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AMS / MAA TEXTBOOKS VOL 16 Deconstructed A Second Course in First-Year Calculus

Zbigniew Nitecki i i “Calculus” — 2011/10/11 — 16:16 — page i — #1 i i

10.1090/text/016

Calculus Deconstructed A Second Course in First-Year Calculus

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Cover Design: Elizabeth Holmes Clark Cover Image: iStock, Izmabel

© 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009923531 Print ISBN: 978-0-88385-756-4 Electronic ISBN: 978-1-61444-602-6 Printed in the United States of America Current Printing (last digit): 10987654321

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Calculus Deconstructed A Second Course in First-Year Calculus

Zbigniew H. Nitecki Tufts University

®

Published and distributed by The Mathematical Association of America

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Council on Publications Paul M. Zorn, Chair MAA Textbooks Editorial Board Zaven A. Karian, Editor George Exner Thomas Garrity Charles R. Hadlock William Higgins Douglas B. Meade Stanley E. Seltzer Shahriar Shahriari Kay B. Somers

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MAA TEXTBOOKS

Calculus Deconstructed: A Second Course in First-Year Calculus, Zbigniew H. Nitecki : A Problem Oriented Approach, Daniel A. Marcus Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological , Robert Edward Lewand Differential Geometry and its Applications, John Oprea Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman Essentials of Mathematics, Margie Hale Field Theory and its Classical Problems, Charles Hadlock Fourier , Rajendra Bhatia and Strategy, Philip D. Straffin Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer Graph Theory: A Problem Oriented Approach, Daniel Marcus Knot Theory, Charles Livingston Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and James W. Daniel Mathematical Modeling in the Environment, Charles Hadlock Mathematics for BusinessDecisionsPart 1: Probability and Simulation (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for BusinessDecisionsPart 2: Calculus and Optimization (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux The Mathematics of Games and Gambling, Edward Packel Math Through the Ages, William Berlinghoff and Fernando Gouvea Noncommutative Rings, I. N. Herstein Non-Euclidean Geometry, H. S. M. Coxeter Number Theory Through Inquiry, David C. Marshall, Edward Odell, and Michael Starbird A Primer of Real Functions, Ralph P. Boas A Radical Approach to , 2nd edition, David M. Bressoud Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson

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Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange about the of the infinite, which they must try to use.

Leonhard Euler Introductio in Analysin Infinitorum (1755) [23, Preface]

Mathematicians are a kind of Frenchmen: whatever you say to them they translate into their own language and it is then something quite different.

Johann Wolfgang von Goethe Maximen und Reflexionen

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In memoriam

MARTIN M.GUTERMAN

1941–2004

Colleague

Coauthor

Friend

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Preface

Calculus is a collection of incredibly efficient and effective toolsforhandlinga widevariety of mathematical problems. Underpinning these methods is an intricate structure of ideas and arguments; in its fullest form this structure goes by the name of real analysis. The present volume is an introduction to the elementary parts of this structure, for the reader with some exposure to the techniques of calculus who would liketo revisit the from a more conceptual point of view. This book was initially developed for the first semester of the Honors Calculus se- quence at Tufts, which gives entering students with a strong background in high school calculus the opportunity to cover the topics of the three-semester calculus se- quence in one year. It could, however, prove equally useful for a course using analysis as the topic for a transition to higher mathematics. The challenge of the Honors Calculus course was twofold. On one hand, the great variability among high school calculus courses required us to cover all the standard topics again, to make sure students end up “on the same page” as their counterparts in the main- stream course. On the other hand, these bright students would be bored out of their by a straight repetition of familiar topics. Keeping in these competing demands, I opted for a course that approaches the tools of calculus through the eyes of a . In contrast to “Calculus Lite”, the present book is “Calculus Tight”: a review of often familiar techniques is presented in the spirit of mathematical rigor (hopefully without the mortis), in a context of ideas, and with some sense of their history. For myself as a mathematician, the intellectual exercise of working through all the theorems of single-variable calculus and constructing an ele- mentary but coherent and self-contained presentation of this theory has been a fascinating . In the process, I have also become fascinated with the history of the subject, and hope that my enthusiasm will help to brighten the exposition.

Idiosyncracies After three centuries of calculus texts, it is probably impossible to offer something truly new. However, here are some of the I have made:

Level: “Calculus” and “analysis”—which for Newton et al. were aspects of the same subject—have in our curriculum become separate countries. I have stationed this book firmly on their common border. On one side, I have discussed at some length topics that would be taken for granted in many analysis courses, in order to fill any “gaps” in the reader’s previous mathematical experience, which I assume to be lim- ited tovarious kinds of calculation. I have also limitedmyself to mathematical topics

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associated with the basics of calculus: advanced notions such as connectedness or compactness, metric spaces and function spaces demand a level of sophistication which are not expected of the reader, although I would like to think that this book will lead to attaining this level. On the other side, I have challenged—and hopefully intrigued—the reader by delving into topics and arguments that might be regarded as too sophisticated for a calculus course. Many of these appear in optional sections at theend of most chapters, and in the Challenge Problems and Historical Notes sec- tions of the problem sets. Even if some of them go over the heads of the audience, they provide glimpses of where some of the basic ideas and concerns of calculus are headed.

Logic: An underlying theme of this book is the way the subject hangs together on the of mathematical arguments. The approach is relatively informal. The reader is assumed to have little if any previous experience with reading definitions and theo- rems or reading and writing proofs, and is encouraged to learn these skills through practice. Starting from the basic, familiar properties of real numbers (along with a formulation of the Completeness )—rather than a formal of axioms— we develop a feeling for the language of mathematics and methods of proof in the concrete context of examples, rather than through any formal study of and . The level of mathematical sophistication, and the demands on the proof- writing skills of the reader, increase as the exposition proceeds. By the end of the book, the reader should have attained a certain level of competence, fluency and self- confidence in using the rhetorical devices of mathematics. A brief appendix at the end of the book (Appendix A) reviews some basic methods of proof: this is intended as a reference to supplement the discussions in the text with an overview.

Limits: The limit at the heart of this treatment is the , rather than the standard ε δ (which is presented in the optional § 3.7). At − the level of ideas, this concept is much easier to absorb, and almost everything we need can be formulated naturally using sequences.

Logarithms and exponentials: Most rigorous treatments of calculus define the logarithm as an and the exponential as its inverse. I have chosen the ahistorical but more natural route of starting with natural powers to define exponentials and then defining logarithms as their inverses. The most difficult step in this “early transcen- dentals” approach, from a rigor point of view, is the differentiabilityof exponentials; see § 4.3.

History: I have injected some of the history of various ideas, in some narrative at the start of each chapter and in exercises denoted History Notes, which work through specific arguments from the initiators of the subject. I make no claims for this as a historical work; much of the discussion is based on secondary sources, although where I have had easy access to the originals I have tried to consult them as well.

Technology: There are no specific references to technology in the text; in my class, the primary tools are paper and pencil (or blackboard and chalk). This does not, of course, preclude the use of technology in conjunction with this book. I would, in

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fact, love to hear from users who have managed to incorporate computer or graphing calculator uses into a course based on the present exposition.

How to use this book Clearly, it is impossible to cover every proof of every result (or anything near it) in class. Nevertheless, I have tried to give the reader the resources to see how everything is sup- ported by careful arguments, and to see how these arguments work. In this sense, the book is part textbook, part background reference. The beginning of each chapter sketches the highlights of the topics and tools to come from a historical perspective. The text itself is designed to be read, oftenwith paper and pencil in hand, balancing techniques and intuition with theorems and their proofs. The exercises come in four flavors: Practice problems are meant as drill in techniques and intuitive exploration of ideas.

Theory problems generally involve some proofs, either to elaborate on a comment in the text or to employ arguments introduced in the text in a “hands on” manner.

Challenge problems require more ingenuityor persistence; in my class they are optional, extra-credit problems.

Historical notes are hybrids of exposition and exercise, designed to aid an active explo- ration of specific arguments from some of the originators of the field. Of course, my actual homework assignments constitute a selection from the exercises given in the book. A note at the start of each set of exercises indicates problems for which answers have been provided in Appendix B; these are almost exclusively Practice problems. Solutions to all problems, including proofs, are given in the Solution Manual accompanying this text. On average, one section of text corresponds to one hour in the classroom. The last section of each chapter after the first is optional (and usually skippedin my class), although the of uniform continuity from § 3.7 appears in the rigorous proof of integrability of continuous functions.

Acknowledgements First of all I want to thank two wonderful classes of students who helped me develop this book, not necessarily by : a small cadre acted as guinea pigs for the “alpha” version of this book—Brian Blaney, Matt Higger, Katie LeVan, Sean McDonald and Lu- cas Walker—and a larger group—Lena Chaihorsky, Chris Charron, Yingting Cheng, Dan Eisenberg, Adam Fried, Forrest Gittleson, Natalie Koo, Tyler London, Natalie Noun, A.J. Raczkowski, Jason Richards, Daniel Ruben, and Maya Shoham—were subjected to the “beta” version, very close to the one here: they enthusiastically rooted out misprints, errors and obfuscations. I would like to particularly single out Jason Richards, who repeatedly questioned and suggested changes to a number of proofs, first as a student in the “beta” class and then as a homework grader in the course for several years.

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I would also like to thank a number of colleagues for helpful conversations, including Fulton Gonzalez, Diego Benardete, Boris Hasselblatt, Jim Propp, George McNinch, and Lenore Feigenbaum. The comments from various anonymous reviewers at the MAA were also very helpful. Second, I want to acknowledge extensive email help with TeXnical issues in the prepa- ration of this manuscript from Richard Koch, who patiently explained features of TeXShop to me, and to numerous contributors to [email protected], especially E. Krishnan, Dennis Kletzing, Lars Madsen, Tom DeMedts, Herbert Voss and Barbara Beaton, who helped me with a cornucopia of special packages. For further help in the final production process I would liketo thank the productionstaff at the MAA, in particular Elaine Pedreira and Beverly Ruedi. Third, I would like to acknowledge the inspiration for the historical approach which I gained from Otto Toeplitz’s Calculus, a Genetic Approach [53]. I have drawn on many sources for historical , above all Charles Edwards’ wonderful exposition, The Historical Development of the Calculus [21]. I have also drawn on Michael Spivak’s Cal- culus [51] and Donald Small & John Hosack’s Calculus, an Integrated Approach [49]; the latter was for many years the text for our course at Tufts. A more complete list of sources and references is given in the Bibliography at the end of this book. One mild disclaimer: when I could, I listed in the Bibliography an original source for some of my historical notes: such a listingdoes not necessarily constitute a claim that I have been able to consult the original directly. Citations in the text of the notes indicate most of my sources for my statements about history. Finally, a word about Marty Guterman, to whose memory this work is dedicated. Dur- ing the thirty or so years that we were colleagues at Tufts, and especially in the course of our joint textbook projects, I learned a great deal from this extraordinary teacher. His untimely death occurred before the present project got underway, but we had frequently discussed some of the ideas that have been incorporated in this text. His reactions and comments would have been highly valued, and he is missed.

Z. Nitecki

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Contents

Preface xi

1 1 1.1 NumbersandNotationinHistory...... 1 1.2 Numbers,Points,andtheAlgebraofInequalities ...... 3 1.3 Intervals...... 7

2 Sequences and their Limits 11 2.1 RealSequences ...... 11 2.2 LimitsofRealSequences ...... 15 2.3 ConvergencetoUnknownLimits ...... 26 2.4 FindingLimits...... 36 2.5 BoundedSets ...... 53 2.6 TheBisectionAlgorithm(Optional) ...... 63

3 Continuity 69 3.1 ContinuousFunctions...... 70 3.2 The Intermediate Theorem and Inverse Functions ...... 83 3.3 ExtremeValuesandBoundsforFunctions ...... 92 3.4 LimitsofFunctions ...... 100 3.5 Discontinuities ...... 115 3.6 ExponentialsandLogarithms ...... 119 3.7 EpsilonsandDeltas(Optional) ...... 126

4 Differentiation 135 4.1 ,SpeedandTangents ...... 136 4.2 Formal Differentiation I: The Algebra of ...... 146 4.3 Formal Differentiation II: Exponentials and Logarithms...... 164 4.4 FormalDifferentiationIII:InverseFunctions ...... 171 4.5 Formal Differentiation IV: The Chain Ruleand Implicit Differentiation . 176 4.6 RelatedRates ...... 185 4.7 ExtremaRevisited...... 192 4.8 GeometricApplicationofLimitsandDerivatives ...... 209 4.9 MeanValueTheorems ...... 217 4.10 L’Hôpital’sRuleandIndeterminateForms ...... 226 4.11 ContinuityandDerivatives(Optional) ...... 237

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5 Integration 247 5.1 AreaandtheDefinitionoftheIntegral ...... 248 5.2 GeneralTheoryoftheRiemannIntegral ...... 267 5.3 TheFundamentalTheoremofCalculus...... 281 5.4 FormalIntegrationI:ManipulationFormulas ...... 294 5.5 FormalIntegrationII:TrigonometricFunctions ...... 306 5.6 Formal Integration III: Rational Functions and Partial Fractions . . . . . 314 5.7 ImproperIntegrals...... 337 5.8 GeometricApplicationsofRiemannSums ...... 348 5.9 Riemann’s Characterization of Integrable Functions (Optional) . . . . . 362

6 367 6.1 Local ApproximationofFunctionsbyPolynomials ...... 368 6.2 ConvergenceofSeries...... 384 6.3 UnconditionalConvergence...... 398 6.4 ConvergenceofPowerSeries ...... 404 6.5 HandlingPowerSeries ...... 411 6.6 Complex Numbers and AnalyticFunctions(Optional) ...... 439

A TheRhetoricofMathematics(MethodsofProof) 445

B Answers to Selected Problems 451

Bibliography 477

Index 481

About the Author 491

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Bibliography

m m(m 1) 2 m(m 1)(m 2) [1] Niels HenrikAbel. Recherches sur la série 1+ 1 x+ 1 2− x + −1 2 3 − +... . In L. Sylow and S. Lie, editors, Œvres Complètes de Niels-Hendrik· Abel· · , volume 1, pages 219–250. Christiana, 1881.

[2] Archimedes. Measurement of a circle. In The Works of Archimedes [32], pages 91–98. Dover reprint, 2002.

[3] Archimedes. On the sphere and cylinder. In The Works of Archimedes [32], pages 1–90. Dover reprint, 2002.

[4] Roger Baker, Charles Christenson, and Henry Orde (translators). Bernhard Riemann Collected Papers. Kendrick Press, 2004.

[5] Margaret E. Baron. The Origins of the Infinitesimal Calculus. Pergamon, 1969. Dover reprint 1987, 2003.

[6] Garrett Birkhoff. A Source Book in Classical Analysis. Harvard Univ. Press, 1973.

[7] Bernard Bolzano. Purely analytic proof of the theorem that between any two values, which give results of opposite sign, there lies at least one real root of the equation. In The Mathematical Works of Bernard Bolzano [47], pages 250–277. Translation of Rein analytischer Beweis des Lehrfasses, dass zwischen ye zwei Werthen, die ein entgegengefestes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (, 1817).

[8] Bernard Bolzano. Theory of functions. In The Mathematical Works of Bernard Bolzano [47], pages 429–589. Translation of Functionenlehre (1830-35, unpub- lished).

[9] Umberto Bottazzini. The Higher Calculus: A history of Real and Complex Analysis from Euler to Weierstrass. Springer Verlag, 1986.

[10] Carl B. Boyer. History of Analytic Geometry. Yeshiva University, 1956. Dover reprint, 2004.

[11] Carl B. Boyer. A History of Mathematics. John & Sons, 1968.

[12] David M. Burton. The History of Mathematics. McGraw-Hill, 5th edition, 2003.

[13] Augustin-Louis Cauchy. Cours d’analyse de l’école royale polytechnique. Paris, 1821.

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478 Bibliography

[14] Augustin-Louis Cauchy. Sur la développement des fonctions en séries, et sur l’integration des équations différentielles ou aux différences partielles. Bulletin de la Societé Philomatique, pages 49–54, 1822.

[15] J. M. Child. The Early Mathematical Manuscripts of Leibniz. Open Court Publ., 1920. Translations by J. M. Child of key works of Leibniz on calculus. Reprinted by Dover, 2005. [16] I. Bernard Cohen and Anne Whitman. :The Principia (Mathematical of Natural Philosophy), A new translation. Univ. of CaliforniaPress, 1999. Translation from the Latin by I. Bernard Cohen and Anne Whitman of Philosophiae Naturalis Principia Mathematica (1687, 1713, 1726).

[17] J. L. Coolidge. The number e. American Mathematical Monthly, 57:591–602, 1950. reprinted in [22, pp. 8-19]. [18] Lewis Carroll (Charles Dodgson). Alice’s Adventures in Wonderland. Macmillan & Co., 1865. Numerous reissues.

[19] William Dunham. The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton Univ. Press, 2005.

[20] Gerald A. Edgar, editor. Classics on . Westview Press, 2004. [21] C. H. Edwards. The Historical Development of the Calculus. Springer-Verlag, 1979.

[22] Tom M. Apostol et. al., editor. Selected Papers on Calculus. Mathematical Associa- tion of America, 1969. [23] . Introduction to the Analysis of the Infinite. Springer-Verlag, 1988. Translation by John D. Blanton of Introductio in Analysin Infinitorum (1748).

[24] Leonhard Euler. Foundations of . Springer-Verlag, 2000. Trans- lation by John D. Blanton of Chapters 1-9 of Institutiones Calculi Differentialis (1755). [25] Giovanni Ferraro. The Rise and Development of the Theory of Series Up To the Early 1820’s. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, 2008.

[26] David Fowler. The Mathematicsof ’s Academy: A New Reconstruction. Claren- don Press, Oxford, second edition, 1999. [27] Judith V. Grabiner. The Origins of Cauchy’s Rigorous Calculus. MIT Press, 1981. Dover reprint, 2005.

[28] I. Grattan-Guinness. The Development of the Foundations of from Euler to Riemann. MIT Press, 1970. [29] Niccolò Guicciardini. Reading the Principia: The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736. Cambridge University Press, 1999.

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Bibliography 479

[30] Jacques Hadamard. Essai sur l’étude des fonctions données par leur développement de Taylor. J. de Math., 8(4):101–186, 1892. [31] E. Hairer and G. Wanner. Analysis by its History. Springer, 1995. [32] Thomas L. Heath. The Works of Archimedes. Cambridge University Press, 1897, 1912. Dover reprint, 2002. [33] Thomas L. Heath. The Thirteen Books of Euclid’s Elements. Cambridge University Press, 1908. Dover reprint, 1956. [34] Victor J. Katz. A History of Mathematics, an introduction. Addison-Wesley, second edition, 1998. [35] Johannes Kepler. Nova stereometria doliorum vinariorum. Linz, 1615. A selection from this appears in [52, pp. 192–7]. [36] Karl Kiesswetter. A simple example of a function, which is everywhere continuous and nowhere differentiable. In Edgar [20], pages 340–9. Translation by Eric Olson of a 1966 article. [37] Morris Kline. Mathematical from Ancient to Modern . Oxford Univ. Press, 1972. [38] Joseph Louis Lagrange. Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits, d’évanouissans, de limitesou de fluxions, et réduitsà l’analysealgébraiquedes quan- tités finies. l’imprimerie de la république, 1797. Second edition, Gauthier-Villars, Paris, 1813. [39] Reinhard Laubenbacher and David Pengelley. Mathematical Expeditions: Chronicles by the Explorers. Undergraduate Texts in Mathematics: Readings in Mathematics. Springer, 1999. [40] GottfriedWilhelm Leibniz. De quadratura arithmeticacirculi ellipseos et hyperbolae cujus corollarium est trignonmetria sine tabulis. Vandenhoeck & Ruprecht, 1993. Edited by E. Knobloch. [41] Jesper Lützen. The foundationsof analysis in the 19th century. In Hans Niels Jahnke, editor, A History of Analysis, chapter 6, pages 155–195. American Mathematical Society, 2003. [42] Isaac Newton. Mathematical principles of natural philosophy. In Isaac Newton: The Principia [16], pages 401–944. Translation from the Latin by I. Bernard Cohen and Anne Whitman of Philosophiae Naturalis Principia Mathematica (1687, 1713, 1726). [43] Fred W. Perkins. An elementary example of a continuousnon-differentiable function. American Mathematical Monthly, 34:476–478, 1927. Reprinted in [22, pp. 137–9]. [44] Daniel Reem. New proofs of basic theorems in calculus. Math Arxiv, 0709.4492v1, 2007.

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[45] Bernhard Riemann. Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe (on the representability of a function by means of a trignon- metric series). In Weber [54], pages 227–264. An English translation of part of this appears in [6, pp. 16–23]. [46] Ranjan Roy. The discovery of the series formula for pi by Leibniz, Gregory and Nilakantha. Mathematics Magazine, 63(5):291–306, 1990. [47] Steve Russ. The Mathematical Works of Bernard Bolzano. Oxford Univ. Press, 2004. [48] George F. Simmons. Calculus Gems: Brief Lives and Memorable Mathematics. McGraw-Hill, 1992. reissued by MAA, 2007. [49] Donald B. Small and John M. Hosack. Calculus, An Integrated Approach. McGraaw- Hill, 1990. [50] David E. Smith. A Source Book in Mathematics. McGraw-Hill, 1929. Dover reprint, 1959. [51] Michael Spivak. Calculus. Publish or Perish, Inc., second edition, 1980. [52] Dirk J. Struik. A Source Book in Mathematics, 1200-1800. Harvard Univ. Press, 1969. [53] Otto Toeplitz. The Calculus, A Genetic Approach. Univ. of Chicago Press, 1963. Translation by Luise Lange of Die Entwicklung der Infinitesimalrechnung, unpub- lished manuscript from 1926–39. Reissued by MAA, 2007. [54] Heinrich Weber, editor. Gesammelte Mathematische Werke und Wissentschaftlicher Nachlass (Collected Mathematical Works of Bernhard Riemann). Dover, 1953. A full translation of this collection into English is [4]. [55] . Über continuerliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotientenbesitzen (on con- tinuous functions of a real argument that do not have a well-defined differential quo- tient). In Edgar [20], pages 2–9. Translated from the German by B. Sawhill. [56] Derek T. Whiteside. The Mathematical Works of Isaac Newton. Johnson Reprint Corporation, 1964. [57] Derek T. Whiteside. The Mathematical Papers of Isaac Newton. Cambridge Univer- sity Press, 1967–1981. 8 vols.

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Index

0 , 227 Archimedes of Syracuse (ca. 287–212 BC) ∞0 ∞ , 228 Measurementof a Circle, 260, 264 0 , 232 On the Sphere and Cylinder, 264 · ∞ , 233 The Method, 357 ∞ − ∞ 00, 233 The Quadrature of the Parabola,45, 289 1∞, 234 area, 247 0, 235 area of circle, 260–261 ∞ calculation of π, 261–263 Abel, Niels Henrik (1802–1829) convex curve, 264 convergenceof power series, 418, 435 , 45 series, 368 method of compression, 11, 45 absolute value, 4 series, 367 abuse of notation, 20, 60 , 135 acc(X), 101 volume of cone, 357 accumulation point arcsine, see function, trigonometric, inverse of a sequence,33 arctangent, see function, trigonometric, inverse of a set, 100 area, 256, 348–353 accuracy, 18 of circle Adelard of Bath (1075–1160) Archimedes, 260–263 translation of Euclid’s Elements,2 Euclid, 259–260 adequating, 208 of polygons, 258 affine polar coordinates, 353–355 approximation, 142 Argand, Jean-Robert (1768–1822) function, 137 Argand diagram, 439 algebraic function, 74 (384–322 BC), 16 Alhazen (ibn-al-Haitham) (ca. 965–1039), 247 asymptote, 209 Al-Khowârizmî (ca. 780–ca. 850 AD) average of f, 274 Hisâb al-jabr Wa’lmuquâbalah,2 axiom, 26 , 47, 50, 66, 385 alternating harmonic series, 47, 387, 398 Barrow, Isaac (1630–1677) sum of, 421 Fundamental Theorem of Calculus, 247 Ampère, André-Marie (1775–1836) integration, 247 , 225 lines, 135 analytic function, 441 Berkeley, George (1685–1753) , 286 The Analyst (1734), 135, 136, 380, 446 approximation Bernoulli, Jacob (1654–1705), 226 accuracy of, 368 Ars conjectandi (1713), 235 affine, 142 of harmonic series, 36 error, 368 Bernoulli, Johann (1667–1748), 226, 367 linear, 372 Alternating Series Test, 396 to a function, 368 L’Hôpital’s Rule, 227 Arbogast, Louis François Antoine (1753–1803) partial fractions, 330 discontiguity, 84 Bh¯askara (b. 1114) arccosine, see function, trigonometric, inverse Pythagoras Theorem, 160

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482 Index

bijection, 87 Résumé des lecons sur le calcul infinitesi- bisection , 65, 67 mal (1823), 237, 277, 383 Bolzano, Bernhard (1781–1848) Cauchy sequence, 62 Functionenlehre(1830–35), 237, 244 continuity, 70 Rein analytischer Beweis (1817), 63, 70, definition of continuity, 126 84, 92, 126 , 136, 140 -Weierstrass Theorem, 32, 84 formulation of basic notions, 446 Cauchy sequence, 63 integration, 248, 277, 362 completeness, 63 Intermediate Value Theorem, 84, 92 continuity, 70 Mean Value Inequality definition of continuity, 126 proof, 225–226 derivative, 136 Mean Value Theorem, 219 formulation of basic notions, 446 Generalized, 222 Intermediate Value Theorem, 63, 84, 92 remainder in , 382 nowhere differentiable function, 136, 237, series, 368 244–245 convergence, 396 supremum, 92 differentiation, 415 Bolzano-Weierstrass Theorem, 32 Divergence Test, 33 Bonnet, Ossian (1819–1892) products of, 430 proof of Rolle’s Theorem, 219 smooth non-analytic function, 237, 383 bounds, see also extrema uniform continuity, 131 for a function, 93 Cavalieri, Bonaventura (1598–1647) for a sequence,21 Cavalieri’s Theorem, 350 for an interval, 9 integration, 277, 289 greatest lower, see infimum volumes, 247, 350 infimum Robert of Chester (12th cent.) of a function, 93 translation of Al-Jabr,2 of a set, 57 , Marcus Tullius (106–43BC) least upper, see supremum finds tomb of Archimedes, 357 closed form for sequence,12 maximum closed interval, 7 of a function, 94 coefficients of a set, 10, 55 of a polynomial, 74 minimum commutative law, 398 of a function, 93 Completeness Axiom, 27 of a set, 10, 55 completing the square, 315 supremum , 439 of a function, 93 argument, 443 of a set, 57 conjugate, 440 Brahmagupta (ca. 628 AD) imaginary part, 439 arithmetic, 2 modulus, 440 Briggs, Henry (1561–1631), 125 x real part, 439 b , 123 component intervals, 250 composition of functions, 78 C, 439 , 213 Cantor, George Ferdinand (1845–1918), 446 conjugate Cardan, Girolamo (1501–1576), 2 complex, 334 cardioid, 354 contact cartesian coordinates, 3 first order, 369 Cauchy, Augustin-Louis (1789–1857) order n, 372 Cours d’analyse (1821), 7, 33, 70, 84, 126, continuity 136, 396, 406 at a point, 115 Leconssur le calculdifférentiel (1829), 222 on an interval, 73

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Index 483

uniform, 131 La Geometrie (1637), 3 contrapositive, 448 circle method, 135 convergence condition for integrability, 267 coordinates, 3 convergenceof a sequence,18, 20 loci, 69 Squeeze Theorem, 45 tangent lines, 135 convergenceof a series, 30, 384–403, see also al- diagonal line, 171 ternating series, geometric series, har- DICE, 217 monic series, p-series, power series differentiation absolute, 63, 385, 387 implicit, 179–181 Comparison Test, 31, 385 logarithmic, 181–183 , 390 Dirichlet, Johann Peter Gustav Lejeune (1805– conditional, 402 1859) Divergence Test, 33, 385 function concept, 70 Integral Test, 340, 385 pathological function, 72 list of tests, 385 succeeds Gauss, 362 , 392 unconditional convergence, 402 , 394 uniform continuity, 131 Generalized Root Test, 395 discontinuity, 115 unconditional, 402 essential, 116 convex jump, 116 curve, 213, 264 point of, 115 function, 213 removable, 116 cos θ disjoint sets, 9 definition, 74 distance series for, 422 between two numbers, 5 critical point (value), 193 divergence of a function D’Alembert, Jean le Rond (1717–1783) to infinity, 111 derivatives, 140 of a sequence,20 wave equation, 70 to infinity, 21 Darboux, Jean-Gaston (1842–1917) to negative infinity, 21 Darboux continuity, 221 of a series Darboux’s Theorem, 221, 224 Divergence Test, 385 integration, 277 domain de Sarasa, Alfons A. (1618–1667), 125 of a function, 71 decreasing natural, 71 function, 85 of convergence, see power series sequence, 26 double-angle formula, 307 definite integral, 257 Du Bois-Reymond, Paul (1818–1896) degree nowhere differentiable function, 237 of a polynomial, 74 of a rational function, 74 e δ, see error controls as a limit, 235 Democritus (ca. 460–370BC) as a series, 422 volume of cone, 357 complex powers, 442 DeMoivre, Abraham (1667–1754) definition, 169 DeMoivre’s Theorem, 443 Euler’s treatment, 438 dense set, 82, 245 irrationality of, 431 derivative, 140 element, 7, 53 nth, 155 empty set, 10 of function defined in pieces, 152–154 ε, see error controls second, 154 epsilon-delta definition of limit, see error con- Descartes, René (1596–1650) trols

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484 Index

error, 18 global, 192 bound, 18, 368 local, 192 estimate, 18 of an approximation, 368 Fermat, Pierre de (1601–1665) ratio, 369 coordinates, 3 error controls, 127–130 Fermat’s (optics), 208 and continuity, 129 Fermat’s Theorem (on extrema), 193 and limits, 130 integration, 247, 289, 290, 347 and uniform continuity, 131 loci, 69 infinite variants, 130 , 135, 193, 208 Euclid of Alexandria (ca. 300 BC) Fibonacci (Leonardo of Pisa) (1175–1250) Elements, 2, 11, 445 Liber Abaci,2 Book I, Prop. 47, 160 Fibonacci sequence,15 Book VII, Prop. 1, 335 formal differential, 294 Book IX, Prop. 20, 448 four-petal rose, 355 Book X, Prop. 1, 52 Fourier, Jean-Baptiste-Joseph de (1768–1830) Book XII, Prop. 1 (areas of polygons), Fourier series, 362 259 heat equation, 70 BookXII, Prop. 2(areas ofcircles), 259– series, 70 260 function, 71 Euclidean algorithm, 332 algebraic, 74 series, 367 analytic, 441 tangents, 135 pole, 441 Eudoxus of Cnidos (408–355 BC) blows up method of exhaustion, 11 at a point, 116 volume of cone, 357 bounded, 93 Euler, Leonard (1707–1783) concave on interval, 213 Institutiones Calculi Differentialis (1755), constant, 73 135 continuous Introductio in Analysin Infinitorum (1748), at a point, 115 69, 83, 125, 235, 438 nowhere differentiable, 242 complex exponents (Euler’s Formula), 442 on an interval, 73 definitions of trig functions, 83 uniformly, 131 e, 125 decreasing, 85 , 125, 438 defined in pieces, 71, 152 formulation of basic notions, 446 interface point, 152 function concept, 69 differentiable, 143 integration, 248 discontinuity partial fractions, 330 essential, 116 power series, 367 jump, 116 series, 365 removable, 116 eventually discontinuous (condition holds), 19 at a point, 115 approximates, 18 divergence increasing sequence, 27 to infinity, 111 , 447 domain, 71 expansion natural, 71 base p, 332 exponential binary, 51, 67 complex exponents(Euler’s Formula), 442 decimal, 16 definition, 123 triadic (base 3), 243 differentiability, 164–168 exponential function, see function, exponential real exponents, 119–125 extrema, see also bounds , 73

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Index 485

increasing, 85 uniform continuity, 131 inverse, 87 Hermite, Charles (1822–1901) limit partial fractions, 330 superior, 394 Heron of Alexandria, 207 limit of, 101 Hilbert, David (1862–1943), 446 monotone, 85 Hippasus of Metapontum (ca. 400 BC), 15 strictly, 211 Hippocrates of Chios (460–380 BC) one-to-one, 87 area of circle, 259 onto, 87 horizontal asymptote, 209 polynomial, 74, 368 rational, 74 i, 439 sign function, 388 (z), 439 trigonometric =image of a set, 93 continuity, 75–77 implicitly defined functions, 86–89, see inverse definition, 74 functions differentiability, 150–151 , 337 inverse, 88 convergence, 338, 342–344 value, 71 divergence, 338, 342, 343 well-defined, 71 Integral Test, 340 Fundamental Theorem of Algebra, 319, 440 increasing FundamentalTheoremof Calculus, 247, 281, 285 function, 85 Leibniz’s proof, 292 sequence, 26 Newton’s proof, 293 increment, 137 indefinite integral, 286 Gauss,Carl Friedrich (1777–1855),319,362, 396, indeterminate forms, 227–235 440 0 , 227 ∞0 Ratio Test, 396 ∞ , 228 geometric series, 44 0 , 232 convergence, 45, 385 · ∞ , 233 ∞ − ∞ Leibniz, 53 00, 233 ∞ (g◦f)(x), 78 1 , 234 Girard, Albert (1590–1663) 0, 235 Invention nouvelle en l’algèbre index,∞ 12 (1629), 3 induction, see proof by induction graph, 136 inequality Gregory of St. Vincent (1584–1667), 125 x>y,3 Gregory, James (1638–1675) x

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486 Index

intersection of sets, 10 Fundamental Theorem of Calculus, 292 interval geometric series, 53 non-trivial, 217 integration, 277 interval of convergence, see power series integration by parts, 305 inverse functions, 87 Leibniz’s Series, 367, 433 Continuous Inverse Function Theorem, 87 partial fractions, 330 derivatives of, 171–174 , 163 inverse trig functions, 88–89 tangents, 136 derivatives, 173–174 Transmutation Theorem, 305, 433 logarithm, 123–125, 169–170 lemma, 445 derivative, 172–173 length of a convex curve, 264 irrational number limit e, 431 of a function definition, 15 definition, 101 irreducible limit at infinity, 110 quadratic polynomial, 315, 319 one-sided limit, 105 of a sequence Jordan, Camille Marie Ennemond (1838–1922) definition, 20 Cours d’analyse (1893), 278 limit superior, 394 jump discontinuity, 116 of a function, 394 limits of integration, 286 Kepler, Johannes (1571–1630) limsup, 395 Nova stereometria dolorium (1615), 193, linear 350 approximation, 372 integration, 350 combination, 147 maxima and minima, 193 equation, 136 volumes, 247 list notation for a set, 53 Khayyam, Omar (ca. 1050–1130), 1, 2 local extremum, 192 locus of an equation, 136 L’Hôpital, Guillaume Françoise Antoine, Mar- logarithm quis de (1661–1704) base b, 123 Analyse des infiniment petitis (1696), 226 complex numbers, 443 Traité analytique des sectionsconiques (1707), derivative of, 172–173 227 natural, 169 L’Hôpital’s Rule lower sum, 251, 257 0 , 227 lowest terms ∞0 ∞ , 228 fraction in, 317 Lagrange, Joseph Louis (1736–1813) rational expression, 320 Théorie des Fonctions Analytiques (1797), 69, 136, 219, 237, 366, 383 Maclaurin, Colin (1698–1746) formulation of basic notions, 446 Treatise on (1742), 367, 380 function concept, 69 Maclaurin series, 380 integration, 248 Mean Value Theorem, 219 Mean Value Theorem, 219, 225 Cauchy, 222 notation for derivative, 141 Integral MVT, 275 power series, 367, 383 Mercator, Nicolaus (1620–1687) remainder in Taylor series, 377 series for logarithm, 125, 367, 421, 437 Lebesgue, Henri Léon (1875–1941), 248, 278 mesh size, 269 left endpoint, 7 method of compression, 11 Leibniz, Gottfried Wilhelm (1646–1714) method of exhaustion, 11 Alternating Series Test, 396 monotone characteristic triangle, 136 function, 85 function concept, 69 sequence, 27

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Index 487

multiplicity for numbers, 317–319 of a root, 319 for rational functions, 319–323 mutual refinement, 255 partial sums, 14 partition, 250 Napier, John (1550–1617), 3, 125 Pascal, Blaise (1623–1662) natural integration, 247, 289 exponential, 169 peak point, 32 logarithm, 169 Peano, Giuseppe (1858–1932), 278 neighborhood, 192 piecewise continuous, 276 Nested Interval Property, 63 Pinching Theorem, see Squeeze Theorem Newton, Isaac (1642–1729) Plato (429–348 BC), 1 De Analysi (1666), 294 Academy, 1 Geometria curvilinea (1680), 163 point Methodus Fluxionum (1671), 294 accumulation Quadrature of Curves (1710), 69 of a set, 100 , 367, 432 isolated, 101 chord-to-arc ratio, 114 of non-differentiability, 193 derivatives, 140 polar coordinates, 353–355 sine, 163 polynomial, 74 Fundamental Theorem of Calculus, 293 coefficients, 74 -Gregory Interpolation Formula, 383 degree, 74, 368 logarithms, 125 postulate, 445 Principia (1687), 2, 3 power series, 404 Book I, Lemma 7, 114 addition, 413 Book II, Lemma 2, 162 basepoint, 404 Book III, Lemma V, 383 coefficients, 404 product rule, 162–163 convergence series for ln 2, 436 interval of, 408 tangents, 135 radius of, 406 Nilakantha, Kerala Gargya (ca. 1500), 433 derivative, 414–418 normal, 144 distributive law, 411 number, see also irrational number negative, 4 domain of convergence, 405 nonnegative, 4 formal derivative, 414 nonpositive, 4 integral, 418 number line, 3 multiplication, 422–428 positive, 4 proof by , 15, 448 rational, 15 proof by induction, 23–24, 50, 322, 449 real, 1 induction hypothesis, 24, 449 induction step, 24, 449 notation, 231 initial case,24, 449 oOnotation, 231 proper one-to-one, 87 fraction, 317 onto, 87 rational function, 320 open interval, 7 property Oresme, Nicole (1323–1382) positive-definite, 5 divergence of harmonic series, 36 symmetric, 5 graphical representation of varying quanti- triangle inequality, 5 ties, 3, 69 , 445 Pythagoras of Samos (ca. 580–500 BC), 1, 15, p-series 445 p-series Test, 341, 385 Theorem of Pythagoras, 160 partial fractions Bh¯askara’s proof, 160

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488 Index

quadrature, 11 monotone, 27 nondecreasing, 27 R,9 nonincreasing, 27 (z), 439 periodic, 24 < radians, 74 real, 11 radius of convergence, see power series series, 14, 30, see also alternating series, geomet- rate of change,139 ric series, harmonic series, p-series, ratio, 44 power series, Taylor series Ratio Test, see convergence of a series divergence, 30 rational partial sums, 14, 30 function, 74 sum of, 30 number, 15 summands, 14 real numbers, 1 terms of, 14, 30 real sequence, 11 set, 53 rectification, 11 bounded, 55 recursive formula, 13 elements, 53 refinement, 254 infimum of, 57 , 185–190 list notation, 53 removable discontinuity, 116 maximum of, 55 revolute, 355 minimum of, 55 Rheticus, George Joachim (1514–1576) set-builder notation, 54 angles, 83 subset, 54 Riemann sum, 269 supremum of, 57 Riemann, Bernhard Georg Friedrich (1826–1866) unbounded, 55 characterization of integrable functions, 362– sign function, 388 364 sin θ conditional convergence, 398 definition, 74 continuity vs. integrability, 364 series for, 422 function concept, 70 slope, 138 Habilitation, 362 Squeeze Theorem integration, 248, 277 for sequences, 45 series, 368 , 193 succeeds Dirichlet, 362 Stevin, Simon (1548–1620) uniform continuity, 131 De thiende (1585), 3 right endpoint, 7 Stieltjes, Thomas-Jean (1856–1894), 248 Roberval, Gilles Personne de (1602–1675) subsequence, 32 integration, 247, 289 subset, 54 Rolle, Michel (1652–1719) summands, 14 Rolle’s Theorem, 218 summation notation, 13 Root Test, see convergence of a series surjection, 87 Sandwich Theorem, see Squeeze Theorem , 141 tangent , 213 line, 142 sequence, 11 map, 142 approximates y, 18 Taylor polynomial bounded, 21 of degree n, 375 Cauchy, 62 Taylor series, 380 constant, 24 remainder decreasing, 26 Cauchy form, 382 divergent, see divergence integral form, 382 eventually increasing, 27 Lagrange form, 377 increasing, 26 Taylor, Brooke (1685–1731)

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Index 489

Methodus incrementorum (1715), 367, 380, Volume 383 of sphere, 355–357 derivation of series, 383 revolutes, 355–360 Taylor’s Theorem, 225 via shells, 359–360 terms, see series via slices, 355–359 Thales of Miletus (ca. 625–ca. 547 BC), 445 Torricelli, Evangelista (1608–1647) Wallis, John (1616–1703) Fundamental Theorem of Calculus, 247 integration, 247, 289–291 integration, 247, 289 Mercator’s series, 437 Transmutation Theorem, see Leibniz negative and fractional exponents, 3 triadic rational, 238 representation of π, 367, 432 order of, 238 Waring, Edward (1734–1793) triangle inequality, 5 Ratio Test, 396 trigonometric formulas Weierstrass, Karl Theodor Wilhelm (1815–1897) double-angle formula, 307 “arithmetization of analysis”, 70 trigonometric functions, 74 Bolzano-Weierstrass Theorem, 32 angle summation formulas, 158 formulation of basic notions, 446 law of cosines, 159 nowhere-differentiable function, 136, 237 twice differentiable, 214 series, 368 uniform continuity, 131, 248 uniform continuity well-defined function, 71 and , 268 word problems on closed interval, 131 strategy union of sets, 9 max-min problems, 200 upper sum, 252, 257 related rates, 186

value x , 4, 440 absolute, 4 |x | R,9 of a function, 71 x-intercept,∈ 136, 209 vertical asymptote, 209 Vièta, François (1540–1603) y-intercept, 136, 209 In artem analyticem isagoge (1591), 3 geometric series, 367 z , 440 | | Volterra, Vito (1860–1940), 278

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About the Author

Zbigniew H. Nitecki is a Professor of Mathematics at Tufts University, where he has been since 1972; previouslyhe was a Gibbs Instructorat Yale (1969–71) and Assistant Professor at CCNY (1971–2). Nitecki has held short-termvisiting positions at Brown, Northwestern, and George-August Universität, Göttingen, Germany, as well as visiting research positions at the Institut des Hautes Etudes Scientifiques (Paris), the Institut Non-Linéaire (Nice), and the University of Warwick (Coventry). In 1982–3, he served as the first program director for the Geometric Analysis Program at the National Science Foundation. Nitecki is a con- tributing editor for Real Analysis Exchange and an associate editor for Qualitative Theory of Dynamical Systems. His mathematical research interests are primarily in dynamical systems theory. In addition to about 40 research and survey articles, Nitecki authored Differentiable Dynamics: An Introduction to the Orbit Structure of Diffeomorphisms (MIT Press, 1971). He co-authored, with M. Guterman, Differential Equations—A First Course (Saunders, 3rd. ed. 1992) and Differential Equations with Linear Algebra (Saunders, 1986). Nitecki has also co-edited the following conference proceedings: with R. C. Robinson, Global Theory of Dynamical Systems, (Springer, Lect. Notes in Math., vol. 819, 1980) and with R. Devaney, Classical Mechanics and Dynamical Systems, (Marcel Dekker, Lect. Notes in Pure & Appl. Math., vol. 70, 1981.)

491

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i i AMS / MAA TEXTBOOKS

Calculus Deconstructed is a thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof. This book is appropriate for a beginning Honors Calculus course assuming high school calculus or a “bridge course” using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combi- nation textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject. No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reductio ad absurdum, induction, contrapositives, etc.) are introduced by example along the way. Between the text and exercises, proofs are available for all the basic results of calculus for functions of one real variable.