MVT: A Most Valuable Theorem

504 Pages·2017·7.164 MB·English

by  Craig Smorynski (auth.)

Craig Smoryński
MVT:
A Most Valuable
Theorem
MVT: A Most Valuable Theorem
ń
Craig Smory ski
MVT:
A Most Valuable Theorem
123
CraigSmoryński
Westmont, IL
USA
ISBN978-3-319-52955-4 ISBN978-3-319-52956-1 (eBook)
DOI 10.1007/978-3-319-52956-1
LibraryofCongressControlNumber:2017931055
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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
writingthisbookthanIhavegenerallydoneinthepast.ThoseIamindebtedto,in
alphabetical order, are Kirsti Andersen, Mark van Atten, John Baldwin, Ádám
Besenyei,ReinhardBölling,MichelBourdeau,RobertBurckel,PeteL.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
hashimselfdelvedintothehistoryoftheMeanValueTheoremandhassharedthe
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.
AdditionallyIwishtothankJohnJ.O’ConnorandEdmondF.Robertson,Jason
Ross, and Lou Talman for permission to quote from their works published online,
andthefollowingcorporationsforpermissiontousecopyrightedmaterialfromthe
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
AnoddcandidateforcontroversyinmathematicseducationistheMeanValueThe-
orem.Itiseasily,ifimprecisely,describedtothemaninthestreet:Drawasmooth
curve and choose two points A and B on the curve. Somewhere on the arc of the
curvebetweenthesetwopointsisanotherpointC atwhichthelinetangenttothe
curvewillbeparalleltothelineconnecting Aand B.Figures1.1and1.2illustrate
thisnicely.
Withsuchpicturesinmind,themethodoffindingsuchapointC andthereason
forthetruthoftheTheoremsuggestthemselves.C isthepoint(oranyofthepoints)
onthecurvefarthestfromtheline AB aspossibleandisfoundbyslidingtheline
ABupward(inFig.1.1)orleftward(inFig.1.2)orinwhateverappropriatedirection
—withoutrotating AB atall—untilitexitsthecurve.Thelastpoint(oranyofthe
lastpointsiftherearemorethanone)atwhich ABhadcontactwiththecurveisthis
farthestpointC.WhyisthetangentatC parallelto AB?ReferringtoFig.1.3,we
see that as we move along the curve from A toC, the tangents are always steeper
thanABbuttheyarebecominglesssteep.AndaswemovefromCtoB,thetangents
continuetogrowlesssteepandnoneareassteepas AB.Thelevelofsteepness,or
slope,ofthetangentatC mustliebetweenthesetwotrendsandthusbearsthesame
levelofsteepnessas AB itself,i.e.,thetangentisparallelto AB.
Theargumentgivenisnotmathematicallyrigorous,butitcanbemadesofirstby
offeringprecisedefinitionsofthetermsinvolved(“curve”,“tangent”)andisolating
the conditions under which the Theorem is to hold, and then appealing to more
advancedtheorytojustifytheindividualsteps.Thiswillbedoneinthecourseofthis
book.Hereletmejustindicatethatsomeconditionsmustbemet.Thehyperbolaof
Fig.1.4demonstratesanobviouscondition:theMeanValueTheorempresupposes
Aand B tolieonthesamebranchofthecurveinquestion.Iftheydon’t,thereare
nopointslyingonthearcofthecurveconnectingthemsimplybecausethereisno
sucharc.Onecouldtrytofindapointmaximallydistantfrom AB elsewhereonthe
curve, but again no such point exists. Indeed, on the hyperbola, if A and B lie on
differentbranches,thereisnopointC atwhichthetangentisparallelto AB.There
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C.Smoryn´ski,MVT:AMostValuableTheorem,
DOI10.1007/978-3-319-52956-1_1