An Introduction to Analytic Geometry and Calculus PDF
Preview An Introduction to Analytic Geometry and Calculus
ACADEMIC PRESS INTERNATIONAL EDITION An Introduction to ANALYTIC GEOMETRY AND CALCULUS A. C. Burdette UNIVERSITY OF CALIFORNIA DAVIS, CALIFORNIA MF) ACADEMIC PRESS New York London ACADEMIC PRESS INTERNATIONAL EDITION This edition not for sale in the United States of America and Canada. COPYRIGHT ©> 1968,BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1X6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 66-30138 PRINTED IN THE UNITED STATES OF AMERICA To my wife Emily Preface The addition of another calculus textbook to the large number already in existence calls for some justification. In this case it is very simple. It was written to satisfy an unfulfilled need for a textbook suitable for an existing course in the University of California, Davis. Since it is plausible that our requirements are not too different from those of others, it is hoped that this book will prove to be a useful contribution to the field. The general goals of the book may best be defined by giving a brief descrip tion of the course for which it was designed. It is a one-year course, meeting three times a week and presupposing a preparation equivalent to three semesters of high-school algebra and one semester of trigonometry. It is expected to provide the student not only with a reasonable degree of under standing of the basic concepts, but also with a working knowledge of the elementary operations of calculus. The presentation is at the freshman level but it serves students of all ranks. Graduate students in various fields find it particularly useful for making up mathematical deficiencies. For many, it is their terminal course in mathematics. Those looking for a completely novel treatment of calculus will be dis appointed. It is true that the mean value theorem plays the central role it should, but seldom does; that the definite integral stands on its own, not as a mere adjunct to the antiderivative; that the choice of subject matter and the level at which it is presented are carefully coordinated with the objectives to be served ; and that there is a wider than usual range of applications. On the whole, however, this is a traditional treatment of the subject. This is true, in particular, of the notation and the terminology, in the belief that this best serves the needs of the students involved. For example, the nonmathematics major, reading in his own field, will find the mathematical models of his problems stated in traditional notation, not the highly sophisticated notation of the modern mathematician. In general, I have held to the conviction that the historical road of develop ment of a discipline is the simplest path to its understanding. This philosophy explains why, for example, I have defined the logarithm function as the inverse of the exponential function, rather than the more direct, and perhaps more logical, definition as an integral. It is my opinion that this approach is more meaningful to the beginning student in spite of its disadvantages. These disadvantages worry him not at all at this point. No attempt has been made to eliminate the need for an instructor, although there are numerous examples and, hopefully, the student can use the text to amplify and clarify classroom discussion. This book is intended to serve as a basic core of material upon which an instructor can develop a course with a considerable degree of latitude. For example, in Chapter 4 limits are defined, a vii vìii PREFACE few proofs of simple cases are given, and then the general theorems are stated without proof. The teacher who wishes to give a more rigorous course can elaborate the work here by presenting proofs of these theorems, while the one who wishes to give a more intuitive course can amplify the motivating examples and soft-pedal the ε, ^-definitions. The analytic geometry has been reduced to bare essentials in order to have more time to devote to calculus. Even so, there is probably somewhat more material than can be covered in the allotted time. Chapters 12, 13, 14 are independent of each other so parts or all of these chapters may be omitted without affecting the others. This permits a certain amount of flexibility to adjust to local needs. At Davis, many of our students expect to take physical chemistry so we cover Chapter 13 completely. On the other hand, we omit Chapter 14 entirely and include only parts of Chapter 12. The results of many informal discussions with my colleagues are to be found in these pages. I thank them for this contribution. Also, I am deeply indebted to Professor S. Saslaw of the United States Naval Academy who read the entire manuscript in its original form and whose suggestions have substantially improved the finished product. Special thanks are due Professor E. B. Roessler, Chairman of the Department of Mathematics, University of California, Davis, for permission to use a preliminary edition of this material in the classroom. Finally, I wish to acknowledge the valuable assistance rendered by the editorial staff of Academic Press in bringing this project to completion. December, 1967 A. C. BURDETTE Davis, California Chapter 1 THE COORDINATE SYSTEM- FUNDAMENTAL RELATIONS l-l. Introduction This chapter, together with the next two, will be devoted to the task of develop ing a modest geometric background from the algebraic point of view. This is usually referred to as analytic geometry in contrast to synthetic geometry already studied in high school or elsewhere. Analytic geometry will serve us well as a basic tool as we proceed with our study of calculus. I-2. Directed Lines In elementary geometry a line segment is a portion of a line defined by the two points which mark its extremities. If the two end points are designated by A and B, we label a line segment AB or BA, making no distinction between the two notations. When the processes of algebra are applied to the problems of geometry, as they are in analytic geometry, we find it useful to define directed lines. DEFINITION l-l. A directed line is a line on which a positive direction has been assigned. A line segment measured in the positive direction is considered positive and the same portion of the line measured in the opposite direction is considered negative. The positive direction may be assigned arbitrarily although, for consistency of notation and convenience in interpreting results, there are some situations in which certain positive directions are used ordinarily. For example, on horizontal and vertical lines the positive directions are usually taken to the right and upward, respectively. DEFINITION I-2. On a directed line, BA = -AB. The following theorem regarding directed line segments will be useful in later work. 2 I. THE COORDINATE SYSTEM—FUNDAMENTAL RELATIONS THEOREM l-l. If A, B, and C are any three points on a directed line, AB + BC = AC. (1-1) The truth of this theorem may be established by considering individually the six possible cases of arrangement. For example, let the points be arranged as indicated in Fig. 1-1, separated by the distances a ^ 0 and b ^ 0, and let a b —i 1 (-► C A B Fig. 1-1 the positive direction on the line be that of the arrow. Thus AB = b, BC=-b-a, AC=-a and (1-1) becomes b + ( — b — a) = —a, which verifies the theorem for this case. The other possibilities can be treated in the same manner. 1-3. Cartesian Coordinates The basis of the cartesian, or rectangular, coordinate system is a pair of mutually perpendicular directed lines, X'X and Y'Y, on each of which a number scale has been chosen with the zero point at their intersection, O. This point is called the origin and the lines are called the x-axis and y-axis. The x-axis is usually taken in a horizontal position and, in that case, it is customary to take the positive directions on the two axes to the right and upward. This will be our choice unless specifically stated to the contrary. Let P be any point in the plane and drop perpendiculars from it to each of the axes, thus determining the projections M and N of P on the x-axis and y- axis, respectively (Fig. 1-2). The measure of the directed segment OM in the units of the scale chosen on the x-axis is called the x-coordinate, or abscissa, of P. Similarly, the measure of the directed segment ON in the units of the scale chosen on the j-axis is called the y-coordinate, or ordinate, of P. Let the measures of OM and ON be Xi and y respectively.! Then the position of u the point P with respect to the coordinate axes is described by the notation (xi,j>i). As indicated, the abscissa is always written first in the parentheses. The pair of numbers (x y^) is referred to collectively as the coordinates of P. 1? When the coordinate axes have been selected and their corresponding number scales chosen, then to each point in the plane corresponds a unique pair of numbers and, conversely, to each pair of numbers corresponds a single point. Thus a coordinate system attaches a numerical property to points and t Hereafter, for simplicity of statement, we will often speak of a segment AB when we mean the measure of AB. This should cause no confusion since the meaning will be clear from the context. 1-3. CARTESIAN COORDINATES 3 Y P(xi,y N -<p O M ' γ· Fig. 1-2 this makes it possible to apply the processes of algebra to the study of geo metric problems. This is the distinguishing characteristic of analytic geometry. There are many physical situations in which the quantity measured by x is entirely different from that measured by y; for example, x and y may represent time and temperature, respectively. If such situations are recorded on a coordinate system, there is no need to use the same scale on the two axes unless inferences are to be drawn from geometric considerations. However, when geometric relationships such as angles between lines and distances between points are involved, it is important that the same scale be used on both axes. Since many physical applications of analytic geometry and calculus hinge on geometric properties, it will be convenient for us to make a blanket assumption on this subject: Unless stated to the contrary, it is to be understood throughout this book that the same scale is used on both coordinate axes. The coordinate axes divide the plane into four quadrants. These are numbered as shown in Fig. 1-3. Such a numbering is convenient for describing certain general situations. Y II I o III IV Fig. 1-3 4 I. THE COORDINATE SYSTEM—FUNDAMENTAL RELATIONS 1-4. Projections of a Line Segment on Horizontal and Vertical Lines Let M and M be the projections of Λ(·*ι>>Ί) and ^2(^5^2)» respectively x 2 (Fig. 1-4), on the x-axis. The directed segment MM is said to be the pro l 2 jection ofPP on the x-axis. Clearly, the projection of PP on any horizontal l 2 l 2 line is equal to MM . Moreover, from (1-1) i 2 MM = M O + OM = OM - ΟΜ. i 2 x 2 2 γ Ρι{Χι Nì \ 1 1 Mx O M2 Fig. 1-4 But by definition OM = x OM = x , x l9 2 2 and we have MM = x -x. (1-2) l 2 2 l Similarly, the projection of PP on any vertical line is equal to its projection l 2 Λ^Λ^ on the >>-axis and NN = N O + ON = ON - ON l 2 x 2 2 u or N N = y-y. (1-3) 1 2 2 i Example l-l. Given the two points A(3, —2), B( — 2, 5), find the projections of AB on the coordinate axes. Choose A as P and B as P and formulas (1-2) and (1-3) give the required l 2 projections : M M = (-2)-(3)=-5, 1 2 N JV = (5)-(-2) = 7. 1 2 Note that the opposite choice of P and P would change the sign of these 1 2 results. This is as it should be since this changes the direction. 1-5. MIDPOINT OF A LINE SEGMENT 5 1-5. Midpoint of a Line Segment Let P(x, y) be the midpoint of the line segment whose end points are P\(xu JiX Pi(x2 > y2)9 and let M, M M , respectively, be the projections of u 2 these points on the x-axis (Fig. 1-5). Then, making use of (1-2), we have P P _M M _x-x X 1 l ¥F ~ MM ~ x - x ' 2 2 2 and, since P is the midpoint of P\P > 2 PP t = 1. PPy P(x, yi) 2 2 P(x,y)\ Λ(Χι,^ι) M, M O Mi Fig. 1-5 Therefore X X\ 1, X7. x from which, solving for x, we obtain the formula Χγ + X 2 (1-4) Similarly, using the projections of P, P P on the >>-axis, we have u 2 y 1 + y 2 y = —^—· (1-5) Example 1-2. Determine the point P(x, y) which bisects the line segment joining the points (2, —3), (4, 5). From (1-4) and (1-5), 2+4 - 3 +5 * = — τ- = 3, }> = — = 1. Thus the required point is (3, 1).
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