Calculus off the Beaten Path: A Journey Through Its Fundamental Ideas

225 Pages·2022·3.035 MB·English

by  Ignacio Zalduendo

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
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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ideas of calculus, with some historical references, and centered on applications.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