Calculus With Analytic Geometry

[-1- r" 16lT - {Ll *, ir a > of du | \/i "' l\/a +Tn + lal -

20. f ---: :r -' " ' J u\ /a+n lZr^n_r r@_ q6 i f a a

In formulas 27 through 38, we may replaceln(z * t/FRl by sinh-lf,lnlu * \trt41 by cosh-'f,. la+ t f i4 lt " l

, I bY sinh- ' f ,

, , I# : rn lu +{F+l+c,8. I t/itTV du:l 1ffif

+ $U" + vFEA + cf _

2e. J uz\/uz + 02 d" -- t (2u' + a2) r/uz

:; . tffil + c

to. I ry : \fr' * nz - " "lLPl*,,t. I =-: tf i ,

- a2 -4 sec-, 1' l ..AA f I/FTF au \FtA , ,-r-.t z ' l - f f : -T*h lu+vTf r \+c

1 u'z i lu u - 1n2zz. J ffi:i t/F tF -;hlu * tffiv1 + c

$,non,ol )v.cerrt.Dn = 7"C "*[n-k

3s. tL:1r".- , lg l * ."" ' J uy@-- oz a" - - la l

' -

,uI+:_YT*,f

tr. J (uz -+ szytrz o":t (2u, +'Saz) l irE7

+{r ' r1"+ tFTTl+cq R I d " : - L L c*'

J (uz-rszlstz -raz1/ff i -

" I# :s in- l i * '*. I 1/o:utur: l tF7+f " in-,I*,

f -nt J uz1ffi

- uz o":t (2u, - sz1 \/F= uz + f; sin-' I*,

42. 1 t/F=F au : {a,77 - a tnl'* Yal *,' - ' J u 1 u I: {a'7 u' - a cosh-l X *,

L ? 1 t l F = T a u : _ l ; ' - u ' _ s i n - l L + c-". J u2 u ---- a

44.

45.i

46.

49.

50.

t % : - t t \ m + o : s i n - r L + cJ t / a z - u z 2 2 n

l L : - 1 , n l o * f f i ! * .J u ! a 2 - u 2 a I u I

: -+cosh-l 1* 'f du \E-u ' , - ,-

J uz t f az - u2 azu

Fsftrl*,: Cont4futing, Lwu - n3

f . - , u - a a ' ^ ^ - - ,I

\ t rau-F du :T t /2au-uz +7 cos- r

[ " r m d u : 2 u 2 - T - t o ' \ f f i r e

47. I oz - uz)stz du : -[ t

" '

- sq')

4sI#:#+c

aAa .

_ , I r , r{az - uz + ? sin-l ;* t

(r-+)*,* t cos - l ( r - ) * ,

s l I ry : t f f i * a cos- ' (r - :)+ csz. Iry:-rY -cos-,(1 -:)*,s3 I# :cos- r (1 -T)* ,

Forms Containing Trigonometric Functions

f

f. '''

5 9 . f s i n u d u c o s u + CJf

6 0 . f c o s u d u : s i n u * CJf

6 1 . I t a t u d u : l n l s e c u l + CJ

62. | .o. u du- lnlsin ul + cJf

63. I t". u du - lnl secu * tan ul + Cr

: lnl tan (*n + iu)l+ C64. [

"r. u du- lnlcsc t/ - cot ul + C

J- lnltan iul + C

6 5 . I r u . ' u d u : t a n u * CJf

5 6 . f c s c 2 u d u c o t u + CJr

5 7 . J

s e c i l t a n u d u : s e c i l + C

58. f . r . u cotu duJf

69. I

sin2 u du : Lrtt - *sin 2u * C

f70. I cos' u du: *u * *sin 2u * C

Jf

71. J tanz u du: tan t t - u+ C

s4 . t+ t t r aua *acos - ' ( r -g ) +cv r ' J l f f i u - u z

' - - \ a /

trtr [ uz du (" | l ' )- \ER +3o-' cos-l55 . 166

F, f du @- ,-/-r o . J f f i : - * T 'f d u u - a , - .: -

J (2au - u2)st2 a2!2au - uzs8 . 1 , udu . -++cv'Lr'

J (Zau - u2)3t2 atffi=E

r72 .

I co t2 u du : - co t u - u+ C

t J t t : - l , i n n - r u c o s t l + n " 1 l . , i r r ' - ' u d u7 3 . J s i n n u t n - r i l c o s u + -

n J u r ^ r

f74. Jcosn u du:+cos ' - r a s in " *+J.o ' " -2 u du

7s. J.ur," u du:#tan'-r "- I tann-2 u i lu

76. / .ot" u du : -i+cotn-'

"

-

I cotu-z u ilu

77.J r" . ' u du :#secn-2 u tan" +#/ '" t ' -2 u du

78. I csc' u du #csc'-z u cot" + #J t""-' u du

7s. / ri^ mu sin nu du -- -W + W+ c

- f , sin (m * n)u , sr\(m --n)u- t c8 0 . l c o s m u c o s n u d u : - ; f f i + @ t L

f a cos( rn * n )u cos(m - n )u -L .81 . J s i nmu cosnu du - -@ r L

f82.

I u sin u du: sin u - tt cos ll + C

f83. I

i l cos u du: cos u * u sin z + C

84. I u'sin z du - 2u srn u * (2 - u2) cos u * CJ

WITH ANALYTIC GEOMETRYthird edition g(

(I

{ {\!\

it

LoulsLeltholdUNIVERSITY OF SOUTHERN CALIFORNIA

HARPER & ROW, PUBLISHERSNew York, Hagerstoutn, San Francisco, London

Sponsoring Editor: George J. TeleckiProject Editor: Karen A. fuddDesigner: Rita NaughtonProduction Supervisor: Francis X. GiordanoCompositor: Progressive TypographersPrinter and Binder: Kingsport PressArt Studio: J & R Technical Services Inc.Chapter opening art: "Study Light,, by patrick CaulfieldTHE CALCULUS WITH ANALYTIC GEOMETRY,Copyright @ 1.968, 1,972, t976 by Louis LeitholdAll rights reserved. Printed in the United States of America. No part of this book may beused or reproduced in any manner whatsoever without written permission except in thecase of brief quotations embodied in critical articles and reviews. For information addressHarper & Row, Publishers, Inc., 10 East 53rd street, New york, N.y. 10022.Library of Congress Cataloging in Publication DataLeithold, Louis.

The calculus, with analytic geometry.

Includes index.L. Calculus. 2. Geometry, Analytic, L Title.

QA303.L428 7975b 515'.1s 75-26639ISBN 0-06-043951-3

Edition

To Gordon Marc

Gontents

Chapter 1REAL NUMBERS,

INTRODUCTION TOANALYTIC GEOMETRY,

AND FUNCTIONSpage 1

Chapter 2LIMITS AND CONTINUIT}

paSe oc

Chapter 3THE DERIVATIVE

page 110

Preface )cu,

1.1 Sets, Real Numbers, and Inequalities 21.2 Absolute Value 141.3 The Number Plane and Graphs of Equations 2L1.4 Distance Formula and Midpoint Formula 281.5 Equations of a Line 331.6 The Circle 43L.7 Functions and Their Graphs 481.8 Function Notation, Operations on Functions, and Types of Func-

tions 56

2.1, The Limit of a Function 662.2 Theorems on Limits of Functions 742.3 One-Sided Limits 852.4 Infinite Limits 882.5 Continuity of a Function at a Number 972.6 Theorems on Continuity 1-0L

3.1 The Tangent Line 1-1-i-3.2 Instantaneous Velocity in Rectilinear Motion 11,53.3 The Derivative of a Function L21-3.4 Differentiability and Continuity L263.5 Some Theorems on Differentiation of Algebraic Functions 1.303.6 The Derivative of a Composite Function L383.7 The Derivative of the Power Function for Rational Exponents L423.8 Implicit Differentiation 1-453.9 The Derivative as a Rate of Change 1.503.10 Related Rates 1543.1L Derivatives of Higher Order 157

-TtI

\

Viii CONTENTS

Chapter 4TOPICS ON LIMITS,

CONTINUITY, AND THEDERIVATIVE

page 164

Chapter 5ADDITIONAT APPLICATIONS

OF THE DERIVATIVEpage 204

Chapter 5THE DIFFERENTIAL AND

ANTIDIFFERENTIATIONpage 243

Chapter 7THE DEFINITE INTEGRAL

page 275

Chapter 8APPLICATIONS OF THE

DEFINITE INTEGRALpage 323

5.1 The Differential 2446.2 Differential Formulas 2495.3 The Inverse of Differentiation 2536.4 Differential Equations with Variables Separable6.5 Antidifferentiation and Rectilinear Motion 2656.6 Applications of Antidifferentiation in Economics

4.L Limits at Infinity 7654.2 Horizontal and Vertical Asymptotes L714.3 Additional Theorems on Limits of Functions 1.744.4 Continuity on an Interval 1774.5 Maximum and Minimum Values of a Function 19L4.6 Applications Involving an Absolute Extremum on a Closed In-

terval L894.7 Rolle's Theorem and the Mean-Value Theorem lgs

5.1 Increasing and Decreasing Functions and the First-DerivativeTest 205

5.2 The second-Derivative Test for Relative Extrema 2i.15.3 Additional Problems Involving Absolute Extrema 2135.4 Concavity and Points of Inflection 2205.5 Applications to Drawing a Sketch of the Graph of a Function 2275.6 An Application of the Derivative in Economics 280

261

269

7.L The Sigma Notation 2767.2 Area 28L7.3 The Definite Integral 2887.4 Properties of the Definite Integral 2gG7.5 The Mean-Value Theorem for Integrals 3067.6 The Fundamental Theorem of the Calculus SLi.

8.1 Area of a Region in a Plane 3248.2 Volume of a Solid of Revolution: Circular-Disk and Circular-Ring

Methods 3308.3 Volume of a Solid of Revolution: Cylindrical-Shell Method 3J68.4 Volume of a Solid Having Known Parallel Plane Sections 3418.5 Work 3448.6 Liquid Pressure 3488.7 Center of Mass of a Rod 35L8.8 Center of Mass of a Plane Region 3568.9 Center of Mass of a Solid of Revolution 3668.10 Length of Arc of a Plane Curve 372

Chapter 9 9.1LOGARITHMIC AND 9.2

EXPONENTIAL FUNCTIONS 9.3page 38L 9.4

9.59.6

CONTENTS

The Natural Logarithmic Function 382The Graph of the Natural Logarithmic Function 391'The Inverse of a Function 395The Exponential Function 405Other Exponential and Logarithmic FunctionsLaws of Growth and DecaY 420

The Sine and Cosine Functions 431Derivatives of the Sine and Cosine Functions 438Integrals Involving Powers of Sine and Cosine 447The Tangent, Cotangent, Secant, and Cosecant Functions 452An Application of the Tangent Function to the Slope of aLine 461,

10.5 Integrals Involving the Tangent, Cotangent, Secant, and Co-secant 466

4L4

ChaPter L0TRIGONOMETRIC

FUNCTIONSpage 430

Chapter LLTECHNIQUES OF

INTEGRATIONpage 497

10.1t0.210.310.410.s

Chapter L2 Lz.LHYPERBOTIC FUNCTIONS I2,2

page 535 12.3

Chapter 13 13.1POLAR COORDINATES T3.2

page 554 13.313.413.5

Chapter 1.4THE CONIC SECTIONS

page 578

InverseTrigonometricFunctions 47LDerivatives of the Inverse Trigonometric FunctionsIntegrals Yielding Inverse Trigonometric Functions

Introduction 492Integration by Parts 493Integration by Trigonometric Substitution 498Integration of Rational Functions by Partial Fractions.2: The Denominator Has Otly Linear Factors 504Integration of Rational Functions by Partial Fractions.4: The Denominator Contains Quadratic Factors 51.2Integration ofRational Functions of Sine and CosineMiscellaneous Substitutions 51-9The Trapezoidal Rule 521,Simpson's Rule 526

The Hyperbolic Functions 536The Inverse Hyperbolic Functions 543Integrals Yielding Inverse Hyperbolic Functions 548

The Polar Coordinate System 555Graphs of Equations in Polar Coordinates 560Intersection of Graphs in Polar Coordinates 567Tangent Lines of Polar Curves 571-Area of a Region in Polar Coordinates

L0.710.810.9

11 .1LL.211 .3Lt.4

11.5

LL.6'Ll..71.L.8'n.9

477484

Cases L and

Cases 3 and

516

LA.l The Parabola 579L4.2 Translation of Axes 58314.3 Some Properties of Conics 588L4.4 Polar Equations of the Conics 592

573

II{:L

INi

1

X CONTENTS

L4.5 Cartesian Equations of the Conics Sgg14.5 The Ellipse 606L4.7 The Hyperbola 61314.8 Rotation of Axes 620

chapter L5 15.1 The Indeterminate Form 0/0 629INDETERMINATE FORMS, 1,5.2 Other Indeterminate Forms 636

IMPROPER INTEGRALS, AND 15.3 Improper Integrals with Infinite Limits of IntegrationTAYLOR'S FORMULA lS.4 Other Improper Integrals 647

Page 628 15.5 Taylor's Formula 6s7

Chapter 15 1,5.1 Sequences 5G0INFINITE SERIES 1'6.2 Monotonic and Bounded Sequence s 667page 65e 16.g Infinite Series of Constant Term s 673

1,6.4 Infinite Series of Positive Terms 6g41,6.5 The Integral Test 69415.6 Infinite series of Positive and Negative Terms 697L6 7 Power Series 707L6.8 Differentiation of Power Series 71316.9 Integration of Power Series 72215.10 Taylor Series 729L6.ll The Binomial Series 738

641

Chapter 17VECTORS IN THE PLANE

AND PARAMETRICEQUATIONS

page 745

Chapter 18VECTORS IN THREE-

DIMENSIONAL SPACE ANDSOLID ANALYTIC

GEOMETRYpage 810

77.1, Vectors in the Plane 74677.2 Properties of Vector Addition and Scalar Multiplication 751L7.3 Dot Product 75617.4 vector-Valued Functions and Parametric Equations 76377.5 Calculus of Vector-Valued Function s 77277.6 Length of Arc 779t7.7 Plane Motion 785I7.8 The Unit Tangent and Unit Normal Vectors and Arc Length as a

Parameter 792L7.9 Curvature 79617.10 Tangential and Normal Components of Acceleration g04

18.1 R3, The Three-Dimensional Number Space 8L1L8.2 Vectors in Three-Dimensional Space 81818.3 The Dot Product in V, 82518.4 Planes 82918.5 Lines in R3 836L8.6 Cross Product 8421,8.7 Cylinders and Surfaces of Revolution BS218.8 Quadric Surfaces 85818.9 Curves in R3 86418.10 Cylindrical and Spherical Coordinates 872

Chapter 19DIFFERENTIAL CALCULUS

OF FUNCTIONS OFSEVERAL VARIABLES

page 880

Chapter 20DIRECTIONAL DERIVATIVES,

GRADIENTS, APPTICATIONSOF PARTIAL DERIVATIVES,

AND LINE INTEGRALSpage 944

ChaPter 2|MULTIPTE INTEGRATION

page 1001

APPENDIXpage A-1

CONTENTS

L9.']-. Functions of More Than One Variable 88Llg.2 Limits of Functions of More Than one Variable 889tg.3 Continuity of Functions of More Than One Variable 90019.4 Partial Derivatives 90519.5 Differentiability and the Total Differential 91'319.6 The Chain Rule 92619.7 Higher-Order Partial Derivatives 934

20.L Directional Derivatives and Gradients 94520.2 Tangent Planes and Normals to Surfaces 95320.3 Extrema of Functions of Two Variables 95620.4 Some Applications of Partial Derivatives to Economics 96720.5 Obtaining a Function from Its Gradient 97520.6 Line Integrals 98120.7 Line Integrals Independent of the Path 989

21.1, The Double Integral L0022'1..2 Evaluation of Double Integrals and Iterated Integrals 100821.3 Center of Mass and Moments of Inertia 1'0L621.4 The Double Integral in Polar Coordinates 1'02221.5 Area of a Surface 102821.6 The Triple Integral 103421.7 The Triple Integral in Cylindrical and Spherical Coordinates L039

Table 1.Table 2Table 3Table 4Table 5Table 6Table 7

Powers and Roots A-2Natural Logarithms A-3Exponential Functions A-5Hyperbolic Functions A-L2Trigonometric Functions A-13Common Logarithms A-1-4The Greek Alphabet A-15

ANSWERS TO ODD-NUMBERED EXERCISES A.1.7

INDEX 4.45

ACKNOWLEDGMENTS Reviewers of rhe CatcutuProfessor William D.Professor Archie D. lProfessor Phillip ClaProfessor Reuben WProfessor Jacob GolilProfessor Robert K. Goodrich, University of ColoradoProfessor Albert Herr, Drexel UniversityProfessor James F. Hurley, University of ConnecticutProfessor Gordon L. Miller, Wisconsin State UniversitvProfessor William W. Mitchell, Jr., phoenix CollegeProfessor Roger B. Nelsen, Lewis and Clark Colle-geProfessor Robert A. Nowlan, southern connectictit state collegeSister Madeleine Rose, Holy Names CollegeProfessor George W. Schultz, St. petersbuig Junior CollegeProfessor Donald R. Sherbert, Universitv of IllinoisProfessor ]ohn V_adney, Fulton-Montgomery community collegeProfessor David Whitman, San Diego State College

Production Staff at Harper & RowGeorge Telecki, Mathematics EditorKaren fudd, Project EditorRita Naughton, Designer

Assistants for Answers to ExercisesJacqueline Dewar, Loyola Marymount UniversityKen Kast, Logicon, Inc.|ean Kilmer, West Covina Unified School District

Cover and Chapter Openins ArtistPatrick Cadlfield, Londin, England

To these _people and to all the users of the first and second editions who have suggested

changes, I express my deep appreciation.L. L.

r l ]FTETAGE

This third edition of THE CALCULUS WITH ANALYTIC GEOMETRY,like the other two, is designed for prospective mathematics majorsas well as for students whose primary interest is in engineering, thephysical sciences, or nontechnical fields. A knowledge of high-schoolalgebra and geometry is assumed.

The text is available either in one volume or in two parts: Part I con-sists of the - first sixteen chapters, and Part II comprises Chapters 16through 21 (Chapter L6 on Infinite Series is included in both parts to makethe use of the two-volume set more flexible). The material in Part I con-sists of the differential and integral calculus of functions of a singlevariable and plane analytic geometrlz, and it may be covered in a one-yearcourse of nine or ten semester hours or twelve quarter hours. The secondpart is suitable for a course consisting of five or six semester hours oreight quarter hours. It includes the calculus of several variables and atreatment of vectors in the plane, as well as in three dimensions, with avector approach to solid analytic geometry.

The objectives of the previous editions have been maintained. I haveendeavored to achieve a healthy balance between the presentation ofelementary calculus from a rigorous approach and that from the older,intuitive, and computational point of view. Bearing in mind that a text-book should be written for the student, I have attempted to keep the pre-sentation geared to a beginner's experience and maturity and to leave nostep unexplained or omitted. I desire that the reader be aware that proofsof theorems are necessary and that these proofs be well motivated andcarefully explained so that they are understandable to the student who hasachieved an average mastery of the preceding sections of the book. If atheorem is stated without proof ,I have generally augmented the discus-sion by both figures and examples, and in such cases I have alwaysstressed that what is presented is an illustration of the content of thetheorem and is not a proof.

Changes in the third edition occur in the first five chapters. The first

xlv PREFACE

section of Chapter I has been rewritten to give a more detailed expositionof the real-number system. The introduction to analytic geometry in thischapter includes the traditional material on straight lines as well as thatof the circle, but a discussion of the parabola is postponed to Chapter 14,The Conic Sections. Functions are now introduced in Chapter 1. I havedefined a function as a set of ordered pairs and have used this idea topoint up the concept of a function as a correspondence between sets ofreal numbers.

The treatment of limits and continuity which formerly consisted often sections in Chapter 2 is now in fwo chapters (2 and 4), with the chap-ter on the derivative placed between them. The concepts of limit and con-tinuity are at the heart of any first course in the calculus. The notion of alimit of a function is first given a step-by-step motivation, which bringsthe discussion from computing the value of a function near a number,through an intuitive treatment of the limiting process, up to a rigorousepsilon-delta definition. A sequence of examples progressively graded indifficulty is included. All the limit theorems are stated, and some proofsare presented in the text, while other proofs have been outlined in theexercises. In the discussion of continuity, I have used as examples andcounterexamples "common, everyday" functions and have avoided thosethat would have little intuitive meaning.

In Chapter 3, before giving the formal definition of a derivative, Ihave defined the tangent line to a curve and instantaneous velocity inrectilinear motion in order to demonstrate in advance that the concept ofa derivative is of wide application, both geometrical and physical. The-orems on differentiation are proved and illustrated by examples. Ap-plication of the derivative to related rates is included.

Additional topics on limits and continuity are given in Chapter 4.Continuity on a closed interwal is defined and discussed, followed bythe introduction of the Extreme-Value Theorem, which involves suchfunctions. Then the Extreme-Value Theorem is used to find the absoluteextrema of functions continuous on a closed interval. Chapter 4 concludeswith Rolle's Theorem and the Mean-Value Theorem. Chapter 5 givesadditional applications of the derivative, including problems on curvesketching as well as some related to business and economics.

The antiderivative is treated in Chapter 6. I use the term "antidif-ferentiation" instead of indefinite integration, but the standard notation! f (x) dx is retained so that you are not given a bizarre new notation thatwould make the reading of standard references difficult. This notation willsuggest that some relation must exist between definite integrals, intro-duced in Chapter 7, and antiderivatives, but I see no harm in this as longas the presentation gives the theoretically proper view of the definiteintegral as the limit of sums. Exercises involving the evaluation of defi-nite integrals by finding limits of sums are given in Chapter 7 to stressthat this is how they are calculated. The introduction of the definite inte-

PREFACE

gral follows the definition of the measure of the area under a curye as alimit of sums. Elementary properties of the definite integral are derivedand the fundamental theorem of the calculus is proved. It is emphasizedthat this is a theorem, and an important one, because it provides us withan alternative to computing limits of sums. It is also emphasized that thedefinite integral is in no sense some special type of antiderivative. InChapter 8 I have given numerous applications of definite integrals. Thepresentation highlights not only the manipulative techniques but alsothe fundamental principles involved. In each application, the definitionsof the new terms are intuitively motivated and explained.

The treatment of logarithmic and exponential functions in Chapter 9is the modern approach. The natural logarithm is defined as an integral,and after the discussion of the inverse of a function, the exponentialfunction is defined as the inverse of the natural logarithmic function. Anirrational power of a real number is then defined. The trigonometricfunctions are defined in Chapter 10 as functions assigning numbers tonumbers. The important trigonometric identities are derived and usedto obtain the formulas for the derivatives and integrals of these functions.Following are sections on the differentiation and integration of the trig-onometric functions as well as of the inverse trigonometric functions.

Chapter LL, on techniques of integration, involves one of the mostimportant computational aspects of the calculus. I have explained thetheoretical backgrounds of each different method after an introductorymotivation. The mastery of integration techniques depends upon theexamples, and I have used as illustrations problems that the student willcertainly meet in practice, those which require patience and persistenceto solve. The material on the approximation of definite integrals includesthe statement of theorems for computing the bounds of the error involvedin these approximations. The theorems and the problems that go withthem, being self-contained, can be omitted from a course if the instructorso wishes.

A self-contained treatment of hyperbolic functions is in Chapter 12.This chapter may be studied immediately following the discussion of thecircular trigonometric functions in Chapter L0, if so desired. The geo-metric interpretation of the hyperbolic functions is postponed untilChapter L7 because it involves the use of parametric equations.

Polar coordinates and some of their applications are given in Chap-ter 13. In Chapter 1.4, conics are treated as a unified subject to stress theirnatural and close relationship to each other. The parabola is discussed inthe first two sections. Then equations of the conics in polar coordinatesare treated, and the cartesian equations of the ellipse and the hyperbolaare derived from the polar equations. The topics of indeterminate forms,improper integrals, and Taylor's formula, and the computational tech-niques involved, are presented in Chapter L5.

I have attempted in Chapter 16 to give as complete a treatment of

xvr PREFACE

infinite series as is feasible in an elementary calculus text. In addition tothe customary computational material, I have included the proof of theequivalence of convergence and boundedness of monotonic sequencesbased on the completeness property of the real numbers and the proofsof the computational processes involving differentiation and integrationof power series.

The first five sections of Chapter L7 on vectors in the plane can betaken up after Chapter 5 if it is desired to introduce vectors earlier in thecourse. The approach to vectors is modern, and it serves both as an intro-duction to the viewpoint of linear algebra and to that of classical vectoranalysis. The applications are to physics and geometry. Chapter 18 treatsvectors in three-dimensional space, and, if desired, the topics in the firstthree sections of this chapter may be studied concurrently with the corre-sponding topics in Chapter 17.

Limits, continuity, and differentiation of functions of several variablesare considered in Chapter L9. The discussion and examples are appliedmainly to functions of two and three variables; however, statements ofmost of the definitions and theorems are extended to functions of nvariables.

In Chaptet 20, a section on directional derivatives and gradients isfollowed by a section that shows the application of the gradient to findingan equation of the tangent plane to a surface. Applications of partialderivatives to the solution of extrema problems and an introduction toLagrange multipliers are presented, as well as a section on applications ofpartial derivatives in economics. Three sections, new in the third edition,are devoted to line integrals and related topics. The double integral of afunction of two variables and the triple integral of a function of threevariables, along with some applications to physics, engineering, andgeometry/ are given in Chapter 2/...

New to this edition is a short table of integrals appearing on the frontand back endpapers. However, as stated in Chapter L1, you are advisedto use a table of integrals only after you have mastered integration.

Louis Leithold

Real numberq, introductionto analytic geometryand functions

REAL NUMBERS, INTRODUCTION TO ANALYTIC GEOMETRY. AND FUNCTIONS

1.1 SETS, REAL NUMBERS, The idea of "set" is used extensively in mathematics and is such a ba-AND INEQUALITIES sic concept that it is not given a formal definition. We can say that a

set is a collection of objects, and the objects in a set are called the elementsof a set. We may speak of the set of books in the New York Public Library,the set of citizens of the United States, the set of trees in Golden GatePark, and so on. In calculus, we are concerned with the set of real numbers.Before discussing this set, we introduce some notation and definitions.

We want every set to be weII defined; that is, there should be somerule or property that enables one to decide whether a given object is oris not an element of a specific set. A pair of braces { } used with wordsor symbols can describe a set.

If S is the set of natural numbers less than 6,we can write the set S as

{ L , 2 , 3 , 4 , 5 }We can also write the set S as

{r, such that r is a natural number less than 6}where the symbol. "x" is called a"variable." A aariable is a symbol usedto represent any element of a given set. Another way of writing the aboveset S is to use what is called set-builder notation, where a vertical bar isused in place of the words "such that." Using set-builder notation todescribe the set S, we have

{rlr is a natural number less than 6}which is read "the set of all r such that r is a natural number less than G."

The set of natural numbers will be denoted by N. Therefore, we maywrite the set N as

{ 7 , 2 , 3 , I. lwhere the three dots are used to indicate that the list goes on and on withno last number. With set-builder notation the set N may be written as{rlr is a natural number}.

The symbol " e " is used to indicate that a specific element belongsto a set. Hence, we may write 8 N, which is read "8 is an element of N.,,The notation a,b e S indicates that both a andb are elements of S. Thesymbol fi is read "is not an element of." Thus, we read *

w as "+ is notan element of N."

We denote the set of all integers by /. Because every element of N isalso an element of / (that is, every natural number is an integer), we saythat N is a "subset" of /, written N E /.

:',.t Definition The set S is a subset of the set T, written S e T,if and only if every element

of S is also an element of T.If , in addition, there is at least one element of T

-J

1.1 SETS, REAL NUMBERS, AND INEQUALITIES

which is not an element of S, then S is a proper subset of T, and it is writ-tenS C T.

Observe from the definition that every set is a subset of itself, but a setis not a proper subset of itself.

In Definition 1.1.1, the "if and only if" qualification is used to com-bine two statements: (i) "the set S is a subset of the set T if every elementof S is also an element of T"; and (ii) "the set S is a subset of set T only ifevery element of S is also an element of 7," which is logically equivalentto the statement"if. S is a subset of.T, then every element of S is also anelement of. T."

o rLLUsrRArroN 1: Let N be the set of natural numbers and let M be theset denoted by {rlr is a natural number less than 1,0}. Because everyelement of M is also an element of N, M is a subset of N and we writeM e N. Also, there is at least one element of N which is not an elementof M, and so M is a proper subset of N and we may write M C N. Further-more, because {5} is the set consisting of the number 5, {6} C M, whichstates that the set consisting of the single element 5 is a Proper subset ofthe set M. We may also write 5 M, which states that the number 6 isan element of the set M. o

Consider the set {xlzx * L:0, and x e I}. This set contains no ele-ments because there is no integer solution of the equation 2x * 1. : 0.Such a set is called the "empty set" or the "null set."

1.t.2 Definition The empty set (or null sef) is the set that contains no elements. The emPtyset is denoted by the symbol A.

The concept of "subset" may be used to define what is meant by twosets being "equal."

1.1.3 Definition Two sets A and B are said to be equal, written A: B, if and only if A e BandB e A .

Essentially, this definition states that the two sets A and B are equalif and only if every element of A is an element of B and every element of Bis an element of A, that is, if the sets A and B have identical elements.

There are two operations on sets that we shall find useful as weproceed. These operations are given in Definitions 1.1.4 and 1.1.5.

1.1.4 Definition Let A and B be two sets. The union of A and B, denoted by A U B and read"A union 8," is the set of all elements that are in A or in B or in both Aand B.

REAL NUMBERS, INTRODUCTION TO ANALYTIC GEOMETRY, AND FUNCTIONS

ExAMPLE 1: Let A: {2, 4, 6,8,10, LzI , B : {1, 4, 9, '1.5} , andC - {2 ,10} . F ind

(a )AuB(c )BUC

SOLUTION:

( a )Au(b )Au(c )BU(d )Au

1..1..5 Definition Let A and Band tead " Aand B.

B - {1 ,2 , 4 , 6 ,9 ,9 , "1 .0 , 1 ,2 , 1 ,6 }C: {2, 4 , 6 ,8, 1 ,0, 12}C : { 1 , 2 , 4 , 9 , 1 , 0 , 1 , 6 }A: {2, 4 , 6 , 8 , I0 ,12} : 4

be two sets. The intersection of A and B, denoted by A n Bintersection 8," is the set of all elements that are in both A

(b )Auc(d )AuA

nxelvrpr.E 2: If A, B, and C arethe sets defined in Example 1,find

@)AnB(c )BnC

SOLUTION:

( a ) A n g : { 4 }( c ) B f i C : A

C: {2 , 1 .0 }A : { 2 , 4 , 6 , 8 , t 0 , 7 2 1 : 4

(b )An(d )An(b )Anc

@)AnA

1.1.6 Axiom(Closure and Uniqueness Laws)

1.1.7 Axiom(Commutatiae Laws)

L.L.8 Axiom(Associatiae Laws)

1."1,.9 Axiom(Distributiae Law)

The real number system consists of a set of elements called real numbersand two operations called addition and multiplication The set of real num-bers is denoted by Rt. The operation of addition is denoted by the symbol"*", and the operation of multiplication is denoted by the symbol',.',.rf a, b c Rt, a * b denotes the sum of a and b, and a - b (or ab) denotestheir product.

We now present seven axioms that give laws governing the operationsof addition and multiplication on the set Rl. The word axiom is used toindicate a formal statement that is assumed to be true without proof.

r f a,b e Rt, then sib is a unique real number, andab is a unique realnumber.

If a, b e Rr, then

a * b : b * a a n d a b : b a

If. a, b, c Rl, then

a + ( b * c ) : ( a + b ) + c a n d a ( b c ) : ( a b ) c

l f a ,b , c C R1, then

a ( b + c ) : a b * a c

1..L.10 Axiom There exist two distinct real numbers 0 and 1 such that for anv real num-J(Existence of Identity Elements) ber A,

a i l : a a n d A . t : a

1.1 SETS, REAL NI.JMBERS, AND INEOUALTTIES

L.1.11 Axiom For every real number a, there exists(Existence of Negatiae or Additiae lnaerse) of a (or additiae inaerse of a), denoted

such that

a * ( - s ) : Q

'1,.1.12 Axiom(Existence of Reciprocal or

Multiplic atia e lnu erse)

1..1..13 Definition

'i,.1.14 Definition

a real number called the negatiaeby -a (read "the negative of a"),

For every real number a, except 0, there exists a real number called thereciprocat of a (or multiplicatiae inaerse of a), denoted by a-t, such that

a ' a - r , : t

Axioms 1,.1,.6 through 1,.I.L2 are called field axioms because if theseaxioms are satisfied by a set of elements, then the set is called afield underthe two operations involved. Hence, the set R1 is a field under additionand multiplication. For the set / of integers, each of the axioms t.1.6through 1.1.11 is satisfied, but Axiom L.1,.1,2 is not satisfied (for instance,the integer 2 has no multiplicative inverse it /). Therefore, the set ofintegers is not a field under addition and multiplication.

lf a, b Rl, the operation of subtraction assigns to a andb areal number,denoted by a - b (read "a minus b"), called the difference of a and b, where

a - b : A * ( - b ) ( 1 )

Equality (1) is read "A minus b equals a plus the negatle of- b."

I f . a , b G R l , a n d b * 0 , t h e o p e r a t i o n o f d i a i s i o n a s s i g n s t o a a n d b a r c a lnumber, denoted by o + b (read "a divided by b"), called the quotient ofa and b, where

a : b : A ' b - r

Other notations for the quotient of a and b ate

! and albb

By using the field axioms and Definitions 1.1.13 and 1.1.14, we canderive properties of the real numbers from which follow the familiaralgebraic operations as well as the techniques of solving equations,factoring, and so forth. In this book we are not concerned with showinghow such properties are derived from the axioms.

Properties that can be shown to be logical consequences of axioms aretheorems. In the statement of most theorems there are two parts: the "if"part, called the hypothesis, and the "then" part, called the conclusion. Theargument verifying a theorem is a proof. A proof consists of showingthat the conclusion follows from the assumed truth of the hypothesis.


L.1..L5 Axiom(Order Axiom)

The concept of a real number being "positive" is given in the fol-lowing axiom.

In the set of real numbers there exists a subset called the positiae numberssuch that

(i) if a Rr, exactly one of the following three statements holds:

a: 0 a is posi t ive -a is posi t ive.(ii) the sum of two positive numbers is positive.

(iii) the product of two positive numbers is positive.

Axiom 1.1.15 is called the order axiom because it enables us to orderthe elements of the set Rr. In Definit ions 'J,.7.77 and 1.1.18 we use thisaxiom to define the relations of "greater than" and "less than" on Rr.

The negatives of the elements of the set of positive numbers form theset of "negative" numbers, as given in the following definition.

The real number a is negatiae iI and only if -a is positive.

From Axiom 1.1.15 and Definit ion 7.1..76 it follows that a real numberis either a positive number, a negative number, or zero. Ary real numbercan be classified as a rqtional number or an irrational number. A rationalnumber is any number that can be expressed as the ratio of two integers.That is, a rational number is a number of the form plq, where p a:nd, qare integers and q + 0.The rational numbers consist of the following:

The integers (positive, negative, and zero). ,

-5 , -4 , -3 , -2 , -L , 0 , 1 , 2 , 3 , 4 , 5 , .

The positive and negative fractions such as

+-#+The positive and negative terminating decimar.s such as

1*1.16 Definition

3,251L,000,000

The positive and negative nonterminating repeating decimals such as0.333. :+ -0549s49s49. . . : - f tThe real numbers which are not rational numbers are called irrational

numbers. These are positive and negative nonterminating, nonrepeatingdecimals, for example,

rt: t.732. n : 3.L4'1,59. tan 140o: -0.839L.

The field axioms do not imply any ordering of the real numbers. Thatis, by means of the field axioms alone we cannot state that 1 is less than

2s6:ffi -0.003251 :


2,2isless than 3, and so on. However, we have introduced the order axiom(Axiom 1.1.15), and because the set Rr of real numbers satisfies the orderaxiom and the field axioms, we say that Rl is an ordered field.

We use the concept of a positive number given in the order axiomto define what we mean by one real number being "less than" another.

11*\7 Def in i t ion I f a,b Rt, then a is less than b (wr i t ten ab) i f andon ly i f b i s lessthan a; with symbols we write

a l b i f a n d o n l y i f b < a

l.l.l9 Definition The symbols = (" is less than or equal to") and = (" is greater than orequal to") are defined as follows:

(1) a - b i f and only i f e i ther a < b or a:b.(ii\ a > b if. and only if either a > b ot a: b-

The statements a 1b, a) b, a 0 if and only if a is positivq.(ii) a < 0 if and only if a is negative.

(1ii) a > 0 if and only if -a < 0.(iv) a < 0 if and only if -a > 0.

1 . 7 . 2 1 T h e o r e m I f . a < b a n d b 1 c , t h e n a 1 c .

o rLLUsrRArroN 3: 3 < 5 and 6


1.1.24 Theorem lf a < b, and c is any positive number, then ac I bc.

o TLLUSTRATToN 6: 2 I 5; so 2 . 4 < s

L.l.25 Theorem rf a < b, and c is any negative number, then ac ) bc.

o rLLUSrRArroN 7: 2 < 5; so 2(-4) > 5(-4).

1 , . 1 . 2 6 T h e o r e m I f 0 < a < b a n d 0 ( c ( d , t h e n a c < b d .

1,.1.27 Theorem

ljl..28 Theorem

1.1.29 Theorem

o r L L U S r R A r r o N 8 : 0 < 4 < 7 a n d 0 < 8 < 9 ; s o 4 ( 8 ) < 7 ( 9 ) .

Theorem 1'.1.24 states that if both members of an inequality are multi-plied by a positive number, the direction of the inequality remainsunchanged, whereas Theorem 1.1.25 states that if both members of aninequality are multiplied by a negative number, the direction of theinequality is reversed. Theorems 7.7.24 and 1.1..25 also hold for division,because dividing both members of an inequality by a number d is equiv-alent to multiplying both members by lld.

To illustrate the type of proof that is usually given, we present aproof of Theorem 1.L.23:

By hypothesis a I b. Then b - a is positive (by Definition 1..1,.17).By hypothesis c I d. Then d - c is positive (by Definition 1.1.17).Hence, (b - a) + (d- c) is posi t ive (by Axiom 1.1.15( i i ) ) .Therefore, (b t d) - (a * c) is posi t ive (by Axioms 1.1.g and l . l .T

and Definit ion 1.1.13).Therefore, A * c < b + d (by Definit ion 1.l. l7).The following theorems are identical to Theorems 1..I.21. to 1.I.26

except that the direction of the inequality is reversed.

If a > b and b ) c, then a ) c.

. r L L U s r R A r r o N 9 : 8 > 4 a n d 4 > - 2 ; s o 8 ) - 2 . .r f a > b, then a* c > b + c, and a- c > b- c i f c is any real number.

o rLLUSrRArroN 10: 3 = -5; so 3 - 4 > -5 - 4.

If a > b and c ) d, then a I c > b + d.

o r L L U S r R A r r o N t ' 1 , : 7 > 2 a n d 3 > - 5 ; s o 7 * 3 > 2 + ( - 5 ) . oIf a > b and if c is any positive number, then ac ) bc.o rLLUsrRArroN 12: -3 > -7; so (-3) > eDA.

L.1.30 Theorem

n1.1.31 Theorem

1.1.32 Theorem

7 9

1.L.33 Definition


lf. a > b and if c is any negative number, then ac I bc.

o rLLUsrRArroN 13: -3 > -7; so (-3)(-4) < (-7)(-4). o

If a > b > }and c > d > 0, then ac > bd.o r L L U S r R A r r o N 1 4 : 4 > 3 > 0 a n d 7 > 5 ) 0 ; s o 4 ( 7 ) > 3 ( 6 ) . o

So far we have required the set Rl of real numbers to satisfy the fieldaxioms and the order axiom, and we have stated that because of this re-quirement Rr is an ordered field. There is one more condition that is im-posed upon the set R1. This condition is called the axiom of completeness(Axiom 16.2.5). We defer the statement of this axiom until Section 16.2because it requires some terminology that is best introduced and dis-cussed later. However, we now give a geometric interpretation to the setof real numbers by associating them with the points on a horizontal line,called an axis. The axiom of completeness guarantees that there is a one-to-one correspondence between the set Rl and the set of points on an axis.

Refer to Figure 1.1.1. A point on the axis is chosen to represent thenumber 0. This point is called the origin A unit of distance is selected.Then each positive number x is represented by the point at a distance of runits to the right of the origin, and each negative number x is representedby the point at a distance of -x units to the left of the origin (it should benoted that if r is negative, then -r is positive). To each real number therecorresponds a unique point on the axis, and with each point on the axisthere is associated only one real number; hence, we have a one-to-onecorrespondence between Rl and the points on the axis. So the points onthe axis are identified with the numbers they represent, and we shalluse the same symbol for both the number and the point representingthat number on the axis. We identify Rl with the axis, and we call Rl thereal number line.

We see that a < b if and only if the point representing the number ais to the left of the point representing the number b. Similarly, a > b ifand only if the point representing a is to the right of the point repre-senting b. For instance, the number 2 is less than the number 5 and thepoint 2 is to the left of the point 5. We could also write 5 ) 2 and saythat the point 5 is to the right of the point 2.

A numberx is be tween a andb i f a ( rand x

aF igu re 1 .1 .3

10 REAL NUMBERS, INTRODUCTION TO ANALYTIC GEOMETRY, AND FUNCTIONS

The "closed interval" from 4 to b is the open interval (a, ,) togetherwith the two endpoints c and b.

1.1.34 Definition The closed interaal from a to b, denoted by lo, bl, is defined by- b , b l : { x l a - x - b }

a

F igu re 1 .1 .2 Figure 1,.1,.2 i l lustrates the open interval (a,b) and Fig. 1.1.3 i l lustratesthe closed interval la, bl.

The "interval half-open on the lef.t" is the open interval (a, b) togetherwith the right endpoint b.

The interaal half-open on the left, denoted by (o, bl, is defined by( a , b l : { x l a < x - b }

we define an "interval half-open on the right" in a similar way.

The interaal half-open on the right, denoted by lo, b), is defined byl a , b ) : { r l a = x < b }

Figure 1'.1.4 i l lustrates the interval (a,bl and Fig. 1.1.5 i l lustrates theinterval la, b).

we shall use the symbol +@ ("positive infinity") and the symbol-m ("negative infinity"); however, care must be taken not to confuse thesesymbols with real numbers, for they do not obey the properties of thereal numbers

7.1.37 Definition (i) (a, +-) : {xlx > a}( i i ) (- .o, b): {xlx < b}

( i i i ) la, +*7 : {xlx > a}( iv) ( -m, b l : {x lx - b}(v) (-*, **) : Rl

a

F igu re 1 .1 .6

-

bF igu re 1 .1 .7

b

" 1.1.35 Definition

'1,.1.36 Definition

-

a bF igu re 1 .1 .4

f---La b

F igu re 1 .1 .5

Figure 1,.1,.6 illustrates the interval (a, *a), and Fig. r.7.T illustratesthe interval (-*, b). Note that (-m,1oo) denotes the set of all real numbers.

For each of the intervals (a,b), [a, b], lq, b), and (a, b] the numbers a

EXAMPLE 3: Find the solutionset of the inequality

2 * 3 x < 5 r * 8

Illustrate it on the real numberline.

1.1 SETS, REAL NUMBERS, AND INEQUALITIES 11

of its endpoints, and a closed interval can be regarded as one which con-tains all of its endpoints. Consequently, the intewalla,ao). is consideredto be a closed interval because it contains its only endpoint a. Similarly,(--, b] is a closed interval, whereas (a, **) and (-oo, b) are oPen. The in-tervals la, b) and (a, bl are neither open nor closed. The interval (--, *-)has no endpoints, and it is considered both oPen and closed.

Intervals are used to represent "solution sets" of inequalities in onevariable. The solution set of such an inequality is the set of all numbersthat satisfy the inequality.

solurroN: lf x is a number such that

2 * 3 x < 5 r * 8

then

2 t 3x - 2 < 5x * 8 - 2 (by Theorem 1'.1'.22)or, equivalently,

3 x < 5 x I 6Then, adding -5r to both members of this inequality, we have

-2x < 6 (by Theorem 7.7.22)Dividing on both sides of this inequality by -2 and reversing the direc-tion of the inequality, we obtain

r ) -3 (by Theorem 1.1.25)What we have proved is that if

2 * 3 x < 5 r * 8then

x > - 3

Each of the steps is reversible; that is, if we start withx > - 3

we multiply on each side by -2, reverse the direction of the inequality,and obtain

- 2 x < 6

Then we add 5x and 2 to both members of the inequality, in which casewe get

2 i 3 x < 5 x * 8

12 REAL NUMBERS; INTRODUCTION TO ANALYTIC GEOMETRY, AND FUNCTIONS

-

- 3 0F igu re 1 .1 .8

ExAMPLE 4: Find the solutionset of the inequality

4 < 3 x - 2 < ' 1 , 0


0F igu re 1 .1 .9

rxauprn 5: Find the solutionset of the inequality

7 2 2x


Therefore, we can conclude that

2 *3x -3So the solution set of the given inequality is the interval (-3, +-;, whichis i l lustrated in Fig. 1.1.8.

solurroN: Adding 2 to each member of the inequalit|, we obtain6 < 3 x < 1 2

Dividing each member by 3, we get

2 1 x < 4

Each step is reversible; so the solution set is the interval (2,41, as is illus-strated in Fig. 7.1,.9.

solurroN: we wish to multiply both members of the inequality by x.However, the direction of the inequality that results will depend uponwhether x is positive or negative. So we must consider two cases.Case 7: r is positive; that is, x ) 0.

Multiplying on both sides by *, we obtain7 > 2 x

Dividing on both sides by 2,we getE > x or, equivalently, x < t

Therefore, since the above steps are reversible, the solution set of Case 1 is{xlx > 0} n {xlx < E} ot, equivalently, {rl0 < 7s E

Again, because the above steps are reversible, the solution set of Case 2 is{ r l * < 0 } n { x > g } : A .

From Cases 1 and 2 we conclude that the solution set of the given in-equality is the open interval (0,2), which is il lustrated in Fig. 1.1.10.

EXAMPLE 6: Find the solutionset of the inequality

x = 1 4x - 5


0

F i g u r e 1 , 1 . 1 1

ExavrplE 7: Find the solutionset of the inequality

( r+3 ) ( x+4 ) > 0

1.1 SETS, REAL NUMBERS, AND INEQUALITIES 13

solurroN: To multiply both members of the inequality by x-3, wemust consider two cases, as in Example 5.Case' l : x- 3 > 0; that is, x ) 3.

Multiplying on both sides of the inequality by x- 3, we getx 1 4 x - 1 2

Adding -4x to both members, we obtain-3x < -12

Dividing on both sides by-3 and reversing the direction of the inequality,we have

x ) 4Thus the solution set of Case 1 is {rlx > 3} n {xlr > 4} or, equivalently,{xlx > 4}, which is the interval (4, +oo;.Case2 : x -3 < 0 ; tha t i s , x < -3 .

Multiplying on both sides by * - 3 and reversing the direction ofthe inequ ality,we have

x ) 4 x - L 2or, equivalently,

-3x > -12

or, equivalently,x 1 4

Therefore, .x must be less than 4 and also less than 3. Thus, the solutionset of Case 2 is the interval (-*, 3).

If the solution sets for Cases 1 and 2 are combined, we obtain(--, 3) U (4, **), which is illustrated in Fig. 1.1.11.

solurroN: The inequality will be satisfied when both factors have thesames ign , tha t i s , i f x t 3 > 0andx + 4> 0 ,o r i f x t 3 < 0andr + 4 < 0 .Let us consider the two cases.C a s e l - : x t 3 > 0 a n d x t 4 ) 0 . T h a t i s ,

x > - 3 a n d x ) - 4Thus, both inequalities hold if x > -3, which is the interval (-3, +*;.C a s e 2 : x - 1 3 < 0 a n d x * 4 < 0 . T h a t i s ,

x 1 -3 and x 1 -4Both inequalities hold if x < -4, which is the interval (-*,-4).

If we combine the solution sets for Cases 1 and 2,we have (-*,-4) U(-3, +-;.


Exercises 7.LIn Exercises 1 through 10, l ist the elements of the given set if A: {0,2,4,6,8}, B :{ 0 , 3 , 6 , 9 } .I , . A U B 2 . C U D 3 .4 . C n D 5 . B U D 6 .7 . B n D 8 . A n C g .

10 . (AuB)n (cuD)

{1 , z , 4 , B} , C : {1 , i , s ,7 ,9 } , and D :

(Bnc )

l1"t*T"-:11 -ilP';3,t"i ,T:tyo.n set of the siven inequatity and illustrate the solution on the real number

A N B

A U C

@n D) u

1 . ' 1 . . 5 x + 2 > x - 6?

, l - x1 4 . 3 x - 5 < i x +- 4 ' - ' 3

1 7 . 2 > - 3 - 3 x > - 7

r )20r i- = 1.\ - r L - x2 3 . ( x - 3 ) ( r + s ) > 02 6 . * * 3 r * 1 > 0

1 2 . 3 - r < 5 * 3 x

1 5 . 1 3 > 2 x - 3 > 5

,f8. I .9

i 1 4

2 1 , . * > 4

2 4 . x 2 - 3 r * 2 > 02 7 . 4 f I 9 x < 9

? n x * ' j , - x" - ' 2 - r - 3 * x

?x - t 02 f - 6 x * 3 < 0

7 43 x - 7 - 3 - 2 x

Prove Theorem 1,.1,.22.

13.

1,6.

{ r : i 1 -3 > Z -7" x x

22.

25.28.

31,.29.

32.35.36.37.38.39.

33. Prove Theorem 1,.1.27. 94.Prove Theorems 1.1,.24 and 1.1.25.lf n > b = O,prove that a2 > W.lf. a and b are nonnegative numbers, and az ) br, prove that a > b.Prove that if a > 0 and b > 0, then A2 : b2 if and only if a: b.Pncve that if b > a > 0 and c ) 0, then

a * c a

b + t ' b

1.2 ABSOLUTE VALUE

40. Prove that if x 1 !, then r < t(x + y) < y.

The concept of the "absolute value" ofa number is used in some importantdefinitions in the study of calculus. Furtherrnore, you will need. tb workwith inequalities involving "absolute value."

The absolute aalue of x, denoted by lrl, is defined byl * l - * * i f x > 0

l t l : - t i f x < o

l0 l :0

1.2|t Definition

1.2 ABSOLUTE VALUE

Thus, the absolute value of a positive number or zero is equal tothe number itself. The absolute value of a nggative numloer is the cor-responding positive number because the negative of a negative numberis positive.. ILLUsTRATIoN 1: l3 l :3 ; l -51: - ( -5) :5; 18 -L4l : l -61: - ( -6) :6.

We see from the definition that the absolute value of a number iseither a positive number or zero; that is, it is nonnegative.

In terms of geometry, the absolute value of a number x is its distancefrom 0, without regard to direction. In general, lo - bl is the distancebetween a and b without regard to direction, that is, without regard towhich is the larger number. Refer to Fig. 1..2.1'.

We have the following properties of absolute values.

l" l < a l f . and only i f -a 1x I a,where a > 0.

Irl = a if. and only if -a < )c < a, where a > 0.ltl > a if andonly if x > a or x I -a,where a > 0.

ltl = a if andonly if x > a or x s -a,where a ) 0.

The proof of a theorem that has ant "if and only if " qualification re-quires two parts, as illustrated in the following proof of Theorem 1.2.2.plnr l.: Prove that lrl < a if.-a < x I a,where a ) O.Here,we havetoconsider two cases: .x > 0 and r < 0.

C a s e l : x = 0 .Then ltl : x. Because r < a,we conclude that lrl < a.

C a s e 2 : x < 0 .Then lrl : -r. Because -a < x, we aPPly Theorem L.1,.25 and obtain

a)-x or, equivalently, -x < a.But because -x: lr l , we have l* l < o.In both cases, then,

< n i f - a < x ( a , w h e r e a > 0panr 2: Prove that lrl < a only if -a 1x I A, where a > 0. Here wemust show that whenever the inequality ltl < a holds, the inequality-a < x 1a also holds. Assume that lrl < a and consider the two casesx > 0 a n d r ( 0 .C a s e T : x > 0 .

Then ltl : r. Because lxl < a,we conclude that r < a. Also, becausea ) 0, it follows from Theorem 1.1.31 that-a < 0. Thus/ we have -a 10< x < a , o t - a < x 1 a .

C a s e 2 : x < 0 .Then ltl :

1 5

l -u - a: la - br |

l , -b= la-b l1l-b

Figure 1.2.1

1.2.2 Theorem

"1..2.3 Corollary

\.2.4 Theorem

7.2.5 Corollary

Because ltl < a, we conclude that -x ( a. Also, be-

1 6 REAL NUMBERS, INTRODUCTION

EXAMPLE l-: Solve for x:

lar + 2l:5.

uxluplr 2: Solve for x:

lz* - r l : l4x + 31.

rxr.vrpu 4: Find the solutionset of the inequality

l x -51

EXAMPLE 5: Find the solutionset of the inequatity

l3=-,,*l = nl2+x l - '

1.2 ABSOLUTE VALUE 17

solurroN By Corollary '1..2.3, the given inequality is equivalent to

- ! , < .3= .2 ' - 4- - 2 * x

If we multiply by 2 * x, we must consider two cases, depending uPonwhether 2 t x is positive or negative.

C a s e ' l - : 2 * r > 0 o r x > . - 2 .Then we have- 4 ( 2 * r ) = 3 - 2 x = 4 ( 2 + x )

or, equivalently,- 8 - 4 x < 3 - 2 x < 8 * 4 x

So, if x ) -2, then also -8 - 4x < 3- 2x andS-2x < 8 + 4x.We solvethese two inequalities. The first inequality is

- 8 - 4 x < 3 - 2 x

Adding 2x * 8 to both members gives-2x < LL

Dividing both members by .2 and reversing the inequality sign, we

obtain

x > - #

The second inequality is

3 - 2 x < 8 t 4 xAdding -4x - 3 to both members gives

- 6 x < 5

Dividing both members by -5 and reversing the inequality sign, weobtain

x > - E

Therefore, if x ) -2, then thex > - # a n d r > - 8 .

inequality holds if and only if

Because all three inequalities r ) -2, )c > -+, and r > -B must besatisfied by the same value of x, we have x > -8, or the interval [-8, +-).C a s e 2 : 2 * x ( 0 o r x < - 2 .

Thus, we have- 4 ( 2 * r ) = 3 - 2 x > 4 ( 2 * x )

or, equivalently,- 8 - 4 x > 3 - 2 x > 8 1 4 x

1 8 REAL NUMBERS, INTROEUCTION TO ANALYTIC GEOMETRY, AND FUNCTIONS

Considering the left inequality, we have- 8 - 4 x > 3 - 2 x

or, equivalently,-2x >'l-.'1,

or, equivalently,rs -+

From the right inequality we have3 - 2 x > 8 * 4 x

or, equivalently,- 5 x > 5

or, equivalently,r s - 8

Therefore, if x 1-2,and r = -8.

the original inequality holds if and only if x = -t

Because all three inequalities must be satisfied by the same value ofx, we have x =-+, or the interval (-*, -u, l.

Combining the solution sets of Cases 1 and 2, we have as the solutionset (-o, -+l u [-8, **).

rxaprprr 6: Find the solutionset of the inequality

l 3 r+21 >5

soLUrIoN' By Theorem L.2.4, the given inequality is equivarent to3 x * 2 > 5 o r 3 x * 2 < - s ( 1 )

That is, the givgn inequality will be satisfied if either of the inequalitiesin (1) is satisfied.

Considering the first inequality, we have3 x * 2 > 5

or, guivalently,

x ) lTherefore, every number in the interval (1, +oo; is a solution.

From the second inequality, we have3 x t 2 < - 5

or, equivalently,x < - r s

Hence, every number in the interval (-*, -+) is a solution.

1.2 ABSOLUTE VALUE 19

1..2.8 TheoremThe Triangle lnequality

The solution set of the given inequality is therefore(1, +*;.

_+) u

You may recall from algebra that the symbol {a, where A > 0,is defined as the unique nonnegfltloe number x such that x2: A. We readtfi as "the principal square root of fl." For example,

f r :2 \6 :0 G:ENorE: tfr + -2; -2 is a square root of. 4, bfi {4 denotes only t]ne posi-tiae square root of 4.

Because we are concerned only with real numbers in this book, \/iis not defined lf. a < 0. From the definition of fr, it follows that

G: lx lFor example, {*: 5 and .... -{5zl':3.

The following theorems about absolute value will be useful later.

1.2.6 Theorem lf. a and b are any numbers, then

lab l : lo l ' lu lExpressed in words, this equation states that the absolute value of theproduct of two numbers is the product of the absolute values of thetwo numbers.

PROOF:

l ab l : '@: { f f i:G ' \E: lo l ' lb l

7.2.7 Theorem lf a is any number and b is any number except 0,

That is, the absolute value of the quotient of two numbers is the quotientof the absolute values of the two numbers.

The proof of Theorem 1.2.7 is left as an exercise (see Exercise 28).

If. a and b are any numbers, then

la+b l = la l + lb l

REAL NUMBERS, ]NTRODUCTION TO ANALYTIC GEOMETRY, AND FUNCTIONS

pRooF: By Definition '1..2.'1., either a :- l o l=a- la l

Furthermore,

o r a : _ l o l ; t h u s

- lb l =b - lb lFrom inequalities (2) and (3) and Theorem 1.1.23, we have

- ( la l + lb l ) = a +b < la l + lu lHence, from Corollary '/...2.3, it follows that

l a+b l = la l+ lb lTheorem r.2.8 has two important corollaries which we

and prove.

1.2.9 Corollary If a and b are any numbers, then

lo -b l = la l + lb l

(3)

I

now state

pRooF: lo - bl : la * (-b) l = lo l + l ( -b) l : la l +1.2.10 Corollary If a and b arc any numbers, then

l o l l b l s l a -b lpRooF: lol: l(r- b) + bl = lo - bl + lbl; thus, subtracting lblboth members of the inequality, we have

lo l - l b l = l o -b l

'.Exercises1,.2 **

st , T -tarL,

In Exercises 1 through L0, solve fo, Jl. l4x * 3l:74. 14+ 3r l :1

7 . lTx l : 4 -x

,olffil :n

3. 15 - 2xl :116. lx - 2 l : 13 -2x l

g. l '+31 :sl x - z l

Q ) l s t - 31 : l 3 r+51g .2x *3 : l a r+51

In Exercises L1. through 14, find all the values of r for which the number is real. At.i s+ ,:: ,j

. t > ) , 2 3v '

U t t t . - l f i - ' ) f i t u 5 i t n

. In Exercises 15 through 26, find the solution set of the given

1.3 THE NUMBER PLANE AND GRAPHS OF EQUATIONS 21

inequality, and illustrate the solution on the real number

17. l3x-41 -z20. 13+2x l< la- r l23. le - zxl =- l4xl

r c I l 1_ l26. !ff iI =1,-rr

line.

1 5 . l r + 4 1 < 718. 16 -2x l > 7

! . l r+41 = lzx-61)a. 1o-sr l =1- - ' l 3+x l - 2

27. Prcve Theorem 1.2.4.

1.3 THE NUMBER PLANE ANDGRAPHS OF EQUATIONS

1..3.1. Definition

16. l2x-sl < 319 . l 2x -s l >322. l3rl > 16 - }xl

l v * ? l2s. lffil .n28. Prove Theorem 1'.2.7.

In Exercises 29 through 32, solve for r and use absolute value bars to write the answer.

31..

29.

33. prcvethatif , andb are any nurnberg, then lc-bl = l4l +lbl.(HINI:Writea-basa'+(-b) and use Theorem 12'8')34. prcve that if a and b are any numben, then lal - lbl < la - bl. (Hrr.rr: Let lal: l(a - b) + bl, and use Theorem 1.2.8.)35. What single inequality is equivatent to the following two inequalities: s>b+ ca d a> b- c?

writing them in parentheses with a comma separating them as (I/ y). Notethat the ordered pair (3, 7) is different from the ordered pair (7,3).

The set of all ordered pairs of real numbers is called the number Plane'and each ordered Pafi (x, D is called a point in the number plane. Thenumber plane is denoted by R2.

fust as we can identify Rl with points on an axis (a one-dimensionalspace), we can identify R2 with points in a geometric plane (a two-dimen-dional space). The method we use with R2 is the one attributed to theFrench mathematician Rene Descartes (1595-1650), who is credited withthe invention of analytic geometry in L637.

A horizontal line is chosen in the geometric plane and is calledthe r axis. A vertical line is chosen and is called lhe y axis. The point ofintersection of the r axis and tlne y axis is called the origin and is denotedby the letter O. A unit of length is chosen (usually the unit length oneach axis is the same). We establish the positive direction on the r axisto the right of the origin, and the positive direction on the y axis abovethe origin.

F igu re 1 .3 .1


We now associate an ordered pair of real numbers (x, y) with a pointP in the geometric plane. The distance of P from the y axis (consideredas positive if P is to the right of the y axis and negative if P is to the leftof the y axis) is called the abscissa (or x coordinate) of P and is denotedby *.The distance of P from the r axis (considered as positive if P is above

r the x axis and negative if P is below the x axis) is called the ordinate (ory coordinate) of P and is denoted by y. The abscissa and the ordinate of apoint are called the rectqngular cartesian coordinates of the point. There isa one-to-one correspondence between the points in a geometric plane

7) and R2; that is, with each point there corresponds a unique ordered pair(x,y), and with each ordered pafu (x, y) there is associated only one point.This one-to-one correspondence is called a rectangular cartesian coordi-nate system, Figure 1.3.1 illustrates a rectangular cartesian coordinatesystem with some points plotted.

The x and y axes are called the coordinate axes, They divide the planeinto four parts, called quadrants, The first quadrant is the one in whichthe abscissa and the ordinate are both positive, that is, the upper rightquadrant. The other quadrants are numbered in the counterclockwisedirection, with the fourth, for example, being the lower right quadrant.

Because of the one-to-one correspondence, we identify R2 with thegeometric plane. For this reason we call an ordered pair (x, y) a point.Similarly, we refer to a "line" in R2 as the set of all points correspondingto a line in the geometric plane, and we use other geometric terms forsets of points in R2.

Consider the equation

A : x 2 - 2 ( 1 )where (x,y) is a point inR2. we call this an equation inR2.

By u solution of this equation, we mean an ordered pair of numbers,one for r and one for y, which satisfies the equation. Foi example, if r isreplaced by 3 in Eq. (1), we see that A:7; thus, x:3 and,y:7 const i -tutes a solution of this equation. If any number is substituted for x inthe right side of Eq. (1), we obtain a corresponding value for y.It is seen,then, that Eq. (1) has an unlimited number of solutions. Table 1.3.1 givesa few such solutions.

Table 1.3.7

x l 0 L 2 3 4 - 1 . - Z - 3 - 4

a : x2 -z l - z - 1 2 7 74 -1 2 7 14

If we plot the points having as coordinates the number pairs (x,y)satisfying Eq. (1), we have a sketch of the graph of the equation. In Fig.1'.3.2 we have plotted points whose coordinates are the number pairs ob-Figure 1 .3 .2

1.3 THE NUMBER PLANE AND GRAPHS OF EQUATIONS

tained frorn Table 1.3.1. These points are connected by u smooth curve.Ary point (x, y) on this curye has coordinates satisfying Eq. (L). Also,the coordinates of any point not on this curve do not satisfy the equation.We have the following general definition.

1.g.2 Definition The graph of an equation in I is the set of all points (x, y) in R2 whosecoordinates are numbers satisfying the equation.

We sometimes call the graph of an equation the locus of the equation.The graph of an equation in R2 is also called a curae. Unless otherrryisestated, an equation with two unknowns, x and y, is considered an equa-tion in R2.

sxelrpr-n L: Draw a sketch of thegraph of the equation'

Y ' - x - 2 : 0 Q )

soLUrIoN: Solving Eq. (2) for !, we havey : ! \ t X + 2

Equations (3) are equivalent to the two equations

Y : l x+2Y-_ \ ' x -2

The coordinates of all points that satisfy Eq. (3)Eq. (4) or (5), and the coordinates of any point that(4) or (5) will satisfy Eq. (3). Table 1.3.2 gives somex and y.

Table 1-.3.2

(3)

(4)(s)

will satisfy eithersatisfies either Eq.of these values of

Note that for any value of r -2 there are two values for y. A sketch of the graphof Eq. (2) is shown in Fig. 1.3.3. The graph is a parabola.

x

v

0 0 1 "1.

\/i -\/52

2

2 3

\/53 - L - 1 - 2

\/2 -\/2 -2 -\tr 1, -L 0

ExAMPLE 2: Draw sketches ofthe graphs of the equations

y : \ / x + 2 ( 5 )and

Y : - 1 / Y t ' 2

F igure 1 .3 .4

rxltrpr.n 3: Draw a sketch of thegraph of the equation

y : l x *31 0 )

Figure 1.3.6

rxetvrplr 4: Draw a sketch of thegraph of the equation

( x - 2 y + 3 ) ( V - x ' ) : 0 ( 8 )

the definition of the absolute value of a number, we

r * 3 > 0

i f r * 3 < 0


soLUrIoN: Equation (5) is the same as Eq. (a). The value of y is non-negative; hence, the graph of Eq. (6) is the upper half of the graph ofEq. (3). A sketch of this graph is shown in Fig. 1..3.4.

Similarly, the graph of the equation

Y : - \E+2a sketch of which is shown in Fig. 1.3.5, is the lower half of the parabolaof Fig. 1.3.3.

F igure 1 .3 .5

soLUTroN: Fromhave

y : x * 3 i fand

y - - ( x+3 )or, equivalently,

y : x * 3 i f x > - 3and

y - - ( x + 3 ) i f x 1 - 3Table 1.3.3 gives some values of x and y satisfying Eq. (Z).

Table 1.3.3

x 0 L 2 3 -L -2 -3 -4 -5 -6 -7 -8 -9

v 3 4 5 6 2 1 0 1, 2 3 4 5 6

A sketch of the graph of Eq. (7) is shown in Fig. 1,.9.6.

soLUTroN: By the property of real numbers that ab:O if anda: 0 or b :0, we have from Eq. (8)

x - 2 y * 3 : 0


y -xz :o

The coordinates of all points that satisfy Eq. (8) will satisfy eitherEq. (9) or Eq. (L0), and the coordinates of any point that satisfies eitherEq. (9) or (10) will satisfy Eq. (8). Therefore, the graph of Eq. (8) will con-sist of the graphs of Eqs. (9) and (10). Table 1..3.4 gives some values of rand y satisfying Eq. (9), and Table 1.3.5 gives some values of x and ysatisfying Eq. (10). A sketch of the graph of Eq. (8) is shown in Fig. 7.3.7.Table L.3.4

x 0 L 2 3 - 1 - 2 - 3 - 4 - 5

v 821r31 ,+o-+-1

Table 1.3.5

x 0 1 , 2 3 - 7 - 2 - 3

v 0 L 4 9 1 4 9

1.3.3 Definition An equation of a graph is an equation which is satisfied by the coordinatesof those, and only those, points on the graph.

o ILLUSTRATIoN 1: In R2, y:8 is an equation whose graph consists ofthose points having an ordinate of 8. This is a line which is parallel to thex axis, and 8 units above the r axis. o

In drawing a sketch of the graph of an equation, it is often helpfulto consider properties of symmetry of a graph.

1.3.4 Definition Two points P and Q are said to be symmetric with respect to a line if andonly if the line is the perpendicular bisector of the line segment PQ. Twopoints P and Q are said to be symmetric with respect to a thirdpoint if andonly if the third point is the midpoint of the line segment PQ.

(3'2\o o rLLUsrRArroN 2: The points (3,2) and (3,-2) are symmetric withrespect to the x axis, the points (3, 2) and (-3, 2) are symmetric withrespect to the y axis, and the points (3,2) and (-3, -2) are symmetricwithrespect to the origin (see Fig. 1.3.8). o

In general, the points (x, y) and (x, -y) are symmetric with respectto the x axis, the points (r, y) and (-x, y) are symmetric with respectto the y axLS, and the points (x,y) and (-r, -y) arc symmetric with respectto the origin.

(10)

F igu re 1 .3 .7

Figure 1 .3 .8

REAL NUMBERS, INTRODUCTION

1.3.5 Definition

L.3.6 Theorem(Tests t 'or Symmetry)

rxlrvrpr.r 5: Draw a sketch of thegraph of the equation

x y : 1 ( 1 1 )

TO ANALYTIC GEOMETRY, AND FUNCTIONS

are symmetric with respect to R.

From Definition 1.3.5 it follows that if a point (x, y) is on a graphwhich is symmetric with respect to the x axis, then the point (x,-y) alsomust be on the graph. And, if both the points (x,y) and (x,-y) are on thegraph, then the graph is symmetric with respect to the x axis. Therefore,the coordinates of the point (x,-y) as well as (x, y) must satisfy an equa-tion of the graph. Hence, we may conclude that the graph of an equationin r and y is symmetric with respect to the r axis if and only if an equiv-alent equation is obtained when y is replaced by -y in the equation. Wehave thus proved part (i) in the following theorem. The proofs of parts(ii) and (iii) are similar.

The graph of an equation in r and y is(i) symmetric with respect to the x axis if and only if an equivalent

equation is obtained when y is replaced by -y in the equation;(ii) symmetric with respect to the y axis if and only if an equivalent

equation is obtained when x is replaced by -x in the equation;(iii) symmetric with respect to the origin if and only if an equivalent

equation is obtained when r is replaced by -x and y is replacedby -y in the equation.

The graph in Fig. 1.3.2 is symmetric with respect to the y axis, andfor Eq. (1) an equivalent equation is obtained when x is replaced by -r.In Example 1 we have Eq. (2) for which an equivalent equation is ob-tained when y is replaced by-y, and its graph sketched inFig. L.3.3 issymmetric with respect to the r axis. The following example gives a graphwhich is symmetric with respect to the origin.

solurroN: We see that if in Eq. (1,1) x is replaced by -x and y is replacedby -y,an equivalent equation is obtained; hence,by Theorem 1.3.6(i i i)the graph is symmetric with respect to the origin. Table L.3.6 gives somevalues of x and y satisfying Eq. (11).

Table L.3.6


From Eq. (11) we obtain y : 'l.lx. We see that as r increases throughpositive values, y decreases through positive values and gets closer andcloser to zero. As x decreases through positive values, y increases throughpositive values and gets larger and larger. As I increases through nega-tive values (i.e., r takes on the values -4, -3, -2, -t, -+, etc.), y takeson negative values having larger and larger absolute values. A sketch ofthe graph is shown in Fig. 1..3.9.

Figure 1 .3 .9

Exercises 1.3In Exercises 1 through 6, plot the dven Point P and such of the following PoinB as may aPPly:(a) The point O suitr ttrit ttre tnJ through Q and P is perpendicular to the r axis and is bisected by it. Give the coordi-

nates of Q.(b) The poiniR such that the line through P and R is perpmdicular to and i6 bisected by the y axis. Give the coordinates

of R.(c) The point s such that the line through P and 5 iE bisected by the origrn. Give the_coordin"F-"-9f s.-1ai fn" poi"t T such that the line *Eough P and 7 is perpendicular to and is bisected by the als' line though the origin

bisecting the first and thtd quadrants. Give the coordinates of T.

In Exercises 7 through2S, draw a sketch of the graph of the equation.

1 . P ( 1 , - 2 )4. P(-2, -2)

7 ' Y : 2 x * 5l0.y : -nf f i13. x : -J1 5 . y : - l x + 2 119 . 4 f *9Y2 :36

2 . P ( -2 ,2 )5. P(-1 , -3)

8 ' Y : 4 x - 31 1 . y ' : x - 3L4. x: y2 * 'I..

1 7 . y : l r l - 520. 4x2 -9Y ' :36

3 . P (2 ,2 )5. P(0, -3)

(c) y ' :2x

(c\ y' : -2N

Y: \81y :5y : l x -51y - - l x l +zA : 4 x 33x2 - L3xy - L0y2: g

xa - 5x2y I 4y2: g22. Y' : 41cs 23. 4x2

- Az :0

2 5 . x z * y ' , : O 2 6 . ( 2 x + V - l ) ( 4 V + r 2 ) : 02 8 . ( y 2 - x * z ) ( V + t E - + ) : O29. Draw a sketch of the graph of each of the following equations:

(a) V: lzx (b\ v : -{2x30. Draw a sketch of the graph of each of the following equations:

(a) Y: !-2x (b\ y : -!-2x


3L. Draw a sketch of the graph of each of the following equations:( a ) r * 3 y : 0 ( b ) r - 3 y : 0 (c) r ' - 9y' :0

32. (a) Write an equation whose gtaPh is the r axis. (b) Write an equation whose graph is the y axis. (c) Write an equafionwhose graph is the set of all points on either the .x axis or the y axis.

33. (a) Write a,n__equ{iol whose graph consists of all points having an abscissa of 4. O) Write an equation whose graphconsistE of all points having an ordinate of -3.

34. Prove that a graPh that i5 symmetric with respect to both coordinate axes is also symmetric with rcspect to the origin.35' Pro-ee that a graPh that is symmeFic with rcspect to any two perpendicular lines is also sym.rretric with respect to their

point of intersection-

1.4 DISTANCE FORMULAAND MIDPOINT FORMULA

D ( - 2 , 4 )

C(-2, -g)

C D : 7(b)

v

C D : - 6(a)

If A is the point (x,.,y) and B is the point (xr,y) (i.e., A and B havethesame ordinate but different abscissas), then the directed distance fromA to B, denoted by AB, is defined ds x2 - x1.

o ILLUSTRATToN 1: Refer to Fig. 1.4.1(a), (b), and (c). u

AB : ' / . , 4(b)

F igure 1 .4 .1

If .4 is the point (3, 4) and B is the point (9, 4), then AB :9 - 3: 6.rf A is the point (-8, 0) and B is the point (6,0), then B :6 - (-s) : 14.If A is the point (4, 2) and B is the point (L,2), then AB : 1. - 4: -3. wesee that AB is positive if B is to the right of A, and AB is negative if B isto the left of A. o

If C is the point (x,yr) and D is the point (x, yr), then the directeddistance from C to D, denoted by CD, is defined as!z- !r.o rLLUsrRArroN 2: Refer to Fig. La.2@) and (b).

If C is the point (L, -2) and D is the point (1, -8), then CD : -g-J-2)--6. If C is the point (-2, -3) and D is the point (-2,4), thenCD:4- (-3) :7. Tlr .e number Cp is posi t ivei f D is above C, and CDis negative if D is below C. o

We consider a directed distance AB as the signed distance traveledby u particle that starts at A(rr, y) and travels to B(x2, !). ln such a case,the abscissa of the particle changes from 11 to x2, dfld we use the notationAr ("delta x") to denote this change; that is,

A ,x : xz - x t

Therefore, AB: Lx.

B(r, z) A(4,2)

Figure 1.4.2

Pr(x", vz)

F igure 1 .4 .3

EXAMPLE l.: If a point P(x, U) issuch that its distance fromA(3,2) is always twice its dis-tance from B(-4, L), find anequation which the coordinatesof P must satisfy.

lF-nl : \MV+WPThat is,

lF:P,l: (1)Formula (1) holds for all possible positions of Pt and P, in all four

qqadrants. The length of the hypotenuse will always be lFfrl, and thelengths of the two legs will always be [Ar[ and lAyl (see Exercises L and2). We state this result as a theorem.

'1,.4.1 Theorem The undirected distance between the two points Pr(xr, Ar) and Pr(xr, yr)is given by

solurroN: From the statement of the problem

lPTl :2lPBlUsing formula (1), we have

@:z@Squaring on both sides, we have

x2 - 6x+ 9 + y ' - 4y I 4 : 4 (xz *8r * t6 + y2 - 2y * 1 )

1.4 DISTANCE FORMULA AND MIDPOINT FORMULA

It is important to note that the symbol Ar denotes the difference be-tween the abscissa of B and the abscissa of A, and it does not mean "deltamultiplied by x."

Similarly, if we consider a particle moving along a line parallel tothe y axis from a point C(x, yr) to a point D(x, yr), then the ordinate ofthe particle changes from Ar to Az.We denote this change by Ly ot

L y : a z - a rThus, CD: Ly.

Now let Pr(r1, yr) and Pr(xr, a) be any two points in the plane. wewish to obtain a formula for finding the nonrtegative distance betweenthese two points. We shatl denote this distance by lPtPrl. W" use abso-lute-value bars because we are concerned only with the length, which isa nonnegative number, of the line segment between the two points P1 andPr. To derive the formula, we note that lffil is the length of the hypot-

"rrrrr" of a right triangle PLMP2. This is illustrated in Fig. 1.4.3 for P1 and

Pr, both of which are in the first quadrant.Using the Pythagorean theorem, we have

EF"f: lArl2 +llvl 'So

or, equivalently,3x2 * 3y ' * 38 r - 4y + 55 :0


ExAMPLE 2: Show that the tri-angle with vertices at A(-2, 4),B(-5, L) , and C(-6,5) isisosceles.

Figure 1.4.4

nxervrrrn 3: Prove analyticallythat the lengths of the diagonalsof a rectangle are equal.

The triangle is shown in Fig. L.4.4.

@: \4+16: f r@: \ t r6+1: { lTW: \ /N: j f i

SOLUTION:

lBel :la-e; :IEAI:

Therefore,

Hence, triangle ABC is isosceles.

SOLUTION: Draw a general rectangle. Because we can choose the coordi-nate axes anywhere in the plane , and because the choice of the positionof the axes does not affect the truth of the theorem, we take the origin atone vertex, the x axis along one side, and the y axis along another side.This procedure simplifies the coordinates of the vertices on the two axes.Refer to Fig. 1..4.5.

Now the hypothesis and the conclusion of the theorem can be stated.Hypothesis: OABC is a rectangle with diagonals OB and AC.C o n clu si o n : I O-B | : lrel .

loBl : f f i : r t+62lACl :W:\ f r ]At

Therefore,

lml : ld-clLet P, (xr, yr) and P, (xr, yr) be the endpoints of a line segment. We

shall denote this line segment by PtPr. This is not to be confused with thenotation PrPr, which denotes the directed distance from P, to Pr. That is,PrP, denotes a number, whereas PrP2 is a line segment. Let P(x, y) be themidpoint of the line segment P,Pr. Refer to Fig. L.4.6.

In Fig. I.4.6 we see that triangles P'RP andPTPrare congruent. There-fore, l-PtRl : lPTl, and so r - xr: xz- x, giving us

A( -2 ,4 )

B(-5, i .)

C(0, b) B(a,b)

(0 ,0)F igure 1 .4 .5

1.4 DISTANCE FORMULA AND MIDPOINT FORMULA

Pz (xz , v z)

P(x,y)

u _ _ _ _ J S ( r r , y r )R(t, yt)

Figure 1.4.6

In the derivation of formulas (2) and (3) it was assumed that x, ) xtand y, ) yr.The same formulas are obtained by using any orderings ofthese numbers (see Exercises 3 and 4).

solurroN: Draw a general quadrilateral. Take the origin at one vertexand the x axis along one side. This method simplifies the coordinates ofthe two vertices on the r axis. See Fig. L.4.7.

Hypothesis: OABC is a quadrilateral. M is the midpoint of OA, N isthe midpoint of CB, R is the midpoint of OC, and S is the midpoint of AB.

Conclusion: MN and RS bisect each other.

pRooF: To prove that two line segments bisect each other, we show thatthey have the same midpoint. Using formulas (2) and (3), we obtain thecoordinates of M, N, R, and S. M is the point Ga, 0), N is the point(+(b+ d) , t r (c* e ) ) , R is the po in t 1 !d , te ) , and S is thepo in t G@+b) , i c ) .

The absc issa o f the midpo in t o f MNis i lLa ++(b + d) l : * (a+b + d) .The ordinate of the midpoint of MN is +[0 + Lk + e)f : ik + e).Therefore, the midpoint of MN is the point (ifu + b + d), ik + e)).The absc issa o f the midpo in t o f RS is i l i a+*@ + b) l : * (a+b + d) .The ordinate of the midpoint of RS is tlie + *cf : Ik + e).Therefore, the midpoint of RS is the point (ifu + b + d), i(c + e)).Thus, the midpoint of MN is the same point as the midpoint of RS.Therefore, MN and RS bisect each other. I

T(xr, Y)

nxaurr,E 4: Prove analyticallythat the line segments ioining themidpoints of the opposite sidesof any quadrilateral bisect eachother.

B(b, c)

Similarly, lRPl : lTPzl. Then A - Ar: Az- A, andtherefore

Hence, the coordinates of the midpoint oftively, the average of the abscissas and thethe endpoints of the line segment.

(3)

a line segment are, respec-average of the ordinates of

Figure 1 .4 .7

9.10.


Exercises 7.41. Derive distance formula (1) if Pr is in the third quadrant and P, is in the second quadrant. Draw a figure.2. Derive distance formula (1) if Pr is in the second quadrant and P, is in the fourth quadrant. Draw a figure.3. Derive midpoint formulas (2) and (3) if Pr is in the first quadrant and p, is in the third quadrant.4. Derive midpoint formulas (2) and (3) iI4 (rr, yr) and &(rr, yr) are both in the second quadrant an d x, > xrand,y, > yr.5. Find the length of the medians of the rriangle having vertices A(2,3),8(3,-g), and C(-1,-1).5. Find the midpoints of the diagonals of the quadrilateral whose vertices are (0,0), (0,4), (3,5), and (3, 1).

(.\Prove that the trianSle with vertices A(3,-6), B(8,-2), and C(-1, -1) is a right triangle. Find the area of the triangle.\'-l(HrNr: Use the converse of the Pythagorean theorem.)8. Prove that the Points A(6, -lS) , B(-2, 2), C(13, 10), and D(21, -5) are the vertices of a square. Find the tength of

a diagonal.By using distance formula (1), prove that the points (-3, 2) , (1, -2), and (9, -10) lie on a line.If one end of a line segment is the point (-4, 2) and the midpoint is (3, -L), find the coordinates of the other end ofthe line segment.

11. The abscissa of a Point iB -6, and its distance from the point (1, 3) is V-74. Find the ordinate of the point.12. Determine whether or not the poifis (14,7) , (2, 2) , and, (-4, -1) lie on a line by using distance formula (1)./;:\

'\3.llf two vertices of an equilateral hiangle are (-4,3) and (0,0), find the third vertex.d Findan equation that must be satisfied by the coordinates of any point that is equidistant from the two points (-3, 2)

and (4 ,6 ) .15. Find an equation that must be satisfied by the coordinates of any point whose distance from the point (5, 3) is atways

two units Feater than it8 distance frcm the point (-4, -2).16. Giver the two Points d(-3,4) andB(2,5), find the coordinates of a point P on the line through A and B such thatp

is (a) twice as far from 4 as frcm B, and (b) twice as far frcm B as from _A.17. Find the coordinates of the three points that divide the tine segment ftom l(-5, 3) to 8(6, g) into four equal parts.18. If rr. and r, are positiv_ intgg:ers,-prove that the coordinates of the point p(r, y), which divides the line segment plp,

in the ratio rr/r2-that i3, IP-,,PUlPrEI : r,/rr-are given by

,: (t" - rt)x, + rrxu

ur4 , : (r, - ,r)y, + ,rv"f z " t 2

In Exercises 19 through 23, use the formulas of Exercise 18 to find the coordinates oI point p.19' The Point P is on the line segment between points Pr (1 ,3) ?flrd, P26,2) and is three times as far from P, as it b ftom pr.20. The Point P is on the line segment between points Pl(1 ,3) and Pr(6,2) and is three times as far from Pr as it is frcmpr.21. The Point P i8 on the line through P' and P, and is three times as far ftom Pr(6, 2) as it is from P, (1,3) but is not be-

tween Pr and Pr.22. The Point P is on the Une though P, and P, and is three times as far from P1(1, 3) as it is ftom Pr(6,2) but is not be-

tween P1 and P2.The point P is on the line through Pr(-3,5) and Pr(-L,2) so that ppr : + . p.,pr.Find an equation whose graph is the circle that is the set of all points that are at a distance of 4 units from the point(1, 3).

23.24.

1.5 EQUATIONS OF A LINE

25. (a) Find an equation whose graph consist8 of all points equidistant frcm the Points (-1,2) and (3,4). (b) Draw a skekhof the gnph of the equation found in (a)'

26. prove analytically that the sum of the squares of the distanc$ of any Point from two opposite vertics of any rcctangleis equal to the sum of the squares of its distances from the other two verticeE'

22. prcve analytically that the line segment ioining the midpoints of two opposite sides of any quadrilateral and the linesegment joining the midpoints of the diagonala of the quadrilateral bieect each other'

28. prove analytically that the midpoint of the hypotenuse of any right triangle is equidistant frcm each of the three vettices.

29. Prove analytically that if the lengths of two of the medians of a triangle arc equal, the triangle is isoeceles.

1..s EQUATIONS OF A LINE Letl be a nonvertical line and Pr(x' U) andP"(xr, A)be any two distinctpoints on l. Figure L.5.1 shows such a line. In the figure, R is the pointQc, Ur), and the points Pr, Pu and R are vertices of a right triangle; further-more, P,.R: xz-xr and R-Pr: Az-Ar The number Az-Ar gives themeasure of the change in the ordinate from Pr to Pr, and it may be posi-tive, negative , or zero. The number x2 - xl gives the measure of the changein the abscissa from P, to Pr, and it may be positive or negative. The num-ber x, - rr may not be zero becaus? x2 * r, since the line I is not vertical.For all choices of the points P, and P, on l, the quotient

Figure 1 .5 .1

A z - U r,

Xz - x r

is constanq this quotient is called the "slope" of the line. Following is theformal definition.

1.S.1 Definition If Pr(xr, Ar) and Pr(xr, yr) arc any two distinct points on line l, which isnot parallel to the y axis, then the slope of l, denoted by m, is given by

; :yE (1)Xz * ?CrIn Eq. (L), x, * x1 since / is not parallel to the y axis.The value of m computed from Eq.(t) is independent of the choice

of the two pointr_P, and P2 on L To show this, suppose we choose two-at different points, tr(n, yr) and Pr(ir,yr), and compute a number nr from

Ee. (1).

7z ,A i

* - _ A z - a rm : - -x z - x t

We shall show tlnilt n - m. Refer to Fig. 1.5.2. Triangles PrRPz and P1RP2are similar, so the lengths of corresponding sides are proportional.Therefore,

V:- Y_t - Az - ArX z - X r X z - X t

or

m : m

R(rz, yJ

P, (xr, y,

Pr(7r,V,)

R(xr ,Yr)

F igure 1 .5 .2

x

34 REAL NUMBERS, INTRODUCTIONTO ANALYTIC GEOMETRY, AND FUNCTIONS

Hence, we conclude that the value of m computed from Eq. (1) is the samenumber no matter what two points on I are selected.

In Fig. 1.5.2, x2 ) x1, Uz ) Ar, x, > n, and y2 > fr. The discussionabove is valid for any ordering of these pairs of numbers since Definition1.5.1 holds for any ordering.

In sec. 1.4 we defined Ly:uz-ur and Ar:xz rr. substi tut ingthese values into Eq. (L), we have

Multiplying on both sides of this equation by Lx, we obtainLy : m Lx e )It is seen from Eq. (2) that if we consider a particle moving along

line l, the change in the ordinate of the particle is proportional to thechange in the abscissa, and the constant of proportionality is the slopeof the line.

If the slope of a line is positive, then as the abscissa of a point on theline increases, the ordinate increases. Such a line is shown in Fig. 1.5.3.In Fig. "1..5.4, we have a line whose slope is negative. For this line, as theabscissa of a point on the line increases, the ordinate decreases. Note thatif the line is parallel to the x axis, then Uz: Ur and so m:0.

If the line is parallel to the y axis, xz: xr, thus, Eq. (1) is meaninglessbecause we cannot divide by zerc. This is the reason that lines paiallelto the y axis, or vertical lines, are excluded

Calculus With Analytic Geometry

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