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Undergraduate Texts in Mathematics

Editors

S. Axler

K.A. Ribet

Undergraduate Texts in Mathematics

(continued after index)Abbott:Understanding Analysis.

Anglin:Mathematics: A Concise History and

Philosophy.

Readings in Mathematics.

Anglin/Lambek:The Heritage of Thales.

Readings in Mathematics.

Apostol:Introduction to Analytic Number

Theory. Second edition.

Armstrong:Basic Topology.

Armstrong:Groups and Symmetry.

Axler:Linear Algebra Done Right. Second

edition.

Beardon:Limits: A New Approach to Real

Analysis.

Bak/Newman:Complex Analysis. Second

edition.

Banchoff/Wermer:Linear Algebra Through

Geometry. Second edition.

Berberian:A First Course in Real Analysis.

Bix:Conics and Cubics: A Concrete

Introduction to Algebraic Curves.

Brèmaud:An Introduction to Probabilistic

Modeling.

Bressoud:Factorization and Primality

Testing.

Bressoud:Second Year Calculus.

Readings in Mathematics.

Brickman:Mathematical Introduction to

Linear Programming and Game Theory.

Browder:Mathematical Analysis: An

Introduction.

Buchmann:Introduction to Cryptography.

Second Edition.

Buskes/van Rooij:Topological Spaces: From

Distance to Neighborhood.

Callahan:The Geometry of Spacetime: An

Introduction to Special and General

Relavitity.

Carter/van Brunt:The Lebesgue- Stieltjes

Integral: A Practical Introduction.

Cederberg:A Course in Modern Geometries.

Second edition.

Chambert-Loir:A Field Guide to Algebra

Childs:A Concrete Introduction to Higher

Algebra. Second edition.

Chung/AitSahlia:Elementary Probability

Theory: With Stochastic Processes and an

Introduction to Mathematical Finance.

Fourth edition.

Cox/Little/O"Shea:Ideals, Varieties, and

Algorithms. Second edition.

Croom:Basic Concepts of Algebraic

Topology.

Cull/Flahive/Robson:Difference Equations.

From Rabbits to Chaos

Curtis:Linear Algebra: An Introductory

Approach. Fourth edition.Daepp/Gorkin:Reading, Writing, and

Proving: A Closer Look at Mathematics.

Devlin:The Joy of Sets: Fundamentals

of Contemporary Set Theory. Second edition.

Dixmier:General Topology.

Driver:Why Math?

Ebbinghaus/Flum/Thomas:Mathematical

Logic. Second edition.

Edgar:Measure, Topology, and Fractal

Geometry.

Elaydi:An Introduction to Difference

Equations. Third edition.

Erdõs/Surányi:Topics in the Theory of

Numbers.

Estep:Practical Analysis on One Variable.

Exner:An Accompaniment to Higher

Mathematics.

Exner:Inside Calculus.

Fine/Rosenberger:The Fundamental Theory

of Algebra.

Fischer:Intermediate Real Analysis.

Flanigan/Kazdan:Calculus Two: Linear and

Nonlinear Functions. Second edition.

Fleming:Functions of Several Variables.

Second edition.

Foulds:Combinatorial Optimization for

Undergraduates.

Foulds:Optimization Techniques: An

Introduction.

Franklin:Methods of Mathematical

Economics.

Frazier:An Introduction to Wavelets

Through Linear Algebra.

Gamelin:Complex Analysis.

Ghorpade/Limaye:A Course in Calculus and

Real Analysis

Gordon:Discrete Probability.

Hairer/Wanner:Analysis by Its History.

Readings in Mathematics.

Halmos:Finite-Dimensional Vector Spaces.

Second edition.

Halmos:Naive Set Theory.

Mathematics.

Readings in Mathematics.

Harris/Hirst/Mossinghoff:Combinatorics and

Graph Theory.

Hartshorne:Geometry: Euclid and Beyond.

Hijab:Introduction to Calculus and Classical

Analysis.

Hilton/Holton/Pedersen:Mathematical

Reflections: In a Room with Many

Mirrors.

Hilton/Holton/Pedersen:Mathematical Vistas:

From a Room with Many Windows.

Iooss/Joseph:Elementary Stability and

Bifurcation Theory. Second Edition.

Sudhir R. Ghorpade

Balmohan V. Limaye

A Course in Calculus

and Real Analysis

With 71 Figures

Sudhir R. Ghorpade

Department of Mathematics

Indian Institute of Technology Bombay

Powai, Mumbai 400076

INDIA srg@math.iitb.ac.inBalmohan V. Limaye

Department of Mathematics

Indian Institute of Technology Bombay

Powai, Mumbai 400076

INDIA bvl@math.iitb.ac.in Mathematics Subject Classification (2000): 26-01 40-XX

Library of Congress Control Number: 2006920312

ISBN-10: 0-387-30530-0 Printed on acid-free paper.

ISBN-13: 978-0387-30530-1

© 2006 Springer Science+Business Media, LLC

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adap- tation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America. (MVY)

987654321

springer.com

Series Editors:

S. Axler

Mathematics Department

San Francisco State University

San Francisco, CA 94132

USA axler@sfsu.eduK. A. Ribet

Department of Mathematics

University of California at Berkeley

Berkeley, CA 94720-3840

USA ribet@math.berkeley.edu

Preface

Calculus is one of the triumphs of the human mind. It emerged from inves- tigations into such basic questions as “nding areas, lengths and volumes. In the third century B.C., Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of “nding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In partic- ular, they showed that the geometric problems of “nding the areas of planar regions and of “nding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today, calculus occupies a central place in mathematics and is an essential component of undergraduate education. It has an immense number of appli- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics. Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notions encountered in calculus, but with greater rigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables. This book attempts to give a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due im- portance is given to computational techniques and applications. In the course of our exposition, we highlight the fact that calculus provides a “rm foun- dation to several concepts and results that are generally encountered in high school and accepted on faith. For instance, this book can help students get a clear understanding of (i) the de“nitions of the logarithmic, exponential and trigonometric functions and a proof of the fact that these are not algebraic functions, (ii) the de“nition of an angle and (iii) the result that the ratio of

VIPreface

the circumference of a circle to its diameter is the same for all circles. It is our experience that a majority of students are unable to absorb these concepts and results without getting into vicious circles. This may partly be due to the division of calculus and real analysis in compartmentalized courses. Calculus is often taught as a service course and as such there is little time to dwell on subtleties and gain perspective. On the other hand, real analysis courses may start at once with metric spaces and devote more time to pathological exam- ples than to consolidating students" knowledge of calculus. A host of topics such as L"Hˆopital"s rule, points of inflection, convergence criteria for Newton"s method, solids of revolution, and quadrature rules, which may have been in- adequately covered in calculus courses, become pass´e when one studies real analysis. Trigonometric, exponential, and logarithmic functions are defined, if at all, in terms of infinite series, thereby missing out on purely algebraic moti- vations for introducing these functions. The ubiquitous role ofπas a ratio of various geometric quantities and as a constant that can be defined indepen- dently using calculus is often not well understood. A possible remedy would be to avoid the separation of calculus and real analysis into seemingly disjoint courses and textbooks. Attempts along these lines have been made in the past as in the excellent books of Hardy and of Courant and John. Ours is another attempt to give a unified exposition of calculus and real analysis and address the concerns expressed above. While this book deals with functions of one variable, we intend to treat functions of several variables in another book. The genesis of this book lies in the notes we prepared for an undergraduate course at the Indian Institute of Technology Bombay in 1997. Encouraged by the feedback from students and colleagues, the notes and problem sets were put together in March 1998 into a bookletthat has been in private circulation. Initially, it seemed that it would be relatively easy to convert that booklet into a book. Seven years have passed since then and we now know a little better! While that booklet was certainly helpful, this book has evolved to acquire a form and philosophy of its own and is quite distinct from the original notes. A glance at the table of contents should give the reader an idea of the topics covered. For the most part, these are standard topics and novelty, if any, lies in how we approach them. Throughout this text we have sought to make a distinction between the intrinsic definition of a geometric notion and the analytic characterizations or criteria that are normally employed in studying it. In many cases we have used articles such as those inACenturyof Calculusto simplify the treatment. Usually each important result is followed by two kinds of examples: one to illustrate the result and the other to show that a hypothesis cannot be dropped. When a concept is defined it appears in boldface. Definitions are not num- bered but can be located using the index. Everything else (propositions, ex- amples, remarks, etc.) is numbered serially in each chapter. The end of a proof is marked by the symbol??, while the symbol?marks the end of an example or a remark. Bibliographic details about the books and articles mentioned in the text and in this preface can be found in the list of references. Citations

PrefaceVII

within the text appear in square brackets. A list of symbols and abbreviations used in the text appears after the list of references. TheNotes and Commentsthat appear at the end of each chapter are an important part of the book. Distinctive features of the exposition are men- tioned here and often pointers to some relevant literature and further devel- opments are provided. We hope that these may inspire many fruitful visits to the library-not when a quiz or the final is around the corner, but per- haps after it is over. TheNotes and Commentsare followed by a fairly large collection of exercises. These are divided into two parts. Exercises in Part A are relatively routine and should be attempted by all students. Part B con- tains problems that are of a theoretical nature or are particularly challenging. These may be done at leisure. Besides the two sets of exercises in every chap- ter, there is a separate collection of problems, called Revision Exercises which appear at the end of Chapter 7. It is in Chapter 7 that the logarithmic, ex- ponential, and trigonometric functions are formally introduced. Their use is strictly avoided in the preceding chapters. This meant that standard examples and counterexamples such asxsin(1/x) could not be discussed earlier. The Revision Exercises provide an opportunity to revisit the material covered in Chapters 1-7 and to work out problems that involve the use of elementary transcendental functions. The formal prerequisites for this course do not go beyond what is normally covered in high school. No knowledge of trigonometry is assumed and expo- sure to linear algebra is not taken for granted. However, we do expect some mathematical maturity and an ability to understand and appreciate proofs. This book can be used as a textbook for a serious undergraduate course in calculus. Parts of the book could be useful for advanced undergraduate and graduate courses in real analysis. Further, this book can also be used for self- study by students who wish to consolidate their knowledge of calculus and real analysis. For teachers and researchers this may be a useful reference for topics that are usually not covered in standard texts. Apart from the first paragraph of this preface, we have not discussed the history of the subject or placed each result in historical context. However, we do not doubt that a knowledge of the historical development of concepts and results is important as well as interesting. Indeed, it can greatly enrich one"s understanding and appreciation of the subject. For those interested, we encourage looking on the Internet, where a wealth of information about the history of mathematics and mathematicians can be readily found. Among the various sources available, we particularly recommend the MacTutor History of Mathematics archivehttp://www-groups.dcs.st-and.ac.uk/history/ at the University of St. Andrews. The books of Boyer, Edwards, and Stillwell are also excellent sources for the history of mathematics, especially calculus. In preparing this book we have received generous assistance from vari- ous organizations and individuals. First, we thank our parent institution IIT Bombay and in particular its Department of Mathematics for providing ex- cellent infrastructure and granting a sabbatical leave for each of us to work

VIIIPreface

on this book. Financial assistance for the preparation of this book was re- ceived from the Curriculum Development Cell at IIT Bombay, for which we are thankful. Several colleagues and students have read parts of this book and have pointed out errors in earlier versions and made a number of useful suggestions. We are indebted to all of them and we mention, in particular, Rafikul Alam, Swanand Khare, Rekha P. Kulkarni, Narayanan Namboodri, S. H. Patil, Tony J. Puthenpurakal, P. Shunmugaraj, and Gopal K. Srinivasan. The figures in the book have been drawn usingPSTricks, and this is the work of Habeeb Basha and to a greater extent of Arunkumar Patil. We appreciate their efforts, and are grateful to them. Thanks are also due to C. L. Anthony, who typed a substantial part of the manuscript. The editorial and TeXnical staff at Springer, New York, have always been helpful and we thank all of them, especially Ina Lindemann and Mark Spencer for believing in us and for their patience and cooperation. We are also grateful to David Kramer, who did an excellent job of copyediting and provided sound counsel on linguistic and stylistic matters. We owe more than gratitude to Sharmila Ghorpade and

Nirmala Limaye for their support.

We would appreciate receiving comments, suggestions, and corrections. These can be sent by e-mail toacicara@gmail.comor by writing to either of us. Corrections, modifications, and relevant information will be posted at http://www.math.iitb.ac.in/≂srg/acicaraand we encourage interested readers to visit this website to learn about updates concerning the book.

Mumbai, IndiaSudhir Ghorpade

July 2005Balmohan Limaye

Contents

1 Numbers and Functions................................... 1

1.1 PropertiesofRealNumbers............................... 2

1.2 Inequalities............................................. 10

1.3 FunctionsandTheirGeometricProperties.................. 13

Exercises ................................................... 31

2 Sequences................................................. 43

2.1 ConvergenceofSequences ................................ 43

2.2 SubsequencesandCauchySequences....................... 55

Exercises ................................................... 60

3 Continuity and Limits..................................... 67

3.1 ContinuityofFunctions .................................. 67

3.2 BasicPropertiesofContinuousFunctions................... 72

3.3 LimitsofFunctionsofaRealVariable...................... 81

Exercises ................................................... 96

4 Differentiation.............................................103

4.1 TheDerivativeandItsBasicProperties....................104

4.2 TheMeanValueandTaylorTheorems.....................117

4.3 Monotonicity,Convexity,andConcavity....................125

4.4 LHopitalsRule.........................................131

Exercises ...................................................138

5 Applications of Differentiation.............................147

5.1 AbsoluteMinimumandMaximum.........................147

5.2 LocalExtremaandPointsofIn"ection.....................150

5.3 LinearandQuadraticApproximations .....................157

5.4 ThePicardandNewtonMethods..........................161

Exercises ...................................................173

XContents

6Integration................................................179

6.1 TheRiemannIntegral....................................179

6.2 IntegrableFunctions.....................................189

6.3 The Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . 200

6.4 RiemannSums..........................................211

Exercises ...................................................218

7 Elementary Transcendental Functions.....................227

7.1 Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . 228

7.2 TrigonometricFunctions..................................240

7.3 SineoftheReciprocal....................................253

7.4 PolarCoordinates .......................................260

7.5 Transcendence ..........................................269

Exercises ...................................................274 RevisionExercises ...........................................284

8 Applications and Approximations of Riemann Integrals. . . . 291

8.1 AreaofaRegionBetweenCurves..........................291

8.2 VolumeofaSolid .......................................298

8.3 ArcLengthofaCurve ...................................311

8.4 AreaofaSurfaceofRevolution ...........................318

8.5 Centroids...............................................324

8.6 QuadratureRules .......................................336

Exercises ...................................................352

9 Infinite Series and Improper Integrals.....................361

9.1 ConvergenceofSeries....................................361

9.2 ConvergenceTestsforSeries..............................367

9.3 PowerSeries............................................376

9.4 ConvergenceofImproperIntegrals.........................384

9.5 ConvergenceTestsforImproperIntegrals...................392

9.6 RelatedIntegrals........................................398

Exercises ...................................................410 References.....................................................419 List of Symbols and Abbreviations.............................423 Index..........................................................427 1

Numbers and Functions

Let us begin at the beginning. When we learn the script of a language, such as the English language, we begin with the letters of the alphabet A, B, C, ...; when we learn the sounds of music, such as those of western classical music, we begin with the notes Do,Re,Mi,.... Likewise, in mathematics, one begins with 1,2,3,...; these are thepositive integersor thenatural numbers. We shall denote the set of positive integers byN.Thus,

N={1,2,3,...}.

These numbers have been known since antiquity. Over the years, the number 0 was conceived 1 and subsequently, the negative integers. Together, these form the setZof integers. 2 Thus,

Z={...,-3,-2,-1,0,1,2,3,...}.

Quotients of integers are calledrational numbers. We shall denote the set of all rational numbers byQ.Thus, Q= m n :m,n?Z,n?=0 Geometrically, the integers can be represented by points on a line by fixing a base point (signifying the number 0) and a unit distance. Such a line is called thenumber lineand it may be drawn as in Figure 1.1. By suitably subdi- viding the segment between 0 and 1, we can also represent rational numbers such as 1/n,wheren?N, and this can, in turn, be used to represent any 1 The invention of 'zero", which also paves the way for the place value system of enumeration, is widely credited to the Indians. Great psychological barriers had For more on this, see the books of Kaplan [39] and Kline [41]. 2 The notationZfor the set of integers is inspired by the German wordZahlenfor numbers.

2 1 Numbers and Functions

Fig. 1.1.The number line

rational number by a unique point on the number line. It is seen that the rational numbers spread themselves rather densely on this line. Nevertheless,quotesdbs_dbs12.pdfusesText_18

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