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TENTH EDITION
CALCULUS ONE VARIABLE
JOHN WILEY & SONS, INC.
CHAPTER I PRECALCULUS REVIEW
1.1 What isCalculus? I
1.2 Review of Elementary Mathematics J
1.3 Review of Inequalities II
1.4 Coordinate I'lane; Analytic Geometry 17
1.5 Functions 24
1.6 The Elementary Functions 32
1.7 Combinations of Functions 41
1.8 A Note on Mnthcmutical Proof; Mathematical Induction 41
CHAPTER 2 LIMITS AND CONTINUITY
2.1 The Limit Process (An Intuitive Introduction) S3
2.2 Definition of Limit 64
2.3 Some Limit Theorems 7J
2.4 Continuity 82
2.5 The Pinching Theorem; Trigonometric Limits 91
2.6 T"'O Basic Theorems 97
Project 2.6 The Bisection Method for Finding the Roots of f (x ) = 0 1(}2
53
xv
xvi • CONTENTS
CHAPTER 3 THE DERIVATIVE; THE PROCESS OF DIFFERENTIATION 105
3. 1 The Derivative 105
3.2 Some Ditl'crentimion Formulas 11 5
3.3 The d/dr Notation; Derivat ives of Higher Order 124
3.4 The Derivative As A Rateor Change 130
3.5 The Chain I{u[e 133
3.6 Differentiating The Trigonomt:trie runClions 142
3 .7 Implicit Difii:rcntiation; Rational Powers 147
CHAPTER 4 THE MEAN-VALUE THEOREM; APPLICATIONS OF THE FIRST AND SECOND DERIVATIVES 154
4.1 The Mean-Value Theorem 154
4 .2 Increasing and Decreasing Functions 160
4.3 Local Extreme Values 167
4.4 Endpoint Extrellll! Values; Absolute Extreme Values 174
4.5 Some Max-M in Problems 182
Projeet4.S Flight Paths ofBird~ 190
4.6 Concavi ly lmd Points of Inflection 190
4.7 Vertical and Horizontal Af.ymptotcs; Vert ical Tangents alld Cusps 11)5
4.8 Some Curve Sketching 201
4.9 Veloci ty and Accelerat ion; Speed 209
Prujet.:t 4.9A Angular Vdoeity; Unifornl Cin:ular Motion 2 17
Project 4.98 Energy of a railing Body (Ncar the Surface of tile Earth) 217
4.10 Related Rates of Change per Unit Time 218
4. 11 Differentials 223
Project 4.11 Mar};inal Cost, Marginal ReV'cllue, Marginal Profi t 228
4.12 Newton-Raph:>on Approximations 229
CHAPTER 5 INTEGRATION
5. I An Area Problem; a Speed·Distance Problem 234
5.2 The Definite Integral ora ContinuoL.!s Function 237
5.3 The Function j(x) = 1~ j(t)dt 246
5.4 The Fundamental Theorem of Integral Ca lculus 254
5 .5 Some Area Problems 260
Project 5.5 In tegrabi lity; Integrat ing Discontinuous Funct ions 266
5 .6 Indefinite Integra ls 268
5 .7 Working l3ack from the Chain R\lle; the II -Substitution 274
5.8 Addi tional Properties of the Definite Integra l 281
5 .9 Mean-Value Theorems for Integrals; Average Value o r a Funct ion 285
234
CHAPTER 6 SOME APPLICATIONS OF THE INTEGRAL
6.1 MoreonArca 292
6 .2 Volume by Para llel Cross Sections; Disks and Washers 296
6.3 Volume by the Shell Method 306
6.4 The Centroid ora Region; j'appus's Theorem on Volumes 312
Project 6.4 Cenlmid ofa Solid o f Revolution 3 19
6.5 ThcNotion of Work 319
"6.6 Fluid Force 327
CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS
7.1 One-la-One runclil)ns; Jnver~es 333
7.2 The Lognrithm Function, Part r 342
7.3 The LAgarithm Function, Pm1 II 347
7.4 The Exp()ncntial Function 356
Project 7.4 Some RationalOounds for the Number e 364
7.5 Arbitmry Po ..... 'Cf'Sj Other Bases 364
7.6 Exponential Growth and Decay 370
7.7 The Inverse Trigonometric Functions 378
Projccl 7.7Refrnction 387
7.8 The Hyperbolic Sine and Cosine 388
'7.9 The Other Hyperbolic Functions 392
CHAPTER 8 TECHNIQUES OF INTEGRATION
8.1 jnlegrdl Tables and Review 398
8.2 Intcgnltion by Parts 402
Project 8.2 Sine Wavesy = sinnx and Cosine Wavesy = cosnx 410
8.3 Powers 1l11d Products ofTrigonomelric Functions 411
8.4 Integrals Featuring .J'ii'f=X!, ..rar+:?, ~ 417
8.5 Rational Functions; Partia l Fractions 422
' 8.6 Somc Rationalizing Substitutions 430
8.7 NU Jllerical lmegration 433
CHAPTER 9 SOME DIFFERENTIAL EQUATIONS
9 .1 First-Order Linear Equ31ions 444
9 .2 Integral Curves; Scp3r.Ib1c Equations 451
Projecl9.2 Orthogonal Trajectories 458
9.3 The Equation y" + ar' + by = 0 459
' Denotcsoptional !.Celion.
CONTENTS • xvii
292
333
398
443
r
xviii. CONTENTS
CHAPTER 10 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS 469
10.1 Geometry of Parubola. Ell ipse, Hyperbola 469
10.2 Polar Coordinates 478
10.3 Sketching Curves in Polar Coordin&les 484
Projttt 10.3 Parabola, Ellip!>e, Hyperbola in Polar Coordinates 491
10.4 Area in PolarCoordinales 492
10.5 Curves Given Parametrically 496
Project 10.5 ParaboHc Trajectories 503
10.6 Tangents to Curves Given Panunctricnlly S03
10.7 Arc Lengtband Speed 509
10.8 The Area of A Surface of Rev oJ uti on; The Centroidofa Curve; Pappus's Theorem on Surface Area 517
Project 10.8 The Cycloid 525
CHAPTER II SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS
11.1 The Least UppcrBound Axiom 521:1
11.2 Sequences or Real Numbers 532
11 .3 Limit of a Sequence 538
Project 11.3 Sequences and the Ncwton-Raphson Method 541
11 .4 Some Important limitS 550
11.5 The Indetenninate Fonn (0/0) SS4
11.6 The Indetenninate Form (00/ 00); Other Indeterminate Forms 560
11.7 Improper Integrals 565
CHAPTER 12 INFINITE SERIES
12.1 Sigma Norutioll 575
12.2 Infinite Series 577
12.3 The Integral Test; Basic Comparison, Limit CompOlrison 585
12.4 The Root Test; the: Ratio Test 593
12.5 Absolute Convergencc and Condit ional Convergence; Alternating Series 597
12.6 Taylor Polynomials in .f; Taylor Series in x 602
12.7 Taylor Polynomials and Taylor Series in;r - a 613
12.8 Power Serics 616
12.9 Differclltiation and Integration of Power Series 623
Project 12.9A The Binomial Serics 633
Project 12.98 Estimating 1l 634
528
575
APPENDIX A SOME ADDITIONAL TOPICS
A. l Rotation of AxeR; El iminating the xy-Term A-I
A.2 Determinants A-3
APPENDIX B SOME ADDITIONAL PROOFS
8 . 1 The Intermediate-Value Theorem A-8
B.2 l3oundcdncss; Extn:me-Valuc Theorem A-9
B.3 Inverses A- \O
B.4 The Integrabil ity of Continuous Functions A-ll
8 .5 The Integral as the Limit of Riemann Sums A-14
ANSWERS TO ODD-NU MBERED EXERCISES A- 15
Index 1-1
Tob ie o f Integra ls Inside Covers
CONTENTS • xix
A - I
A-8
I
TABLES OF INTEGRALS
POWERS
J u·" 1. ~du=--+C.n:;f':"" l
n+1 J du 2. -;; = In lui + C
EXPONENTIALS AND LOGARITH MS
3. J e"du=e"+C
5. f ue"du=ue"-e"+C
7. J u"e" du = u"e" - n J Un-leu du
9. ! (l nU)2 du = U(lnu)2- 2ulnu+2u+C
11 . u" ln udu=u"+ ----- + C J ,[ In" I 1 11 + 1 (n + I f
4. / P" du=i!....+C In p
6. f u2ell du = u 2e" - 2ue" + 2e" + C
8. !ln udu=Ulnu - u +C
10. J ulnudu =4u21n u - *u2+C
/
d" 12. - = inllnul+ C
u Inu
SINES AND COSINES
13. ! sinudu = -cosu + C
15. J sin2udu= ~ u - ~sin211 +C
17. f sinJ udu = f cosJ u - cosu + C
J. sin,,-luc05 11 n- I f .. 2 19. sm" udu =---,,--+ - n-j sin - IIdu
/
cos"-I usinu 11-1 / 2 20. cos" II du = --,,-- + -n- cos" - II du
21. J II sinuciu = - /I casu + sinu + C
23. ju"sinudu= - ullcosu+n ! u,,-lcosudu
14. f cosudu = sinu + C
16. f cos2 udu = 4u +* sin lu + C
18. J cosJ udu = s in u - t sin3 ,t + C
22. f ucosudu=usinu+cosu+C
24. f u"cosudu=u"sinu - n J un - Is inudu
/
.. sin [em - n)u] sin [em + n)u] 2 2 25. smll/usmlllldu=~-~+C, m #n
26 J d sin[(m - n)u] sin [(m +n)u] C 2,# 2 . cosmucosnu u=~+~+ ,m n
/
. cos [(m- n)u] cos [(m+n)u] 2 2 27. smmucosnudu = 2(m - n) ~+c, 111 ":f:.n
28. f (!"usi n budu= a2:Jil (aSin bu-bCOSbU) + C
/
e"" 29. ellU cosbu du = -'--2(acosbu + bsinbu) + C
a +b
TANGENTS AND SECANTS
30. J lanudu = ln lsec ul + C
32. f tan2udu = lan u - u + C
34. f seeu lan udu =seeu +C
36. J sec3 lldll= ~sec ll tanu + ~ ln l seeu+tan ll l+C
37. lanl/udll = - - - tann- 1udu /
lann-I II / n - 1
31. f seeudu= ln lsecu+ tanu l + C
33. f seel udll = tanu + C
35. f tan3 udu = 4 tan2 u + In leosu l + C
38. sec" II dll = ---- + - - sec - 2 u du /
sec"-2I1tan ll n- 2/ " n - 1 n - 1
CONTANGENTS AND COSECANTS
39. J cOll/du = In I sinll l + C
4J. ! co~ u du = - col u - u + C
43.! cscucotudu =- cscu+C
45. f ese3 udu = -4esc li COl u + i In Icscll - cm u l + C
46. / COt"Udu=_cot"- l u -/ eorr-2 udll 11 -1
47. cscnutiu = ----- + - - esc"-2 Il du /
CSCn-21lCotu n-2/ n - 1 n - I
40. f eseu du = In Icsc u - cot ul + C
42.! csc2udll = - cot u+C
44. J eOl3 1ldu = -~ COl2
II - In Isin ul + C
HYPERBOLIC FUNCTIONS
48. J si nh udu=eoshu+C
50. J tanh u du = In (cosh u) + C
52. J scchu du = arctan (sinhu)+C
54. J scch 2udu = tanh II + C
56. J sech u tanh II du = - scch II + C
58. f sinhl udu = ~sinh 2u - 4" + C
60. f lanh2II du = u - Ianhu + C
62. f us inhudu = ucoshu - sinhu + C
49. f coshudu = sinh u + C
51. J cothu du = In I sinh II I + C
53. ! cschudll = In I tanh 4111 + C
55. J csch 2udu = - cath 1/ + C
57. f cseh ueolil udll= -csch ll+C
59. f cosh2udll= ~sinh2u + 4u +C
61 . f eoth2 udu =u- cothu+C
63. f u cosh udu = usinhu - cosh u + C
(rabiecofilinued<JIlhehadr;)
(continued/rom/herronl)
INVERSE TRIGONOMETRIC FUNCTIONS
64. f arcsinudll=lIarcsinu+~+C 65. f arccosudu=lIarccosu -~+C
66. f arctanu du = u arctanu - 4 In(1 + u 2) + C 67. f arccotu du = u arccotu + 4 1n(1 + u2
) + C
68. f arcsecudu = uarcsecu -In lu +~I +C 69. f arccscudu= uarccscu+ln lu+~1 +C
70. f u arcsinudu = ~[(2u2-I)arcsinu + u~ + C] 71. f uarctanlldu = 4(u2 + l)arctanu - 4u + C
72. f lIarccosudu = ~ [(2U2 - 1) arccos u - u~ + c]
J I [ . J u"+' ] 73. unarCSinlldu= - - un+1 arCSlTIU- ~du ,n#- -i n+1 v1 - u2
J I [ J u"+' ] 74. u" arccosu du = - - un+1 arccosu + "----------.,, du ,n #- - I n+l vl-u2
J 1 [ J u"+' 1 75. unarctanudu=-- u,,+l arctanu- --2du ,n#- -l n + 1 1 +u
~,a>O
76. -- = - arctan - + C J du 1 u
a2 +u2 a a 77. f .ja~:u2 =In (u+Ja2
+u2) +C
78. f ..,Ja2 +u2 du = ~.Ja2 +u2 + ~In(u +.Ja2 +u2) +C
79.! 1I2 Ja2 +u2 du=
J.Ja2+ u2 la+../a2+u2 1 80. --u-du =.Ja2+u2 - aln --u-- +C
J~ ,/a2+u' ( ~) 81. -u-,- du= ---u- +1n u+va2 +u2 +C
J u2du u a2 ( ) 82. ~=-..,J02+u2- - 1n u+.Ja2 + u2 +C
v0 2 +u2 2 2
. ---_ -- In ---- + 83 J du _ 1 la+~1 C· u~ a u J
du .Ja2 + u2
84. --- ~ ---- + C u2../a2 + u2 a2u
85. ---~ --- +C J du 11
(a 2 + u2)3/2 a2 ../ 02 + u2
~,a>O
86. - - - = arcsin - + c J du u
../a2 _ u2 a J u a2 u
87. /a 2 -u2du = Z.Ja2 _u2 + 2arcsin~ + C
88. u2.Ja2 - u2 du = _(2u 2 - ( 2))02 - u2 + - arcsin - + C J u a4 u
8 8 a
j~ r;------, /a+v'a' -"'/ 89. --,,- dl/= ..;02-u2 -0In --,, __ +C
j~ lr=;-----:; .11 90. --- dll= --va 2 _112 -arcstn - + C
I12 II a j /l2du U r=;-----:; 0 2 II 91. --- = --va2 _112 + -arcsin- +C ~ 2 2 a
j du 1 /a+~/ 92. u./02 _ U2 =-;; In --u-- +C j d" 1 r;------, 93. ---=--va2- u2+C
u2~ a1u
j dll II 95. ---=---+C
(a 2 _ 1/2)3/2 a2.ja2 _ 11 2 j du 1 la +"1 96. --=- In -- +C 0 2 - u l 20 a - II
J u2 _ a2 , u > 0
j till I lui 97. 11./11 2 _ 02 = -;:; arcsec -;; + C j~ r=;-----:; II
98. --- dll = 'l/U2 _ 0 2 - aarcsec - + C " a
99. f Ju2 -a1du = ~ju2 _ 0 2 - ~ln ) u +/112 - a2) +C
j ~ -I,,' _a' I r;------,I 101. -,-,,- dU=---,,-+ ln u+ ..;u2 _02 +C
j 112 dll /I r=;-----:; 02 I r;------,I 102. --- = -V1l2 _ 0 2 + - In II + "u2 _02 +C
JIl2 _02 2 2
103. j du ./112 - 02 j du u
-",-.;-,,-, --a' = -a-',-, - + C 104. -(u-, --a'-)'-/' = --a'-';-"-' --a- ' + C
l05.j~-~+1 I +~I + C (u 2 _a 2)3/ 2 -~ n II II a
a + bu,...;a+bii
j udu I 106. a+bu =/J.(a+bu - aln1a+bul)+ C
j u2du 1 2 107. o+bll = W[(a+bu) - 4a(a+bu)+2a21n la+bul] +C
1 08.j ~=.!. ln l-"_ I +C u(a+bu) a a + bu 109. ---=--+-In -- +C j du 1 b la+b" l
u2(a + bll) au a2 /I
j udu Q I 110. (a+bu)' = b'(a+bu) +p- ln1a+bul+ C Il l. ---=----- In -- +C j du 1 1 la+ bu l
u(a + bull a(a + bll) a2 II
j u'dU I( a' ) Ill. ---~ =- a+bu - - --2a ln la+bul +C (a ~bur OJ a+bu
j u du 2,--,,- j r-:-;- 2 113. ~=----;(bu-2a)",a +bu +C 114. uva+blldu=~(3bu - 2a)(a+buiI2+C ,,0 - bu 3tr ISb
This text is devoted to the study of single variable calculus. Whil e applications from the sciences, engineering, and economics are often used to motivate or ill ustrate mathematical ideas, the emphasis is on the three basic concepts of calculus: limit, derivative, and integral.
This edition is the rcsult of a collaborative effort with S.L. Salas, who scrutinized every single sentence in the text for possible improvement in precis ion and readability. His gift for writing and his uncompromising standards of mathematical accuracy and clarity jJl uminate the beauty or the subject while increasing its accessibility to studenls. It has been a pleasure for me to work with him.
FEATURES OF THE TENTH EDITION
Precision and Clarity
The emphasis is on mathematical exposition; the topics are treated in a cl ear and understandable manner. Mathematical statements are careful and precise; the basic concepts and important points are nOI obscured by excess verbiage.
Balance of Theory and Applications
Problems drawn trom the physical sciences are often used to introduce basic concepts in calculus. In turn, the concepts and methods of calculus are applied to a variety of probl ems in the sciences, engineering, business, and the social sciences through text examples and exercises. Because the presentation is fl exible, instructors can vary the balance of theory and applications according to the needs of their students.
Accessibility
This tex t is designed to be completely accessible to the beginning calculus student without sacrificing appropriate mathematical rigor. The important theorems are explained
vii
viii • PREFACE
and proved, and the mathematical techniques are justified . These may be covered or omitted according to the theoretical level desired in the course.
Visualization
TIle importance of visualization cannot be over-emphasized in developing students' understanding of mathematical conccpts. !-o r that reason, over 1200 illustrations accompany the text examples and exercise sets.
Technology
The technology component of the text has been strengthened by revising existing exercises and by developing new exercises. Well over halfof the exercise sets have problems requiring either a graphing util ity or a computer algebra system (CAS). Technology exercises (designated by~) are designed to illustrate or expand material developed within the sections .
Projects
Projects with an emphasis on problem solving orrcr students the opportunity to investigate a variety of special topics that supplement the text material. The projects typically require an approach that involves both theory and applications, including the usc o f technology. Many of the projects are sui tablc for group-learning activities.
Early Coverage of Differential Equations
Different ial equations are fonnally introduced in Chapter 7 in connection with applications to exponential growth and decay. Firsl-order li near equations, separable equations, and second linear equations wi th constant coefficients, plus a variety of appl ications, arc treated in a separate chapter immediate ly fo llowing the techniques o f integration materi al in Chapter 8.
CHANGES IN CONTENT AND ORGANIZATION
In our effort to produce an even more effective text, we consulted with the users of the Ninth Edition and with other calculus instructors. Our primary goals in preparing the Tenth Edition were the following:
1. Improve the exposition. As notcd above, every topic hlls been examined for possible improvement in the clarity and aeeumeyo f its presentation. Essentially every section in the text underwent some revision; a number of sections and subscctions were completely rewritten.
2. Improve the illustntti\'c examples. Many of the ex isting examples have been modified to enhance students' understanding of the material. New examples have been added to sections that were rewritten or substantially revised.
3. Revise th e exercise sets. Every exercise sct was examined for balance between drill problems, midlevel problems, and more challenging applications and conceptual problems. In many lnstances, the number of routine problems was reduced and new midleve! to challenging problems were added.
Specific changes made to achieve these goals and meet the needs of today's students and instructors include:
Comprehensive Chapter-End Review ExerCise Sets
The Skill Mastery Review Exercise Sets introduced in the Ninth Edition have been expanded into chapter-end exercise sets. Each chapter concludes wi th a comprehensive set of problems designed to test and to re-enforce students' understanding of basic concepts and methods devcloped within the chapter. These review excreise sets average over 50 problems per set.
Precalculus Review (Chapter 1)
The subj ect matter of lhis chapter- inequali ties, basic analytic geometry, the func tion concept and the elementary func tions- is unchanged. However, much of the material has been rewritten and simplified.
limits (Chapter 2)
The approach to limits is unchanged, but many of the explanations have been revised . The illustrati ve examples throughout the chapter have been modified, and new examples have been added.
Differentiation and Applications (Chapters 3 and 4)
There are some sign ifi cant changes in the organization of this material. Realizing that a m treatments of lin car motion, rates of change per unit time, and the Newton-Raphson method depended on an understanding of increasingldeereasing functions and the concavity of gmphs, we movcd these topics from Chapter 3 (the derivative) to Chaptcr 4 (applications of the derivative). Thus, Chapter 3 is now a shorter chapter which focuses solely on the derivative and the processes of differentiation, and Chapter 4 is ex panded to encompass all of the standard applications of the derivative---curve-sketch ing, optimization, linear motion, rates of change, and approximation. As in all previous editions, Chapter 4 begins with the mean-value theorem as the theoretical basis for all the applicatlOns.
Integration and Applications (Chapters 5 and 6)
In a brief introductory section, area and distance are used to motivate the definite integral in Chapter S. Whil e the defin ition of the definite integral is based on upper and lower sums, the connection with Riemann sums is also given. Explanations, examples, and exercises throughout Chapters 5 and 6 have been modified, blll the topics and organi zation remain llS in the Ninth Edition.
The Transcendental Functions, Techniques of Integration (Chapters 7 and 8)
The coverage of the inverse trigonometric functions (Chapter 7) has been reduced slightly. The treatment of powers of the trigonometric functions (Chapter 8) has been completely rewri tten. The optional sections on first-order linear differential equations and separable differential equations have been moved to Chapter 9, the new chapter on differential equations.
Some Differential Equations (Chapter 9)
Th is new chapter is a brief introduction to differential equations and their applications. In addition to the coverage of fi rst-order linear equations and separabl e equations noted
PREFACE • ix
x • PREFACE
above, we have moved the section on second-order linear homogeneous equations with constant coefficients from the Ninth Edition 's Chapter 18 to tltis chapter.
Sequences and Series (Chapters 11 and 12)
Efforts were made to reduce the overall length of these chapters through rewriting and eliminating peripheral material. Eliminating extmneous problems reduced severa1 exercise sets. Some notations and terminology have been modified to be consistent with common usage.
SUPPLEMENTS
An Instructor's Solutions Manual, ISBN 0470127309, includes solutions for all problems in the text.
A Student Solutions Manual, ISBN 0470105534, includes solutions for selected problems in the text.
A Companion Web site, www.wiley.comJcollegc/salas, provides a wealth ofresources for students and instructors, including:
• Power Point Slides for important ideas and graphics for study and note taking.
• Online Review Quizzes to enable students to test their knowledge of key concepts. For further review diagnostic feedback is provided that refers to pertinent sections of the text.
• Animations comprise a series of interactive Java applets that allow students to explore the geometric significance of many major concepts of Calculus.
• Algebra and Trigonometry Refreshers is a self-paced, guided review of key algebra and trigonometry topics that are essential for mastering calculus.
• Personal Response System Questions provide a convenient source of questions to use with a variety of personal response systems.
• Printed Test Bank contains static tests which can be printed for quick tests.
• Co mputerized Test Bank includes questions from the printed test bank with algorithmically generated problems.
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PREFACE • xl
CHAPTER
In this chapter we gather together for reference and review those parts of elementary mathematics that are necessary for the study of calculus. We assume that you are familiar with most of this material and that you don't require detailed explanations. But first a few words about the nature of calculus and a brief outline of the history of the subject.
• 1.1 WHAT IS CALCULUS?
To a Roman in the days of the empire, a "calculus" was a pebble used in counting and gambling. Centuries later, "calculare" came to mean "to calculate," "to compute," "to figure out." For our purposes, calculus is e lementary mathematics (algebra, geometry, trigonometry) enhanced by the limit process.
Calculus takes ideas from elementary mathematics and extends them to a more general situation. Some examples are on pages 2 and 3. On the left-hand side you will find an idea from elementary mathematics; on the right, this same idea as extended by calculus.
It is fin ing to say something about the history of calculus. The origi ns can be traced back to ancient Greece. The ancient Greeks raised many questions (often paradoxical) about tangents, motion, area, the infinitely smail, the infinitely large-qucstions that today are clarified and answered by calculus. Here and (here the Greeks themselves provided answers (some very elegant), but mostly they provided only questions.
Elementary Mathematics
slope of a line y = mx +b
Calculus
slope of a curve Y ~ f(x)
(Table continues)
- - ----
------- -
mass of an object of constant density
center ofa sphere
1.2 REVIEW OF ELEMENTARY MATHEMATI
mass of an object of varying density
center of gravity of a more general solid
A fter the Greeks, progress was slow. Communication was limited, and each scholar was obliged to start almost from scratch. Over the centuries, some ingenious solutions to calculus·type problems were devised, but no general techn iques were put forth. Progress was impeded by the lack of a convenient notalion. Algebra, fowlded in the ninth century by Arab sc holars, was not fu lly systematized until the sixteenth century. Then, in the seventeenth century, Descartes established analytic geometry, and the stage wassel.
The actual invention of calculus is credi ted to Sir Isaac Newton (1642- 1727), an Engli shman, and to Gottfried Wilhelm Leibniz (1646-171 6), a German. Newton 's invention is one of the few good tums that the great plague did mankind. The plague forced the closi ng ofCrunbridge University in 1665, and young Isaac Newton of Trinity College returned to his home in Lincolnshire fo r eighteen months of meditation, out of which grew his method offtllxion~·, his theory of gravitation, and his theolY aflight. The method of t1uxions is what concerns us here. A treatise with this title was written by Newton ill 1672, but it remained unpublished until 1736, nine years after his death. The new method (calculus to us) was first announced in 1687, but in vague general terms without symbolism, formulas, or applications. Newton himsel f seemed reluctant to publish anything tangible about his new method, and it is not surprising that its development on the Continent, in spite of a late start, soon overtook Newton and went beyond him.
Leibniz started his work in 1673 , eight years after Newton. In 1675 he initiated the basic modern notalion: dT and f. His first publications appeared in 1684 and 1686. These made li tt le stir in Germany, but the two brothers Bernou lli of Basel (Switzerland) took up the ideas and added profusely to them. From 1690 onward, calculus grew rapidly and reached roughly its present state in about a hundred years. Certain theoretical subtleties were not fu lly resolved until the twentieth century .
• 1.2 REVIEW OF ELEMENTARY MATHEMATICS
In this section we review the terminology, notation, and formulas of elementary math· cmatics.
Sets
A set is a collection of distinct objects. The objcels in a set are called the elements or members of the sct. We will denote sets by capital letters A , B, C, . . unduse !o\\'ercase letters a, b, c, . .. to denote the elements.
CHAPTER 1 PRECALCULUS REVIEW
For a collection of objects to be a set it must be well-defined; that is, given any object x, it must be possible to determine with certainty whether or not x is an element of the set. Thus the collection of a ll even numbers, the collection of aU lines parallel to a given line I, the solutions of lhe equation x 2 = 9 are a ll sets. The collection of all inlelligent adults is not a sel. It 's not clear who should be included.
Noltons and Notation
the object x is in the set A x E A the object x is not in the set A x r;. A
the set of all x which satisfy property P Ix : PJ ({x ,,' ~ 9) ~ (- 3, 3))
A is a subset of S, A is contained in BAS; B B contains A B :2 A
the union of A and B A U B (A U B = {x : x E A or x E B I)
the intersection of A and BAn B (A n B = {x : x E A and x E B))
the empty set
These arc thc only notions from set theory that y Oll will nced at this point.
Real Numbers
CIo$$fflcatton
positive integerst 1,2,3, . integers 0, I , - I , 2, - 2, 3, -3,.
rational numbers p / q. with p , q integers, q '" 0; for example, 5/2, - 19/ 7, - 4 / 1 = - 4
irrational numbers real numbers which are not rational; for example ./i, -Y7, 1r
Decimal Representation
Each real number can be expressed as a decimal. To express a rational number p/q as a decimal, we divide the denominator q into the numerator p. The resulting decimal either terminates or repeats :
3 5 ~ 0.6 ,
are tcnninating decimals;
~ ~ 0.6666 .. ~ 0.6,
116
27 W~1.35,
J7 ~ 3.135135· . ~ 3.135
43 8" ~5.375
and
arc rcpeating decimals. (The bar over the sequence of digits indicatcs that the sequence repeats indefinitely.) The converse is also true; namely, every terminating or repeating decimal represents a rational number.
l AlsocallednafUrolnumberJ'
1.2 REVIEW OF ELEMENTARY MATHEMATIC
The decimal expansion of an irrational number can neither terminate nor rcpeal. The expansions
h ~ 1.4142 13562 ·· and rr = 3.141592653 ·
do not terminate and do not develop any repeating pattern. If we stop the decimal expansion of a given number at a certain decimal place,
then the result is a rational number that approximates the g iven number. For instance, 1.414 = 14141!OOO is a rational number approximation to ..J2 and 3.1 4 = 314/ 100 is a rat ional number approximation to Jf. More accurate approx imations can be obtained by using more decimal places from the expansions.
The Number Line (COOfdinale Line. Reclline)
On a hori zontal line \\'e choose a point O. We call this point the orig in and assign to it coordinate O. Now we choose a point U to the right of 0 and assign to it coordinate I. See Figure 1.2.1. The d istance between 0 and U determi nes a scale (a unit length). We go on as follows: the point a units to the right of a is ass igned coordinate a; the point a units to the left of 0 is assigned coordinate -a.
In this manner we establish a one· to-one correspondence between the points o f a line and the numbers of the real number system. Figure 1.2.2 shows some real numbers represented as points on the number li ne. Positive numbers appear to the right o f 0, negative numbers to the left of O.
I I 1 1
- 2 :f 1::/f
O,der Properties
(I) Either a < b,b < a, Ofa = b.
(ii) If a < bandb <c,then a <c.
Figure 1.2.2
(iii) If a < b, then a + c < b + c fOf all real numbers c.
(iv) If a < band c > 0, then ac < be.
(v) If a < band c < 0, then ae > be.
(Techniques for solving inequali ties arc reviewed in Section 1.3.)
Density
II
(trichotomy)
(transitivity)
Between any two real numbers there are infinitely many rational numbers and infinitely many irrational numbers. In particular, there is no smallest posirive realllumber.
Absolute Value
lal = ( a, ~ r o::: ° - 0, lfa < O.
other characterizal;ons lal = maxta, -a); lal = N. geometric interpretation lal = distance between a and 0;
10 - cl = distance between a and c .
Figure 1.2.1
• CHAPTER 1 PRECALCULUS REVIEW
properties (i) la l = 0 iff a = O.t (II) I -al ~ Ial.
(iii) labl ~ lallbl. (Iv) la + hi s ial + Ibl. (v) Ilol - lbll s la - bl.
(vi) la l2 = 1021 = ,, 2 .
(lhetriangkinequali ty)1t (a variantofthetriungleinequali ty)
Techniques for solving inequali ties that feature absolute value are reviewed in Section 1.3.
Intervals
Suppose that a < b. The open inten'al (a , b) is the sel o fal! numbers between a and b:
(a. b) = Ix : a < x < b).
The closed interval [a, b] is the open interval (a . b) together with the endpoints a and b:
[a. b] = Ix : a :s x :s hI.
There are seven other types of intervals:
(a, bJ = Ix : a < x :s bJ.
[a. b) = Ix : a :s x :s h},
(a . oo)={x : a < x).
[a, 00) = {x : a :s x l.
(- 00, b) ~ Ix 'x < b} .
(-00, b] ~ Ix 'x s bl,
(-00, 00) = the sct of real numbers.
Interval notation is easy to remember: we use a square bracket to include an endpoint and a parenthesis to exclude it. On a number line, inclusion is indicated by a solid dot, excl usion by an open dot. The symbols 00 and - 00, read "infin ity" and "negative infinity" (or "minus infinity"), do not represellt real numbers. In the intervals listed above, the symbol 00 is used to indie3le that the interval extends indefi nitely in the positive direction; thc symbol - 00 is used to indicate that the interval extends indefinitely in the negativc direction.
Open and Closed
Any interVtl1 thai contains no endpoims is called open: (0, b), (a, 00), (-00, b), (-00, (0) are open. Any interval th<lt contains each of its endpoints (there may be one or two) is culled closed: [a, b], [a, 00), (- 00, b] are closed. The intervals (a, b] and [a . b ) are called half-open (ha lf-closed): (a, h] is open on the left and closed on the right; [a, b) is closed on the le ft and open on the right. Points o ran interval which are not endpoints are called interior points of the interval.
t l3y "iff"· \\'C mean "if ami only ie This expression is used so oft en in mathematics thai ii's eonvt nie:ntto have: an abbreviation for it.
t1The: absolute \l3.1uc of the sum of two numbers cannot exceed the sum of their absolute va lues. This is analogous to the fact that in a triangle tht lenb"h of one ~ide cannot e;.:ceed the: sum of the lengths of the other t .... o sides.
