Craig Smoryński MVT: A Most Valuable Theorem MVT: A Most Valuable Theorem Craig Smoryński MVT: A Most Valuable Theorem 123 Craig Smoryński Westmont, IL USA ISBN 978-3-319-52955-4 ISBN 978-3-319-52956-1 (eBook) DOI 10.1007/978-3-319-52956-1 Library of Congress Control Number: 2017931055 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Further Reading: Books by the Same Author University of Illinois at Chicago Circle: Notes on Hilbert’s Tenth Problem; An Introduction to Unsolvability 1972 Springer-Verlag: Self-Reference and Modal Logic 1985 Logical Number Theory I; An Introduction 1991 History of Mathematics; A Supplement 2008 College Publications: Adventures in Formalism 2012 Chapters in Mathematics 2012 Chapters in Probability 2012 A Treatise on the Binomial Theorem 2012 v Acknowledgements For various reasons I have relied more heavily on the help and advice of others in writing this book than I have generally done in the past. Those I am indebted to, in alphabetical order, are Kirsti Andersen, Mark van Atten, John Baldwin, Ádám Besenyei, Reinhard Bölling, Michel Bourdeau, Robert Burckel, Pete L. Clark, Dirk van Dalen, P.P. Divakaran, Enrico Giusti, Prakash Gorroochurn, Eckart Menzler, David Mumford, K. Ramasubramanian, Fred Thulin, and Peter Ullrich. Of these, special mention should be made of Ádám Besenyei and Robert Burckel. Besenyei has himself delved into the history of the Mean Value Theorem and has shared the results of his research with me, sent me materials, and made helpful comments. Burckel brought to my attention the fascinating results on the Mean Value Theorem in Complex Analysis, sent me materials, and made many suggestion improving my exposition on this and other subjects as well. Additionally I wish to thank John J. O’Connor and Edmond F. Robertson, Jason Ross, and Lou Talman for permission to quote from their works published online, and the following corporations for permission to use copyrighted material from the indicated sources: American Mathematical Society: Notices of the AMS A Century of Mathematics III Cengage: Dictionary of Scientific Biography Dover Publications Inc.: La Geometrie Mathematical Association of America: American Mathematical Monthly College Mathematics Journal Mathematics Magazine Selected Papers on Calculus Oxford University Press: Mathematics Emerging: A Sourcebook 1540–1900 vii viii Acknowledgements The Oxford Handbook of the History of Mathematics Princeton University Press: Proclus, A Commentary on the First Book of Euclid’s Elements Crest of the Peacock, 3rd edition Springer Science+Business LLC: Archive for History of Exact Sciences Cauchy’s Cours d’analyse: An Annotated Translation Conflicts between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17–19th Century France and Germany Denkweise grosser Mathematiker Giuseppe Peano between Mathematics and Logic Mathematical Conversations Peano, Making a Career The Historical Development of the Calculus University of Wisconsin Press: The Science of Mechanics in the Middle Ages Contents 1 Introduction.............................................. 1 2 Curves and Tangents ...................................... 7 2.1 Curves ............................................. 7 2.2 Continuous Curves .................................... 45 2.2.1 Defining Continuity ............................. 45 2.2.2 Properties of Continuity.......................... 59 2.2.3 Peano’s Space-Filling Curve ...................... 72 2.3 Smooth Curves ....................................... 79 2.3.1 Traditional Views of Tangents .................... 79 2.3.2 Isaac Barrow .................................. 90 2.3.3 Transition to Newton and Leibniz .................. 98 2.3.4 Newton ...................................... 99 2.3.5 Leibniz ...................................... 109 2.3.6 Bumps in the Road ............................. 112 2.3.7 The Derivative Defined .......................... 123 3 The Mean Value Theorem .................................. 151 3.1 The Mean Value Theorem and Related Results .............. 151 3.1.1 Variants of the Mean Value Theorem ............... 151 3.1.2 The Mean Value Theorem and Integration ........... 159 3.1.3 Applications of the Mean Value Theorem ............ 168 3.2 Precursors to the Mean Value Theorem .................... 180 3.2.1 Generalities ................................... 180 3.2.2 Conic Sections in Classical Greece ................. 182 3.2.3 The Analytic Approach in Mediæval India ........... 188 3.2.4 The Mean Speed Theorem in Mediæval Europe ....... 216 3.2.5 Valerio and Cavalieri............................ 222 3.2.6 Rolle ........................................ 228 3.2.7 Taylor’s Theorem .............................. 235 ix x Contents 3.3 Lagrange and the Mean Value Theorem .................... 239 3.4 Transition to Cauchy: Ampère ........................... 264 3.5 Cauchy and the Mean Value Theorem ..................... 274 3.6 Bolzano and the Mean Value Theorem..................... 291 3.7 More Textbooks but No Progress ......................... 312 3.8 Weierstrass, Bonnet, Serret, and the Mean Value Theorem ..... 333 3.9 Peano and the Mean Value Theorem ...................... 351 3.10 Gilbert Revisited...................................... 360 3.10.1 Gilbert’s Final Words ........................... 360 3.10.2 Thomas Flett and the Vindication of Gilbert .......... 364 3.10.3 Generalisations of the Mean Value Theorem.......... 375 3.11 Acker and the Mean Value Theorem ...................... 381 3.12 Loose Ends.......................................... 386 3.12.1 Flett’s Theorem ................................ 386 3.12.2 Finding the Mean Value ......................... 391 3.12.3 Complex Considerations ......................... 405 3.12.4 An Anticlimax................................. 441 4 Calculus Reform .......................................... 445 4.1 The Great Debate ..................................... 445 4.2 Beyond Polemics ..................................... 465 4.3 Constructive Thoughts on the Subject ..................... 469 Appendix A: Short Bibliography ................................ 485 Index ...................................................... 489 Chapter 1 Introduction An odd candidate for controversy in mathematics education is the Mean Value The- orem. It is easily, if imprecisely, described to the man in the street: Draw a smooth curve and choose two points A and B on the curve. Somewhere on the arc of the curve between these two points is another point C at which the line tangent to the curve will be parallel to the line connecting A and B. Figures 1.1 and 1.2 illustrate this nicely. With such pictures in mind, the method of finding such a point C and the reason for the truth of the Theorem suggest themselves. C is the point (or any of the points) on the curve farthest from the line AB as possible and is found by sliding the line AB upward (in Fig. 1.1) or leftward (in Fig. 1.2) or in whatever appropriate direction — without rotating AB at all — until it exits the curve. The last point (or any of the last points if there are more than one) at which AB had contact with the curve is this farthest point C. Why is the tangent at C parallel to AB? Referring to Fig. 1.3,we see that as we move along the curve from A to C, the tangents are always steeper than AB but they are becoming less steep. And as we move from C to B, the tangents continue to grow less steep and none are as steep as AB. The level of steepness, or slope, of the tangent at C must lie between these two trends and thus bears the same level of steepness as AB itself, i.e., the tangent is parallel to AB. The argument given is not mathematically rigorous, but it can be made so first by offering precise definitions of the terms involved (“curve”, “tangent”) and isolating the conditions under which the Theorem is to hold, and then appealing to more advanced theory to justify the individual steps. This will be done in the course of this book. Here let me just indicate that some conditions must be met. The hyperbola of Fig. 1.4 demonstrates an obvious condition: the