A Course in Calculus and Real Analysis

547 Pages·2018·6.91 MB·English

by  Sudhir R. Ghorpade

Undergraduate Texts in Mathematics
Sudhir R. Ghorpade
Balmohan V. Limaye
A Course in
Calculus and
Real Analysis

Second Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
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Undergraduate Texts in Mathematics are generally aimedat third- and fourth-year
undergraduate mathematicsstudentsatNorthAmericanuniversities.These texts strive
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More information about this series at http://www.springer.com/series/666
Sudhir R. Ghorpade • Balmohan V. Limaye
A Course in Calculus
and Real Analysis
Second Edition
Sudhir R. Ghorpade Balmohan V. Limaye
Department of Mathematics Department of Mathematics
Indian Institute of Technology Bombay Indian Institute of Technology Dharwad
Powai, Mumbai 400076, India Dharwad, Karnataka 580011, India

ISSN 0172-6056 ISSN2197- 5604 (electr onic)
Undergraduate Texts in Mathematics
ISBN 978-3-030-01399-8 ISBN978 -3-030-01400-1 (eB ook)
https://doi.org/10.1007/978-3-030-01400-1

Library of Congress Control Number: 2018959752

Mathematics Subject Classification (2010): 26-01, 26A06, 40-01

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Preface
IthasbeenmorethanadecadesinceACourseinCalculusandRealAnalysis,
or in short, ACICARA, was first published. The response from students and
teachersalikehasbeengratifying.Overtheyears,wehavealsoreceivedseveral
commentsandsuggestionsfromreaders.Thuswekeptadding corrections that
werepointedoutbyothersornoticedbyourselvestothedynamicerrataonthe
web page of the book. It was also felt that the inclusion of some additional
topics would enhance the utility of the book. So when the publisher proposed
that we bring out a new edition of ACICARA, we thought the time was ripe
and accepted the suggestion. We then worked for about a year and a half to
put together this second edition, which contains a substantial amount of new
material.Thegoalsofthisbookwereenunciatedintheprefacetotheoriginal
edition1 and essentially remain the same. We briefly recall these below and
also mention some salient features of the book.
Our primary goal is to give a self-contained and rigorous introduction to
the calculus of functions of one variable, and in fact, a unified exposition of
calculus and classical real analysis. At the same time, we have attempted to
givedueimportancetocomputationaltechniquesandapplicationsofcalculus.
Thetopicscoveredaremostlystandard,andthenovelty,ifany,liesinhowwe
approach them. Throughout this text we have sought to make a distinction
between the intrinsic definition of a geometric notion and its analytic charac-
terization.Usuallyeachimportantresultisfollowedbytwokindsofexamples:
one to illustrate the result and the other to show that a hypothesis cannot be
dropped.Whenaconceptisdefineditappearsinboldface.Definitionsarenot
numbered but can be located using the index. Everything else (propositions,
examples,remarks,etc.)isnumberedseriallyineachchapter.Thenumbering
of exercisesnowindicates the number of thechapter wherethey occur,or the
symbol R, which corresponds to the Revision Exercises given at the end of
Chapter7.Theendofaproofismarkedbythesymbol(cid:2)(cid:3),whilethesymbol(cid:2)
1 Prefaceandthetableofcontentsforthefirsteditionareavailableonthewebpage
of the book: http://www.math.iitb.ac.in/∼srg/acicara/.
V
VI Preface
marks the end of an example or a remark. Citations to other books and arti-
cles appear as a number in square brackets, and the bibliographic details can
be found in the list of references. A list of symbols and abbreviations used in
the text is given, in the order of their appearance, after the list of references.
In the Notes and Comments at the end of each chapter, distinctive features
of the exposition are mentioned and pointers to some relevant literature and
furtherdevelopmentsareprovided.Exercisesforeachchapteraredividedinto
two parts: Part A contains problems that are relatively routine, while Part B
has problems that are of a theoretical nature or particularly challenging.
Themajoradditioninthiseditionisanewchapterthatdiscussessequences
and series of real-valued functions of a real variable, and their continuous
counterpart, namely, improper integrals depending on a parameter. Another
important addition consists of two appendices, of which the first outlines the
constructionofrealnumbers,whilethesecondprovidesaself-containedproof
of the fundamental theorem of algebra. Also, a section on cluster points of
sequencesisaddedtoChapter2,andoneonRiemannintegralsoverbounded
sets is added to Chapter 6. Besides these, a number of minor revisions have
been made in Chapters 1–9. Wherever appropriate, we have given references
to A Course in Multivariable Calculus and Analysis, or in short, ACIMC,
which is a sequel to ACICARA, published by Springer in 2010.
We thank IIT Bombay and IIT Dharwad for enabling us to work on this
book. The figures in the book were drawn using PSTricks. We are grateful to
ArunkumarPatilandNirmalaLimayeforcreatingthefiguresforthefirstand
the second editions. Jonathan Lewin read large parts of preliminary versions
ofChapters9and10,andmademanyvaluablecommentsandsuggestions,for
whichwearegratefultohim.WearealsoverythankfultoAnjanChakrabarty
and Venkitesh Iyer, who read the entire manuscript and pointed out several
corrections and provided useful suggestions. The editorial staff at Springer,
New York, have always been helpful, and we thank all of them, especially
Loretta Bartolini and Dimana Tzvetkova for their interest and kind cooper-
ation. We are also grateful to David Kramer for his excellent copyediting.
Last, but not least, we would like to thank our families for their support and
understanding, without which this book would not have been possible.
We wish to express our collective gratitude to all those who took time
to write to us with their comments, suggestions, and corrections in ACI-
CARA.Wewouldappreciatereceivingcommentsonthiseditionaswell.These
can be sent by e-mail to either of us at [email protected] and
[email protected],modifications,andrelevantinfor-
mationwillbepostedat http://www.math.iitb.ac.in/∼srg/acicara,and
we expect to maintain and periodically update this website.
Mumbai and Dharwad, India Sudhir Ghorpade
March 2018 Balmohan Limaye
Contents
1 Numbers and Functions ................................... 1
1.1 Properties of Real Numbers............................... 2
1.2 Inequalities ............................................. 10
1.3 Functions and Their Geometric Properties.................. 13
Exercises ................................................... 32
2 Sequences ................................................. 41
2.1 Convergence of Sequences ................................ 41
2.2 Subsequences and Cauchy Sequences....................... 54
2.3 Cluster Points of Sequences............................... 59
Exercises ................................................... 62
3 Continuity and Limits ..................................... 67
3.1 Continuity of Functions .................................. 67
3.2 Basic Properties of Continuous Functions................... 73
3.3 Limits of Functions of a Real Variable...................... 83
Exercises ................................................... 98
4 Differentiation.............................................105
4.1 Derivative and Its Basic Properties ........................106
4.2 Mean Value Theorem and Taylor Theorem..................119
4.3 Monotonicity, Convexity, and Concavity....................127
4.4 L’Hˆopital’s Rule.........................................133
Exercises ...................................................141
5 Applications of Differentiation .............................149
5.1 Absolute Minimum and Maximum.........................149
5.2 Local Extrema and Points of Inflection .....................152
5.3 Linear and Quadratic Approximations. .....................159
5.4 Picard and Newton Methods..............................163
Exercises ...................................................175
VII
VIII Contents
6 Integration ................................................181
6.1 Riemann Integral........................................181
6.2 Integrable Functions .....................................191
6.3 Fundamental Theorem of Calculus.........................202
6.4 Riemann Sums..........................................210
6.5 Riemann Integrals over Bounded Sets ......................216
Exercises ...................................................225
7 Elementary Transcendental Functions .....................233
7.1 Logarithmic and Exponential Functions ....................234
7.2 Trigonometric Functions..................................246
7.3 Sine of the Reciprocal....................................258
7.4 Polar Coordinates .......................................265
7.5 Transcendence ..........................................274
Exercises ...................................................279
Revision Exercises ...........................................289
8 Applications and Approximations of Riemann Integrals....295
8.1 Area of a Region Between Curves..........................295
8.2 Volume of a Solid .......................................302
8.3 Arc Length of a Curve ...................................314
8.4 Area of a Surface of Revolution ...........................322
8.5 Centroids...............................................328
8.6 Quadrature Rules .......................................340
Exercises ...................................................357
9 Infinite Series and Improper Integrals .....................365
9.1 Convergence of Series ....................................365
9.2 Convergence Tests for Series ..............................372
9.3 Power Series ............................................381
9.4 Convergence of Improper Integrals.........................391
9.5 Convergence Tests for Improper Integrals...................399
9.6 Related Improper Integrals ...............................406
Exercises ...................................................417
10 Sequences and Series of Functions,
Integrals Depending on a Parameter.......................425
10.1 Pointwise Convergence of Sequences .......................426
10.2 Uniform Convergence of Sequences ........................429
10.3 Uniform Convergence of Series ............................438
10.4 Weierstrass Approximation Theorems ......................448
10.5 Bounded Convergence....................................458
10.6 Riemann Integrals Depending on a Parameter...............466
10.7 Improper Integrals Depending on a Parameter ..............471
Exercises ...................................................492
Contents IX
A Construction of the Real Numbers.........................503
A.1 Equivalence Relations....................................503
A.2 Cauchy Sequences of Rational Numbers ....................505
A.3 Uniqueness of a Complete Ordered Field ...................512
B Fundamental Theorem of Algebra .........................517
B.1 Complex Numbers and Complex Functions .................517
B.2 Polynomials and Their Roots .............................519
References.....................................................523
List of Symbols and Abbreviations.............................527
Index..........................................................533