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- 1. Alternate EditionCalculus with analytk: geometry Earl W Swokowski Marquette University ~ Prindle, Weber & Schmidt Boston, Massachusetts
- 2. Dedicated to the memory of my mother and father; Sophia and john SwokowskiF=WS F=UBLISHEF=tSPnndle Weber & Schmrdt ·II.· Wollard Granl Press · ooc: · Duxbury Press · •Statler Off1ce Bu•ld.ng · 20 Prov•dence Street · Boston Massachusetts 02116Copyright© 1983 by PWS PublishersAll rights reserved. No part of this book may be reproduced ortransmitted in any form or by any means, electronic or mechanical,including photocopying, recording, or by any information storageand retrieval system, without permission, in writing, from thepublisher.PWS Publishers is a division of Wadsworth, Inc.Portions of this book previously appeared in Calculus with Ana-lytic Geometry, Second Edition by Earl W. Swokowski. Copyright© 1979 by Prindle, Weber & Schmidt.87 86 85 84 83 - 10 9 8 7 6 5 4 3 2ISBN 0-87150-341-7Library of Congress Cataloging in Publication DataSwokowski, Earl W. Calculus with analytic geometry. Includes index. I. Calculus. 2. Geometry, Analytic. I. TitleQA303.S94 1983 515.15 82-21481ISBN 0-87150-341-7Cover image courtesy of General Motors Research Laboratories.The computer graphic image depicts the location of valence elec-trons trapped near the surface of rhodium. Quantum mechanicalcalculations using the Schrodingerequation were employed to gen-erate the image. The two sets of peaks in the foreground reveal apreferential accumulation of electrons around the surface atoms.Production and design: Kathi TownesText composition: Composition House LimitedTechnical artwork: Vantage Art, Inc.Cover printing: Federated Lithographers-Printers, Inc.Text printing/binding: Von Hoffmann Press, Inc.Printed in the United States of America
- 3. Preface Most students study calculus for its use as a tool in areas functions leads to some nontrivial applications of the Chain other than mathematics. They desire information about why Rule and enlarges the scope of applications of the derivative. calculus is important, and where and how it can be applied. I In Chapter4testvalues are used to determine intervals in kept these facts in mind as I wrote this text. In particular, which derivatives are positive or negative. This pedagogical when introducing new concepts I often refer to problems device is also employed to help obtain graphs of rational that are familiar to students and that require methods of functions. calculus for solutions. Numerous examples and exercises Chapters 5 and 6, on properties and applications of defi- have been designed to further motivate student interest, not nite integrals, include exercises on numerical integration only in the mathematical or physical sciences, but in other that require reference to graphs to approximate areas, vol- disciplines as well. Figures are frequently used to bridge the umes, work, and force exerted by a liquid. gap between the statement of a problem and its solution. Inverse functions are discussed in the first section of In addition to achieving a good balance between theory Chapter 7 and are used in Section 7. 3 to define the natural-and applications, my primary objective was to write a book exponential function as the inverse of the natural log- that can be read and understood by college freshmen. In each arithmic function. section I have striven for accuracy and clarity of exposition, Chapters 8- 10 contain material on transcendental func- together with a presentation that makes the transition from tions, techniques of integration, and improper integrals. precalculus mathematics to calculus as smooth as possible. Infinite series are presented in a precise manner in Chapter The comments that follow highlight some of the features II. Chapter 12 consists of a detailed study of conic sections. of this text. Chapters 13-15 deal with curves, vectors and vector- A review of the trigonometric functions is contained in valued functions. There are many examples and exercises the last section of Chapter I. It was placed there, instead of pertaining to parametric and polar equations, and a strong in an appendix, to alert students to the fact that trigonometry emphasis is placed on geometric and applied aspects of is, indeed, a prerequisite for calculus, as indicated by the vectors. title of the chapter. Tests for symmetry are also introduced Functions of several variables are discussed at length in early, so that they can be used throughout the text. Chapter 16. The relevance of level curves and surfaces to In Chapter 2 limits involving the sine and cosine func- practical situations is illustrated in examples and exercises. tions are considered after limits of algebraic functions, and The approach to increments and differentials is motivated by thus are readily available for use in obtaining derivative for- analogous single variable concepts. The definition of direc- mulas in Chapter 3. The early introduction of trigonometric tional derivative does not require the use of direction angles ;;;
- 4. iv Prefaceof a line, and considerable stress is given to the gradient of versity of Georgia; Stanley M. Lukawecki, Clemson Uni-a function. The study of maxima and minima includes an versity; Louise E. Moser, California State University,examination of boundary extrema. The final section, on Hayward; Norman K. Nystrom, American River College;Lagrange multipliers, includes a proof that indicates David A. Petrie, Cypress College; William Robinson, Ven-the geometric nature of why the method is valid. tura College; JohnT. Scheick, Ohio State University; Jon W. Properties and applications of multiple integrals are con- Scott, Montgomery College; Monty J. Strauss, Texas Techsidered in Chapter 17. University; Richard G. Vinson, University of South Ala- Vector fields are discussed in Chapter 18, and special bama; Loyd Wilcox, Golden West College; and T. J.attention is given to conservative fields. The physical sig- Worosz, Metropolitan State College, Denver.nificance of divergence and curl is brought out by using the I also wish to express my gratitude to Christian C.theorems of Gauss and Stokes. The last two sections con- Braunschweiger of Marquette University, who provided an-tain results on Jacobians and change of variables in multiple swers for exercises; Thomas A. Bronikowski of Marquetteintegrals. University, who authored the student supplement containing Chapter 19, on differential equations, includes two sepa- detailed solutions for one-third of the exercises; Stephen B.rate sections on applications. Rodi of Austin Community College, who developed a com- There is a review section at the end of each chapter con- plete solutions manual; Michael B. Gregory of the U niver-sisting of a list of important topics and pertinent exercises. sity of North Dakota, who supplied a number of challengingThe review exercises are similar to those that appear exercises; and Christopher L. Morgan, California State U ni-throughout the text and may be used by students to prepare versity at Hayward, and Howard Pyron, University of Mis-for examinations. Answers to odd-numbered exercises are souri at Rolla, who prepared the computer graphics. Specialgiven at the end of the text. Instructors may obtain an answer thanks are due to Stephen J. Merrill of Marquette Universitybooklet for the even-numbered exercises from the publisher. for suggesting several interesting examples, including one Portions of this text are based on material that appears in that indicates how infinite sequences and series may be em-my book Calculus withAnalyticGeometry, Second Edition. ployed to study the time course of an epidemic, and anotherThis second edition is available for courses where a later that illustrates the use of exponential functions in the field ofintroduction of the trigonometric functions is desired. radiation therapy. I wish to thank the following individuals, who received I am grateful for the valuable assistance of the staff ofall, or parts of, the manuscript and offered many helpful PWS Publishers. In particular, Mary LeQuesne and Joesuggestions: Alfred Andrew, Georgia Institute of Technol- Power were very helpful with exercise sets; Kathi Townesogy; Jan F. Andrus, University of New Orleans; Robert M. did a superlative job as copy editor; and David Pallai, whoBrooks, University of Utah; Dennis R. Dunniger, Michigan supervised the production of this large project, was a con-State University; Daniel Drucker, Wayne State University; stant source of information and advice.Joseph M. Egar, Cleveland State University; Ronald D. In addition to all of the persons named here, I express myFerguson, San Antonio State College; Stuart Goldenberg, sincere appreciation to the many unnamed students andCalifornia Polytechnic State University; Theodore Guinn, teachers who have helped shape my views on how calculusUniversity of New Mexico; Joe A. Guthrie, University of should be presented in the classroom.Texas, El Paso; David Hoff, Indiana University; AdamHulin, University of New Orleans; W. D. Lichtenstein, Uni- Earl W. Swokowski
- 5. Table of Contents Introduction: What Is Calculus? ix 3. 7 Implicit Differentiation 126 3.8 Derivatives Involving Powers of Functions 131 3.9 Higher Order Derivatives 135 3.10 NewtonsMethod 1381 Prerequisites for Calculus 1 3.11 Review 141 1.1 Real Numbers 1 1.2 Coordinate Systems in Two Dimensions 9 1.3 Lines 18 4 Applications of the Derivative 144 1.4 Functions 24 1.5 Combinations of Functions 33 4.1 Local Extrema of Functions 144 1.6 The Trigonometric Functions 37 4.2 Rolles Theorem and the Mean Value Theorem /52 1.7 Review 47 4.3 The First Derivative Test 156 4.4 Concavity and the Second Derivative Test 162 4.5 Horizontal and Vertical Asymptotes 170 4.6 Applications of Extrema 1822 Limits and Continuity of Functions 49 4. 7 The Derivative as a Rate of Change 192 2.1 Introduction 49 4.8 Related Rates 201 2.2 Definition of Limit 54 4. 9 Antiderivatives 206 2.3 Theorems on Limits 60 4.10 Applications to Economics 214 2.4 One-Sided Limits 68 4.11 Review 22/ 2.5 Limits of Trigonometric Functions 72 2.6 Continuous Functions 76 2.7 Review 86 5 The Definite Integral 223 5.1 Area 223 5.2 Definition of Definite Integral 2323 The Derivative 87 5.3 Properties of the Definite Integral 239 3 .I Introduction 87 5.4 The Mean Value Theorem for Definite 3.2 The Derivative of a Function 92 Integrals 244 3.3 Rules for Finding Derivatives 98 5. 5 The Fundamental Theorem of Calculus 246 3.4 Derivatives of the Sine and Cosine Functions 106 5.6 Indefinite Integrals and Change of Variables 254 3.5 Increments and Differentials 111 5.7 Numerical Integration 263 3.6 The Chain Rule 119 5.8 Review 271 v
- 6. vi Table of Contents6 Applications of the Definite Integral 274 10 Indeterminate Forms, Improper Integrals, and 6.1 Area 274 Taylors Formula 457 6.2 Solids of Revolution 283 6.3 Volumes Using Cylindrical Shells 291 10.1 The Indeterminate Forms 0/0 and xfx 457 6.4 Volumes by Slicing 296 10.2 Other Indeterminate Forms 464 6.5 Work 299 I 0.3 Integrals with Infinite Limits of Integration 468 6.6 Force Exerted by a Liquid 306 10.4 Integrals with Discontinuous Integrands 473 6.7 Arc Length 31/ 10.5 Taylors Formula 479 6.8 Other Applications 317 10.6 Review 488 6.9 Review 323 11 Infinite Series 490 11.1 Infinite Sequences 4907 Exponential and Logarithmic Functions 325 11.2 Convergent or Divergent Infinite Series 500 7.1 Inverse Functions 325 11.3 Positive Term Series 5// 7.2 The Natural Logarithmic Function 329 11.4 Alternating Series 519 7.3 The Natural Exponential Function 337 11.5 Absolute Convergence 523 7.4 Differentiation and Integration 345 11.6 Power Series 530 7.5 General Exponential and Logarithmic II. 7 Power Series Representations of Functions 536 Functions 352 11.8 Taylor and Maclaurin Series 541 7.6 Laws of Growth and Decay 359 11.9 The Binomial Series 550 7.7 Derivatives of Inverse Functions 366 11.10 Review 553 7.8 Review 370 12 Topics in Analytic Geometry 556 12.1 Conic Sections 5568 Other Transcendental Functions 372 12.2 Parabolas 557 12.3 Ellipses 565 8.1 Derivatives of the Trigonometric Functions 372 12.4 Hyperbolas 571 8.2 Integrals of Trigonometric Functions 378 12.5 Rotation of Axes 577 8.3 The Inverse Trigonometric Functions 382 12.6 Review 581 8.4 Derivatives and Integrals Involving Inverse Trigonometric Functions 387 8.5 The Hyperbolic Functions 393 8.6 The Inverse Hyperbolic Functions 399 13 Plane Curves and Polar Coordinates 583 8.7 Review 403 13. I Plane Curves 583 13.2 Tangent Lines to Curves 59/ 13.3 Polar Coordinate Systems 594 13.4 Polar Equations of Conics 6039 Additional Techniques and Applications 13.5 Areas in Polar Coordinates 608 13.6 Lengths of Curves 611 of Integration 405 13.7 Surfaces of Revolution 615 9.1 Integration by Parts 406 13.8 Review 6/9 9.2 Trigonometric Integrals 412 9.3 Trigonometric Substitutions 418 9.4 Partial Fractions 423 9.5 Quadratic Expressions 430 14 Vectors and Solid Analytic Geometry 621 9.6 Miscellaneous Substitutions 433 14.1 Vectors in Two Dimensions 621 9.7 Tables of Integrals 437 14.2 Rectangular Coordinate Systems in Three 9.8 Moments and Centroids of Plane Regions 440 Dimensions 631 9.9 Centroids of Solids of Revolution 447 14.3 Vectors in Three Dimensions 635 9.10 Review 453 14.4 The Vector Product 645
- 7. Table of Contents vii 14.5 Lines in Space 652 18 Topics In Vector Calculus 829 14.6 Planes 654 14.7 Cylinders and Surfaces of Revolution 661 18.1 Vector Fields 829 14.8 Quadric Surfaces 665 18.2 Line Integrals 836 14.9 Cylindrical and Spherical Coordinate 18.3 Independence of Path 847 Systems 670 18.4 Greens Theorem 854 14.10 Review 673 18.5 Surface Integrals 862 18.6 The Divergence Theorem 869 18.7 Stokes Theorem 875 18.8 Transformations of Coordinates 88215 Vector-Valued Functions 676 18.9 Change of Variables in Multiple Integrals 885 15.1 Definitions and Graphs 676 18.10 Review 891 15.2 Limits, Derivatives, and Integrals 680 15.3 Motion 688 15.4 Curvature 692 15.5 Tangential and Normal Components of 19 Differential Equations 893 Acceleration 700 19. I Introduction 893 15.6 Keplers Laws 705 19.2 Exact Differential Equations 898 15.7 Review 710 19.3 Homogeneous Differential Equations 902 19.4 First-Order Linear Differential Equations 906 19.5 Applications 90916 Partial Differentiation 713 19.6 Second-Order Linear Differential Equations 914 16.1 Functions of Several Variables 713 19.7 Nonhomogeneous Linear Differential 16.2 Limits and Continuity 72/ Equations 920 16.3 Partial Derivatives 727 19.8 Vibrations 926 16.4 Increments and Differentials 733 19.9 Series Solutions of Differential EquatiDns 931 16.5 The Chain Rule 742 19.10 Review 934 16.6 Directional Derivatives 750 16.7 Tangent Planes and Normal Lines to Surfaces 758 16.8 Extrema of Functions of Several Variables 764 16.9 Lagrange Multipliers 770 Appendices 16.10 Review 778 Mathematical Induction Al II Theorems on Limits and Definite Integrals A817 Multiple Integrals 780 III Tables A Trigonometric Functions A 18 17. I Double Integrals 780 B Exponential Functions Al9 17.2 Evaluation of Double Integrals 785 C Natural Logarithms Al9 17.3 AreasandVolumes 794 IV Formulas from Geometry A20 17.4 Moments and Center of Mass 798 17.5 Double Integrals in Polar Coordinates 804 17.6 Triple Integrals 809 17.7 Applications of Triple Integrals 816 17.8 Triple Integrals in Cylindrical and Spherical Answers to Odd-Numbered Exercises A21 Coordinates 820 17.9 Surface Area 824 17.10 Review 827 Index A57
- 8. Introduction:What is Calculus? Calculus was invented in the seventeenth century to provide a tool for solving problems involving motion. The subject matter of geometry, algebra, and trigonometry is applicable to objects which move at constant speeds; how- ever, methods introduced in calculus are required to study the orbits of planets, to calculate the flight of a rocket, to predict the path of a charged particle through an electromagnetic field and, for that matter, to deal with all aspects of motion. In order to discuss objects in motion it is essential first to define what is meant by velocity and acceleration. Roughly speaking, the velocity of an object is a measure of the rate at which the distance traveled changes with respect to time. Acceleration is a measure of the rate at which velocity changes. Velocity may vary considerably, as is evident from the motion of a drag-strip racer or the descent of a space capsule as it reenters the Earths atmosphere. In order to give precise meanings to the notions of velocity and acceleration it is necessary to use one of the fundamental concepts of calculus, the derivative. Although calculus was introduced to help solve problems in physics, it has been applied to many different fields. One of the reasons for its versatility is the fact that the derivative is useful in the study of rates of change of many entities other than objects in motion. For example, a chemist may use derivatives to forecast the outcome of various chemical reactions. A biologist may employ it in the investigation of the rate of growth of bacteria in a culture. An electrical engineer uses the derivative to describe the change in current in an electrical circuit. Economists have applied it to problems involving corporate profits and losses. The derivative is also used to find tangent lines to curves. Although this has some independent geometric interest, the significance of tangent lines is of major importance in physical problems. For example, if a particle moves along a curve, then the tangent line indicates the direction of motion. If we restrict our attention to a sufficiently small portion of the curve, then in a ix
- 9. X Introduction: What is Calculus? certain sense the tangent line may be used to approximate the position of the particle. Many problems involving maximum and minimum values may be attacked with the aid of the derivative. Some typical questions that can be answered are: At what angle of elevation should a projectile be fired in order to achieve its maximum range? If a tin can is to hold one gallon of a liquid, what dimensions require the least amount of tin? At what point between two light sources will the illumination be greatest? How can certain corporations maximize their revenue? How can a manufacturer minimize the cost of producing a given article? Another fundamental concept of calculus is known as the definite integral. It, too, has many applications in the sciences. A physicist uses it to find the work required to stretch or compress a spring. An engineer may use it to find the center of mass or moment of inertia of a solid. The definite integral can be used by a biologist to calculate the flow of blood through an arteriole. An economist may employ it to estimate depreciation of equipment in a manufacturing plant. Mathematicians use definite integrals to investigate such concepts as areas of surfaces, volumes of geometric solids, and lengths of curves. All the examples we have listed, and many more, will be discussed in detail as we progress through this book. There is literally no end to the applications of calculus. Indeed, in the future perhaps you, the reader, will discover new uses for this important branch of mathematics. The derivative and the definite integral are defined in terms of certain limiting processes. The notion of limit is the initial idea which separates calculus from the more elementary branches of mathematics. Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) discovered the connection between derivatives and integrals. Because of this, and their other contributions to the subject, they are credited with the invention of calculus. Many other mathematicians have added a great deal to its develop- ment. The preceding discussion has not answered the question "What is calculus?" Actually, there is no simple answer. Calculus could be called the study of limits, derivatives, and integrals; however, this statement is meaning- less if definitions of the terms are unknown. Although we have given a few examples to illustrate what can be accomplished with derivatives and inte- grals, neither of these concepts has been given any meaning. Defining them will be one of the principal objectives of our early work in this text.
- 10. Prerequisites for CalculusThis chapter contains topics necessary for the study of attention to one of the most important concepts incalculus. After a brief review of real numbers, coordinate mathematics-the notion of function.systems, and graphs in two dimensions, we turn our J.J Real Numbers Real numbers are used considerably in precalculus mathematics, and we will assume familiarity with the fundamental properties of addition, subtraction, multiplication, division, exponents and radicals. Throughout this chapter, unless otherwise specified, lower-case letters a, b, c, ... denote real numbers. The positive integers 1, 2, 3, 4, ... may be obtained by adding the real number 1 successively to itself. The integers consist of all positive and negative integers together with the real number 0. A rational number is a real number that can be expressed as a quotient ajb, where a and bare integers and b -1= 0. Real numbers that are not rational are called irrational. The ratio of the circumference of a circle to its diameter is irrational. This real number is denoted by n and the notation n ~ 3.1416 is used to indicate that n is ap- proximately equal to 3.1416. Another example of an irrational number is .)2. Real numbers may be represented by nonterminating decimals. For example, the decimal representation for the rational number 7434/2310 is found by long division to be 3.2181818 ... , where the digits I and 8 repeat indefinitely. Rational numbers may always be represented by repeating decimals. Decimal representations for irrational numbers may also be obtained; however, they are non terminating and nonrepeating. It is possible to associate real numbers with points on a line lin such a way that to each real number a there corresponds one and only one point, and I
- 11. 2 1 Prerequisites for Calculus conversely, to each point Pthere corresponds precisely one real number. Such an association between two sets is referred to as a one-to-one correspondence. We first choose an arbitrary point 0, called the origin, and associate with it the real number 0. Points associated with the integers are then determined by considering successive line segments of equal length on either side of 0 as illustrated in Figure 1.1. The points corresponding to rational numbers such as 253 and -tare obtained by subdividing the equal line segments. Points associated with certain irrational numbers, such as fi, can be found by geometric construction. For other irrational numbers such as rc, no con- struction is possible. However, the point corresponding to n can be approxi- mated to any degree of accuracy by locating successively the points corres- ponding to 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, .... It can be shown that to every irrational number there corresponds a unique point on I and, con- versely, every point that is not associated with a rational number corresponds to an irrational number. 0 B A • -3 • -2 ~~r • 0 17; .,fi ; 1f • 4 ; b • • a 1 3 23 -2 2 5 FIGURE 1.1 The number a that is associated with a point A on I is called the coordinate of A. An assignment of coordinates to points on I is called a coordinate system for /, and I is called a coordinate line, or a real line. A direction can be assigned to I by taking the positive direction to the right and the negative direction to the left. The positive direction is noted by placing an arrowhead on I as shown in Figure 1.1. The real numbers which correspond to points to the right of 0 in Figure 1.1 are called positive real numbers, whereas those which correspond to points to the left of 0 are negative real numbers. The real number 0 is neither positive nor negative. The collection of positive real numbers is closed relative to addition and multiplication; that is, if a and b are positive, then so is the sum a + band the product ab. If a and bare real numbers, and a - b is positive, we say that a is greater than h and write a > b. An equivalent statement is h is less than a, written b < a. The symbols > or < are called inequality signs and expressions such as a > b or b < a are called inequalities. From the manner in which we con- structed the coordinate line I in Figure 1.1, we see that if A and B are points with coordinates a and b, respectively, then a > b (orb < a) if and only if A lies to the right of B. Since a - 0 = a, it follows that a > 0 if and only if a is positive. Similarly, a < 0 means that a is negative. The following properties of inequalities can be proved. If a > b and b > c, then a > c. If a > b, then a + c > b + c. (J.J) If a > b and c > 0, then ac > be. If a > b and c < 0, then ac < be. Analogous properties for" less than" can also be established.
- 12. Real Numbers 1.1 3 The symbol a ~ b, which is read a is greater than or equal to h, means that either a > bora = b. The symbol a < b < c means that a < band b < c, inwhichcasewesaythathishetweenaandc. Thenotationsa :s; b,a < b :s; c, a :s; b < c, a :s; b :s; c, and so on, have similar meanings. Another property, called completeness, is needed to characterize the real numbers. This property will be discussed in Chapter 11. If a is a real number, then it is the coordinate of some point A on a co- ordinate line /, and the symbol Ia I is used to denote the number of units (or 1-41=4 141=4 distance) between A and the origin, without regard to direction. Referring ,------"-...~ to Figure 1.2 we see that for the point with coordinate -4 we have I -41 = 4. + I I I + I I I + I• Similarly, 141 = 4. In general, if a is negative we change its sign to find Ia I,-5 -4 -3 -2 -1 0 1 2 3 4 5 I whereas if a is nonnegative then Ia I = a. The nonnegative number Ia I is FIGURE 1.2 called the absolute value of a. The following definition of absolute value summarizes our remarks. if a~ 0 Definition (1.2) lal = { a -a if a< 0 Example 1 Find 131, 1-31,101, IJ2- 21, and 12- fll. Solution Since 3, 2 - J2, and 0 are nonnegative, we have 131 = 3, 12 - J21 = 2 - J2, and I0 I = 0. Since -3 and J2 - 2 are negative, we use the formula Ia I = -a of Definition (1.2) to obtain 1-31 = -( -3) = 3 and IJ2- 21 = -<J2- 2) = 2- J2. • The following three general properties of absolute values may be established. (1.3) lal = 1-al, labl = lallbl, -lal ;S; a ;S; lal It can also be shown that if b is any positive real number, then Ia I < b if and only if - b < a < b (1.4) Ia I > b if and only if a > b or a < - b Ia I = b if and only if a = b or a = -b. It follows from the first and third properties stated in (1.4) that Ia I :s; b if and only if -b :s; a :s; b. The Triangle Inequality (1 .5) Ia + bl ;S; lal + lbl
- 13. 4 1 Prerequisites for Calculus Proof From (1.3), -lal ~a~ lal and -lbl ~ b ~ lbl. Adding corresponding sides we obtain -(lal + lbi) ~a+ b~ lal + lbl. Using the remark preceding this theorem gives us the desired conclusion. D 5= 17-21= 12-71 We shall use the concept of absolute value to define the distance between ~ any two points on a coordinate line. Let us begin by noting that the distance I I I I + I I I I + I• between the points with coordinates 2 and 7 shown in Figure 1.3 equals -2 -1 0 1 2 3 4 5 6 7 8 1 5 units on /. This distance is the difference, 7 - 2, obtained by subtracting FIGURE 1.3 the smaller coordinate from the larger. If we employ absolute values, then, since 17 - 21 = 12 - 71, it is unnecessary to be concerned about the order of subtraction. We shall use this as our motivation for the next definition. Definition (1.6) Let a and b be the coordinates of two points A and B, respectively, on a coordinate line 1. The distance between A and B, denoted by d(A,B), is defined by d(A, B) = lb- al. The number d(A, B) is also called the length of the line segment AB. Observe that, since d(B, A)= Ia- bl and lb- al = Ia- bl, we may write d(A, B) = d(B, A). Also note that the distance between the origin 0 and the point A is d(O, A)= Ia- Ol = Ia I, which agrees with the geometric interpretation of absolute value illustrated in Figure 1.2. Example 2 If A, B, C, and D have coordinates -5, -3, 1, and 6, respectively, find d(A, B), d(C, B), d(O, A), and d(C, D). A B OC D Solution The points are indicated in Figure 1.4. l•ltllttlllltl. By Definition (1.6), -5 -3 0 1 6 1 FIGURE 1.4 d(A, B) = I - 3 - (- 5) I = I - 3 + 51 = 121 = 2. d(C, B)= 1-3- 11 = 1-41 = 4. d(O, A)= l-5- 01 =I-51= 5. d( C, D) = 16 - 11 = 151 = 5. •
- 14. Real Numbers 1.1 5 The concept of absolute value has uses other than that of finding distances between points. Generally, it is employed whenever one is interested in the magnitude or numerical value of a real number without regard to its sign. In order to shorten explanations it is sometimes convenient to use the notation and terminology of sets. A set may be thought of as a collection of objects of some type. The objects are called elements of the set. Throughout our work ~will denote the set of real numbers. If Sis a set, then a E S means that a is an element of S, whereas a ¢ S signifies that a is not an element of S. If every element of a set Sis also an element of a set T, then Sis called a subset ofT. Two sets Sand Tare said to be equal, written S = T, if Sand Tcontain precisely the same elements. The notation S #- T means that Sand Tare not equal. If Sand Tare sets, their unionS u T consists of the elements which are either in S, in T, or in both S and T. The intersection S n T consists of the elements which the sets have in common. If the elements of a set Shave a certain property, tpen we write S = {x : ... } where the property describing the arbitrary element x is stated in the space after the colon. For example, {x: x > 3} may be used to represent the set of all real numbers greater than 3. Of major importance in calculus are certain subsets of~ called intervals. If a < b, tpe symbol (a, b) is sometimes used for all re!!l numbers between a and b. This set is called an open interval. Thus we have: (1.7) (a, b) = { x: a < x < b}. The numbers a and b are called the endpoints of the interval. The graphofa set Sofreal numbers is defined as the points on a coordinate ( ) line that correspond to the numbers inS. In particular, the graph of the open a b interval (a, b) consists of all points between the points corresponding to a and b. In Figure 1.5 we have sketched the graphs of a general open interval (a, b) ( I ) and the special open intervals (- 1, 3) and (2, 4). The parentheses in the figure -1 0 3 indicate that the endpoints of the intervals are not to be included. For convenience, we shall use the terms interval and graph of an interval inter- ( ) changeably. 0 2 4 If we wish to include an endpoint of an interval, a bracket is used insteadFIGURE 1.5 Open intervals (a, b), (-I, 3), of a parenthesis. If a < b, then closed intervals, denoted by [a, b], and half-and (2, 4) open intervals, denoted by [a, b) or (a, b], are defined as follows. [a, b] = { x: a ~ x ~ b} (1.8) [a, b) = {x: a ~ x < b} (a, b] = {x: a< x ~ b} Typical graphs are sketched in Figure 1.6, where a bracket indicates that the corresponding endpoint is part of the graph. [ 3 [ ) ( 3 a b a b a b FIGURE 1.6
- 15. 6 1 Prerequisites for Calculus In future discussions of intervals, whenever the magnitudes of a and b are not stated explicitly it will always be assumed that a < b. If an interval is a subset of another interval I it is called a subinterval of/. For example, the closed interval [2, 3] is a subinterval of [0, 5]. We shall sometimes employ the following infinite intervals. (a, oo) = {x: x >a} [a,oo) = {x:x ~a} (1.9) (-oo,a) = {x:x <a} (-oo,a] = {x:x ~a} (-oo,oo)=IR For example, (1, oo) represents all real numbers greater than 1. The symbol oo denotes" infinity" and is merely a notational device. It is not to be interpreted as representing a real number. As indicated in this section, we frequently make use of letters to denote arbitrary elements of a set. For example, we may use x to denote a real numbei, although no particular real number is specified. A letter that is used to represent any element of a given set is sometimes called a variable. Through- out this text, unless otherwise specified, variables will represent real numbers. The domain of a variable is the set of real numbers represented by the variable. To illustrate, given the expression JX, we note that in order to obtain a real number we must have x ~ 0, and hence in this case the domain of x is assumed to be the set of nonnegative real numbers. Similarly, when working with the expression 1/(x - 2) we must exclude x = 2 (Why?), and con- sequently we take the domain of x as the set of all real numbers different from 2. It is often necessary to consider inequalities that involve variables, such as x2 - 3 < 2x + 4. If certain numbers such as 4 or 5 are substituted for x, we obtain the false statements 13 < 12 or 22 < 14, respectively. Other numbers such as 1 or 2 produce the true statements -2 < 6 or 1 < 8, respectively. In general, if we are given an inequality in x and if a true statement is obtained when x is replaced by a real number a, then a is called a solution of the inequality. Thus 1 and 2 are soluti9ns of the inequality x 2 - 3 < 2x + 4, whereas 4 and 5 are not solutions. To solve an inequality means to find all solutions. We say that two inequalities are equivalent if they have exactly the same solutions. A standard method for solving an inequality is to replace it with a chain of equivalent inequalities, terminating in one for which the solutions are obvious. The main tools used in applying this method are properties such as those listed in (1.1), (1.3), and (1.4). For example, ifx represents a real number, then adding the same expression in x to both sides leads to an equivalent inequality. We may multiply both sides of an inequality by an expression containing x if we are certain that the expression is positive for all values of x under consideration. If we multiply both sides of an inequality by an expres- sion that is always negative, such as -7 - x 2 , then the inequality sign is reversed.
- 16. Real Numbers 1.1 7 The reader should supply reasons for the solutions of the following inequalities. Example 3 Solve the inequality 4x +3> 2x - 5. Solution The following inequalities are equivalent: 4x +3> 2x- 5 4x > 2x- 8 2x > -8 X> -4 Hence the solutions consist of all real numbers greater than -4, that is, the numbers in the infinite interval (- 4, oo ). • E xamp le 4 So Ive the mequa 1 -5 < - - -3x < 1. . . 1ty 4 -2 Solution We may proceed as follows: 4- 3x -5<--<1 2 -10 < 4- 3x < 2 -14<-3x<-2 14 2 ->x>- 3 3 2 14 -<x <- 3 3 Hence the solutions are the numbers in the open interval (2/3, 14/3). • Example 5 Solve x 2 - 7x + 10 > 0. Solution Since the inequality may be written (x - 5)(x - 2) > 0,Signofx-2: - - - + + + + + + + + + + it follows that x is a solution if and only if both factors x - 5 and x - 2 areSign of x - 5: - - - - - - - - - + + + + positive, or both are negative. The diagram in Figure 1.7 indicates the signs of I I I I ) I I ( I • these factors for various real numbers. Evidently, both factors are positive -2 -1 0 1 2 3 4 5 6 X if xis in the interval (5, oo) and both are negative if xis in (- oo, 2). Hence FIGURE 1.7 the solutions consist of all real numbers in the union (- oo, 2) u (5, oo ). • Among the most important inequalities occurring in calculus are those containing absolute values of the type illustrated in the next example.
- 17. 8 I Prerequisites for Calculus Example 6 Solve the inequality Ix - 31 < 0.1. Solution Using (1.4) and (1.1), the given inequality is equivalent to each of the following: - 0.1 < X - 3 < 0.1 -0.1 + 3 < (x - 3) +3< 0.1 +3 2.9 <X< 3.1. Thus the solutions are the real numbers in the open interval (2.9, 3.1). • Example 7 Solve 12x - 71 > 3.. Solution By (1.4), x is a solution of 12x - 71 > 3 if and only if either 2x- 7 > 3 or 2x- 7 < -3. The first of these two inequalities is equivalent to 2x > 10, or x > 5. The second is equivalent to 2x < 4, or x < 2. Hence the solutions of 12x - 71 > 3 are the numbers in the union (- oo, 2) u (5, oo ). • 1.1 ExercisesIn Exercises 1 and 2 replace the comma between each pair of 5 If A, B, and C are points on a coordinate line with co-real numbers with the appropriate symbol <, >, or =. ordinates -5, -I, and 7, respectively, find the following distances. 1 (a) -2, -5 (b) -2, 5 (c) 6- I, 2 +3 (d) !. 0.66 (a) d(A, B) (b) d(B, C) (e) 2, J4 (f) 1!:, ¥ (c) d(C, B) (d) d(A, C) 2 (a) -3,0 (b) -8, -3 6 Rework Exercise 5 if A, B, and C have coordinates 2, -8, and -3, respectively. (c) 8, -3 (d)i-i,-fs (e) Ji, 1.4 (f) tm. 3.6513 Solve the inequalities in Exercises 7-34 and express the solutions in terms of intervals.Rewrite the expressions in Exercises 3 and 4 without using 7 5x- 6 >II 8 3x- 5 < 10symbols for absolute values. 9 2- 7x ~ 16 10 7- 2x ~ -3 3 (a) 12-51 (b) I-51+ 1-21 11 12x + 11 > 5 12 lx+21<1 (c) 151 + 1-21 (e) In - 22/11 .• (d) 1-51-l-21 (f) ( -2)/l-21 13 3x + 2 < 5x- 8 14 2 + 7x < 3x- 10 (g) It- o.s1 (h) I< -WI 15 12 ~ 5x- 3 > -7 16 5 > 2- 9x > -4 (i) 15 - X I if X > 5 (j) Ia - b I if a < b 3- 7x 17 -1 < -4 <-6 - 18 0 ~ 4x- I~ 2 4 (a) 14- 81 (b)l3-nl (c) 1-41-1-81 (d) 1-4 + 81 5 4 19 - - > 0 20 -2--9 > 0 (e) 1-W (f) 12- fil 7- 2x X + (g) l-0.671 (h) -l-31 21 lx- 101 < 0.3 22 12x + 31 <2 -- (i) lx 2 + 11 (j) l-4- x 2 1 5
- 18. Coordinate Systems in Two Dimensions 1.2 923 - 3x 17-- <1 2 - I 24 13- llxl ~ 41 ~lf0ll5b1llf0) Natural length25 125x- 81 > 7 26 12x+11<027 3x 2 + 5x- 2 < 0 28 2x 2 - 9x + 7 < 029 2x 2 + 9x + 4 ~ 0 30 x 2 - !Ox::::; 200 FIGURE FOR EXERCISE 37 I31 2 < 100 32 5+Jx<1 X 38 Boyles Law for a certain gas states that pv = 200, where p denotes the pressure (lb/in. 2) and v denotes the volume 3x + 2 3 2 (in. 3 ). If 25 ::::; v ::::; 50, what is the corresponding range for33 - - < 0 34 - - > - - 2x- 7- x-9 x+2 p?35 The relationship between the Fahrenheit and Celsius 39 If a baseball is thrown straight upward from level ground temperature scales is given by C = (~)(F - 32). If 60 ::::; with an initial velocity of 72 ft/sec, its altitude s (in feet) F ::::; 80, express the corresponding range for C in terms after t seconds is given by s = -16t 2 + 72t. For what of an inequality. values oft will the ball be at least 32 feet above the ground?36 In the study of electricity, Ohms Law states that if R 40 The period T (sec) of a simple pendulum of length l (em) denotes the resistance of an object (in ohms), E the is given by T = 2n$g, where g is a physical constant. potential difference across the object (in volts), and I If, for the pendulum in a grandfather clock, g = 980 and the current that flows through it (in amperes), then 98 ::::; l ::::; 100, what is the corresponding range for T? R = E/1 (see figure). If the voltage is 110, what values of the resistance will result in a current that does not exceed 41 Prove that Ia- bl ~ Ia I- lbl. 10amperes? (Hint: Write lal = l(a- b)+ hi andapply(l.5).) 42 If n is any positive integer and a 1 , a 2 , ••• , an are real Resistance R numbers, prove that Iat+ a2 + .. ·+ani::::; latl + la2l + .. ·+I ani· ~urrent/ (Hint: By (1.5), Iat + a2 + ... +ani::::; latl + la2 + ... +ani.) 43 If 0 < a < b, or if a < b < 0, prove that {1/a) > (1/b ). Voltage£ 44 If 0 < a < b, prove that a2 < b2 • Why is the restriction 0 < a necessary? FIGURE FOR EXERCISE 36 45 If a < b and c < d, prove that a + c < b + d.37 According to Hookes Law, the force F (in pounds) 46 If a < band c < d, is it always true that ac < bd? Explain. required to stretch a certain spring x inches beyond its 47 Prove (1.3). natural length is given by F = (4.5)x (see figure). If 10::::; F::::; 18, what is the corresponding range for x? 48 Prove (1.4). J.2 Coordinate Systems in Two Dimensions In Section 1.1 we discussed how coordinates may be assigned to points on a line. Coordinate systems can also be introduced in planes by means of ordered pairs. The term ordered pair refers to two real numbers, where one
- 19. 10 1 Prerequisites for Calculus is designated as the "first" number and the other as the "second." The symbol (a, b) is used to denote the ordered pair consisting of the real numbers a and b where a is first and b is second. There are many uses for ordered pairs. They were used in Section 1.1 to denote open intervals. In this section they will represent points in a plane. Although ordered pairs are employed in different situations, there is little chance for confusion, since it should always be clear from the discussion whether the symbol (a, b) represents an interval, a point, or some other mathematical object. We consider two ordered pairs (a, b) and (c, d) equal, and write (a, b) = (c, d) if and only if a = c and b =d. This implies, in particular, that (a, b) =F (b, a) if a # b. The set of all ordered pairs will be denoted by IR x IR. A rectangular, or Cartesian,* coordinate system may be introduced in a plane by considering two perpendicular coordinate lines in the plane which intersect in the origin 0 on each line. Unless specified otherwise, the same unit of length is chosen on each line. Usually one of the lines is horizontal with positive direction to the right, and the other line is vertical with positive y direction upward, as indicated by the arrowheads in Figure 1.8. The two I lines are called coordinate axes and the point 0 is called the origin. The hori- 1(a, b) -------,-- zontal line is often referred to as the x-axis and the vertical line as they-axis, PI and they are labeled x and y, respectively. The plane is then called a co- II I I I ordinate plane or, with the preceding notation for coordinate axes, and xy- I plane. In certain applications different labels such as d, t, etc., are used for the I coordinate lines. The coordinate axes divide the plane into four parts called -4 -3 -2 the first, second, third, and fourth quadrants and labeled I, II, III, and IV, respectively, as shown in (i) of Figure 1.8. III IV Each point Pin an xy-plane may be assigned a unique ordered pair. If vertical and horizontal lines through P intersect the x- and y-axes at points with coordinates a and b, respectively (see (i) of Figure 1.8), then Pis assigned (i) the ordered pair (a, b). The number a is called the x-coordinate (or abscissa) of P, and b is called they-coordinate (or ordinate) of P. We sometimes say that P has coordinates (a, b). Conversely, every ordered pair (a, b) determines y (4, 6) a point Pin the xy-plane with coordinates a and b. Specifically, Pis the point 6 • of intersection of lines perpendicular to the x-axis and y-axis at the points 5 having coordinates a and b, respectively. This establishes a one-to-one 4 correspondence between the set of all points in the xy-plane and the set of all (-5, 3) (5, 3) • 3 • ordered pairs. It is sometimes convenient to refer to the point (a, b) meaning 2 the point with x-coordinate a andy-coordinate b. The symbol P(a, b) will (-4, 0) denote the point P with coordinates (a, b). To plot a point P(a, b) means to locate, in a coordinate plane, the point P with coordinates (a, b). This point -6 -5 -4 -3 -2 X is represented by a dot in the appropriate position, as illustrated in (ii) of Figure 1.8. -2 The next statement provides a formula for finding the distance d(P, Q) -3 • (5, -3) between two points P and Q in a coordinate plane. •(-6, -4) -4 (ii) * The term "Cartesian" is used in honor of the French mathematician and philosopher FIGURE 1.8 Rene Descartes (1596-1650), who was one of the first to employ such coordinate systems.
- 20. Coordinate Systems in Two Dimensions 1.2 II Distance Formula (1.10) The distance between any two points P 1(x 1, y 1) and Pz(x 2 , Yz) in a coordinate plane is given by y Proof If x 1 =f. x 2 and y 1 =f. Yz, then as illustrated in Figure 1.9, the points P 1 , P 2 , and Pix 3 , y 1 ) are vertices of a right triangle. By the Pythag- orean Theorem, ,~2 -y1P 1 (x~o y 1) _____ _.1 P3(x2,y 1 ) Using the fact that d(P 1 , P 3 ) = lx 2 - x 1 1 and d(P 3 , P 2 ) = ly 2 - y 1 1 gives us the desired formula. If y 1 = y 2 , the points P1 and P 2 lie on the same horizontal line and IX2 -x11 d(P 1 , P 2 ) = lx 2 - x 1 1 = j(x 2 - x 1) 2 . FIGURE 1.9 Similarly, if x 1 = x 2 , the points are on the same vertical line and These are special cases of the Distance Formula. Although we referred to Figure 1.9, the argument used in this proof of the Distance Formula is independent of the positions of the points P 1 and P 2 • 0 Example 1 Prove that the triangle with vertices A( -1, - 3), B(6, 1), and C(2, - 5) is a right triangle and find its area. Solution By the Distance Formula, d(A, B)= j(-1- 6) 2 + (-3- 1) 2 = j49 + 16 = j65 d(B, C) = j(6 - + (1 + 5) 2 = j16 + 36 = j52 2) 2 d(A,C) = j(-1- 2) 2 + (-3 + 5) 2 = )9+4 = fo. Hence [d(A, B)] 2 = [d(B, C)] 2 + [d(A, C)Y; that is, the triangle is a right triangle with hypotenuse AB. The area is !J52fi3 = 13 square units. • It is easy to obtain a formula for the midpoint of a line segment. Let P 1(x 1,y 1 ) and P 2 (x 2 ,Yz) be two points in a coordinate plane and let M be y the midpoint of the segment P 1 P 2 • The lines through P 1 and P 2 parallel to they-axis intersect the x-axis at A 1(x 1 ,0) and A2 (x 2 ,0) and, from plane geometry, the line through M parallel to the y-axis bisects the segment A 1 A 2 (see Figure 1.10).1fx 1 < x 2 , then x 2 - x 1 > 0, and hence d(A 1 , A 2 ) = x 2 - x 1 • Since M 1 is halfway from A 1 to A 2 , the x-coordinate of M 1 is x1 + !(x 2 - x 1) = x 1 + !x 2 - !x 1 X = !x 1 + !x 2 x 1 + x2 FIGURE 1.10 2
- 21. 12 1 Prerequisites for Calculus It follows that the x-coordinate of M is also (x 1 + x 2 )/2. It can be shown in similar fashion that they-coordinate of M is (y 1 + Y2)/2. Moreover, these formulas hold for all positions of P 1 and P 2 . This gives us the following result. Midpoint Formula (1.11) Example 2 Find the midpoint M of the line segment from P 1( - 2, 3) to P 2 (4, -2). Plot the points P 1 , P 2 , M and verify that d(P" M) = d(P 2 , M). Solution Applying the Midpoint Formula (1.11), the coordinates of M are ( -2 2+ 4 , 3 + ( -2)) or ( 1, ~) . 2 2 y The three points P 1 , P 2 , and Mare plotted in Figure 1.11. Using the Distance Formula we obtain P 1 (-2, 3) d(P 1 , M) = j(-2- V + (3- !) 2 = J9 + (1}-) = j61;2 d(P2, M) = j(4- 1? + ( -2- !) 2 = J9 + eJ) = fo/2 X Hence d(P 1 , M) = d(P 2 , M). • If W is a set of ordered pairs, then we may consider the point P(x, y) in a coordinate plane which corresponds to the ordered pair (x, y) in W. The graph of W is the set of all points that correspond to the ordered pairs in W. FIGURE 1.11 The phrase" sketch the graph W" means to illustrate the significant features of the graph geometrically on a coordinate plane. y Example 3 Sketch the graph of W = {(x, y): lxl s 2, IYI s 1}. 3 Solution The indicated inequalities are equivalent to -2 s x s 2 and 2 - 1 s y s 1. Hence the graph of W consists of all points within and on the boundary of the rectangular region shown in Figure 1.12. • Example 4 Sketch the graph of W = {(x, y): y = 2x- 1}. {(x, y): lx I ,;;; 2, IY I ,;;; 1} -2 Solution We begin by finding points with coordinates of the form (x, y) where the ordered pair (x, y) is in W. It is convenient to list these coordinates FIGURE 1.12 in the following tabular form, where for each real number x the corresponding value for y is 2x - 1.
- 22. Coordinate Systems in Two Dimensions 1.2 13 y (3, 5) X ~-2 -1 0 2 3 y -5 -3 -1 3 5 After plotting, it appears that the points with these coordinates all lie on a line and we sketch the graph (see Figure 1.13). Ordinarily the few points we have plotted would not be enough to illustrate the graph; however, in this elementary case we can be reasonably sure that the graph is a line. In the next X section we will prove that our conjecture is correct. • {(x, y): y = 2x -1} The x-coordinates of points at which a graph intersects the x-axis are called the x-intercepts of the graph. Similarly, they-coordinates of points at which a graph intercepts they-axis are called they-intercepts. In Figure 1.13, there is one x-intercept 1/2 and one y-intercept - 1.FIGURE /.13 It is impossible to sketch the entire graph in Example 4 since x may be assigned values which are numerically as large as desired. Nevertheless, we often call a drawing of the type given in Figure 1.13 the graph of W or a sketch of the graph where it is understood that the drawing is only a device for visualizing the actual graph and the line does not terminate as shown in the figure. In general, the sketch of a graph should illustrate enough of the graph so that the remaining parts are evident. The graph in Example 4 is determined by the equation y = 2x - 1 in the sense that for every real number x, the equation can be used to find a number y such that (x, y) is in W. Given an equation in x andy, we say that an ordered pair (a, b) is a solution of the equation if equality is obtained when a is sub- stituted for x and b for y. For example, (2, 3) is a solution of y = 2x - 1 since substitution of 2 for x and 3 for y leads to 3 = 4 - 1, or 3 = 3. Two equations in x andy are said to be equivalent if they have exactly the same solutions. The solutions of an equation in x and y determine a set S of ordered pairs, and we define the graph of the equation as the graph of S. Notice that the solutions of the equation y = 2x - 1 are the pairs (a, b) such that b = 2a - 1, and hence the solutions are identical with the set W given in Example 4. Consequently the graph of the equation y = 2x - 1 is the same as the graph of W (see Figure I.i3). For some of the equations we shall encounter in this chapter the technique used for sketching the graph will consist of plotting a sufficient number of points until some pattern emerges, and then sketching the graph accordingly. This is obviously a crude (and often inaccurate) way to arrive at the graph; however, it is a method often employed in el~mentary courses. As we progress through this text, techniques will be introduced that will enable us to sketch accurate graphs without plotting many points. Example 5 Sketch the graph of the equation y = x 2 • Solution To obtain the graph, it is necessary to plot more points than in the previous example. Increasing successive x-coordinates by t, we obtain the following table. X -3 ! -2 -~ -1 t 0 t ~ 2 ! 3 y 9 ll 4 4 *0 *
- 23. 14 1 Prerequisites for Calculus Larger numerical values of x produce even larger values of y. For example, the points (4, 16), (5, 25), and (6, 36) are on the graph, as are ( -4, 16), (- 5, 25), and (- 6, 36). Plotting the points given by the table and drawing a smooth curve through these points gives us the sketch in Figure 1.14, where we have labeled several points. • The graph in Example 5 is called a parabola. The lowest point (0, 0) is called the vertex of the parabola and we say that the parabola opens upward. If the graph were inverted, as would be the case for y = - x 2 , then the para- bola opens downward. They-axis is called the axis of the parabola. Parabolas and their properties will be discussed in detail in Chapter 12, where it will be shown that the graph of every equation of the form y = ax 2 + bx + c, FIGURE 1.14 with a =f. 0, is a parabola whose axis is parallel to they-axis. Parabolas may also open to the right or to the left (cf. Example 6). If the coordinate plane in Figure 1.14 is folded along they-axis, then the graph which lies in the left half of the plane coincides with that in the right half. We say that the graph is symmetric with respect to they-axis. As in (i) of Figure 1.15, a graph is symmetric with respect to the y-axis provided that the point ( -x, y) is on the graph whenever (x, y) is on the graph. Similarly, as in (ii) of Figure 1.15, a graph is symmetric with respect to the x-axis if, whenever a point (x, y) is on the graph, then (x, - y) is also on the graph. In this case if we fold the coordinate plane along the x-axis, the part of the graph which lies above the x-axis will coincide with the part which lies below. Another type of symmetry which certain graphs possess is called symmetry with respect to the origin. In this situation, whenever a point (x, y) is on the graph, then (- x, - y) is also on the graph, as illustrated in (iii) of Figure 1.15. y y ~ X X (i) y-axis (ii) x-axis (iii) origin FIGURE 1.15 Symmetries The following tests are useful for investigating these three types of symmetry for graphs of equations in x and y. Tests for Symmetry (1.12) (i) The graph of an equation is symmetric with respect to they-axis if substitution of- x for x leads to an equivalent equation. (ii) The graph of an equation is symmetric with respect to the x-axis if substitution of - y for y leads to an equivalent equation. (iii) The graph of an equation is symmetric with respect to the origin if the simultaneous substitution of -x for x and - y for y leads to an equivalent equation.
- 24. Coordinate Systems in Two Dimensions 1.2 15 If, in the equation of Example 5, we substitute - x for x, we obtain y = ( -x) 2 , which is equivalent toy= x 2 • Hence, by Test (i), the graph is sym- metric with respect to the y-axis. If symmetry with respect to an axis exists, then it is sufficient to determine the graph in half of the coordinate plane, since the remainder of the graph is a mirror image, or reflection, of that half. Example 6 Sketch the graph of i = x. Solution Since substitution of - y for y does not change the equation, the graph is symmetric with respect to the x-axis. (See Symmetry Test (ii).) It is sufficient, therefore, to plot points with nonnegative y-coordinates and then reflect through the x-axis. Since i = x, they-coordinates of points x above the x-axis are given by y = Jx. Coordinates of some points on the graph are tabulated below. A portion of the graph is sketched in Figure 1.16. The graph is a parabola that opens to the right, with its vertex at the origin. In this case the x-axis is the axis of the parabola.FIGURE 1.16 X 0 2 3 4 9 y 0 j2j323 • Example 7 Sketch the graph of the equation 4y = x 3 . Solution If we substitute - x for x and - y for y, then 4(- y) = ( -x) 3 or -4y = -x 3 • Multiplying both sides by -1, we see that the last equation has the same solutions as the given equation 4y = x 3 . Hence, from Symmetry Test (iii), the graph is symmetric with respect to the origin. The following table lists some points on the graph. X 0 I 2 t 2 5 2 By symmetry (or substitution) we see that the points ( -1, -!), ( -2, -2),FIGURE 1.17 etc., are on the graph. Plotting points leads to the graph in Figure 1.17. • If C(h, k) is a point in a coordinate plane, then a circle with center C and radius r > 0 may be defined as the collection of all points in the plane that are r units from C. As shown in (i) of Figure 1.18, a point P(x, y) is on the circle if and only if d( C, P) = r or, by the Distance Formula, if and only if j(x - h) 2 + (y - k) 2 = r. The equivalent equation (1.13) (x - h)z + (y - k)z = ,z
- 25. /6 1 Prerequisites for Calculus y is an equation of a circle of radius rand center C(h, k). If h = 0 and k = 0, this equation reduces to x 2 + y 2 = r 2 , which is an equation of a circle of radius r with center at the origin (see (ii) of Figure 1.18). If r = l, the graph of (1.13) is called a unit circle. X Example 8 Find an equation of the circle with center C(- 2, 3) which passes through the point D( 4, 5). (i) (x- h)2 + (y- k)2 = ,2 Solution Since Dis on the circle, the radius r is d(C, D). By the Distance Formula, y r = j(-2- 4) 2 + (3- 5) 2 = )36+4 =flO. Using ( 1.13) with h = - 2 and k = 3, we obtain (x + 2) 2 + (y - 3? = 40, (-r, 0) (r, 0) x or x2 + y 2 + 4x - 6y - 27 = 0. • Squaring terms in (1.13) and simplifying, we obtain an equation of the (ii) x2 + y2 = ,2 form FIGURE 1.18 x2 + y2 + ax + by + c = 0 where a, b, and c are real numbers. Conversely, if we begin with such an equation, it is always possible, by completing the squares in x andy, to obtain an equation of the form (x - h) 2 + (y - k) 2 = d. The method will be illustrated in Example 9. If d > 0, the graph is a circle with center (h, k) and radius r = Jd. If d = 0, then, since (x - h) 2 :::::: 0 and (y - k) 2 :::::: 0, the only solution of the equation is (h, k), and hence the graph consists of only one point. Finally, if d < 0, the equation has no real solutions and there is no graph. Example 9 Find the center and radius of the circle with equation x2 + y2 - 4x + 6y - 3 = 0. Solution We begin by arranging the equation as follows: (x 2 - 4x) + (y 2 + 6y) = 3. Next we complete the squares by adding appropriate numbers within the parentheses. Of course, to obtain equivalent equations we must add the numbers to both sides of the equation. In order to complete the square for an expression of the form x 2 + ax, we add the square of half the coefficient of x, that is, (a/2) 2 , to both sides of the equation. Similarly, for y 2 + by, we add

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