[-1- r" 16lT
- {Ll
*, ir a > o
f
du
| \/i
"' l\/a +Tn + lal
-20. f ---::r
-' " -'
J u \ / a + n
l Z r ^ n _ r r @ _
q 6 i f a < o
l ! - a | - a f itu \/;T6 b(2n - 3) f-
du L"J ,nytfifi n(n - 7\yn-r 2a(n - l) J un-r la * bu
Lo**ryotl i.J '-' t l
* I#F:+'"\#l
Tql"c
{&)
,4.
Ih:]tan-'1*,
,u.
Ih:*'nl=;l*c=
i f l u l < a if lul> a[ 1 t " r , r , - , L + c
l a al l c o t r , - , L + c
l a a i f l u l < a i f l u l > aIn formulas 27 through 38, we may replace ln(z * t/FRl by sinh-lf, lnlu * \trt41 by cosh-'f, . l a + t f i 4 l t " l , I b Y s i n h - ' f ,
, ,
I # : r n l u
+ { F + l + c
,8.
I
t/itTV du:l 1ffif +
$U"
+ vFEA + c
f _
2e.
J uz\/uz + 02 d" -- t (2u' + a2) r/uz
:; . tffil + c
to.
I ry
: \fr'
* nz
-
" "lLPl*,
,t.
I =-:
tfi, - a2
-4 sec-,
1'l
..
AA f I/FTF au \FtA , ,-r-.t z '
l - f f : - T * h l u + v T f r \ + c
1 u'z ilu u - 1n2zz.
J ffi:i
t/F tF -;hlu
* tffiv1 + c
$,non,ol
)v.cerrt.Dn
=
7"C "*[n-k
3 s .
t L : 1 r " . - ,
l g l
* .
" " ' J uy@-- oz a"-- lal '
-,uI+:_YT*,
f
tr.
J (uz -+ szytrz o":t (2u, +'Saz) lirE7
+ { r ' r 1 " + t F T T l + c
q R I d " : - L L c *' J (uz-rszlstz -raz1/ffi-" I # : s i n - 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 ---- a45. i 46. 49. 50.
l L : - 1
, n l o * f f i !
* .
J u ! a 2 - u 2 a I u I: -+cosh-l
1* '
f-
d u \ E - u ' , - , J u z t f a z - u 2 a z uFsftrl*,:
Cont4futing,
Lwu
- n3
f . - , u - a a ' ^ ^ - - , I \ t r a u - F d u : T t / 2 a u - u z + 7 c o s - r [ " r m d u : 2 u 2 - T - t o ' \ f f i r e4 s I # : # + c
(r-+)*,
* t c o s - l ( r - ) * ,
sl
I ry
: tffi
* a cos-'
(r -:)+ c
s z .
I r y : - r Y
- c o s - , ( 1
- : ) * ,
s 3
I # : c o s - r ( 1
- T ) * ,
Forms Containing Trigonometric Functions
f f. ''' 5 9 . f s i n u d u c o s u + C J f 6 0 . f c o s u d u : s i n u * C J f 6 1 . It a t u d u : l n l s e c u l + C J
62. | .o. u du- lnlsin ul + c J f 63. I t". u du - lnl secu * tan ul + C r : lnltan (*n + iu)l+ C 64. [ "r. u du- lnlcsc t/ - cot ul + C J - lnltan iul + C 6 5 . I r u . ' u d u : t a n u * C J f 5 6 . f c s c 2 u d u c o t u + C J r 5 7 . J s e c i l t a n u d u : s e c i l + C 5 8 . f . r . u c o t u d u J f 69. I sin2 u du : Lrtt - *sin 2u * C f
70. I cos' u du: *u * *sin 2u * C
J f 7 1 . J t a n z u d u : t a n t t - u+ C
s 4 .
t +
t t r a u a * a c o s - ' ( r - g ) + c
v r ' J l f f i u - u z ' - - \ a / trtr [ uz du (" |l')- \ER +3o-' cos-l5 5 .
1 6 6
F, f du @-,-/-r o .
J f f i : -
*
T '
f d u: -
u - a , - . J ( 2 a u - u 2 ) s t 2 a 2 ! 2 a u - uzs 8 .
1 ,
u d u . - + + c
v'Lr'J (Zau - u2)3t2 atffi=E
r
7 2 . I c o t 2 u d u : - c o 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 u 7 3 . J s i n n u t n - r i l c o s u + - n J u r ^ r f7 4 .
J c o s n
u d u : + c o s ' - r
a s i n
"
* + J . o ' " - 2
u d u
7 s .
J . u r , "
u d u : # t a n ' - r
" - I t a n n - 2
u i l u
76.
/
.ot" u du
: -i+cotn-'
"
-
I
cotu-z
u ilu
7 7 . J
r " . ' u d u
: # s e c n - 2 u tan"
+ # / ' " t ' - 2
u d u
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 c o s ( r n * n ) u c o s ( m - n)u - L . 8 1 . J s i n m u c o s n u d u - - @ r L f 82.
I u sin u du: sin u - tt cos ll + C f
83.
I il cos u du: cos u * u sin z + C
84. I u'sin z du - 2u srn u * (2 - u2) cos u * C J
WITH ANALYTIC GEOMETRY
third edition
g
(
(
I{
{
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!
\
i tLoulsLelthold
UNIVERSITY OF SOUTHERN CALIFORNIAHARPER & ROW, PUBLISHERS
Sponsoring Editor: George J. Telecki Project Editor: Karen A. fudd Designer: Rita Naughton
Production Supervisor: Francis X. Giordano Compositor: Progressive Typographers Printer and Binder: Kingsport Press Art Studio: J & R Technical Services Inc.
Chapter opening art: "Study Light,, by patrick Caulfield THE CALCULUS WITH ANALYTIC GEOMETRY, Copyright @ 1.968, 1,972, t976 by Louis Leithold
All rights reserved. Printed in the United States of America. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information address Harper & Row, Publishers, Inc., 10 East 53rd street, New york, N.y. 10022.
Library of Congress Cataloging in Publication Data Leithold, Louis.
The calculus, with analytic geometry. Includes index.
L. Calculus. 2. Geometry, Analytic, L Title. QA303.L428 7975b 515'.1s 75-26639 ISBN 0-06-043951-3
Gontents
Chapter 1
REAL NUMBERS,
INTRODUCTION TO
ANALYTIC GEOMETRY,
AND FUNCTIONS
page 1Chapter 2
LIMITS AND CONTINUIT}
paSe oc
Chapter 3
THE DERIVATIVE
page 110
Preface
)cu,
1.1 Sets, Real Numbers, and Inequalities
2
1.2 Absolute Value
14
1.3 The Number Plane and Graphs of Equations
2L
1.4 Distance Formula and Midpoint Formula
28
1.5 Equations of a Line
33
1.6 The Circle
43
L.7 Functions and Their Graphs
48
1.8 Function Notation, Operations on Functions, and Types of
Func-tions
56
2.1, The Limit of a Function
66
2.2 Theorems on Limits of Functions
74
2.3 One-Sided Limits
85
2.4 Infinite Limits
88
2.5 Continuity of a Function at a Number
97
2.6 Theorems on Continuity
1-0L
3.1 The Tangent Line
1-1-i-3.2 Instantaneous Velocity in Rectilinear Motion
11,5
3.3 The Derivative of a Function
L21-3.4 Differentiability and Continuity
L26
3.5 Some Theorems on Differentiation of Algebraic Functions
1.30
3.6 The Derivative of a Composite Function
L38
3.7 The Derivative of the Power Function for Rational Exponents
L42
3.8 Implicit Differentiation
1-45
3.9 The Derivative as a Rate of Change
1.50
3.10 Related Rates
154
t
I\
Chapter 4
TOPICS ON LIMITS,
CONTINUITY, AND THE
DERIVATIVE
page 164Chapter 5
ADDITIONAT APPLICATIONS
OF THE DERIVATIVE
page 204Chapter 5
THE DIFFERENTIAL AND
ANTIDIFFERENTIATION
page 243
Chapter 7
THE DEFINITE INTEGRAL
page 275
Chapter 8
APPLICATIONS OF THE
DEFINITE INTEGRAL
page 323 5.1 The Differential 244 6.2 Differential Formulas 2495.3 The Inverse of Differentiation 253
6.4 Differential Equations with Variables Separable 6.5 Antidifferentiation and Rectilinear Motion 265 6.6 Applications of Antidifferentiation in Economics
4.L Limits at Infinity
765
4.2 Horizontal and Vertical Asymptotes
L71
4.3 Additional Theorems on Limits of Functions
1.74
4.4 Continuity on an Interval
177
4.5 Maximum and Minimum Values of a Function
19L
4.6 Applications Involving an Absolute Extremum on a Closed
In-terval
L89
4.7 Rolle's Theorem and the Mean-Value Theorem
lgs
5.1 Increasing and Decreasing Functions and the First-Derivative
Test
205
5.2 The second-Derivative Test for Relative Extrema
2i.1
5.3 Additional Problems Involving Absolute Extrema
213
5.4 Concavity and Points of Inflection
220
5.5 Applications to Drawing a Sketch of the Graph of a Function
227
5.6 An Application of the Derivative in Economics
280
261
269
7.L The Sigma Notation 276 7.2 Area 28L
7.3 The Definite Integral 288
7.4 Properties of the Definite Integral 2gG 7.5 The Mean-Value Theorem for Integrals 306 7.6 The Fundamental Theorem of the Calculus SLi.
8.1 Area of a Region in a Plane 324
8.2 Volume of a Solid of Revolution: Circular-Disk and Circular-Ring Methods 330
8.3 Volume of a Solid of Revolution: Cylindrical-Shell Method 3J6 8.4 Volume of a Solid Having Known Parallel Plane Sections 341 8.5 Work 344
8.6 Liquid Pressure 348
8.7 Center of Mass of a Rod 35L
8.8 Center of Mass of a Plane Region 356 8.9 Center of Mass of a Solid of Revolution 366 8.10 Length of Arc of a Plane Curve 372
Chapter 9
9.1
LOGARITHMIC AND
9.2
EXPONENTIAL FUNCTIONS
9.3
page 38L 9.49.5
9 . 6
CONTENTS
The Natural Logarithmic Function
382
The Graph of the Natural Logarithmic Function
391'
The Inverse of a Function
395
The Exponential Function
405
Other Exponential and Logarithmic Functions Laws of Growth and DecaY 420
The Sine and Cosine Functions 431
Derivatives of the Sine and Cosine Functions
438
Integrals Involving Powers of Sine and Cosine
447
The Tangent, Cotangent, Secant, and Cosecant Functions
452
An Application of the Tangent Function to the Slope of a
Line
461,
10.5 Integrals Involving the Tangent, Cotangent, Secant, and
Co-secant
466
4L4
ChaPter L0
TRIGONOMETRIC
FUNCTIONS
page 430Chapter LL
TECHNIQUES OF
INTEGRATION
page 49710.1
t0.2
10.3
10.4
10.s
Chapter L2 Lz.L
HYPERBOTIC FUNCTIONS I2,2
page
535 12.3
Chapter 13 13.1
POLAR COORDINATES
T3.2
page
554 13.3
13.4
13.5
Chapter 1.4
THE CONIC SECTIONS
page 578
InverseTrigonometricFunctions
47L
Derivatives of the Inverse Trigonometric Functions
Integrals Yielding Inverse Trigonometric Functions
Introduction
492
Integration by Parts
493
Integration by Trigonometric Substitution
498
Integration of Rational Functions by Partial Fractions.
2: The Denominator Has Otly Linear Factors
504
Integration of Rational Functions by Partial Fractions.
4: The Denominator Contains Quadratic Factors
51.2
Integration ofRational Functions of Sine and Cosine
Miscellaneous Substitutions
51-9
The Trapezoidal Rule
521,
Simpson's Rule
526
The Hyperbolic Functions
536
The Inverse Hyperbolic Functions
543
Integrals Yielding Inverse Hyperbolic Functions
548
The Polar Coordinate System
555
Graphs of Equations in Polar Coordinates
560
Intersection of Graphs in Polar Coordinates
567
Tangent Lines of Polar Curves
571-Area of a Region in Polar Coordinates
L0.7
10.8
10.9
1 1 . 1
LL.2
1 1 . 3
Lt.4
11.5
LL.6'
Ll..7
1.L.8
'n.9
477
484
Cases L and
Cases 3 and
5 1 6
LA.l The Parabola
579
L4.2 Translation of Axes
583
14.3 Some Properties of Conics
588
L4.4 Polar Equations of the Conics
592
I{
:LI
N
i 1L4.5 Cartesian Equations of the Conics
Sgg
14.5 The Ellipse
606
L4.7 The Hyperbola
613
14.8 Rotation of Axes
620
chapter L5 15.1 The Indeterminate Form 0/0
629
INDETERMINATE FORMS, 1,5.2 Other Indeterminate Forms
636
IMPROPER INTEGRALS, AND
15.3 Improper Integrals with Infinite Limits of Integration
TAYLOR'S FORMULA
lS.4 Other Improper Integrals
647
Page
628 15.5 Taylor's Formula
6s7
Chapter 15 1,5.1 Sequences 5G0
INFINITE SERIES 1'6.2 Monotonic and Bounded Sequence
s
667
page
65e 16.g Infinite Series of Constant Term s
673
1,6.4 Infinite Series of Positive Terms
6g4
1,6.5 The Integral Test
694
15.6 Infinite series of Positive and Negative Terms
697
L6 7 Power Series
707
L6.8 Differentiation of Power Series
713
16.9 Integration of Power Series
722
15.10 Taylor Series
729
L6.ll The Binomial Series
738
641
Chapter 17
VECTORS IN THE PLANE
AND PARAMETRIC
EQUATIONS
page 745
Chapter 18
VECTORS IN
THREE-DIMENSIONAL SPACE AND
SOLID ANALYTIC
GEOMETRY
page 810
77.1, Vectors in the Plane 746
77.2 Properties of Vector Addition and Scalar Multiplication 751 L7.3 Dot Product 756
17.4 vector-Valued Functions and Parametric Equations 763 77.5 Calculus of Vector-Valued Function s 772
77.6 Length of Arc 779 t7.7 Plane Motion 785
I7.8 The Unit Tangent and Unit Normal Vectors and Arc Length as a Parameter 792
L7.9 Curvature 796
17.10 Tangential and Normal Components of Acceleration g04 18.1 R3, The Three-Dimensional Number Space 8L1
L8.2 Vectors in Three-Dimensional Space 818 18.3 The Dot Product in V, 825
18.4 Planes 829 18.5 Lines in R3 836 L8.6 Cross Product 842
1,8.7 Cylinders and Surfaces of Revolution BS2 18.8 Quadric Surfaces 858
18.9 Curves in R3 864
Chapter 19
DIFFERENTIAL CALCULUS
OF FUNCTIONS OF
SEVERAL VARIABLES
page 880Chapter 20
DIRECTIONAL DERIVATIVES,
GRADIENTS, APPTICATIONS
OF PARTIAL DERIVATIVES,
AND LINE INTEGRALS
page 944ChaPter 2|
MULTIPTE INTEGRATION
page 1001APPENDIX
page A-1 CONTENTS L9.']-. Functions of More Than One Variable 88Llg.2 Limits of Functions of More Than one Variable 889 tg.3 Continuity of Functions of More Than One Variable 900 19.4 Partial Derivatives 905
19.5 Differentiability and the Total Differential 91'3 19.6 The Chain Rule 926
19.7 Higher-Order Partial Derivatives 934 20.L Directional Derivatives and Gradients 945 20.2 Tangent Planes and Normals to Surfaces 953 20.3 Extrema of Functions of Two Variables 956
20.4 Some Applications of Partial Derivatives to Economics 967 20.5 Obtaining a Function from Its Gradient 975
20.6 Line Integrals 981
20.7 Line Integrals Independent of the Path 989 21.1, The Double Integral L002
2'1..2 Evaluation of Double Integrals and Iterated Integrals 1008 21.3 Center of Mass and Moments of Inertia 1'0L6
21.4 The Double Integral in Polar Coordinates 1'022 21.5 Area of a Surface 1028
21.6 The Triple Integral 1034
21.7 The Triple Integral in Cylindrical and Spherical Coordinates L039
Table 1.
Table 2
Table 3
Table 4
Table 5
Table 6
Table 7
Powers and Roots
A-2
Natural Logarithms
A-3
Exponential Functions
A-5
Hyperbolic Functions
A-L2
Trigonometric Functions
A-13
Common Logarithms
A-1-4
The Greek Alphabet
A-15
ANSWERS TO ODD-NUMBERED EXERCISES A.1.7
Professor Archie D. l Professor Phillip Cla Professor Reuben W Professor Jacob Golil
Professor Robert K. Goodrich, University of Colorado Professor Albert Herr, Drexel University
Professor James F. Hurley, University of Connecticut Professor Gordon L. Miller, Wisconsin State Universitv Professor William W. Mitchell, Jr., phoenix College Professor Roger B. Nelsen, Lewis and Clark Colle-ge
Professor Robert A. Nowlan, southern connectictit state college Sister Madeleine Rose, Holy Names College
Professor George W. Schultz, St. petersbuig Junior College Professor Donald R. Sherbert, Universitv of Illinois
Professor ]ohn V_adney, Fulton-Montgomery community college Professor David Whitman, San Diego State College
Production Staff at Harper & Row George Telecki, Mathematics Editor Karen fudd, Project Editor
Rita Naughton, Designer Assistants for Answers to Exercises
Jacqueline Dewar, Loyola Marymount University Ken Kast, Logicon, Inc.
|ean Kilmer, West Covina Unified School District Cover and Chapter Openins Artist
Patrick 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.
r l ]
FTETAGE
This third edition of THE CALCULUS WITH ANALYTIC GEOMETRY, like the other two, is designed for prospective mathematics majors as well as for students whose primary interest is in engineering, the physical sciences, or nontechnical fields. A knowledge of high-school algebra 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 16 through 21 (Chapter L6 on Infinite Series is included in both parts to make the 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 single variable and plane analytic geometrlz, and it may be covered in a one-year course of nine or ten semester hours or twelve quarter hours. The second part is suitable for a course consisting of five or six semester hours or eight quarter hours. It includes the calculus of several variables and a treatment of vectors in the plane, as well as in three dimensions, with a vector approach to solid analytic geometry.
The objectives of the previous editions have been maintained. I have endeavored to achieve a healthy balance between the presentation of elementary 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 no step unexplained or omitted. I desire that the reader be aware that proofs of theorems are necessary and that these proofs be well motivated and carefully explained so that they are understandable to the student who has achieved an average mastery of the preceding sections of the book. If a theorem is stated without proof ,I have generally augmented the discus-sion by both figures and examples, and in such cases I have always stressed that what is presented is an illustration of the content of the theorem and is not a proof.
section of Chapter I has been rewritten to give a more detailed exposition of the real-number system. The introduction to analytic geometry in this chapter includes the traditional material on straight lines as well as that of the circle, but a discussion of the parabola is postponed to Chapter 14, The Conic Sections. Functions are now introduced in Chapter 1. I have defined a function as a set of ordered pairs and have used this idea to point up the concept of a function as a correspondence between sets of real numbers.
The treatment of limits and continuity which formerly consisted of ten 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 a limit of a function is first given a step-by-step motivation, which brings the discussion from computing the value of a function near a number, through an intuitive treatment of the limiting process, up to a rigorous epsilon-delta definition. A sequence of examples progressively graded in difficulty is included. All the limit theorems are stated, and some proofs are presented in the text, while other proofs have been outlined in the exercises. In the discussion of continuity, I have used as examples and counterexamples "common, everyday" functions and have avoided those that would have little intuitive meaning.
In Chapter 3, before giving the formal definition of a derivative, I have defined the tangent line to a curve and instantaneous velocity in rectilinear motion in order to demonstrate in advance that the concept of a 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 by the introduction of the Extreme-Value Theorem, which involves such functions. Then the Extreme-Value Theorem is used to find the absolute extrema of functions continuous on a closed interval. Chapter 4 concludes with Rolle's Theorem and the Mean-Value Theorem. Chapter 5 gives additional applications of the derivative, including problems on curve sketching 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 that would make the reading of standard references difficult. This notation will suggest that some relation must exist between definite integrals, intro-duced in Chapter 7, and antiderivatives, but I see no harm in this as long as the presentation gives the theoretically proper view of the definite integral as the limit of sums. Exercises involving the evaluation of defi-nite integrals by finding limits of sums are given in Chapter 7 to stress that 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 a limit of sums. Elementary properties of the definite integral are derived and the fundamental theorem of the calculus is proved. It is emphasized that this is a theorem, and an important one, because it provides us with an alternative to computing limits of sums. It is also emphasized that the definite integral is in no sense some special type of antiderivative. In Chapter 8 I have given numerous applications of definite integrals. The presentation highlights not only the manipulative techniques but also the fundamental principles involved. In each application, the definitions of the new terms are intuitively motivated and explained.
The treatment of logarithmic and exponential functions in Chapter 9 is the modern approach. The natural logarithm is defined as an integral, and after the discussion of the inverse of a function, the exponential function is defined as the inverse of the natural logarithmic function. An irrational power of a real number is then defined. The trigonometric functions are defined in Chapter 10 as functions assigning numbers to numbers. The important trigonometric identities are derived and used to 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 trigtrig-onometric functions.
Chapter LL, on techniques of integration, involves one of the most important computational aspects of the calculus. I have explained the theoretical backgrounds of each different method after an introductory motivation. The mastery of integration techniques depends upon the examples, and I have used as illustrations problems that the student will certainly meet in practice, those which require patience and persistence to solve. The material on the approximation of definite integrals includes the statement of theorems for computing the bounds of the error involved in these approximations. The theorems and the problems that go with them, being self-contained, can be omitted from a course if the instructor so wishes.
A self-contained treatment of hyperbolic functions is in Chapter 12. This chapter may be studied immediately following the discussion of the circular trigonometric functions in Chapter L0, if so desired. The geo-metric interpretation of the hyperbolic functions is postponed until Chapter L7 because it involves the use of parametric equations.
Polar coordinates and some of their applications are given in Chap-ter 13. In ChapChap-ter 1.4, conics are treated as a unified subject to stress their natural and close relationship to each other. The parabola is discussed in the first two sections. Then equations of the conics in polar coordinates are treated, and the cartesian equations of the ellipse and the hyperbola are 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.
infinite series as is feasible in an elementary calculus text. In addition to the customary computational material, I have included the proof of the equivalence of convergence and boundedness of monotonic sequences based on the completeness property of the real numbers and the proofs of the computational processes involving differentiation and integration of power series.
The first five sections of Chapter L7 on vectors in the plane can be taken up after Chapter 5 if it is desired to introduce vectors earlier in the course. 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 vector analysis. The applications are to physics and geometry. Chapter 18 treats vectors in three-dimensional space, and, if desired, the topics in the first three sections of this chapter may be studied concurrently with the corre-sponding topics in Chapter 17.
Limits, continuity, and differentiation of functions of several variables are considered in Chapter L9. The discussion and examples are applied mainly to functions of two and three variables; however, statements of most of the definitions and theorems are extended to functions of n variables.
In Chaptet 20, a section on directional derivatives and gradients is followed by a section that shows the application of the gradient to finding an equation of the tangent plane to a surface. Applications of partial derivatives to the solution of extrema problems and an introduction to Lagrange multipliers are presented, as well as a section on applications of partial derivatives in economics. Three sections, new in the third edition, are devoted to line integrals and related topics. The double integral of a function of two variables and the triple integral of a function of three variables, along with some applications to physics, engineering, and geometry/ are given in Chapter 2/...
New to this edition is a short table of integrals appearing on the front and back endpapers. However, as stated in Chapter L1, you are advised to use a table of integrals only after you have mastered integration.
Real numberq, introduction
to analytic geometry
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 elements of 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 Gate Park, 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 some rule or property that enables one to decide whether a given object is or is not an element of a specific set. A pair of braces { } used with words or 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 used to represent any element of a given set. Another way of writing the above set S is to use what is called set-builder notation, where a vertical bar is used in place of the words "such that." Using set-builder notation to describe 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 may write the set N as
{ 7 , 2 , 3 ,
I . lwhere the three dots are used to indicate that the list goes on and on with
no 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 belongs
to 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. The
symbol fi is read "is not an element of." Thus, we read * € w as "+ is not
an element of N."
We denote the set of all integers by /. Because every element of N is
also an element of / (that is, every natural number is an integer), we say
that 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
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-t e n S C T .
Observe from the definition that every set is a subset of itself, but a set
is 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 element
of S is also an element of T"; and (ii) "the set S is a subset of set T only if
every element of S is also an element of 7," which is logically equivalent
to the statement"if. S is a subset of.T, then every element of S is also an
element of. T."
o rLLUsrRArroN
1: Let N be the set of natural numbers and let M be the
set denoted by {rlr is a natural number less than 1,0}. Because every
element of M is also an element of N, M is a subset of N and we write
M e N. Also, there is at least one element of N which is not an element
of 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, which
states that the set consisting of the single element 5 is a Proper subset of
the set M. We may also write 5 € M, which states that the number 6 is
an 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 emPty
set is denoted by the symbol A.
The concept of "subset" may be used to define what is meant by two
sets 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 B
a n d B e A .
Essentially, this definition states that the two sets A and B are equal if and only if every element of A is an element of B and every element of B is 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 we proceed. 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 A and B.
E x A M P L E 1 : L e t A : {2, 4, 6,8,
10, LzI , B : {1, 4, 9,
'1.5}
, and
C - { 2 , 1 0 } .
F i n d
( a ) A u B
( c ) B U C
SOLUTION:( a ) A u
( b ) A u
( c ) B U
( d ) A u
1..1..5
Definition
Let A and B
and tead " A
and B.
B - { 1 , 2 , 4 , 6 , 9 , 9 , " 1 . 0 ,
1 , 2 ,
1 , 6 }
C : {2, 4, 6,8, 1,0,
1 2 }
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 B
intersection 8," is the set of all elements that are in both A
( b ) A u c
( d ) A u A
nxelvrpr.E
2: If A, B, and C are
the sets defined in Example 1,
find
@ ) A n B
( c ) B n C
S O L U T I O N :( 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 ) A n
( d ) A n
( b ) A n c
@ ) A n A
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 numbers and 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) denotes their product.
We now present seven axioms that give laws governing the operations of addition and multiplication on the set Rl. The word axiom is used to indicate a formal statement that is assumed to be true without proof. r f a , b e R t , t h e n sib i s a u n i q u e r e a l number, andab is a unique real number. 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 R 1 , th e n 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,
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 negatiae by -a (read "the negative of a"),
For every real number a, except 0, there exists a real number called the reciprocat 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 these axioms are satisfied by a set of elements, then the set is called afield under the two operations involved. Hence, the set R1 is a field under addition and multiplication. For the set / of integers, each of the axioms t.1.6 through 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 of integers 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 l number, denoted by o + b (read "a divided by b"), called the quotient of a and b, where
a : b : A ' b - r
Other notations for the quotient of a and b ate ! and alb
b
By using the field axioms and Definitions 1.1.13 and 1.1.14, we can derive properties of the real numbers from which follow the familiar algebraic operations as well as the techniques of solving equations, factoring, and so forth. In this book we are not concerned with showing how such properties are derived from the axioms.
Properties that can be shown to be logical consequences of axioms are theorems. In the statement of most theorems there are two parts: the "if" part, called the hypothesis, and the "then" part, called the conclusion. The argument verifying a theorem is a proof. A proof consists of showing that 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 numbers such that
(i) if a € Rr, exactly one of the following three statements holds: a : 0 a i s p o s i t i v e - a is positive.
(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 order the elements of the set Rr. In Definitions 'J,.7.77 and 1.1.18 we use this axiom to define the relations of "greater than" and "less than" on Rr. The negatives of the elements of the set of positive numbers form the set 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 Definition 7.1..76 it follows that a real number is either a positive number, a negative number, or zero. Ary real number can be classified as a rqtional number or an irrational number. A rational number 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, q are 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
3,251
L,000,000
The positive and negative nonterminating
repeating
decimals
such as
0.333.
: +
- 0 5 4 9 s 4 9 s 4 9 . . . : - f t
The real numbers which are not rational numbers are called irrational numbers. These are positive and negative nonterminating, nonrepeating decimals, for example,
rt: t.732.
n : 3.L4'1,59.
t a n 1 4 0 o : - 0 . 8 3 9 L .The field axioms do not imply any ordering of the real numbers. That is, by means of the field axioms alone we cannot state that 1 is less than
1.1 SETS, REAL NUMBERS, AND INEQUALITIES
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 order axiom and the field axioms, we say that Rl is an ordered field.
We use the concept of a positive number given in the order axiom to define what we mean by one real number being "less than" another. 1 1 * \ 7 D e f i n i t i o n I f a , b € R t , t h e n a i s l e s s th a n b ( w r i t t e n a < b ) i f a n d o n l y i f b - a r s
positive.
o rt,r,usrRArIbrs 2: (a) 3 < I because 8 - 3:5, and 5 is positive. (b) -L0 < -6 because -6 - (-10) :4, anrd 4 is positive. .
1 . 1 . 1 8 D e f i n i t i o n I f . a , b C R l , t h e n a i s g r e a t e r t h a n b ( w r i t t e n a > b ) i f a n d o n l y i f b i s l e s s than 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 or equal to") are defined as follows:
( 1 ) a - b i f a n d o n l y i f e i t h e r a < b o r a : b . (ii\ a > b if. and only if either a > b ot a:
b-T h e s t a t e m e n t s a 1 b , a ) b , a < b , a n d a = b a r e c a l l e d i n e q u a l i t i e s . S o m e e x a m p l e s a r e 2 < 7 , - 5 < 6 , - S 1 - 4 , " ] . ' 4 > 8 , 2 > - 4 , - 3 > - 6 , 5 = 3, -10 < -7.In particular, a 1b and a > b are called strict inequal-ities, whereas a -b anda= b are called nonstrict inequalities..
The following theorems can be proved by using L.L.6 through 1.1'.19. "1..1.20 Theorem (i) 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 <'1.4; so 3 < 14.
1 . 1 . 2 2 T h e o r e m l f a < b , t h e n a * c < ' b + c , a n d a - c < b - c i f c i s a n y r e a l n u m b e r . o rLLUsrRATroN 4: 41
s o 4 * 5 < 7 * 5 ; a n d 4 - 5 < 7 - 5 .
1.1.23 Theorem If. a < b and c I d, then a * c < b + d.
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 remains unchanged, whereas Theorem 1.1.25 states that if both members of an inequality are multiplied by a negative number, the direction of the inequality 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 a proof 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). H e n c e , ( b - a ) + (d- c) is positive (by Axiom 1.1.15(ii)).
T h e r e f o r e , ( b t d ) - (a * c) is positive (by Axioms 1.1.g and l.l.T and Definition 1.1.13). Therefore, A * c < b + d (by Definition 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 , t h e n a * c > b + c, and a- c > b- c if 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 ) . o If a > b and if c is any positive number, then ac ) bc.
o rLLUsrRArroN 12: -3 > -7; so (-3) > eDA.
n
1.1.31 Theorem
1.1.32 Theorem
7 9
1.L.33 Definition
1.1 SETS, REAL NUMBERS, AND INEQUALITIES
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 field axioms 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 16.2 because it requires some terminology that is best introduced and dis-cussed later. However, we now give a geometric interpretation to the set of 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 the number 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 r units to the right of the origin, and each negative number x is represented by the point at a distance of -x units to the left of the origin (it should be noted that if r is negative, then -r is positive). To each real number there corresponds a unique point on the axis, and with each point on the axis there is associated only one real number; hence, we have a one-to-one correspondence between Rl and the points on the axis. So the points on the axis are identified with the numbers they represent, and we shall use the same symbol for both the number and the point representing that number on the axis. We identify Rl with the axis, and we call Rl the real number line.
We see that a < b if and only if the point representing the number a is to the left of the point representing the number b. Similarly, a > b if and 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 the point 2 is to the left of the point 5. We could also write 5 ) 2 and say that the point 5 is to the right of the point 2.
A n u m b e r x i s b e t w e e n a a n d b i f a ( r a n d x < b . W e c a n w r i t e t h i s as a continued inequality as follows:
a l x l b ( 2 )
The set of all numbers x satisfying the continued inequality (2) is called an "open interyal."
The open interaal from a to b, denoted by (o, b), is defined by ( a , b ) : { r l a l x < b }
t z 2 4
l l l t t r t t l l r l l
- 4 t - 2 0 | z t 4 F i g u r e 1 . 1 . 1
a
F i g u r e 1 . 1 . 3
The "closed interval" from 4 to b is the open interval (a, ,) together with 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 i g u r e 1 . 1 . 2 Figure 1,.1,.2 illustrates the open interval (a,b) and Fig. 1.1.3 illustrates
the closed interval la, bl.
The "interval half-open on the lef.t" is the open interval (a, b) together with 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 by l a , b ) : { r l a = x < b }
Figure 1'.1.4 illustrates the interval (a,bl and Fig. 1.1.5 illustrates the interval la, b).
we shall use the symbol +@ ("positive infinity") and the symbol -m ("negative infinity"); however, care must be taken not to confuse these symbols with real numbers, for they do not obey the properties of the real numbers
7.1.37 Definition
(i) (a, +-) : {xlx > a}
(ii) (-.o, b): {xlx < b}
(iii) la, +*7 : {xlx > a}
( i v ) ( - m , b l : {xlx - b}
(v) (-*, **) : Rl a F i g u r e 1 . 1 . 6 -b F i g u r e 1 . 1 . 7 b " 1.1.35 Definition'1,.1.36
Definition
-a b F i g u r e 1 . 1 . 4 f---L a b F i g u r e 1 . 1 . 5Figure 1,.1,.6 illustrates the interval (a, *a), and Fig. r.7.T illustrates the 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 solution
set of the inequality
2 * 3 x < 5 r * 8
Illustrate it on the real number
line.
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 considered to 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 one variable. The solution set of such an inequality is the set of all numbers that 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 6
Then, 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 * 8
then
x > - 3
Each of the steps is reversible; that is, if we start with
x > - 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 case
we get
-- 3 0
F i g u r e 1 . 1 . 8
ExAMPLE 4: Find the solution set of the inequality
4 < 3 x - 2 < ' 1 , 0
Illustrate it on the real number line.
0 F i g u r e 1 . 1 . 9
rxauprn 5: Find the solution
set of the inequality
7 2 2
x
Illustrate it on the real number
line.
Therefore, we can conclude that
2 * 3 x < 5 r * 8
i f a n d o n l y i f x > - 3
So the solution set of the given inequality is the interval (-3, +-;, which
is illustrated in Fig. 1.1.8.solurroN: Adding 2 to each member of the inequalit|, we obtain 6 < 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 upon
whether 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 obtain
7 > 2 x
Dividing on both sides by 2,we get
E > 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}, which is the
interval (0, g).
Cqse
2: x is negative; that is, x ( 0.
Multiplying on both sides by * and reversing the direction of the
inequality, we find
7 < 2 x
Dividing on both sides by 2,we have
E < x or, equivalently, x > 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 illustrated in Fig. 1.1.10.
EXAMPLE 6: Find the solution set of the inequality
x = 1 4
x - 5
Illustrate it on the real number
line.
0 F i g u r e 1 , 1 . 1 1
ExavrplE 7: Find the solution set 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, we
must 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 get
x 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 ) 4
Thus the solution set of Case 1 is {rlx > 3} n {xlr > 4} or, equivalently,
{xlx > 4}, which is the interval (4, +oo;.
C a s e 2 : x - 3 < 0 ; t h a t is , x < - 3 .
Multiplying on both sides by * - 3 and reversing the direction of
the inequ ality,we have
x ) 4 x - L 2
or, equivalently,
-3x > -12
or, equivalently,
x 1 4
Therefore, .x must be less than 4 and also less than 3. Thus, the solution
set 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 the
s a m e s i g n , t h a t i s ,
i f x t 3 > 0 a n d x + 4 > 0 , o r if x t 3 < 0 a n d r + 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 ) - 4
Thus, 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
a n d x 1 - 4
Both 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.L
In Exercises 1 through 10, list 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 .
1 0 . ( A u B ) n ( c u D )
{ 1 , z , 4 , B } , C : { 1 , i, s , 7 , 9 } , a n d D :( B n c )
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 - x 1 4 . 3 x - 5 < i x +- 4 ' - ' 3 1 7 . 2 > - 3 - 3 x > - 7 r ) 20r i- = 1. \ - r L - x 2 3 . ( x - 3 ) ( r + s ) > 0 2 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 > 0 2 7 . 4 f I 9 x < 9 ? n x * ' j , - x " - ' 2 - r - 3 * x? x - t < a
2 < 5 - 3 x < l l f < g l - x - 2 x 2 > 0 2 f - 6 x * 3 < 0 7 4 3 x - 7 - 3 - 2 x Prove Theorem 1,.1,.22. 13. 1,6.{ r : i
1 - 3 > Z - 7
" x x22.
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 important
definitions in the study of calculus. Furtherrnore, you will need. tb work
with inequalities involving "absolute value."
The absolute
aalue of x, denoted by lrl, is defined by
l * l - * *
i f x > 0
l t l : - t
i f x < o
l 0 l
: 0
1.2 ABSOLUTE VALUE
Thus, the absolute value of a positive number or zero is equal to the number itself. The absolute value of a nggative numloer is the cor-responding positive number because the negative of a negative number
is positive.
. I L L U s T R A T I o N
1 : l3l:3; l-51:-(-5):5;
1 8
- L 4 l :
l - 6 1 : - ( - 6 ) : 6 .
We see from the definition that the absolute value of a number is
either a positive number or zero; that is, it is nonnegative.
In terms of geometry, the absolute value of a number x is its distance
from 0, without regard to direction. In general, lo - bl is the distance
between a and b without regard to direction, that is, without regard to
which is the larger number. Refer to Fig. 1..2.1'.
We have the following properties of absolute values.
l"l < a lf. and only if
-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 haveto
consider 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:
lrl, we have l*l < o.
In both cases,
then,
< n i f - a < x ( a ,
w h e r e a > 0
panr 2: Prove that lrl < a only if -a 1x I A, where a > 0. Here we must show that whenever the inequality ltl < a holds, the inequality -a < x 1a also holds. Assume that lrl < a and consider the two cases x > 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, because a ) 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 5l - u - a : l a
- br|
l , - b = l a - b l 1
l-b F i g u r e 1 . 2 . 11.2.2 Theorem
"1..2.3
Corollary
\.2.4 Theorem
7.2.5 Corollary
be-EXAMPLE l-: Solve for x:
lar + 2l:5.
uxluplr 2: Solve for x:
lz*
- rl: l4x + 31.
rxr.vrpu 4: Find the solution set of the inequality
l x - 5 1 < +
Illustrate it on the real number line.
cause x I 0, it follows from Theorem l.r.zs that 0 < -x. Therefore, we
have -a < 0 ( -r < a, or -a <-x < a, which by applying Theorem
1..7.25
yields -a < x 1a.
In both cases,
l * l < o
o n l y i f - a < x 1 n ,
w h e r e a > 0
I
The proof of Theorcm7.2.4 is left as an exercise (see Exercise 27).
The following examples illustrate the solution of equations and
in-equalities involving absolute values.
solurroN: This equation will be satisfied if either
3 x * 2 : 5
o r - ( 3 r * 2 ) : 5
Considering each equation separatellr w€ have
x : L a n d x : - &
which are the two solutions to the given equation.
solurroN: This equation will be satisfied if either
2 x - L - 4 x * 3
o r 2 x - 1 - - G x + j \
Solving the first equation, w€ have x: -2; solving the second, we get
x: -*, thus giving us two solutions to the original equation.
solurroN: If r is a number such that
l r - 5 1 < a
then
- 4 < x-S < 4 (by Theorem'1,.2.2)
Adding 5 to each member of the preceding inequality, we obtain 7 < x < 9
Because each step is reversible, we can conclude that l x - 5 1 < i f a n d o n l y i f L < x < 9
So, the solution set of the given inequality is (1, 9), which is illustrated in Fig. 1,.2.2.
Exarnrpt,n 3: Solve f.or x:
l 5 r + 4 1
: - 3 .
soLUTroN: Because the absolute value of a number may never be nega-tive, this equation has no solution.
EXAMPLE
5: Find the solution
set of the inequatity
l3=-,,*l
= n
l 2 + 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 uPon
whether 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 solve
these 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 x
Adding -4x - 3 to both members gives
- 6 x < 5
Dividing both members by -5 and reversing the inequality sign, we
obtain
x > - E
Therefore, if x ) -2, then the x > - # a n d r > - 8 .
inequality holds if and only if
Because all three inequalities r ) -2, )c > -+, and r >
-B must be
satisfied 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,
Considering the left inequality, we have
- 8 - 4 x > 3 - 2 x
or, equivalently,
-2x >'l-.'1,
or, equivalently,
r s - +
From the right inequality we have
3 - 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 of x, we have x =-+, or the interval (-*, -u, l.
Combining the solution sets of Cases 1 and 2, we have as the solution
