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Calculus and its Applications PDF

541 Pages·1963·14.949 MB·English
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CALCULUS and its APPLICATIONS By P. Mainardi and H. Barkan Department of Mathematics Newark College of Engineering Newark, New Jersey PERGAMON PRESS Oxford . London . New York . Paris PERGAMON PRESS LTD. Heading ton Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. / PERGAMON PRESS INC. 122 East 55th Street, New York 22 N. Y. t GAUTHIER-VILLARS ED. 55 Quai des Grands-A u% us tins, Paris 6e PERGAMON PRESS G.m.b.H Kaiserstrasse 75, Frankfurt am Main Distributed in the Western Hemisphere by THE MACMILLAN COMPANY—NEW YORK pursuant to a special arrangement with Pergamon Press Inc. Copyright® 1963 PERGAMON PRESS INC. Library of Congress Catalog Card Number 63-20798 PREFACE Throughout this book on Calculus, emphasis has been placed on the understanding of ideas. In keeping with this goal we have frequently taken applications from physics, engineering, economics and other areas, whenever we felt that these applications would help clarify the the mathematical concepts under study. We have not hesitated, however, to employ mathematical rigor where that rigor also contributed to increased understanding. We have systematically used the "trapping" technique in defining geometrical and physical entities that are usually regarded as limits of sums. Its use in defining length of curve and surface of revolution is believed to be somewhat novel, as is the simplifying symbolism em­ ployed. Also, there may be some novelty in the treatment of average value, where we give a more general definition than is usual. Our inclusion of the concepts of external and internal parameters may serve to increase the studentfs appreciation of parametric functions. In the treatment of separable differential equations, we stress with more detail than usual their special ill suitability in describing physical laws. Among the many colleagues and friends who helped in the preparation of this text, we would like to extend our thanks particularly to Dona Hausser who typed the entire manuscript, to Professor Robert Salamon who prepared many of the drawings and to Murray Lieb who worked a large number of problems and read the entire manuscript. P. M. H. B. CHAPTER I FUNDAMENTAL IDEAS 1.1 The Concept of Function It is a commonplace that many of the facts of physics, •geometry, biology, economics, etc., are expressible by simple mathematical equations involving numerical quanti­ ties. The distance s traversed in the time t by a body falling from rest in a vacuum is given by s • |gt^ where g is a constant. In geometry we have for the volume V of a sphere of radius r: V - ^ TTI^. The existence of such relationships has lead mathematicians to define the con­ cept of function* Prior to 1950, the usual textbook definition of function was given more or less as follows: We consider the totality of numbers x which lie be­ tween two numbers a and b. The symbol x is regarded as denoting any of the numbers in this interval* If to each value of x in this interval there corresponds a single definite value y, where x and y are connected by any rule whatsoever, we say that y is a function of x and write symbolically, y • f(x)« We call the x the independent variable and the y the dependent variable* More recent writers have objected to this definition as too vague and imprecise. Consistent with the current movement in mathematics toward increased abstraction, they have formulated the following definition of function* First we define an ordered pair of numbers as a pair in which one is designated as the first and the other as the second* Thus (2,3) is an ordered pair with 2 first and 3 second* A set of ordered pairs is called a fraction if no two different ordered pairs have the same first element* The set of first elements is called the domain of the function, and the set of second elements is called the range of the function* From this definition it follows that the set of ordered pairs (2,7), (3,U), (7,7) constitutes a function* The domain of the function is the set 2, 3, and 7# The range of the function is the set 7,U« I 2 Sec. LI It is customary to call a symbol which represents an . arbitrary element of the domain the independent variable. A symbol which represents an arbitrary'element of ' tkie range is called the dependent variable. If x is the in­ dependent variable and y is the dependent variable, and f is a symbol standing for the function (the set of ordered pairs), this information is represented symbol- cally by y - fix). Similarly, the symbolical expression, z * g(u)> implies that g stands for a set of ordered pairs; u is the symbol denoting the set of first elements; and z is the symbol denoting the set of second elements. Fur­ ther, none of the ordered pairs has the same first element with differing second elements. This latter requirement merely assures the function will be single-valued. Further, note that the symbol f(x), read Rf of x,f, stands for y, the dependent variable. The student should note the difference in the meanings of f and f (x). Thus in a specific instance, f(2) would denote the element in the range of the dependent variable y corresponding to the element 2 in the domain of the independent variable x. Frequently, the correspondence between the independ­ ent variable x and the dependent variable y is established by requiring that values of y be determined by specific rules of operation on the corresponding values of x, as for example in y • x2 - Ijx + 2 where x - 2, 3, U# Here the dependent variable is denoted by both the symbol y and the expression x2 - Ux + 2. The dependent variable plays a dual role. It not only stands for the set of numbers which constitute the range of the dependent varia­ ble, but it also tells us specifically how to compute the values of the dependent variable y which correspond to values of the independent variable x. Further we will call * symbol used to represent an arbitrary element of a set (even though no function is in­ volved) a variable. A symbol which is to be assigned one and only one value during a discussion will be called a constant. Thus we have for x - 2, x2 - Ux • 2 - 22 - U(2) + 2 - -2; for x - 3, x2 - Ux + 2 - 32 - U(3) • ? - - !; for x - U, x2 - kx + 2 - U2 - U(U) + 2 - +2. Sec. 1.2 3 We have seen that the dependent variable may be de­ noted* by a single letter, say y, or, for example, by the more informative analytical expression x^ - Ux + 2. We could of course have denoted y by the previously dis­ cussed bu£ less specific symbol f(x)* Thus, if we now write f(x) • x^ - Ux + 2, we may view f(x) as a command to perform certain operations on x which will yield the corresponding value of y. The symbol f(2) thus may be taken as a command to perform certain operations on the number 2 which will provide the corresponding value of the dependent variable, namely -2. Before ending the discussion of function we should point out that to Leibniz, who was the first to use the word "function11 and to the mathematicians of the 18th century, the function referred to a dependent variable given by an analytical expression in the independent variable, such as y • x^ - Ux + 2. In other words, in Leibniz1 s definition the law of correspondence had to be given by an analytical expression* The definition given in this book places no such restriction on the law of correspondence* Physicists found only too often that their empirical data which exhibited a relationship be­ tween physical quantities resisted analytical formulation* A.s the ideas they tried to describe became more complex, the physicists were often at a loss to find analytic functions which their empirical data would "fit" and which would predict the correspondence between hitherto unexamined values of the Variables* In the absence of any analytic function which would describe an obvious correspondence existing between the variables representing physical quantities, the physicist turned to the more general definition of function which places no restric­ tions on the form of the correspondence* 1*2 Exercises 1. Which of the following sets of ordered pairs constitute functions? a* (2,3), (2,U), (3,7). b. (2,3), (3,3), (U,3). -1 c T d. (0,1), (2,3), (U5). f \ Sec. L2 2. Indicate, where appropriate, the range and domain for each of the sets in Exercise 1. Some sets of ordered pairs (x,y) may be conveniently designated by the notation C j j seme statement 1 x y \ 9 I about x and y.J This notation is taken to stand for the set of all ordered pairs (x,y) for which ths statement about x and y is true. In each of the Exercises 3 through 12, sketch in Cartesian coordinates the sets indicated. I 3. x,y) y = x and x £ 0^ . h. x,y) | 7 * 4* + 2 and y > oj . 5. x,y) | y = Ux + ? and - • < x < <•> } . 6. { x,y) j y • Ux + 2 .and y - 3x + 2 and - oo < x< • } . 7. { x,y) | — <» <x< oo and — • <y< 00 J { I 0. x,y) y "/x and x k 0 } . { 9. ,y) | y = x 2 and y > x and x iO) . 10. ,y) I x2 + y2 < h} . 11. ,y) 1 x2 + y2 £ U and y £ -5 } . 12. ,y) I y - (x 2 - 2) } . 13. The table exhibits corresponding values X f(x) of the variables x and f(x). By guessing, 2 h formulate a simple analytic expression 3 9 for f(x) and use it to find the missing h 16 values of f(x). 5 25 6 7 7 7 Ik. Will f(x) = x + (x - 2)(x - 3)(x - U)(x - 5) satisfy the set of values giver, in the table of exercise 1? Find the Hissing entries using this rule of correspondence. Sec. 1.2 15* Plot in the Cartesian coordinate system the values of f(x) versus selected values of x, given fx • 2 when xx ^- 0, f(x) - 4 0 ^ xr when x < 0. Describe the domain and range* 16* Plot in Cartesian coordinates corresponding values of g(x) versus selected values of x, given , f2x + 3 when x ^ 0, x •s(x> a i I U when x > 0. Describe the domain and range. 17. Plot in Cartesian coordinates corresponding values of f(t) versus selected values of t, given t|_L_l when t / 1, f(t) 2 when t • 1* Describe the domain and range* 18. Given f (x) * x2 when x > 0, g(x) • x when - « < x < + • Do f and g represent the same function? Explain* 19. Given f(x) » |x| (The symbol |x| means the numeri­ cal or absolute value of x*) fx when x A 0, g ( x) " \-x when x < 0 # Do f and g represent the same function? Explain* 20* Given a (t) - cos(-t) for all t, 0 (t) * cos t for all t. Do ot and represent the same function? Explain* 21. Given f(x) - x2 for all x, g(x) - (-x)2 for all x. Do f and g represent the same function? Explain. 22. Given f(x) » x2 for -2 < x < 3, g(x) • 3x for x > -1. What is the domain of x for which f(x) • g(x) is defined? o Sec. 1.3 23. Given <X(t) - t2 - U fcr t<5, 0(t) - t • 2 for t<3. What is the domain of t for which fi^f is defined? 1.3 Introduction to the Limit Concept Starting from vague sense impressions, man has had to invent, perhaps over a period of thousands of generations, such concepts a3 space, object, distance, time interval, motion, and speed in order to acquire his existing store of knowledge with which he tries to control his environ­ ment. It would not be an exaggeration to say as Conant has said, "Science advances not by the accumulation of new facts, but by the continuous development of new and fruitful concepts.n One of the most fruitful concepts invented by man is that of instantaneous speed. However, we shall see that this is a special case of a more general concept, namely, that of limit. Let us put ourselves in the position of an earlier generation of men to whom the concept of speed had not yet by the long process of usage and familiarity become trivial and obvious. We might in this position of early primitive man have become aware of the futility of attempting to catch certain animals by direct oursuit. We would have developed a vague concept of elusiveness which included without distinction a conglomeration of the ideas of swiftness, cnange of direction, change of pace, and feinting, but excluded such ideas as physical power and terrain conditions. With additional experience, the need for a more refined concept might have asserted itself. No doubt, some animals, relying almost exclusive­ ly on their fleetness, would have run in a fixed direction from their pursuers. This could have led eventually to the recognition of some men as eminent hunters because of their ability to overtake these animals. Hence a superior hunter might have evoked in the minds of his tribesmen a dynamic image of rapidly moving feet, and of diminishing distance between him and his quarry. At some time we would have recognized that the pursuit of one animal by another was vaguely similar to our own pursuit of or by animals. Slowly over the ages distinctions be­ tween similar events would have become common sense

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