SUMS Readings Ignacio Zalduendo Calculus off the Beaten Path A Journey Through Its Fundamental Ideas Springer Undergraduate Mathematics Series SUMS Readings AdvisoryEditors MarkA.J.Chaplain,StAndrews,UnitedKingdom AngusMacintyre,Edinburgh,UnitedKingdom SimonScott,London,UnitedKingdom NicoleSnashall,Leicester,UnitedKingdom EndreSüli,Oxford,UnitedKingdom MichaelR.Tehranchi,Cambridge,UnitedKingdom JohnF.Toland,Bath,UnitedKingdom SUMS Readings isacollection of books thatprovides students withopportunities to deepen understanding and broaden horizons. Aimed mainly at undergraduates, the series is intended for books that do not fit the classical textbook format, fromleisurely-yet-rigorousintroductionstotopicsofwideinterest,topresentations of specialised topics that are not commonly taught. Its books may be read in parallelwithundergraduatestudies,assupplementaryreadingforspecificcourses, backgroundreadingforundergraduateprojects,oroutofsheerintellectualcuriosity. Theemphasisoftheseriesisonnovelty,accessibilityandclarityofexposition,as wellasself-studywitheasy-to-followexamplesandsolvedexercises. Ignacio Zalduendo Calculus off the Beaten Path A Journey Through Its Fundamental Ideas IgnacioZalduendo DepartmentofMathematics TorcuatodiTellaUniversity BuenosAires,Argentina ISSN1615-2085 ISSN2197-4144 (electronic) SpringerUndergraduateMathematicsSeries ISSN2730-5813 ISSN2730-5821 (electronic) SUMSReadings ISBN978-3-031-15764-6 ISBN978-3-031-15765-3 (eBook) https://doi.org/10.1007/978-3-031-15765-3 MathematicsSubjectClassification:26-01,26A06,40-01,26Dxx ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Cálculo para Iñaki Preface After the publication of Matemática para Iñaki [13], I proposed to keep the promise expressed in its Prologue and write Cálculo para Iñaki. My intention was to give a reasoned, very accessible, and colloquial explanation of the main ideas of calculus, with some historical references, and centered on applications. Having taught calculus courses for over 40 years, I know that some aspects of the matter can be hopelessly arid and boring to students and teacher alike. Thus Ialsoproposedtowriteabookemphasizingwhattomearethemoreconceptually importantaspects,andinterestingapplications,leavingaside—wheneverpossible— thetechnicalitiesandthepurelycomputational.ItsoonbecameapparentthatIhad ledmyselfintoacomplicatedsituation.TheintentionsIexpressedaboveresultedat timesincongruentandverydifficulttoreconcile. ThefactisthatmanyoftheinterestingapplicationswhichIinsistedonincluding (such as the Basel problem and the sum of Gregory’s series) required more and deeper concepts, slowly distancing me from my original purpose of extreme accessibility. Thus, this is not the very elementary book that I set out to write, butratherthebestIcoulddowithnon-elementarysubjectmatter.However,Ihave strivedforclarityandcolloquiality,andintheend,Iamhappywithboththecontent andthetoneofthetext. Andso,withadifferenttitleandinanotherlanguage,hereitis.Toshowthespirit inwhichthebookiswritten,perhapsitisconvenienttolistheresomeofthetopics and applications which I did not want to leave out, and which are not commonly includedincalculuscourses: (cid:2) Aconstructionoftherealnumbers (cid:2) Riemann’sseriestheorem(re√arrangementtheorem) (cid:2) Proofsoftheirrationalityof 2,eandπ (cid:2) Pythagoreantriples (cid:2) TheconceptoflimitinAncientGreece (cid:2) Snell’slawandtheBrachistochrone (cid:2) Buffon’sneedle (cid:2) Growthoftheharmonicseries (cid:2) Gregory’sseries (cid:2) Stirling’sformula (cid:2) Curvature vii viii Preface (cid:2) Convexity (cid:2) Randomwalkandthebellcurve (cid:2) Theisoperimetricinequality (cid:2) Classicalinequalities(AG,Jensen,Young) (cid:2) TheBaselproblem (cid:2) Densityfunctions,barycenter,andexpectation (cid:2) Pappus’theorem (cid:2) TheMethodofArchimedes (cid:2) Thecatenary (cid:2) TheGammafunction The educated reader may notice varying levels of informality and formality in different parts of the text. My personal inclination is more towards informality: I would much rather be accused of mathematical incompleteness than of lack of expository clarity. I will strengthen the hypotheses if this does away with an inessential technicality. However, at times, formality has permitted the discussion of important notions. For example, summing well Gregory’s series gave me the opportunitytotalkaboutuniformconvergence,whichismuchmoreimportantthan Gregory’sseries. Iamfarfrombeinganexpertonthehistoryofmathematics.Butonre-reading some of the older sources for this book, I could not help but think that many of the underlying ideas of Calculus have been developing for 2400 years, certainly since before the time of Archimedes, although they come of age in the XVIIth Century.Ihavetriedtopointouttheoriginofsomeoftheseideasinthetext,without pretendingthatthisisahistorybook. Finally, my personal views on some of the subject matter included here, and whichneednotbesharedbyothers,havealsoshapedthetext.Amongthem,Imust confess the following: Taylor polynomials of order one and two seem to me the more important, just as the first and second derivatives are those with immediate applicability and a clear geometrical significance. Integration is an area where I do not find formality particularly useful. Although I do address the difficulties of defining the integral, and I refer to Riemann’s and Lebesgue’s definitions, I adopt the intuitive idea of integral as the area under a continuous curve which was so productiveuntiltheXIXthCentury.Also,Ifindinequalitiesarecentraltoanalysis, andIincludedachapterdiscussingafewofthem. This book was transformed from illegible manuscript to elegant LATEXtext by PabloSanches.IhavereceivedvaluablesuggestionsandcommentsfromFederico Poncio, Lara Sánchez Peña, Guillermo Ranea, Vicky Venuti, Damián Pinasco, AngelinesPrieto,andMaiteFernándezUnzueta.Finally,RobinsondosSantoshas been a kind and understanding Editor to this rather stubborn author. To all, my deepestgratitude. BuenosAires,Argentina IgnacioZalduendo June10,2022 Contents 1 TheRealNumbers........................................................... 1 TheRationalLine ............................................................. 1 DensityofQ............................................................. 2 SomeBasicNo√tions .................................................... 3 Irrationalityof 2 ...................................................... 4 FromEudoxustoDedekind ............................................ 5 TheRealLine.................................................................. 8 DyadicSeries—AConstructionofR .................................. 11 TheScarcityofQ ....................................................... 13 TheCompletenessofR................................................. 14 Cardinality..................................................................... 16 Exercises....................................................................... 19 2 SequencesandSeries ........................................................ 21 Sequences...................................................................... 21 LimitsofSequences..................................................... 22 Cantor’sNestedIntervalsTheorem .................................... 24 Subsequences............................................................ 25 Series........................................................................... 28 TheHarmonicSeries.................................................... 29 SeriesofPositiveTerms ................................................ 30 SerieswithPositiveandNegativeTerms .............................. 32 TheRiemannSeriesTheorem.......................................... 34 AbsoluteandUnconditionalConvergence............................. 36 Exercises....................................................................... 36 3 Functions...................................................................... 39 TheElementaryFunctions.................................................... 39 Polynomials ............................................................. 40 CircularFunctions ...................................................... 40 TheExponentialFunction:Bernoulli’sInequality..................... 43 Irrationalityofe(cid:2)........................(cid:3)................................. 49 Convergenceof ∞ (1+a )andof ∞ a ....................... 50 k=1 k k=1 k ix x Contents HyperbolicFunctions................................................... 51 InjectivityandInverseFunctions....................................... 52 CurvesinthePlane:ParametrizedCurves ................................... 55 TheCycloid ............................................................. 56 PythagoreanTriples..................................................... 57 Continuity ..................................................................... 59 BolzanoandWeierstrass................................................ 60 Limits.......................................................................... 62 LimitsinAncientGreece:TheAreaofaCircle....................... 62 ThreeImportantLimits................................................. 65 Exercises....................................................................... 67 4 TheDerivative................................................................ 71 Derivative...................................................................... 71 Tangents ................................................................. 72 Newton–Raphson ....................................................... 75 DerivativesoftheElementaryFunctions .............................. 77 TheChainRule.......................................................... 79 DerivativeoftheInverseFunction ..................................... 80 TheDerivativeofaParametrizedCurve............................... 82 FirstDerivative,TangentLine,andGrowth.................................. 84 TheMeanValueTheorems............................................. 84 L’Hôpital’sRule......................................................... 87 Snell’sLaw.............................................................. 89 TheBrachistochrone.................................................... 92 Exercises....................................................................... 94 5 TheIntegral................................................................... 99 MeasureandIntegral.......................................................... 99 TheFundamentalTheoremofCalculus...................................... 105 APauseforComments ................................................. 108 Buffon’sNeedle......................................................... 111 Irrationalityofπ ........................................................ 113 ImproperIntegrals............................................................. 114 IntegrationandSums:LinearityoftheIntegral.............................. 118 UniformConvergence—TheWeierstrassM-Test..................... 118 Gregory’sSeries......................................................... 123 IntegrationandProducts:IntegrationbyParts............................... 124 Stirling’sFormula....................................................... 125 IntegrationandComposition:IntegrationbySubstitution................... 126 ANoteonNotation ..................................................... 127 LengthofCurves.TheCatenary ....................................... 128 AreaEnclosedbyaSimpleClosedCurve............................. 131 Exercises....................................................................... 133