point the body reaches in three seconds. We wish to arrive at a definition of the speed of the body at the point PI. The distance OPi is given by s\ = 16(3)2 = 144. Hence the length of OP\ is 144 ft. In 3.1 seconds after falling t = /2 = 3.1 and 52 = 16(3.1)2 = 153.76. The average speed for the distance P\P^ is the measure of this distance, Sz — si, divided by the measure of the time it takes to fall through this distance which is ti — t\:

Hence the average speed for the distance PiP2 is 97 ft. per sec. Let the time of falling be 3.01 seconds. Then i3 = 3.01, s3 = 16(3.01)2 = 144.9616,

1.2 SPEED AT A POINT

13

and it follows that the average speed for the distance PiPz is 96.16 ft. per sec. If the time of falling is 3.001 seconds then ¿4 = 3.001, 54 = 16(3.001)2 = 144.096016, 54-5t .096016 /«-*! ~ -001 ~ 96'016' and the average speed for the distance PiP4 is 96.016 ft. per second. Continue this procedure with ¿ 6 , ¿e, • • • taking on the values 3.0001, 3.00001, . . . If the results are checked for the distances PiP2, PiP3, PiPít it is seen by the way the computations turn out that the average speeds for the distances PiP6, PiPe, . . . are 96.0016 ft. per sec., 96.00016 ft. per sec It thus becomes evident that as the interval of time after passing PI gets shorter and shorter, the average speed for the distance covered in the interval of time gets nearer and nearer to 96 ft. per sec. This suggests that the speed of the falling body at the point PI be defined as 96 ft. per sec. It also gives a clue to a formal definition of speed at a point. 1.2 Speed at a point. // a moving body passes a point PI the measure of the speed of the body at the point PI is the number to which the measure of the average speed for the distance P\Pn gets nearer and nearer as the point Pn gets nearer and nearer to the point PI. Note that in the case of the falling body there is no distance P\Pn for which the average speed is exactly what has been defined to be the speed at Pi. 1.3 The idea of limit. In the foregoing a sequence of ratios

was considered, where n took on greater and greater positive integer values. The statement was made that the ratio got nearer and nearer to 96. Equivalent statements are: As n increases the ratio approaches 96; the ratio tends to 96. The symbol for "approaches," or "tends to" is —>. Thus, as n increases The number 96 is called the limit of the ratio (sn — Si)/(tn—ti) as n increases indefinitely, or briefly, as n increases. If n increases through the set of positive integers (decreases through the set of negative integers) this behaviour of n is described by the symbolism,

14

SPEED AND LIMITS

"n —> a» " (n —» -co). The final abbreviated symbolism for the behaviour of the ratio (sn — s^)/(tn— /i) in relation to the number 96 is

This is read as follows: The limit as n increases of The limit concept has many variations and applications. It is a concept which is not grasped easily nor at once. A discussion of the idea in the abstract is not profitable at this stage. A more satisfactory procedure is, for a time, to study carefully each application of the idea that comes up. For the sake of completeness we give, in precise language, a formal definition of the limit of a function A (n) which depends on integer values of n. The function A (n) tends to a limit L as n increases,

if to every positive number «, no matter how small, there corresponds an integer n( which is such that when n > n(. In the foregoing A(n) is the ratio (sn — Si)/(/ n — ¿i) and L = 96. Other examples are (a)

In Example (a) the function A(n) = ! + !/« which obviously tends to unity as n —-» °°. If e > 0 is given, is there nc for which We show that nf may be taken as the first integer which is greater than 1/e. For such a determination of nt we have for any n > nf

Thus no matter how near zero is the positive number e there is always an integer n which satisfies the requirements of the formal definition of the limit of (« + !)/« as n —» °°. In this example it is obvious that the limit as n increases of («+l)/ra is unity. One naturally asks what is the point of the formal definition? A partial answer is that it is often necessary to make use of the fact that a limit exists when it is not possible to know exactly what the limit is. Further-

1.4 PROPERTIES OF LIMITS

15

more, to show that a limit exists, it is often the case that the simplest way to do it is to show that the conditions of the formal definition are satisfied. The expression {(«+!)/«}" tends to a limit as n —» oo and the limit lies between 2 and 3 but these facts cannot be determined by inspection (see §6.8). Find an nf for each of (b) and (c)? The idea of a limit also comes up in terms of variables other than the integers n. For example let A (x) = x*. Then if x takes on values which get nearer and nearer to 3 the function A (x} takes on values which get nearer and nearer to 9. In the limit notation,

There is a formal definition which covers this case. The function A (x) tends to a limit L as x tends to a if to any positive number e, no matter how close to zero, there corresponds a positive number S e which is such that |a(x) - L} e •whenever 0 3 as x —> 9.

1.4 Properties of limits. We state some properties of limits which we shall use in what follows without further reference to them. Let A and B be variables which tend to limits. Let k be a constant. Then lim (kA) = k lim A ; lim (A+E) = lim A+lim B;

lim (AB) = (lim A)(lim B); lim (A/E) = lim .4/lim B provided lim 5 ^ 0 .

These follow immediately from the definí' >< n of limit. EXERCISE l(a; 1. Estimate the speed of a body falling from rest when (i) t = 1, (ii) / = 2. 2. A body moves in a straight line so that the measure of its distance from a fixed point 0 on a line is given by 5 = 3/2+5i where t is the measure of the time since it started from 0. Estimate its speed when t = 1. 3. A body moves in a straight line according to the law 5 = ¿ 2 +5£+l, where 5 is the measure of the distance of the body from a fixed point 0 on the line and t is the measure of the time since the body started moving (i = 0, 5 = 1). Find the speed of the body when t = 3.

16

SPEED AND LIMITS 4. Find the limit as n increases of the following

(a) 5. Determine the following limits.

(a)

Ans.

(e)

(b)

Ans.

(f)

(v) (d)

(g) (h)

Ans.

6. Let the circle x 2 +;y 2 = V intersect the y-axis at the point P and the circle (x — 3)2+;y2 = 9 at the point Q, Q in the first quadrant. Let PQ intersect the x-axis at R. How do you think the point R will change as h —» 0? Prove your conjecture. 7. Given the functions x sin - and sin -, x 7* Q. Show that as x —» 0 the x x first tends to a limit while the second does not. 8. By the use of tables, or other methods of computation, estimate the following limits.

1.5 Improvements in notation. The speed of a body falling from rest has been calculated at the point reached in 3 seconds. The speed at a point reached in 2 seconds, 5 seconds or any definite number of seconds can be estimated in a similar way. It would, however, be necessary to make a different set of calculations for each number of seconds. A notation and procedure will now be worked out which leads to an expression for the speed at a point in terms of the time at which the falling body reaches the point. Let PI, Figure 1.2, be the point reached in t\ seconds. Then ji, the measure of the distance OP\, is 51 =

16¿J2.

Let P be the point reached at a later time. Denote this later time by 0 of AV/Ax is the rate of change of the volume with respect to x; i.e. with respect to the height or width. If y = /(*) is represented by a graph in the xy-p\ane, then the limit ot Ay/Ax as Ax—>0 may be interpreted as the slope of the tangent line to the graph at the point [x,f(x)] on the graph. If y is the work done in moving a body, and x is the distance through which the body moves, then the limit of Ay /Ax is the force acting on the body. If the amount of consumer goods produced by a company is x, and the price is a function of x, then the gross revenue R is a function of x. The lim AR/Ax as Ax—>0 is known in economic theory as marginal revenue. We have said sufficient to indicate that in the general functional relation y = f(x) the limit of the ratio Ay /Ax,

has considerable interest and importance. For this reason it has been given a special name, the derivative of y with respect to x. This limit is represented by any one of the symbols 19

20

DERIVATIVE OF A FUNCTION

2.2 The derivative as the slope of the tangent line to a curve. The

graph of a function y = f(x) is shown in Figure 2.1. P(x,y) is a point on the curve. Q(x-\-&x, y+Ay) is a second point on the curve. If there is a tangent

Fig. 2.1

line to the curve at the point P then, by definition, the tangent line is the limiting position assumed by the secant line through the points P and Q as Q moves along the curve towards P. If the secant line does not approach a limiting position as Q approaches P then the curve does not have a tangent at P. Furthermore the limiting position which the secant line approaches must be the same regardless of the manner in which Q approaches P. The point Q as it approaches P may take positions on the left of P as well as on the right, or it may take positions sometimes on the left and sometimes on the right. In Figure 2.1 it is assumed that there is a tangent line T to the curve at P and that the line T makes an angle 6 with the x-axis. Let 6' be the angle that the secant line through PQ makes with the x-axis. The line PH is parallel to the x-axis. Consequently the angle HPQ is equal to the angle B'. Also

It is assumed to be intuitively evident that as Q moves towards P, 6' —> 6, and tan 6' —> tan 6. Also Ax —> 0.

2.3 THE FOUR STEP RULE

21

Hence

The limit on the left is, by definition, the derivative of the function J — f ( x ) at the point P(x, y ) , and tan 6 is, by definition, the slope of the tangent to the curve at P. Thus it has been shown that if the graph of the function y = f(x) has a tangent at a point (x, y) on the curve then the slope of the tangent is the value of Dxy at this point. 2.3 The four step rule. Because of the wide use made of the derivative it is essential that careful consideration be given to the formalities of its determination for various types of functions. The procedure so far developed consists of the following four steps. I. Let a value of x be held constant and write II. Subtract the first of these relations from the second to get III. Divide both sides of this equation by Ax to get

IV. Determine the limit of the right side as Ax tends to zero to get

If this limit exists it is the derivative of y with respect to x,

In determining speed when the distance 5 is a function of the time / there has been developed a technique which may be used in the case of any function of a single variable, y = f(x). One more example of this technique is given using x and y in place of s and t. It is an application of the four step rule. Example 2.1 Find Dxy if

Take a fixed value of x and write (1)

22

DERIVATIVE OF A FUNCTION

Now use a second value of x denoted by x+Ax. This second value x+Ax when used in place of x in the relation defining y in terms of x gives a second value for y, which we denote by y+Ay. Thus (2)

The subtraction of (1) from (2) gives

Now dividing both sides by Ax gives (3)

What is finally wanted is the number to which Ay/Ax gets nearer and nearer as Ax gets nearer and nearer to zero. In other words what is required is From (3)

If we honour the agreement to keep x fixed, it results that only Ax can change and the way this changes is to get nearer and nearer to zero. Hence 4Ax approaches zero, (x+Ax) 2 approaches x 2 , and consequently

Thus

Once this process becomes familiar it will not take as many steps to find Dxy. The work should appear about as in the following example.

2.4 LIMIT OF A RATIO

23

Example 2.2 Find

EXERCISE 2 (a) In each of the following find Dxy by finding the limit as Ax—»0 of Ay/Ax.

1.

4.

2.

5.

3. 6. Find Dts if 5 = - — t. What sign has Dts for all values of / except zero? What does the sign of Dts indicate about the behaviour of s as t increases? Plot s against t to verify this behaviour. 7. Draw a graph of the curve whose equation is y = x* — 2x. Verify that the point (2, 2) is on the curve and determine the slope of the tangent to the curve at this point. At what point on the curve does the tangent make an angle of 45° with the oc-axis? 2.4 The limit of a ratio when both numerator and denominator tend to zero. The value of the limit of the ratio Ay/A» is not always as easy to determine as it is in the preceding examples and exercises. This is illustrated by the following :

24

DERIVATIVE OF A FUNCTION Example 2.8 If y = V* find the derivative of y with respect to x.

Divide both sides by Ax.

Now ask to what number does the ratio

get nearer and nearer as Ax gets nearer and nearer to zero? If we fix our attention on the numerator we note that as Ax—>0 the numerator tends to zero, and we are tempted to say that the ratio tends to zero. On the other hand, if we fix our attention on the denominator, noting that it gets nearer and nearer to zero, we are tempted to say that the ratio gets greater and greater. At this point the essential question to be settled is: If we know that a variable u tends to zero, and that a second variable v tends to zero, what conclusion can we draw from this about the behaviour of the ratio u/v? A few examples should help us to see what is involved in arriving at an answer. Let u = x2, v = x. Then as x—>0 both u and v tend to zero and the ratio u/v = x tends to zero. Let u = x, v = x3. Then as x—»0 both u and v tend to zero and u/v = 1/x2 becomes greater and greater. Let u vary over the sequence and v over the sequence Then both u and v tend to zero and u/v varies as follows : Thus the ratio u/v does not tend to a limit but oscillates between 1 and — 1. Let u vary over the sequence and v over the sequence

EXERCISE

25

Both u and v tend to zero and for u/v we have - : 2, 2, 2, 2, 2, ... and the ratio u/v remains constant. From these examples it is obvious that the answer to the question, "What do we know about the behaviour of the ratio u/v when we know that M—>0 and v—»0?" is that we know nothing. The behaviour of the ratio u/v depends on the relative manner in which u and v tend to zero. Each case has to be considered individually. The case we are studying is

To obtain the answer make use of the special properties of the function y = -Vx. Write and multiply both numerator and denominator by \/x-\-Ax-\-\/x- This gives

From this it is seen that as A;t—>0

EXERCISE 2(b) Find Dxy in each of the following: 1.

3.

2.

In each of the ratios in the following problems both the numerator and the denominator tend to zero as x tends to the indicated limit. Determine the values of the indicated limits.

4.

Ans.

7.

Ans.

5.

Ans.

8.

Ans.

6.

Ans.

9.

Ans.

CHAPTER III

RULES AND FORMULAS FOR DIFFERENTIATION 3.1 Rules for differentiation. Some system has now been brought into the methods for finding the limit of the ratio Ay/Ax as Ax—»0. There are, however, some rules which make for further simplification. These will first be stated in general terms, and you are urged to memorize them as they are stated. 1. The derivative of a constant is zero. 2. The derivative of x with respect to x is unity. 3. The derivative of a constant times a function is the constant times the derivative of the function. 4. The derivative of the sum of two functions is the sum of their derivatives. 5. The derivative of the product of two functions is the first times the derivative of the second plus the second times the derivative of the first. 6. The derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 7. The derivative of a function raised to a constant power is the index of the power times the function to the index of the power minus one all multiplied by the derivative of the function. 8. If y = f(u) and u = (x) then the derivative of y with respect to x is the derivative of y with respect to u times the derivative of u with respect to x. 3.2 Formulas for differentiation. The rules stated in §3.1 may be written as formulas.

I dxc = 0.

V dx(uv) = udxv+vdxu. VI

II dxx =1. iii dx(cv) = cDxv. iv dx(u+v) = dxu+dxv.

vii dxvn = nvn-1 Dxv.

viii dxy = duy dxu.

3.3 Proofs of formulas for differentiation. Formal proofs are now given of I, II, VI, VII and VIII. The proofs of the remaining three are left as exercises. 26

3.3 PROOFS OF FORMULAS

27

Let y = /(x) = c, where c is a constant. Then y+Ay = /(x+Ax) = c, and Ay = 0. Hence Ay/Ax = 0 for every value of Ax ^ 0, and consequently,

For the proof of II, y = /(x) = x, and y+Ay = /(x+Ax) = x+Ax, from which it follows that Ay = Ax, Ay/Ax = Ax/Ax = 1, and consequently

The remaining formulas involve functions «(x) and v(x). It is assumed that AM = w(x+Ax)— w(x)—>0 as Ax—>0 and that exists. The function »(x) also has these properties. In formula VI,

which is VI. Formula VII will now be proved for n a positive integer.

28

RULES FOR DIFFERENTIATION

as A*—»0, Av/Ax-+Dxv and, since Az>—>0 as Ax—»0, each term in the bracket except the first tends to zero. Hence Dxy = nvn~l Dxv. In formula VIII, y = /(w) where M is a function of x, u = u(x). Consider further that M is a function having the same properties as the functions u and v in VI. In particular this means that if xa is on an interval (a, b) contained in the domain of definition of the function u then AM = u(x0-\-Ax) — U(XD) —» 0 as Ax —* 0. It also means that AM/Ax —> Dxu at x = x0. The statement that y = f(u) where M is a function of x, u = u(x), implies that y is a function of x, y = f[u(x)] = F(x). It is assumed that for x — x0, Dxy = DXF exists and that for u = u0 = u(x0), Duy exists. The problem is to show that Dxy = Duy Dxu. This is easy if xM;y = 2. For x0 = 0, w(0+A B x)—M(U) = 0 when Anx = 1/vW. Also Dxu exists for x = 0, for

3.3 PROOFS OF FORMULAS

29

Formula VIII is now established for all possible cases. It may be used to prove formula VII when n is a rational fraction, n = p/q, p and q integers. Let (a, b) be an interval on which Dxv exists and on which [z>(x)]1/s takes on real values according to the assumptions of §0.9. For p an integer let y = vv/ 0 is given there exists

56

THE DEFINITE INTEGRAL

Fig. 5.3

5 > 0 such that (x"k) — 4>(x'k) < y when \x"k — x'k < 5. Hence if the maximum of (xk+i—x/c), denoted by max (xk+i — Xk), is sufficiently close to zero

It is thus seen that Sn — sn—>0 as max (xk+\— xk~) —> 0. Then, because sn < A < Sn, it follows that both sn and Sn tend to the required area A as n —» » and max (x/c+i — x^) —> 0. We now take a further step in pointing out that if £4 is any point with xk < |fc < xk+i then $(x'k) < 0. The area between y = f ( x ) , the x-axis, and the ordinates through x = a and x = b is revolved about the x-axis. Prove that the volume of the solid thus obtained is given by

14. Let/(x) be defined on [a, b] and let m, M be two numbers such that m < f(x) < M, x on [a, b]. If the definite integral oîf(x) over [a, b] exists, show that

In forming the sum whose limit is the definite integral there is no restriction on the manner of subdividing [a, b] except that max(xk+i — x*)—>0. There is no restriction whatever on the choice of £t on [xk, xk+i]. Use these facts to form sums in a way to establish the results in Problem 15 and 16. 15. If c is a point between a and b then

16. If f(x) is an even function,/(x) = f ( — x ) , the definite integral

and this definite integral is zero if f(x) is odd, f(x) = —f( — x). 17. Two spheres in contact have a common tangent cone. These three surfaces divide the space into three parts only one of which is bounded by all three surfaces; it is "ring-shaped." Being given the radii of the spheres r

5.19

DEFINITE INTEGRAL OF A CONTINUOUS FUNCTION

67

and R, find the volume of the "ring-shaped" part (the desired expression is a rational function of r and R}(PC, 1948).

18. Let the integral of f(x~) over [a, b] exist and let G > 0 be a number such that \f(x)\ < G for x on [a, b]. Show that

5.19 The existence of the definite integral of a continuous function. Let the domain of definition of the continuous function f be the closed interval [a, b]. Then the definite integral of f over [a, b] exists. Let D be any subdivision of [a, b] by the points XQ = a < Xi < . . . < xn = b, and let d be the maximum of x^+i — xk, k = 0, 1, . . . , n—1. The number d is called the norm of D. Let m, M be respectively the minimum and the maximum of f ( x ) for x on [a, b], and let mK, Mk be the same for the interval [xk, xk+i]. Because m < mk < Mk < M it follows that B-l

n-l

(1) m(b—a) < s = X) mk(xk+i-x) < 5 = ^ Mk(xk+i-x¿) < M(b-a). t-o

*=o

Let DI, DZ, . . . , be a sequence of subdivisions of [a, b] for which the points of Dn+i include the points of Dn and for which dn, the norm of £>„, tends to zero as n increases. Each interval of Dn is divided into a finite number of intervals by the points of Dn+i- A term M¿* (xk+in — xkn) of Sn is not less than the sum of the terms of Sn+i arising from the intervals [x"+1, xj+in+1] of Dn+i which are on the interval [xkn, xk+in] of Dn. This is because the maximum of f ( x ) on each of these intervals is not greater than Mk". It then follows that Sn+i < Sn and in a similar way it follows that sn < sn+i. It now follows from (1) that (2)

m(b-a) 0 be given, and fix j/ > 0 so that ?j < e/3. Fix 5 > 0 so that if x, x' are any two points on [a, b] with \x'— x\ < S then (3)

The existence of ô follows from §5.5. Now fix n = n' so that dn> < 8 and so that, if SH> = S', then (4)

\S'-I\ and D, Then, as in (2),

(6) From (5) and the first set of relations in (6) we get |5' — s" < 77 and this combines with (4) to give (7)

|s"- 1| < 2,.

From 5— 5 < 17 and the second relation in (6) we get |5-s"| < r,,

\S-s" < r,

and these combine with (7) to give 7-377 < 5 < 5 < /+377.

Now let £i; be any point on the intervals [#t, Xn+i] of the subdivision D. Then m k < /(?s) < Mk and it follows that the sum (8)

]£/(&) (*«-!-**) i-=0

is between 5 and S or equal to one of them. Hence this sum differs from / by not more than 877 = t. It has thus been shown that the number / determined above is such that to any e > 0 there corresponds S > 0 for which the sum (8) differs from / by less than t if the maximum of xk+i — xk < o, k = 0, 1, . . . , n—l. Thus I qualifies under the definition of §5.10 to be the definite integral of f(x) over [a, b]. 5.20 The indefinite integral. Let f be a function which is defined on an interval [a, b] and which is integrable on every interval [a, x], a < x < b. Then,

(1) has one value for each x on a < x < b and, except in special cases, f(x) is identically zero for example, G (x) changes with x. For this reason it is called the indefinite integral of f. G(x) is not defined for x = a for nothing has been said about the integral of a function over a point. The definition of integrability implies that there

5.21 THE INTEGRAL CALCULUS

69

is a number M > 0 such that |/(x)| < M for x on [a, b]. It then follows from the result of Problem 18, Exercise 5(a), that

It follows from this that G(x) —» 0 as x —> a and it is, therefore, reasonable to define G(a) to be zero, G(x) is now defined on [a, b]. If the function f is continuous on [a, b] then G'(x) = f ( x ) . It follows from §5.19 that G(x) exists, a < x ^ b, and G(a) = 0 by the definition just given. Let x 0 and x 0 +Ax, Ax > 0 be points on a < x < b. By Problem 15, Exercise 5(a),

Let m (àx), M(Ax) be the minimum and maximum respectively of f(x) for x on [x0, Xo+Ax]. Then

Hence (1)

Because/(x) is continuous it follows that both w(Ax) and M(AX) tend to f (XD) as Ax —» 0. Consequently as Ax —> 0 the middle term of (1) tends to/(x 0 ). This result may be obtained in the same way if xa is any point on a < x < b and Ax is negative. It then follows that G'(x) =/(x), a < x < b. 5.21 The fundamental theorem of the integral calculus, ///(x) is continuous on the interval a < x < b and if F(x) is any function whose differential is f(x}dx then (1)

If the differential of F(x) isf(x)dx then, by the definition of a differential, §4.1, F'(x) =/(*). If

then, by §5.20, G'(x) = /(x). Hence F(x) and G(x) have the same derivative and it follows from §4.4 that F(x) = G(x) + C, C a constant. By §5.20 G (a) = 0. Hence C = F (a) and

70

THE DEFINITE INTEGRAL

When x = b this gives (1), the desired result. It is to be emphasized that in (1) F(x) is any function whose differential is f(x)dx.

EXERCISE 5(b) 1. If f(x) is bounded on an interval [a, b] and has only a finite number of discontinuities on [a, b] show that the definite integral of f(x} over [a, b] exists. 2. If f(x) is continuous on [a, b] and f(x) > 0, a < x < b show that Hf(x}dx > 0. 3. Let f(x) be defined on the interval [0, 1] by the rules: f(x) = 1, x rational (x = p/q, q ^ 0, p < q) ; f(x) = 0, x irrational. Show that f(x) is discontinuous at each point of [0, 1] and that the integral of f(x) over [0, 1] does not exist. 4. Let a function/(x) be defined on [0, 1] by the rules :f(x) = l/ s when x is rational (x = p/q, q ^ 0, p < q); f(x) = 0 when x is irrational. Show that f(x) is continuous at the irrational points on [0, 1]. Also show that $lf(x)dx does not exist. 5. Let the functions f(x) and 0(x) be continuous on an interval [a, b]. Show that f+, f are continuous at each point Xa in [a, b], and that//0 is continuous at Xo if «^(aco) ^ 0.

CHAPTER VI

THE TRANSCENDENTAL FUNCTIONS 6.1 Transcendental functions. To this point we have been dealing with polynomials which are expressions of the form acos v as Az>—>0, which implies that the derivative of sin v with respect to v is cos v. Then, by Formula VIII, there is obtained IX

A;(sin v) = (cos v) Dxv.

To get the derivative of cost* write cosv = sin Or/2 — v), then set TT/2-V

= U.

74

THE TRANSCENDENTAL FUNCTIONS

Again using formula VIII, we get X

Dx(cosv) = ( — smv)Dxv.

If the remaining trigonometric functions are each expressed in terms of sin v and cos v it is possible to use formulas from among those numbered I to X inclusive to obtain XI Dx(tan ») = (sec2 v) Dxv. XII Dx(cotv) = - (cosec2 v) Dxv. XIII .Dj^seczO = (sec v tan v} Dzv. XIV Dx(cosecv) = (—cosec v cot») Dxv. We now give some examples illustrating the use of the formulas. Example 6.1 If y = 4 cos (*3+3) find Dxy. In this example, a;3+3 = v. By formula X Dxy = -4sin(x s +3)(3x 2 ) = -12a;2 sin (*»+3). Example 6.2 Find Dxy if In this case we have a function raised to a power,

Hence the derivative will be the index of the power times the function to the index of the power minus one, all multiplied by the derivative of the function.

You will wonder perhaps where the factor | comes from. Tofindout set x/3 = v. Then

EXERCISE

75

EXERCISE 6(b) Differentiate the following functions: 1. sin* + cosx.

Ans. cos x — sin x.

2. sin Sx.

Ans. 3 cos 3#.

3.

Ans.

4.

Ans.

5. tan Sx.

Ans. 3 sec2 Sx.

6.

Ans.

7. sec x -\- cosec x.

Ans.

8. sec 2x cosec 2x.

Ans. 2(sec22x — cosec22^)

9. sin(4i+l).

Ans. 4cos(4/+l).

10. 11. 12.

Ans.

13.

Ans.

14. tan4*.

Ans. 4 sec2x tan3*.

15. \ sin 2x — x cos 2x.

Ans. 2x sin 2x.

16. £ x 2 —x sin x — cos x.

Ans.

17. 13 tan x + 12 sec x — Qx.

Ans. (2 sec x + 3 tan x) 2 .

In the following problems find Dzy:

18. sin xy = x + y.

Ans.

19. sec je + cosec y = xy.

Ans.

20. x tan xy = 1.

Ans.

76

THE TRANSCENDENTAL FUNCTIONS

6.5 The exponentialfunction f(x) = ax. If a is a real number greater than unity, then the function y = ax is never negative. For if x is positive then y is obviously positive, and y increases as x increases. If x is negative, x = — M, M>0, then y = cTu = I/a" which again is positive. Furthermore as x decreases, u increases and y = I/a"—»0. When x = O, y = a° = 1. Also as x—> oo, y—» °o. From these considerations it follows that the graph of y — ax appears as in Figure 6.3. It is apparent from the graph that y is a function with domain of definition — œ < x < °° and range 0 < y < °°.

Fig. 6.3

6.6 The logarithmic function. If y = loga x then by definition x = ay and the curve representing this relation is obtained by interchanging

Fig. 6.4

6.7 THE FUNCTION (1 + 1/x)1

77

1

x and y for the curve y = a , as in Figure 6.4. An examination of the graph, or of the relation x = ay, shows that there is no logarithm for a negative number. It is seen from the graph that y is a function with domain 0 < x < =° and range — °° < y < ». 6.7 The function (l + l/x)*. To obtain formulas for the derivatives of the exponential and logarithmic functions it is necessary to study the behaviour of the expression

as \x\ increases without limit. First let x—»«> through integer values 1, 2, ... re If x takes on the values 2, 10, 100, 1000, (1+ !/«)" takes on the values 2.2500, 2.5937, 2.7048, 2.7169. This seems to indicate that the value of (1+ !/»)" increases as n increases. If this is so, does it increase indefinitely, or does it tend to a limit? As a matter of fact, this expression does tend to a limit which, correct to four decimal places, is 2.7183. It will now be shown that the limit exists. 6.8 The limit of the function (1 + 1/x)1 as |a;|—»». For « a positive integer, (1)

By the binomial theorem

Hence

This relation, used in (1), gives

from which it follows that

Thus (l + l/w)"+1 is the nth term of a sequence Si, Si, . . . which is such that 5B0 for all values of n. Hence, by §5.6, Sn tends to a limit. Call this limit e, and write

78

THE TRANSCENDENTAL FUNCTIONS

fí

As n increases the numerator on the right tends to e and the denominator tends to unity. Hence the left side tends to e. Thus it has been shown that (1+ !/«)" tends to a limit as n increases through integer values. This limit is denoted by e. The next step is to show that

where x takes on all values. Note first that if x> 1 there is an integer n such that M < x < w + l . It is then easily verified that

This can be written as follows

As x increases n increases. But as n increases the second factor in each of the left hand and right hand members of this inequality tends to unity and the first factor in the left hand member and in the right hand member tends to e. Hence the middle member must tend to e as x increases. This disposes of the case in which x increases without limit. What happens when x decreases without limit? For x negative and decreasing set x = — t. Then t is positive and increasing and

As t increases the first factor in the right hand member tends to e and the second factor tends to unity. But as t increases x—» — œ. As noted above the value of e to three decimal places is 2.718. In chapter xn it will be shown how to find the value of e to any required number of decimal places.

6.9 The derivatives of the logarithmic and exponential functions.

We first obtain D,y where y = logav, a > 1.

6.9 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS

79

Here again we have on the right a ratio whose numerator and denominator both tend to zero, and it is necessary to make use of the special properties of the function Iog0z; to determine the behaviour of this ratio. If we note that log A — log B = log (A/B), p log A = log Av we may write

Now set v/Av = t. Then

Since v is fixed and v > 0 (logav has no meaning when v < 0) we see from the relation / = v/¿a> that \t\ —» °o as Av tends to 0. Hence as Av approaches zero (1 + l/t)' tends to e, and Iog0 (1 + 1/0* tends to logae. Consequently

In this way the number e intrudes itself into mathematics. The only restriction on the number a is that a > 1. Hence the number e = 2.7183 + qualifies as one value of a. This choice of a gives the simplest possible form for the derivative. If it is agreed that \ogev = Inv, then we have XV

We may now use Formula XV to determine the derivative of the exponential function This gives

Thus D~av = av In a, and consequently XVI

Dza" = avDxv In a.

Example 6.3 Find Dxy if y

80

THE TRANSCENDENTAL FUNCTIONS A simpler way of doing this example is to write

Then

Example 64 Find Dxy if y = 5COS x\ By Formula XVI Dxy = 5COS x'(~2x sin x2) In 5 = -2 In 5(x5cos *'sin x 2 ). Example 6.5 Find Dxy if y = (¿c2-*)21. In this example we have a function raised to a variable power. It can best be handled by taking the natural logarithm of both sides of the equation. In y = 2xln(xí-x),

EXERCISE 6(c) Find Dxy if 1. y = x In x— 1.

Ans.

2. y = ex°.

Ans.

3. y = In cot x.

Ans.

4.

Ans.

5. y - e2l-l, and for x< — 1, y is not defined; but to each x on the range — lw as n—> =°. The limit of the left side is what we have agreed to call e*". Hence eto = cos M + i sin w. For the general case we have

The first factor on the right tends to ec. The methods used above can easily be adapted to show that

Thus our definition of e* has led us to the convenient expression

EXERCISE 6(h) 1. Show that e

to

= cos u — i sin co.

2. If Zi and z¡ are two complex numbers show that eV. = e *.+*._

6.14 Anti-differentials involving involving transcendental transcendental functions. functions.

The

important formulas for anti-differentials which are not the direct inverses of differential formulas are now listed. They can be verified by showing that the differential of the right side is the expression under the anti-differential sign on the left. Later there will be given a systematic development of these and other anti-differential formulas. I

II

6.14 ANTI-DIFFERENTIALS

89

III IV V VI VII VIII

IX X XI XII

For convenient reference these formulas should be copied on a suitable card. Example 6.7 Evaluate

This example suggests Formula XII. Accordingly, set \/5 x = v, which gives dx = dv/\/5. Then

90

THE TRANSCENDENTAL FUNCTIONS Example 6.8 Evaluate

Notice that cos x dx is the differential of sin x and set sin x = v. This gives cos x dx = dv and

Example 6.9 Evaluate J sin3 2x dx. Set 2x — v. Then dx = dv/2 and

Example 6.10 Evaluate Jcos2 3xdx. Set Sx = v. Then dx = dv/3. Then

EXERCISE

91

EXERCISE 6(i) 1.

14.

27.

2.

15.

28.

3.

16.

29.

4.

17.

30.

5.

18.

31.

6.

19.

32.

7.

20.

. 33.

8.

21.

34.

9.

22.

35.

10.

23.

36.

11.

24.

37.

12.

25.

38.

13.

26.

39. 40.

41.

Use the methods of Examples 6.9 and 6.10 to evaluate (a)

(c)

(b)

(d)

92

THE TRANSCENDENTAL FUNCTIONS ANSWERS FOR EXERCISE 6(i)

1.

20.

2.

21.

3.

22.

4.

23.

5.

24.

6.

25.

7.

26.

8.

27.

9.

28.

10.

29.

11.

30.

12.

31.

13.

32.

14.

33.

15.

34.

16.

35.

17.

36.

18.

37.

19.

38.

EXERCISE

93

42. By inspection or by suitable substitutions find (a)

(b)

(c)

43. Find the area between the graph of y = sin x and the part of the x-axis between x = 0 and x = TT. 44. Show that

45. Find the area bounded by the curve of y — cos x the x-axis, x = 0, x = jr. Note that the answer is not

46. Find the area bounded by the graph of y — tan x, the x-axis and x = 7T/4. Ans. In V2.

CHAPTER VII

RELATED RATES, MAXIMA AND MINIMA, CURVE PLOTTING 7.1 Related rates. In the examples and problems of chapter i we considered functions of the variable t, s = /(/), where s is the distance of a moving point from a fixed point and / is time measured from t = 0. In all of these problems the limit of as/'àt as Ai—>0 turned out to be positive for ¿>0, which were the only values of / considered. If the law of motion is s = 1 + 1/t then

which is negative for all values of /. Speed is understood to be the rate at which a moving body is changing position relative to a change in time and is, therefore, positive or zero. Hence when s = l + 1/í the limit of the ratio As/A/ cannot be interpreted directly as speed. To resolve this problem we go, as in chapter II, to a study of the general functional relation y = f(x). 7.2 Increasing and decreasing functions. // y = f(x) is such that Dzy = f ' ( x ) is positive when x = Xo, then f(x) increases as x increases through XQ. If f ' ( x ) is negative for x = XD thenf(x) decreases as x increases through XQ. Let/'(*o) = ¿>0. Then {/(x0+Ax)-/(x0)}/Ax tends to d>0 as Ax-»0. Hence for Ax—»0 and sufficiently close to zero, Ax > 0,

If Ax0. Then, since Ax is negative,/(x 0 +Ax)—/(xo) is negative from which it follows that f(xo) >/(xo+Ax). Hence for any two points Xi, x2 sufficiently close to Xo with Xi0 is negative or zero. But the limit of this ratio as Ax—->0 is f'(x0. Since f'(x0) cannot be both greater than zero and at the same time less than or equal to zero it follows that /'(*) = 0. That /'(^o) = 0 when xc makes f(x) a minimum follows by similar reasoning. The significance of this theorem is that the values of x which make /(x) a maximum or a minimum are among those values which make/'(x) = 0. Such values of x are called critical values of x. More than this we cannot say. The fact that/'(xo) = 0 does not insure that x0 makes/(x) either a maximum or a minimum. For example /(x) = x3 has a derivative at every point of any open interval containing the origin and f'(x) = 3x2 = 0 when x = 0. However, it is obvious that x = 0 makes /(x) neither a maximum nor a minimum. It will now be shown that a function can have a maximum value at a point x0 on an open interval without even having a derivative at x0 and consequently y =/(*) tends to the line y = x. This suggests that the line y = x be drawn in the diagram. The relation of y = f(x) = x + l/(x — 1) to the line y = x is also described by saying that y/x —> 1 as \x\ —» axis. For the area of the present diagram this leads to complications in determining the limits of integration. It seems advisable to clear this point by means of an example. Example 8.6 Determine the centroid of the area bounded by the graph of y = 4x —x3, and the segment of the #-axis between x = 0 and x = 2.

Fig. 8.7

For the ^-coordinate,

where l(x) is the length of the rectangle PQ, l(x) = y = 4x—x3, and

If the same general idea is used to determine y then

where l(y) is the length of the rectangle BR, l(y)=Xî—Xi.

EXERCISE

131

This means that x¡ and Xi must be determined in terms of y from the equation It is not practicable to get x in terms of y from this equation. We return to the rectangle PQ, determine its moment about the line y = y and take this as the element of moment. By §8.10 the centroid of the rectangle PQ is at its centre which is marked c in the diagram. By §8.11 where dA is the area of rectangle PQ, dA = (4x — x3)dx = ydx. Also if ye is the ordinate of the point c then yc = y/2. Hence

where y = 4x—x3. If the line y = y is so placed that the moment of the area about it is zero then

EXERCISE 8(e) Find the position of the centroids of the following uniform plates. The answers given depend on the choice of coordinate axes. 1. An equilateral triangle with side a units. Ans. a/2 V3 from base. 2. Any triangle. Ans. Intersection of medians. 3. A semicircle of radius a. Ans. áa/Sir. 4. A parabolic segment in which the base is at right angles to the axis, the length of the base being 26 units and the height h units. Ans. 3h/5. 5. The area cut from the parabola y2 = 4x by the line y = x. 6. The area, in the first quadrant, of the ellipse xz/a?-}-y'2/b* = 1. Ans. x = 4a/3?r, y = 46/3r. 7. The area between two equal parabolas with a common vertex, axes at right angles, and latus rectum of 4a. Ans. x = y = 9a/5.

132

DISPLACEMENT, VELOCITY AND ACCELERATION

8. The area bounded by y = x3, y = 0 and x — 2. Ans. x = 8/5, 9 = 16/7. 9. The area bounded by the x-axis and the curve y = 16 — x*. Ans.x = 0,y = 64/9. Find the position of the centroids of the following solid figures of uniform density. 10. A pyramid of height h units on a square base of side a units. Ans. h/'4. 11. A segment of a paraboloid of revolution in which the base is at right angles to the axis. (Height h units, base radius a units.) Ans. z = 2h/3, x = y = 0. 12. A hemisphere of radius a units. Ans. x = 3o/8, y = z = 0. 13. The solid bounded by one half the ellipsoid xí/aí+yí/bí+zí/cí = 1 and the yz-plane. Ans. x = 3o/8, y = z = 0. 14. The solid formed when the area above the x-axis bounded by the parabola y2 = x = z = 0. 18. One half of a conoid which has a circular base of radius a units and height h units. The conoid is bisected by a plane at right angles to its straight edge. Ans. x = 4a/3ir, y = 0, z = h/3. 19. Prove that the necessary and sufficient condition that a triangle inscribed in an ellipse have maximum area is that its centroid coincide with the centre of the ellipse (PC, 1952). 20. (First theorem of Pappus.) If an area A is revolved about a line L which does not intersect the area (it may be tangent to the area) then the volume generated is 2irlA where / is the distance of the centroid of the area from the line L. Let the x-axis coincide with the line L. Let l(y) be a section of the area perpendicular to the y-axis. Then

Use the first theorem of Pappus to find the volume generated in each of the following problems:

8.13 WORK 21. 22. 23. 24. 25. 26. 27.

133

The triangle of Problem 1 about one side of the triangle. Any triangle with base a and altitude h, revolved about its base. The semicircle of Problem 3 about its diameter. Ans. 4a?ir/3. The parabolic segment of Problem 4 about its base. Ans. 128x/5. The area in Problem 5 about the x-axis. Ans. 32x/3. The area in Problem 7 about the y-axis. Ans. 4ir2a2. The volume of a torus of diameter 6a generated by a circle of radius a.

8.13 Work. If a force of F pounds acts in the direction of a displacement of x feet the work done is Fx foot-pounds. This definition of work for a constant force leads to a unique determination of work done by a varying force. Example 8.7 Determine the work done in pumping the water from a full cylindrical cistern of radius a and depth h. Take sections of the water perpendicular to the axis of the cylinder at

Fig. 8.8

Fig. 8.9

depths Xk and x/e+j, Figure 8.8. Let w be the weight of a cubic foot of water, and let W¡¡ be the work done in lifting the water between these sections to the top of the cistern. Then and it easily follows from this that

Example 8.8 Determine the work done if the cistern in Example 8.7 is an inverted right circular cone, and the water is pumped b feet above the level of the top of the cistern.

134

DISPLACEMENT, VELOCITY AND ACCELERATION

From Figure 8.9 it is seen that where

Hence

EXERCISE 8(f) 1. A conical cistern 8 ft. across the top and 10 ft. deep is filled with water to a depth of 7 ft. Find the work done in pumping out the cistern, if the water is discharged 2 ft. above the top. Ans. 1247rw ft.-lb. 2. A hemispherical cistern 5 ft. in radius is filled with water. Find the work done in pumping it out if the water is discharged one foot above the top. Ans. 2875irw/12 ft.-lb. 3. A cistern in the form of a paraboloid of revolution 6 ft. across the top and 12 ft. deep is filled with water to a depth of 8 ft. Calculate the work done in pumping all the water to a height of one ft. above the top. Ans. 18Í1TW ft.-lb. 4. A cistern in the form of a frustum of a pyramid, 4 ft. square at the bottom, 8 ft. square at the top, and 12 ft. deep, is filled with water. Calculate the work necessary to raise the water to a height of 2 ft. above the top. Ans. 3008w ft.-lb. 5. A reservoir 10 ft. deep has a rectangular top 20 ft. by 40 ft., a rectangular bottom 10 ft. by 30 ft., and uniformly sloping sides. Calculate the work done in pumping out the water to a depth of 5 ft. if the water is discharged 6 inches above the top. Ans. lll,625w/12 ft.-lb. 6. According to Hooke's law the force required to hold an elastic rod or spring of natural length a inches to the length a-\-x inches is kx/a pounds, where k is constant. Calculate the work done in stretching the rod from length a to length b inches. Ans. k(b—a)2/2a in.-lb. 7. A gas is confined in a cylinder by a movable piston. Assuming that the pressure p pounds per sq. in. and the volume v cu. in. occupied by the gas satisfy Boyle's law, pv = k, where k is constant, find the work required to compress 100 cu. in. of air at atmospheric pressure (14.7 Ib. per sq. in.) to a volume of 10 cu. in. Ans. 1470 In 10 in.-lb. 8. Inside the earth a particle is attracted toward the centre with a force proportional to its distance from the centre. Find the work required to

8.14 HYDROSTATIC PRESSURE

135

lift a 10 Ib. body from the centre to the surface of the earth. The radius of the earth is 3960 mi. Ans. 19800 mi.-lb. 9. Determine the work done in pumping the water from a hemispherical cistern of radius 10 ft. Ans. 4166.7w ft.-lb. Also 2500irw ft.-lb. 10. The force required to hold a spring in compression is given by F = kx, where x is the distance the spring has been compressed and k is a constant which depends on the type of spring. How much work is done in compressing a given spring 6 in.? Ans. k/8 ft.-lb. 11. A bucket of water weighing 80 Ib. is raised at the rate of 4 ft. per min. The water leaks out at a constant rate so that it is half lost when the bucket has been raised 20 ft. If the weight of the bucket is neglected, how much work is done when the bucket is raised the 20 ft.? Ans. 1200 ft.-lb. 12. Determine the work done in Problem 2 if the water is raised 10 ft. above the level of the top of the bowl. Ans. 11875wir/12 ft.-lb. 13. A rectangular cistern 10 ft. deep has a square base 6 ft. on a side. The cistern is filled with water. The bottom is so arranged that it can move upward as a piston. If it moves to the top of the cistern so that all the water is spilled over the sides how much work is done? Ans. 1800w ft.-lb. 14. A cylindrical tank has its axis horizontal and is filled with oil which weighs 60 Ib. per cu. ft. The radius of the tank is 15 ft. and the length 15 ft. How much work is done if the oil is pumped 10 ft. above the level of the top of the tank? Ans. 100(15)47r ft.-lb. 8.14 Hydrostatic Pressure. Figure 8.10 shows two containers filled with water to the same depth. Also the areas of the bottoms of the two are

Fig. 8.10

the same. It is readily accepted that the water exerts pressure on the bottom of each ; but it is not easy to believe that the pressure on the bottom of each is the same. This is, nevertheless, true. The pressure depends only on the depth. If w is the weight of a cubic foot of water in pounds, h the depth of the water in feet, then the pressure on the bottom of each container is given by P = wh, pounds per square foot. This is a direct result of the fact that at any point below the surface of a liquid the pressure is the same in all directions.

136

DISPLACEMENT, VELOCITY AND ACCELERATION

If a submerged plane area is horizontal the force on the area is equal to the weight of a column of water whose base is the area and whose height is the depth of water over the area. This is no longer the exact force if the area is not horizontal. Example 8.9 A triangle is submerged in a vertical position with its vertex above the base and two feet below the surface of the water. Find the total force on the triangle if its base is 5 feet and its altitude is 4 feet. The procedure here is that of dividing the area of the triangle into narrow strips parallel to the base. If AF is the force on the strip shown in Figure 8.11 and A^4 is the area of this strip then

whA < F w(h+dh) A. This leads to an element of force

dF = whldh, where / is the length and dh the width of a rectangle based on the section at depth h. Also Fig. 8 11

Hence

In this example,

If A is the depth of the centroid of the triangle then

Hence if an area and h, the depth of its centroid, are known the force can be computed from the relation

F = whA.

EXERCISE

137

Now consider the turning moment of the force on this triangle about the base line.

At what point above the base would the total force on the triangle be applied to produce this moment? The total force is 140w/3 Ib. Let d be the distance from the base line at which this force is applied to give the same moment. Then

This point, at a distance of 8/7 ft. above the base, is called the centre of pressure of the triangle. EXERCISE 8(g) Calculate the force due to the pressure of the liquid (which is water unless otherwise specified) on one side of the surfaces described below. Water weighs w Ib. per cu. ft., w = 62.5. Also find the depth of the centre of pressure in 1, 2, 3, 4, 5, 7. 1. A rectangle with sides a and b units submerged vertically in water with side of a units in the surface. Ans. wab2/2 Ib.; 20/3 ft. 2. Each half of this rectangle, formed by drawing a diagonal. Ans. wab*/6, wab*/3; b/2, 3b/4. 3. The same rectangle submerged so that its top is c feet below the surface. Ans. wab(c+b/2) Ib.; 2(62+3oc+3c2)/3(&+2c) ft. 4. A trapezoid with depth 40 ft., length in the surface 200 ft. and length at the bottom 160 ft. Ans. 416000w/3 Ib.; 340/13 ft. 5. A semicircle with radius a units submerged vertically with bounding diameter in the surface. Ans. 2wa3/3 Ib.; 3ira/16 ft. 6. A circular bulkhead in a water-main in which the radius is 2 ft. and the centre is 50 ft. below the level of water in the standpipe. Ans. 2007TTO Ib. 7. The end of a parabolic trough in which the depth is 3 ft. and the width across the top is 4 ft., when the trough is full of water. Ans. 600 Ib.; 12/7 ft. 8. A rectangle of sides a and b units, which has an edge of b units in the surface and which is inclined at 30° to the vertical. Ans. wa?b\/3/4 Ib.; 2a/3 ft.

138

DISPLACEMENT, VELOCITY AND ACCELERATION

9. A rectangular tank is half filled with water and above this is oil. If the oil is half as heavy as water, show that the force on the sides is one fourth greater than it would be if the tank were filled with oil. 10. A tank in the form of a frustum of a cone is 5 ft. deep and 8 ft. across the top. The vertical angle of the cone is 60°. Find the force against the conical surface of the frustum, when it is filled with water. Ans.

11. A semi-circular plate with radius 5 ft. is submerged with its plane vertical and its diameter in the surface of the water. Find the force on one face of the plate. Ans. 83.3w Ib. 12. An ellipse with major axis 10 ft. and minor axis 8 ft. is submerged with its minor axis vertical and the upper end of the minor axis 12 ft. below the surface of the water. What is the force on the ellipse? Ans. 32Û7TW Ib. 13. A vertical floodgate is a trapezoid with parallel sides 8 ft. and 4 ft., and width 4 ft. The short side is upward and 12 ft. below the surface of the water. Find the total force on the gate. Ans. 341.3w. 14. A plate in the shape of a segment of a parabola has a base of 4 ft. and an altitude of 5 ft. It is submerged in a vertical position with vertex downward and base in the surface of the water. Find the force on the plate. Ans. 80w/3 Ib. 15. The end of a large reservoir is a vertical plane surface in the shape of a trapezoid 40 ft. wide at the top, 30 ft. wide at the bottom, and 12 ft. deep. The reservoir is filled with water weighing 62.5 Ib. per cu. ft. Find the force due to the pressure of the water against the end of the reservoir. Ans. 150,000 Ib.

CHAPTER IX

FURTHER METHODS OF DETERMINING ANTI-DIFFERENTIALS 9.1 The anti-differential of udv. This involves two functions, u(x) and v(x). The formula for the differential of a product gives

Take the anti-differential of both sides of this last equation.

Thus we have Formula (A)

It often happens that an anti-differential of udv cannot be determined by known methods while that of vdu is easily resolved. Example 9.1 Determine / xexdx. If we set dv = exdx, u = x, then v = ex, du = dx and

Note that in determining v from dv = e*dx we have v = e* + C. We took C = 0. This is the usual practice in the use of Formula (A). It is easily verified that the result is the same if C is a number other than zero. Example 9.2 Determine J xV* dx.

Apply Formula (A) to the anti-differential on the right. 139

140

DETERMINING ANTI-DIFFERENTIALS

Finally,

It is to be noted that some discretion must be exercised in the choice of u and dii. In Example 9.1, why not set u = exl Then and

This is a correct result but it gets us nowhere. The anti-differential on the right is more complicated than the original. Example 9.3 Determine f eax sin bx dx.

Repeat the operation on the anti-differential on the right to get

EXERCISE

141

Example 9.4 Determine f sec3 BdB.

EXERCISE 9 (a) Determine the following anti-differentials: 1.

Ans.

2.

Ans.

3.

Ans.

4.

Ans.

5.

Ans.

6.

Ans.

7.

Ans.

8.

Ans.

9.

Ans.

10.

Ans.

Use formulas such as 2 sin A sin B = cos (^4 —B) —cos (A +-B) to evalu ate Problem 8 and also the following: 11.

12.

142

DETERMINING ANTI-DIFFERENTIALS

9.2 Anti-differentials containing Trigonometric substitution.

trigonometric

functions.

Example 9.5 Determine J tan x dx. This is Formula II, which we have already verified and used.

Now note that under the last anti-differential sign we have the differential of cos x divided by cos x, which is the differential of In cos x. Hence

Similarly,

Example 9.6 Determine / sec x dx.

Similarly,

Example 9.7 Determine

Figure 9.1 shows a right-angled triangle, x and a being the sides about the right angle. Then

Fig. 9.1

9.2 TRIGONOMETRIC FUNCTIONS

143

Examples 9.5-9.6 inclusive give direct methods of proofs of Formulas II-V. The method of Example 9.7 can be used to evaluate a wide class of anti-differentials which involve factors of the form n an odd integer. This method is known as trigonometric substitution. Example 9.8 Evaluate

Fig. 9.2 In Figure 9.2,

EXERCISE 9(b) Prove Formulas VI-XII inclusive by using trigonometric substitution, as in Examples 9.7 and 9.8, and the preceding Formulas I-V. Example 9.9 Determine / sin6 x dx.

144

DETERMINING ANTI-DIFFERENTIALS

This method can be used to find the anti-differential of any odd power of the sine or cosine. For example Example 9.10 Determine J cos4 x dx.

The method of Example 9.10 may be used to find the anti-differential of any even power of the sine or cosine. For example,

The methods of Examples 9.9 and 9.10 can also be adapted to find the antidifferentials of products of sines and cosines. Example 9.11 Determine J sin2 x cos6 x dx.

9.2 TRIGONOMETRIC FUNCTIONS 6

145

3

Example 9.12 Determine J tan x sec x dx.

The device in Example 9.12 is that of expressing the function under the anti-differential sign as powers of the secant multiplied by sec x tan x dx which is the differential of the secant. Another device which works in many cases is that of turning the expression under the anti-differential sign into powers of the tangent multiplied by sec2 x dx which is the differential of the tangent. These remarks also hold with cotangent and cosecant replacing tangent and secant. Example 9.13 Determine

Set x — \ = v. Then dx = dv and

The method of Example 9.13 can be used for a wide variety of problems. It involves the process of completing the square. For example, the antidifferential of (3x2+2x)*dx is the anti-differential of

which is the anti-differential oî\/3(v'! — l/9)^dv where v = x+J, and dv = dx.

146

DETERMINING ANTI-DIFFERENTIALS

EXERCISE 9(c) Evaluate the following anti-differentials and check your results by differentiating the answers.

1.

9.

2.

10.

3.

11.

4.

12.

5.

13.

6.

14.

7.

15.

8.

16.

EXERCISE 9(d) Use the methods developed in chapter ix and the Formulas I-XII of chapter VI to evaluate the following anti-differentials. Check by differentiating the answers. 1.

8.

2.

9.

3.

10.

4.

11.

5.

12.

6.

13.

7.

14.

9.3 IMPROPER INTEGRALS 15.

28.

16.

29.

17.

30.

18.

31.

19. 20. 21.

147

32. 33. 34. 35.

22. 23. 24. 25. 26.

27.

36. 37. 38. 39. 40.

41. Use the method of Examples 9.7 and 9.8 to obtain formulas XIV XV

9.3 Improper integrals. Let/(x) be a function defined on a < x < b. Then the integral of f(x) over [a, b] in the sense of the definition of §5.10 does not exist. In that definition it is required that there be a number M > 0 such that \f(x~)\ < M, a < x < b. Suppose the conditions of §5.10 are satisfied for every interval [a, h], a < h < b. Then

exists. If this integral tends to a limit as h —» b, this limit is the improper integral of f(x) over [a, b]. Example 9.14 Find

Example 9.IS Find

EXERCISE 9(e) 1. Show that 2. Show that 3. Evaluate

(a)

(c)

(b)

(d)

9.4 RATIONAL FUNCTIONS

149

4. Show that

does not exist. 5. Use the fact that for x> 1, 0< e~x' < e~x, and Problem 1 to show that

exists. 6. Let f(x) be integrable on every interval (a, b) and let f ( x ) = a) is revolved about the x-axis to form a torus. Use the first theorem of Pappus to find the volume of the torus, and use the second theorem of Pappus to find the area of the torus.

170

MEAN VALUE THEOREM, PARAMETRIC EQUATIONS

10.10 Differential equations of the first and second order. Problems in arc length and curvature often give rise to differential equations involving first and second derivatives. One such problem is that of determining the curve which passes through the point (0,1) with zero slope, and is such that the reciprocal of its curvature at any point is equal to the derivative of arc length at that point. If 5 denotes arc length, then by §10.8

By the formula for curvature of §10.6, the condition of the problem is

Which gives

This is a second order differential equation because the highest order derivative it contains is the second. Furthermore, this equation is such that both the variables x and y are absent. There is a procedure for solving equations of this kind which we now state. If in a second order differential equation one of the variables x, y is absent, or both are absent, set If y is absent, and the second order equation becomes a first order equation in p, x, dp/dxIf x is absent but y not absent use the relation

The second order equation then becomes a first order equation in p, y, dp/dy. Example 10.8 Solve the differential equation of §10.10,

10.10 DIFFERENTIAL EQUATIONS

171

According to the conditions of the problem, when x = 0 the slope is zero. Hence

When x = O, y = 1, cos x = 1. Consequently c = 1 and y = —In cos x + 1 = In sec x + 1. It is easily verified that this function satisfies the differential equation of Example 10.8. Example 10.9 Find the curve that passes through the point (0, 1) with slope equal to zero and is such that at any point x on the curve the reciprocal of the curvature is the ordinate times the derivative of arc length. This leads to the equation

Let dy/dx = p. Then

The equation then becomes

When y = 1, p = 0, c-\ = 0, c2 = 1 and

so that Thus

172

MEAN VALUE THEOREM, PARAMETRIC EQUATIONS

It is easily verified that this function satisfies the requirements of Example 10.9. EXERCISE 10(d) Solve the following differential equations: 1.

Ans.

2.

Ans.

Use the method of §8.4 to solve the resulting equation in p = dy/dx. 3.

Ans. y = In(Sex x+tan x). Ans. y = X1, or y2 = c2x + c3.

4.

Ans.

5. 6.

Ans. 7. 8.

Ans.

9. Find the equation of the curve along whose normal the centre of curvature and the x-axis are equidistant from the curve. 10. Find the equation of the curve that goes through the point (0, 3) with the slope zero, and whose curvature is always equal to the sine of its angle of inclination. 11. Determine the curve which goes through the point (0, 3) with slope 0 and for which the curvature is constant. Ans. xí-\-(y — c)2 = (3 —c) 2 . 12. Determine the curve which is tangent to the x-axis at the origin and for which the length of arc from a fixed point P0 to a variable point P is proportional to the slope at P. Ans. 2(y + k) = k(ex/*+e~x/*). 13. Determine the curve for which the curvature at any point is equal to the slope of the normal. Ans.

14. Determine the curve which at the point (0, 2) is tangent to the line

10.11 THE CATENARY

173

y = 2 and for which the radius of curvature at any point is equal to one half the square of the ordinate of the point. Ans. y = é*'"+e-x/a. 15. Show that the equation

characterizes a conic touching the four sides of a fixed square (PC, 1952). 16. Find all solutions of the differential equation yy" — 2(y')2 = 0 which pass through x = 1, y = 1 (PC, 1938). 10.11 The catenary. If the ends of a flexible chain are fixed at two points the curve formed by the hanging chain is called a catenary. Let A, Figure 10.5, be the lowest point of the curve, P(x, y) a point on the right hand branch, 0 the slope angle of the tangent at P. Take the y-axis through A. The forces on the segment AP are the weight of the segment and the tensions TO and T at A and P respectively. Since these forces are in equilibrium the horizontal forces must balance. Hence Also the vertical forces must balance. Consequently

Fig. 10.5

where 5 is the length of the section AP, and w is the weight of a unit length of chain. Hence

We thus get the differential equation

where a = TQ/W. If dy/dx = p this becomes

When x = 0, p = 0. Hence ec = 1

174

MEAN FALUE THEOREM, PARAMETRIC EQUATIONS Hence or

We did not indicate the position of the x-axis with reference to the

Curve in the diagram. Ta the x-axis so that y = a when x = 0. Then C = 0, and the equation of the catenary is

10.12 Hyperbolic functions.

The

function

Fig. 10.6

which represents the catenary is a member of a class of functions called hyperbolic functions. They are so named because they are associated with the hyperbola x2—y1 = 1 in the following way. Let u be the area of the sector OPQP', Figure 10.6. Then

It then follows that whence Then

Thus x and y, the coordinates of P, are expressed in terms of the area u. These functions of u are known respectively as the hyperbolic cosine and hyperbolic sine of u, cosh u and sinh u.

EXERCISE

175

The hyperbolic tangent is defined by the relation The functions cosech u, sech u, cotanh u are defined as the reciprocals of sinh u, cosh u, tanh u respectively. If x = cosh M then It was shown above that u = ln(x-\-^/x2— 1). Then by X, p. 89,

Hence

Similarly

We thus have alternatives for the anti-differential Formulas X, p. 89. EXERCISE 10(e) 1. Derive the formulas cosh2 A — s i n h 2 . 4 = 1 . 1 — tan2 A = sech2 A. coth2 A — 1 = cosech2 A. 2. Derive the formulas sinh(^4±5) = sinh A cosh B ± cosh A sinh B. cosh (A ±B) = cosh A cosh B ± sinh A sinh B. sinh 2A = 2 sinh A cosh A,

cosh 2A = cosh2 ^4+sinh 2 ^4.

3. Show that — sinh u = (cosh M) — and derive formulas for the dx dx derivatives of the remaining hyperbolic functions, 4. Obtain the following formulas.

176

MEAN VALUE THEOREM, PARAMETRIC EQUATIONS

5. Draw the curves y = sinh x, y = cosh x and from a study of these curves determine the values of u for which the formulas in Problem 4 are valid. 6. Replace the hyperbola x2—y1 = 1 in §10.12 by the circle x2-\-y2 = 1 and show that if u is the area of the circular sector OPP' then x = cos u, y = sin u. 10.13 Linear differential equations of the second order with constant coefficients. These are equations of the form (1)

where a, b and c are constants. These equations are said to be of the second order because the derivative of highest order is the second. They are said to be linear because the derivatives and the dependent variable are to a degree no higher than the first. We shall seek what is called the general solution of equations of this type. We have seen that the general solution of a first order differential equation unavoidably contains one arbitrary constant. In much the same way, and for the same reason, a general solution of a second order equation contains two arbitrary constants. A simple example is Then which is the general solution of the equation. It is called the general solution because all possible solutions of the equation can be obtained by assigning suitable values to A and B. There are equations for which the so-called general solution does not contain every possible solution. These are exceptional cases which will not be considered. We begin this study by considering some particular examples. Example 10.10 Solve the differential equation (2)

One way of seeking a solution is that of asking ourselves what is the function whose second derivative is co2 times the function? We recall immediately that the first and second derivatives of e? are e?, and this leads us to the answer y = e"x. A more general function is y = Ae"x, A any constant. Also if B is any constant y = Bé~"x satisfies (2) and, finally, the function (3) y = Ae^+Bë-"* satisfies (2). It turns out that every possible solution of (2) is contained in

10.13

LINEAR DIFFERENTIAL EQUATIONS

177

(3) for a proper choice of the constants A, B. (3) is, therefore, the general solution of (2). We have obtained (3) by the guess-and-try method. Is there a more direct approach? Let us ask if there is a function of the form y = emx which satisfies (2)? Then and (2) becomes This is satisfied if This gives two functions y = e"x, y = e

ux

which satisfy (2). The function

x

y = Ae" +Be-"x

(3)

also satisfies (2) and is the general solution. Example 10.11 Solve the differential equation (4)

This suggests that we seek functions whose second derivative is -co2 times the function. In due time we discover two functions, y = sin co.v and y = cos owe, both of which satisfy this requirement and consequently satisfy (4). Similarly, (5) y = A sin wx-\-B cos ux satisfies (4) and is the general solution of (4). The constants A and B may take on complex values. What happens if we use the general procedure suggested in Example 10.10, that of assuming y = emx is a solution? Then

and (4) becomes miemx-\- l/2-\/x as Ax —» 0, nor that (sin x)/x —> 1 as x —» 0. But the evidence came from special devices rigged to suit these particular problems. We shall now set up some general methods for determining limits of the types we have already met and many other types. 10.15 1'Hospital's rule. Let the domain of definition of the functions / and g be a < x < c, c < x < b, a < c < b. At each point of the domain of definition let f(x) and g(x) be continuous, let/' and g' exist, and for x near c let g(x) and g'(x) be different from zero. We note that under the conditions stated neither / nor g is defined at x = c. Hence neither/' nor g' exists at x = c. There are several cases to consider. First let/(r) and g(x) tend to zero as x —> c. In this case if

Let e > 0 be given. Fix 5 > 0 such that if x is on the interval (c — d, c+5), x j¿ c, then g(x), ¿(x) are both different from zero and (1)

Fix a point x ^ c on (c — o, c+5) and let x' be a point between x and c and sufficiently near c to ensure that g(x) —g(x') ^ 0. Then for the interval determined by the points x and x' relation (1) of the mean value theorem, §10.3, is valid. Consequently, since g(x)—g(x') j6- 0,

10.15 L'HospiTAL's RULE (2)

/(x) -/(>') = /gl g(*)-g(*) g (£) and it follows from (1) above that

185

^ between * and *',

(3)

Relation (3) holds no matter how near is x' to c, £ in (2) changing with x' , but remaining on (c — S, c+S), £ ^ c. Then because f(x') and g(x') tend to zero as x' —> 0 we can conclude that (4)

Hence, since e is arbitrary and x was any point on (c—o, c+S), by the second definition of limit in §1.3, it follows that

Example 10.16

For the next case f(x) and g(x) both increase without limit as x —» c. As in the first case it is assumed that/'(*)/g'(^) —* L. Let e > 0 be given and take «i > 0 and such that ei2 + ei(Z,+ l) < e. Then fix ô > 0 so that if x is on (c — o, c+S) then g(x) and g'(#) are both different from zero and (5)

Now fix KO on (c — S, c+5). Then fix 5i > 0 with Si < 5 and so that if x is on (c — oi, c+Si) and between xa and c, x ^ c, then g(x)— g(x0) > 0 and (6)

That (6) is possible follows from the fact that f(x) and g(x) increase indefinitely as x —* c. Again by the mean value theorem we have

186

MEAN VALUE THEOREM, PARAMETRIC EQUATIONS

£ between KO and x. Then

and it follows from (5) and (6) that

Hence for \x — c\ < Si, x 7* c,

Because t is arbitrary it again follows from the second definition of §1.3 that

Example 10.17 Find

and both numerator and denominator of the ratio in brackets tends to co as x —» 0, x > 0. Hence

Example 10.18 Find

if the limit on the right exists. We note that sin 2x and x itself both tend to zero. Hence we cannot at once say whether or not the limit on the right exists. But the rule we are using is applicable to this ratio (sin 2x)/x, and

Hence

EXERCISE

187

Finally consider the case for which both f(x) and g(x) become infinite as x —* oo . It is assumed that at all points x greater than some fixed point Xo the functions/ and g are continuous, /' and g' exist and g'(x) 7^ 0. It is also assumed that f ( x ) / g ' ( x ) —> L as x —» °o. Fix Xi > XQ such that for x > xi (7)

Now fix Xí > x\ such that for x > #2, g(x)—g(xi) > 0 and (8)

Then for x > xt, by relation (1) of §10.3

As in the previous case this leads to

and finally to

It can therefore be said that to an arbitrary e > 0 there corresponds xe for which \f (x)/g(x) — L\ < e, x > x,, which is the precise formulation of the statement that the limit as x —> œ of f(x)/g(x) is L. Example 10.19

EXERCISE 10(g) Determine the limits indicated in the problems listed below. 1.

Ans. 32.

4.

Ans. 1.

2.

Ans. 1.

5.

Ans. 3.

3.

Ans. 2.

6.

Ans. 0.

188

MEAN VALUE THEOREM, PARAMETRIC EQUATIONS

7.

Ans. 5.

16.

8.

Ans. 0.

17.

Ans. 0.

9.

Ans. 0.

18.

Ans. 0.

10.

Ans. 1.

19.

.¿ws. 2.

11.

Ans. 2.

20.

12.

13.

21.

Ans.

14. 15.

Ans.

.4ws. 1.

22. 23.

Ans.

24.

Ans. 1.

25.

CHAPTER XI

FUNDAMENTAL THEOREMS 11.1 Introduction. In §0.9 we agreed to assume the real number system with all its properties and to assume the operations that have come to be associated with this system. In chapter v, §§5.5, and 5.6 it became necessary to make further assumptions concerning properties of continuous functions and the behaviour of sequences of numbers, before proceeding to the development of the definite integral. In this chapter, among other things, there are proofs of these assumptions based on those of §0.9. Why go to the trouble of working out proofs? Why can we not continue the study of mathematics using assumptions of §§5.5, 5.6 as we have to this point? The answer to the latter question is that we could. There are two things to be said in regard to the former. One is that in building any system of mathematics it is aesthetically desirable to make the basic assumptions few and simple. The other is that while from a practical point of view the proofs of the assumptions of §§5.5, 5.6 may not be important in themselves, the concepts and methods used in developing the proofs are as important as any in mathematics. Indeed without a thorough understanding of these concepts and methods, advance in the study of mathematics soon comes to an end. This brief chapter contains not only the proofs in question but also the foundation on which much of the further work in analysis depends. 11.2 Sets and their properties. Collections or assemblages of numbers are called sets of numbers, or simply sets. Thus the collection of even integers 2, 4, 6, . . . , is a set. Other examples are the collection of rational numbers between zero and unity; the numbers x for which —5 < x < — 2; all numbers of the form n\/2, n an integer. The symbol S will denote a set. That a number s is a member of 5 will be denoted by 5 € 5 which is read "s is an element of S," or "s is contained c* »» m 6. If 5 is a set, and if there is a number G such that if 5 € 5 then 5 < G the set 5 is said to be bounded above and G is called an upper bound for S. If there is a number g such that ii s £ S then s > g the set S is said to be bounded below and g is called a lower bound for 5. If a set is bounded above and below it is bounded and there is a number G such that if s Ç. S then s\ < G. Among all possible bounds for a set there are two which play a special role. Let 5 be the set of rational numbers on the open interval (0, 1). Then 189

190

FUNDAMENTAL THEOREMS

17, or 2, or any number not less than unity is an upper bound for 5. If 5 6 5 then s < 1; but if a is any number less than unity then there are numbers s (E S for which s > a. The number unity is called the supremum, or least upper bound of the set 5. The numbers —1000, —17, —112 are lower bounds for the set S. Any number which is not greater than zero is a lower bound for 5. If a is any number greater than zero there are numbers 5 Ç 5 for which s < a. The number zero is called the infimum or greatest lower bound of the set S. 11.3 The supremum and infimum of a set. Let 5 be a set of numbers. If there is a number M which is an upper bound for S and which is such that M is less than any other upper bound for 5 then M is the supremum of the set 5. If there is a number m which is a lower bound for 5 and if m is greater than any other lower bound for 5 then m is the infimum of the set 5. There is another description of the supremum and infimum of a set which is useful in applications. It features the elements of 5. The number M is the supremum of a set S if for all s (E S, s < M, and if for any positive t, no matter how near zero, there is at least one element 5 € 5 for which 5 > M—e. There is a corresponding description for the infimum of S. These special bounds for sets may or may not belong to the set. If 5 is the set of rational numbers on (0, 1) neither unity, the supremum, nor zero, the infimum, belong to the set. If S is the set of rational numbers on the half closed interval (0, 1] the supremum is unity which does belong to the set. The infimum is zero and does not belong to the set. Does every set have these special bounds? The set of rational numbers on (0, 1) has both a supremum and an infimum. Can this be said of the set consisting of numbers of the form p\/l$/q where p and q are integers and p < ql The set of positive integers does not have a supremum for there is no number M for which n < M for all elements n of the set 1,2, 3 . . . . This set is not bounded above. Does every set which is bounded above have a supremum? These questions are answered by the following theorem. 11.4 The existence of suprema and ínfima of sets. // a set of real numbers is bounded above it has a supremum; if bounded below it has an infimum. First let 5 be such that it has at least one positive element. Because the set is bounded, there is an integer k such that s < k for every element s in 5. Then there is a greatest integer no of the set 0, 1, 2, . . . , k—1 such

11.5 SEQUENCES

191

that s > MO for at least one element s of S. Likewise, there is a greatest digit MI of the set 0, 1, 2, . . . , 9 such that s > n0. HI for at least one element s oí S (note that the dot in MO . MI represents the decimal point). Again, there is a greatest digit M 2 of the set 0, 1, 2, . . . , 9 such that s > M O . MiM 2 for at least one element 5 of 5. This process can be continued indefinitely. We next show that the resulting number M = MO . M i M 2 . . . is the supremum of the set S. If s is an element of 5 then s < M. For suppose there is some 5 with s > M. Let b be a number such that M < b < s. Then b < MO+!, for otherwise MO would not be the greatest number of the set 0, 1, 2, . . . , k — 1 for which there was an element s > MO. Hence b can be written MO . 6162 • • • where each bk is one of the digits 1, 2, . . . , 9, 0. Since b > M = MO . n\n-i. . . there is a first subscript r with b, > nr. Since there is an element s > b > MO . 61. . . br, it follows that n, is not the greatest digit in the set 1, 2, . . . , 9, 0 for which there is an element 5 > MO . n\ . . . nr. We again have a contradiction, and conclude that s < M = MO • M i M 2 . . . for every s in 5. Let e > 0 be given. Since M O . MiM 2 . . . nr —> M as r —» °° , it follows that for r sufficiently great, MO . n\n-i. . . nT > M— t. But nr was so chosen that there is an element s of 5 with s > MO . n\n%. . . nr. Hence there is an element 5 of 5 with 5 > M— e. Thus the number M is the supremum of the set 5. If the set 5 contains no positive elements take K > 0 and such that for some 5 in 5 the number s-\-K is positive. Then the set whose elements are s-\-K has a supremum M and M—K is the supremum of the set 5. We now conclude that if a set 5 is bounded above, then 5 has a supremum. The proof of the second part of the theorem is similar. 11.5 Sequences. In the introductory chapter and in §5.6 the term sequence was used with the expectation that its meaning would be clear from the context, as it no doubt was. It is, perhaps, in order to point out that there is a distinction between sequences and sets. The collection 3, 2, 5, 7, 1, 6 is a set 5. It is also a sequence with 3 in the first place, 2 in the second place, and so on. The collection 5, 7, 1, 2, 6, 3 constitutes the same set, but not the same sequence. A sequence is a set of numbers assigned to places, a first place, a second place, and so on. A change in places among two or more numbers changes the sequence. The collection of positive integers 1, 2, 3, 4, . . . , is a set and a sequence. The collection 2, 1, 4, 3, . . . , is the same set but not the same sequence. Thus it may be said that a sequence is an ordered set. If a sequence sit s2 is such that JB+I ^ sn,n = 1 , 2 , . . . , the sequence is non decreasing; if sn+i < sn the sequence is nonincreasing; in both cases the sequence is monotone. If a sequence s\, 52, . . . , is such that there is a number G > 0 for which \sn\ < G, n = 1 , 2 , . . . , the sequence is bounded.

192

FUNDAMENTAL THEOREMS

11.6 The limit of a monotone sequence. // a monotone sequence slt Si, . . . , is bounded then sn tends to a limit as n increases. The proof is given for sn non-decreasing as n increases. The proof is similar when sn is non-increasing. The set of numbers Si, 52, . . . , is bounded and has, by Theorem 11.1, a supremum M. Then sn < M for all n, and for e > 0 there exists a value N of n such that SN > M—t. Because sn does not decrease as n increases it follows that SN < sn < M for n ~^> N and consequently \sn — M\ < e for n > N. Then since e is any positive number, by the definition of §1.3 it follows that sn tends to the limit M. 11.7 Functions and their properties. Let the domain of definition of the function / be the set X. If there is a number G such that f(x) < G, x Ç. X then f(x) is bounded above and G is an upper bound for / on X. If there is a number g such that f(x) > g, x Ç X then f(x) is bounded below and g is a lower bound for / on X. If f(x) is bounded above and below it is bounded and there is a positive number G for which \f(x)\ < G, x Ç X. A number M is the supremum of a function / if M is an upper bound for / and if G > M whenever G is any other upper bound for /. A number m is the infimum of the function / if m is a lower bound for / and if g < m whenever g is any other lower bound for /. As in the case of sets there is an alternative description for the supremum of a function / which features its domain of definition. Let X be the domain of definition of a function /. If there is a number M such that f(x) < M, x £ X and such that for each positive number t, no matter how near zero, there is at least one number x 6 X for which f(x) > M — € then M is the supremum of the function /. There is a similar description of the infimum of the function /. 11.8 The suprema and ínfima of bounded functions. // a function f is bounded above it has a supremum. If it is bounded below it has an infimum. This result is a consequence of §11.4. If a function/is bounded above then the set Y which constitutes the range of / is bounded above and has, therefore, a supremum. This supremum of the set Y is easily shown to satisfy either of the descriptions given above for the supremum of the function /. The existence of the infimum of a function which is bounded below follows in a similar way. 11.9 Properties of continuous functions. The concept of continuity as it applies to functions is defined in §5.3. It is probably no exaggeration to say that "continuity" is the most important single concept in analysis. Let the domain of definition of a function / be the interval / and let Xo

11.10 BOUNDS OF CONTINUOUS FUNCTIONS

193

be a point on /. If to each positive number e, no matter how near zero, there corresponds a positive number 8 = 8(xo, t) which is such that (1)

l/(*)-/(*o)| < e

whenever \x— XQ < ô then the function/ is continuous at the point Xo. If / is continuous at each point on an interval / then we say it is continuous on /. This is the statement of §5.3. We repeat it here in order to make a closer examination of its implications. Let f(x) = i/x with domain of definition the half open interval /, 0 < x < 1. Let Xo be a point on /. Let e > 0 be given. Choose 8 = 5(¡co, t) any positive number less than the smaller of tXo/2 and tXo2/2. Thus ô = 8(xo, e) is determined in terms of the point Xo and the number e, and we note that if \x — Xo\ < ô < Xo/2 then x > Xo/2. Also

Thus f(x) is continuous at each point Xo of / and is, therefore, continuous on the half open interval /. But no matter how great is the number G there are points x £ / for which f(x) = l/x > G. Thus the function f(x) = l/x is continuous on the half open interval (0, 1] but it is not bounded on this interval. In this example things went wrong near the point x = 0 which is not on the domain of /. Can a function be continuous on a closed interval and fail to be bounded? We next raise a second question in regard to continuous functions and then give answers to both. Let the function / be defined by the relations f(x) = x, 0 < x < 1, /(I) = 0. Thus the domain of definition of / is the closed interval [0, 1] and the supremum of / is unity. But for no value of x does f(x) take on its supreme value. This function / is not continuous. Can a function be continuous on a closed interval and fail to take on its extreme values? 11.10 The bounds of functions which are continuous on closed intervals. Let the domain of definition of the function f be the closed interval [a, b]. I f / is continuous on [a, b] then f is bounded. For d > 0 there is 5 > 0 such that \f(x)—f(a)\ < d when \x — a\ < o. Hence there are points x\ > a such that/(x) is bounded on [a, Xi\. Let Xo

194

FUNDAMENTAL THEOREMS

be the supremum of points x which are such that f(x) is bounded on [a, x]. There is ô > 0 such that (1) |/(*)-/(*o)| M—l/(n+l) > M—l/n, and consequently x is in An. It then follows that xn+i, the infimum of An+\, is not less than xn, the infimum of An. Consequently #,, < xn+\. Also xn < b. It then follows from §11.6 that xn tends to a limit XQ. It will now be shown that f(xn] is not less than M— l/n. Suppose f(%n) — M—l/n — rj, i\ > 0. Because f(x) is continuous there exists ô > 0 such that f(x) < M— l/n — r?/2 for all points x on the interval (xn — 6, *.+«). Again, since xn is the infimum of the set An, there is at least one point x of An on [xn, xn+5). Then since x is a point of An, f(x) > M—l/n > M — l/n —ti/2. Thus again there is a contradiction, and we conclude that /(«.) > M-l/n. We now have M—l/n M as n —> oo. But xn —>XQ, and because f(x) is continuous at x 0 , f(xn) —>/(x0). Hence f(x0) = M. The second part of the theorem may be proved in a similar way. 11.12 The oscillation of a continuous function. If the domain of definition of a function / is the interval / the oscillation of / on 7 is the supremum of \f(x)—f(x')\ for x, x' any two points on /. I f / is continuous

11.12 OSCILLATION OF A CONTINUOUS FUNCTION on I and xo is a point of / then \f(x)—f(x')\ and x' are sufficiently near x. For

195

is arbitrarily near zero if x,

and because f(x) is continuous at XQ both terms on the right are as near zero as desired if \x'— x\ < \x'— Xt\-\-\Xt¡ — x\ is sufficiently near zero. The general question with which we are concerned is that of finding conditions under which \f(x) —f(x')\ is as near zero as desired if only \x—x'\ is sufficiently near zero without reference to a fixed point XQ. We give two illustrations to show that continuity on an interval is not sufficient. The first is the example of §11.9,/(x) = \/x, 0 < x < 1. Let ô be fixed, as near zero as desired. Fix x' with 0 < x' < 6. If x is such that 0 < x < x' < ô then \x — x'\ < ô and

can be made arbitrarily great by taking x sufficiently near zero.

The second illustration is based on the function f(x) = sin (I/a;), 0 < x < 1/T, the graph of which is indicated in Figure 11.1. The domain of definition 7 is the interval 0 < x < 2/7r.

Fig. 11.1 The function / is continuous at each point xo € /. For

196

FUNDAMENTAL THEOREMS

which tends to zero as x —» x0. Hence \f(x)—f(xn)\ —> 0 as x—>;to, which implies that f(x) is continuous at x 0 be fixed, as near zero as desired. There is a point x' with 0 < x' < 5 for which f(x) = — I , and a point x with 0 < x < x' < ô for which /(x) = 1. Then !/(*)-/(*') I = |l-(-l)| = 2 . Thus for this function, no matter how near zero is the positive number, 5, there are points x, x', with \x — x' < 5 and \f(x)—f(x')\ = 2, x, x' on the interval of definition /. The function / in the first illustration is unbounded. For this reason we hardly expected it to have the properties for which we are looking. But the function in the second illustration is bounded. Yet it does not behave in the desired way. Hence no distinction can be made on the basis of boundedness. In both illustrations the domain of definition of the function is a half open interval. Can a function which is continuous on a closed interval fail to have the desired properties? This is answered by the following theorem. 11.13 Functions continuous on a closed interval. Let the function f be continuous on the closed interval I. To 0 there corresponds 5 > 0 which is such that if x, x' are any two points on I with \x—x' a. It then follows from (2) that which contradicts (1). Hence the theorem is true. It is worth noting that this proof fails if J is not closed for it would not necessarily follow that the point Xo, the supremum of the set xn, is on /. 11.14 Values assumed by continuous functions. Let the function f be continuous on the closed interval [a, b]. If f(a) ¿¿ f(b] and N is a number

EXERCISE

197

between f(a) and f(b) there is at least one point c between a and b for which f(c) = N, a < c < b. It is no restriction to assume/(a) N on [b-5, b]. Hence a+S < c < b-5. Let €1 > e2 > . . . be a sequence of positive numbers with tn —> 0 as n —•* oo. By the definition of a supremum, §11.3, there is xn > c— tn for whichf(x n ~) < N. Because xn—>c and f(x) is continuous at c it follows that

(1)

Hm f(xn) = /(c)< N. Xn->c

If .T > c, f(x) > AT. Let x\ > x¿, . . . , xn' > c, be a sequence of points such that xn' —» c as n —-> oo. Then, as in (1), (2)

lim /(«i) = f(c) > N.

Xit'-^C

From relations (1) and (2) it follows that/(c) = N. EXERCISE 11 1. Determine the supremum and infimum of the following sets: (a)

(b)

(c)

(d)

(e) (f) fe) (h) (i) (j)

2. Let si, Í2, be a sequence of numbers. If there is a number s such that for e > 0 there is n( for which 5re < s + t, n > n,, and if to each integer n there is n' > n such that $„' > s — e, then s is the limit superior of sn as n —> œ. There is a corresponding definition for the limit inferior s of the sequence. The notations for these respective limits are

198

FUNDAMENTAL THEOREMS

State the definition for s and show that s < s. Also show that il s = s = s then 5 is the limit of sn as n —* °°. 3. Determine 5 and s for the sequences (a)—(j) of Problem 1. 4. If un > 0 and sn = Ui+u2+ . . . +un, show that if sn tends to a limit as n increases then un tends to zero. 5. If sn in Problem 4 tends to a limit as n increases show that for all integers p > 0, tends to zero as n —» ». 6. Show that

and use this to show that Sn —» °° as w —» °o where

7. Let

Note that

and use this to show that Sn tends to a limit and that this limit is unity. 8. Given a sequence Si, 5 2 , . . . , show that if Sn tends to a limit as n —» °°, then \Sn—Sm\ —» 0 as n, m —» °°. 9. Show that

increases as n increases. 10. Show that for n > 3

Hence show that as n —* &, In n/n tends to a limit. What is this limit? 11. Let {an} be a decreasing sequence of positive numbers with limit zero and such that for all n. Prove that

EXERCISE

199

12. If {an\ is a sequence of numbers such that for n > 1 (2-a.n)an+i = 1, prove that as n —> °°, an tends to a limit and this limit is unity (PC, 1947). 13. Let f i ( x ) , fï(x) be two functions which are defined on the interval ( a , b ) . Let MI = sup/!(X), Mt = sup/ 2 (ac), M = sup {/i(*)+/2(*)} x on (a, b). Show that M < Mi+M2. Construct examples for which If < Mi+M 2 , M = Mi + TI^, and for which M = MI = M2. 14. The function f(x) = a(x), f(x) (x) are continuous at *0. Also show that if 0(xo) ^ 0 then/(#)/$(#) is continuous at XQ. 16. Let f(x) be continuous on the closed interval [a, b] and be such that f(x) > 0 for x on [a, b]. Show that there is a number d > 0 such that/(x) > d for x on [a, i]. 17. Let x be a real number greater than unity, n a member of the set of positive integers. Show that xn —* °° as n —* . If x > 1 and w > 1 then xn+l = xx™ > re". Hence x" is an increasing sequence and tends to a finite limit L by §11.6, or becomes infinite, as n —> oo. Suppose x" —» L as w —> œ. Then xn~l —» L as w —» oo. But then x™ = x je""1 —> xL which is a contradiction when x > 1. 18. If x is a real number between zero and unity, 0 < x < 1, show that x" —> 0 as n —» 0°. 19. Let a be a positive real number, n a positive integer greater than unity. There are numbers a for which a™ < a, (!/&)" for & sufficiently great, for example. The set a for which an < a is bounded above, a < a + 1 for example. Hence the set a for which an < a is not empty, and has a supremum «o by §11.4. Show that aon = a. This number ao is called the rath root of the number a and is denoted by al/n. If m is an integer (al/m)n is denoted by an'n. 20. Let a be a real number greater than unity and let x be a positive real number. Let S be the set of real numbers ap/t for all rational numbers p/q < x. The set 5 is bounded above. For if h/r is a rational number greater than x, h/r > x and p/q < x, then h/r > p/q, hq > pr and

200

FUNDAMENTAL THEOREMS h/T

Hence a is an upper bound for the set 5 and by §11.4 the set 5 has a supremum. This supremum is denoted by ax and is called "a to the power x," or the exponential function ax. If x\ and xi are positive real numbers with Xz > Xi and a > 1 show that a*2 > a11. 21. Let X be a set of real numbers which contains an infinite number of elements, or members, x. The number xa is a limit of the set X if in every interval containing xo there is an infinite number of elements x with x ¿¿ XQ. The number XQ may or may not be a member of X. Show that every bounded set X which contains an infinite number of elements has at least one limit. This is known as the Bolzano-Weierstrass theorem after the German mathematicians Bernard Bolzano and Karl Weierstrass who first proved the theorem. Let S be the set of all real numbers 5 where s is such that there is an infinite number of elements x of the set X with x > s. Because X is bounded the set is bounded above. Let Xo be the supremum of 5. Let e > 0 be given. By §11.3 there is an element s of 5 with s > x^ — e. Hence there is an infinite number of elements x of X with x > Xo— t. There are only a finite number of elements of x with x > x0+e. Otherwise x0 would not be the supremum of the set 5. Hence between XQ — e and Xo+e there is an infinite number of elements of X, and because e is arbitrary XD qualifies as a limit point of the set J^. The set X may have many limit points. If X is the set of rational numbers on the interval (0, 1) every point on the interval [0, 1] is a limit of the set X. Prove this and show that x = 1 corresponds to the number XQ of the proof given above. 22. Let Si, is, . . . , be an infinite sequence of distinct real numbers which is such that \sn — sm\ —» 0 as n —» °o. It is then called a Cauchy sequence after the French mathematician A. L. Cauchy. For any Cauchy sequence sn tends to a limit as n —> ». Let e > 0 be given. Fix n\ so that (1)

There is only a finite number of terms of the sequence with n < n\. Hence the terms of the sequence constitute a bounded set with an infinite number of elements. Accordingly this set has at least one limit point So by Problem 21. Because SQ is a limit point of the set there exists nt > n\ and such that Then if n > nf > n\ it follows from (1) that Because e is arbitrary the number So is, according to the definition of §1.3, the limit of sn as n —» oo.

CHAPTER XII

INFINITE SERIES 12.1 The meaning of sequences and series. We have already used the word 'sequence' in its ordinary sense of 'one after the other'. If a set of numbers is so arranged that there is a first, a second, and so on, as in the set 1, I, | , . . . , we have called it a sequence. We shall continue to use the term sequence of numbers in this sense. If MI, M2, ... is a sequence of numbers then the array u1+u2+ ... is called a series. If the number of terms different from zero is finite, then the array takes the form «l + «2+ • • • +Un,

which has a definite sum, and this sum is said to be the value of the series. If there is an infinite number of terms, «l+«2 +

is an infinite series and ordinary addition does not apply to assign a value to the series. It has been found useful to assign values to an infinite series in other ways. A study of these is now made. 12.2 Convergence, divergence and value of infinite series. Given the series ut+u2 + . . . , let Sn = «I + MÜ + • • • +M B .

The number sn is the sum of the first n terms of the series. If sn tends to a limit L as n increases, the series converges and L is the value of the series. If sn does not tend to a limit, the series diverges. The sums sn are called partial sums of the series. Example 12.1 Show that the series converges, and determine its value.

201

202

INFINITE SERIES

It appears that as n increases sn approaches the value 2. To see that this is actually the case, we note that the series is a geometrical progression, and conseauentlv

As n increases -- tends to zero and sn tends to the value 2. Hence the series 2" converges and has the value 2. Example 12.2 Show that the series converges and that its value is 2.

and sn tends to 2 as M increases. Example 12.3 Show that the series diverges. To show that the series diverges we group together certain terms of Sin, n a power of 2. The value of each bracket is greater than J ; and as n increases the number of brackets increases. Hence sin —> °o. Therefore it can be said that sn does not tend to a finite limit, and this is what we mean by saying the series diverges. The series of this example is known as the harmonic series. Example 12.4 The series 1-1 + 1-1+... diverges. In this case sn = 1 or zero, according as n is odd or even. Consequently sn does not tend to a limit and the series diverges. 12.3 A second method of assigning a value to a series. Consider the series of Example 12.4. Si = 1, s2 = 0, s3 = 1, 54 = 0, . . . .

12.4

TESTS FOR CONVERGENCE

203

It follows that 0 and vn < un. Then the series i>i+z>2+ .. . converges. Set Sn = «1 + M2 +

• • • +Un

l+*>2+ • • • +Z>n

The partial sums sn tend to a limit L. Then, since un > 0 it follows that sn < L for all n, since vn < «„. It also follows that ffn < Sn < L.

Now let us notice that because vn > 0 the partial sums an do not decrease

204

INFINITE SERIES

as n increases. Hence, by §11.6, 2+ • • • converges. Let UI+HZ + . . . , un > 0 be a series which diverges. Let »i+»2 + . . . . . »„ > 0 be a series for which vn > un. Then the series vt+v2+ . . . also diverges. If the positive term series %+%+ • • • diverges then sn = Wi4-«2 + • • -+w n must be such that sn —> °° as n —» «o. Then, since vn > un, crn = ^1+^2 + • • • -\-Vn is greater than or equal to sn = Mi+«2+ . . . +un, and it follows that 1. Another series often used for comparison is the ^-series which diverges if p < 1. To show this we note that if p = 1 it is the harmonic series of Example 12.3. Now if p < 1, l/n" > l/n, and, by the comparison test for divergence, the ^-series diverges. To show that the series converges for p > 1 note that

It then follows that for

If p > 1 so that 1/2P~1 < 1, then the right side of this inequality is the first n terms of a geometric series with r = l/2p~l < 1. Consequently for all k

12.5 THE RATIO TEST

205

Then, by §11.6, sn tends to a limit. This completes the proof that the p-series diverges for p < 1 and converges for p > 1. 12.5 The ratio test. term series. If (1)

This test, in its direct application, is to positive «1 + M2+ . . . .

is a positive terms series and the ratio un-\.\/un tends to a limit L < 1 the series converges. If L > 1 the series diverges. Let r be any number such that L < r < 1. Then, since un+i/un —» L as n —» co , there is a positive integer TV such that It then follows that

Now consider the series (2) MI+ . . . +Uff+ruN+riuN+

....

The sum of the first p terms of this series, p > N, may be written as follows 1, choose a number p with 1 < p < L. Then there is a positive integer N with

and by proceeding as in the case for L < 1, the comparison test for divergence can be used to show that series (1) diverges. When L = 1 there are some series which converge and others which diverge. The series

206

INFINITE SERIES

is the ^-series with p = 2, and is, therefore, convergent. But un+i/un = wViw + l) 2 -» 1 as »-> oo. On the other hand, the series is such that un+\/un = «/(w + 1) —» 1 as n—» , and it was shown in Example 12.3 that this series diverges. Example 12.5 Does the series

converge or diverge? If we try the ratio test, we find that un+i/un —> 1, which does not help. Note that But 1/M3/2 is the general term of the ^-series with p = 3/2. It then follows from the comparison test that the series of this example converges. EXERCISE 12 (a) Determine whether or not the following series converge: Ans. 1.

2.

Ans.

3.

Ans.

4.

Ans.

5.

Ans.

6.

Ans.

7.

(PC, 1942)

12.6 Series of positive and negative terms. Absolute convergence. Associated with a series (1) is the positive-term series (2) which is useful when (1) contains an infinite number of both positive and negative terms. Since (2) is a positive term series, the comparison tests

12.6 ABSOLUTE CONVERGENCE

207

and the ratio test apply to determine its convergence or divergence. In this way the following information about (1) may be obtained. If the series (2) converges so does the series (1) which is then said to converge absolutely. Suppose (2) converges and has the value L. Write

and let

Then £„ is non-decreasing and Consequently £„ tends to a limit by §11.6. Then, because where £„ and 0, and sn(x~) = 1 +x+. . . +xn tends to f(x) = 1/(1— x). Thus, in the language of the preceding sections, the series (1) l+x+x*+... converges to the function l/(l—x} for all x on the interval —1 < x < 1. If \x\ > 1 then \xn+1\ —> 1. Consequently the series (1) does not converge when x\ > 1. When x = 1 the function f(x) = 1/(1— x) is not defined and the series (1) is 1 + 1 + 1+ . . .which obviously diverges. When x = — 1, f(x) = \, but for x = —1 the series (1) is 1 — 1 + 1 — 1+ . . . which again diverges. We have now considered all values of x and have found that for \x\ < 1 the function

12.8 FUNCTIONS REPRESENTED BY SERIES

211

is represented by the series (1), and that this series does not represent f(x) for |*| > 1. Is there any other series which represents/^) = \/(\—x) for any values of x with \x\ > 1? We shall answer this by considering the more general question of the conditions under which a function /(x) denned on an interval (a, b) may be represented by a series on this interval. We begin our study of this problem by experimenting with a polynomial /(*) = ao*"+o^"~1+ - - - +an. In this we set x = a+h to get f(a+h) = a 0 (a+A)"+ai(o W^ . . . +an. If the various powers of a-\-h are expanded by the binomial theorem and the results rearranged in increasing powers of h there is obtained

If h is now replaced by its value x — a this becomes

We thus have a formula for expressing f(x) as a finite sum of increasing powers of x — a. If, for example, f(x) = x4 — 2x+l and a = 2, this formula gives x*-2x+l = 13+30(x-2) + 12(x-2) 2 +4( :K -2) 3 + (x-2) 4 . Because f(x) is a polynomial _f (n+1) (x) and all successive derivatives are zero. Consequently the formula terminates with the (w + l)s< term. If f(x) is some function which for x = a has an indefinite number of derivatives different from zero, the formula becomes an infinite series. We now go on to show that under some circumstances, the value of this infinite series

is the function/^). Let f(x) be defined and have derivatives of all orders on some interval containing the point x = a, and let x = b be a second point on such an interval. Write (1)

This relation defines Rn as the difference between /(&) and the sum of the terms other than Rn on the right. With Rn defined in this way let the number A be defined by the relation

212

INFINITE SERIES

We can now show that A = /"+1(£) where £ is a value of * between a and b. We first take an arbitrary step for which we have no defence except that it serves our purpose. We define an arbitrary function (j>(x) by the relation

It is obvious that 0(6) = 0, and comparison with (1) shows that (a) Since all the derivatives of f(x) exist it follows that 4>'(x) exists for (a, b). Hence (x) satisfies Rolle's theorem, and a3 + b3, a^+bt) c(ai, a 2) as, a4) = (ca\, ca2, ca3, cat)

is a set of elements of a vector space with quadruples as vectors. The elements of a general vector space can be infinite sequences or even functions. EXERCISE 16 1. Show that the vectors 2i+4j+k, j+k, 2i+2j-k are linearly independent. 2. Determine which of the following sets of vectors is a basis for R$ (a) i+2j+3k, j+k, 3i+2j+k. (b) 2k, 2Í+2J, 2j+2k. (c) 3i+3j+k, i+j,k. 3. Determine which of the following sets of vectors are linearly independent. (a) i+j+k, (b) 21+j,

j+2k, 2i-2k.

i-k.

(c) i+2j+3k, i+3j+5k. (d) i+j+k, j+k, i+2j+4k. 4. Consider number sets as vectors and show that the vectors (1000),

(0100),

(0010),

(0001)

form a basis for Rt. 5. Show that any five vectors in RI are linearly dependent.

CHAPTER XVII

VECTOR PRODUCTS APPLICATIONS OF VECTORS 17.1 Vector products. The problem of finding a vector that is perpendicular to each of two other vectors is of such frequent occurrence that it merits special consideration. If A = aii+û^j+ffsk, B = bii+bzj+b^k. are two vectors then by § 15.8 the vector

is perpendicular to A and B. This vector N is called the vector product or cross product of the vectors A and B. The notation is

Note that

N = AXB.

It then follows formally that

Unless N = O the vector — N is not the same as the vector N. Hence vector multiplication is not commutative. If 9 is the angle not greater than 180 degrees from A to B then

If we put A = a1i+a2JTa3k, B = 01i+02j+&3k in the numerator of the expression representing sin 6 it is easily verified that

281

Fig. 17.1

282

VECTOR PRODUCTS AND APPLICATIONS

The product AXB is usually defined as a vector N perpendicular to both A and B with magnitude |A||B| sin 6, with direction so that right handed rotation about N takes A into B. The determinant definition which we have given fixes the direction of N. Note that |N| is twice the area of the triangle two of whose sides are A and B. From the definition we have given of a cross product it follows that

Because AxB = -(BxA) it follows that

Thus vector multiplication obeys the distributive law. 17.2 Curvilinear motion. In §8.2 we studied motion along a straight line. This involved only one direction and its inverse. If a force F acts on a mass particle which is free to move in a plane, or in three-dimensional space, the direction of action of the force, and of the velocity, is much less restricted. The fact that both force and motion have magnitude and direction leads us to associate them with vectors. We now study this problem relative to motion along a plane curve. When this is done the extension of the results to a curve in three-dimensional space is a formality. Let a force act on a mass particle P in such a way that it moves along a curve in the xy-plane whose parametric representation in terms of time t is x =