Paradoxes and Sophisms in Calculus

113 Pages·2013·1.907 MB·English

by  Sergiy Klymchuk, Susan G. Staples

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Classroom Resource Materials
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Paradoxes and Sophisms
in
Calculus
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(cid:13)c 2013bytheMathematicalAssociationofAmerica,Inc.
LibraryofCongressCatalogCardNumber2013934605
PrinteditionISBN978-0-88385-781-6
ElectroniceditionISBN978-1-61444-110-6
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
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Paradoxes and Sophisms
in
Calculus
Sergiy Klymchuk
AucklandUniversity of Technology
and
Susan Staples
Texas Christian University
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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CouncilonPublicationsandCommunications
FrankFarris,Chair
CommitteeonBooks
GeraldM.Bryce,Chair
ClassroomResourceMaterialsEditorialBoard
GeraldM.Bryce,Editor
MichaelBardzell
JenniferBergner
DianeL.Herrmann
PaulR.Klingsberg
MaryMorley
PhilipP.Mummert
MarkParker
BarbaraE.Reynolds
SusanG.Staples
PhilipD.Straffin
CynthiaJWoodburn
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CLASSROOMRESOURCEMATERIALS
ClassroomResourceMaterialsisintendedtoprovidesupplementaryclass-
room material for students—laboratory exercises, projects, historical in-
formation,textbookswithunusualapproachesforpresentingmathematical
ideas,careerinformation,etc.
101CareersinMathematics,2ndeditioneditedbyAndrewSterrett
Archimedes:WhatDidHeDoBesidesCryEureka?,ShermanStein
Calculus:AnActiveApproachwithProjects,StephenHilbert,DianeDriscoll
Schwartz,StanSeltzer,JohnMaceli,andEricRobinson
CalculusMysteriesandThrillers,R.GrantWoods
ConjectureandProof,Miklo´sLaczkovich
CounterexamplesinCalculus,SergiyKlymchuk
CreativeMathematics,H.S.Wall
EnvironmentalMathematicsintheClassroom,editedbyB. A.Fusaroand
P.C.Kenschaft
Excursions inClassicalAnalysis: PathwaystoAdvanced Problem Solving
andUndergraduateResearch, byHongweiChen
ExplorationsinComplexAnalysis,MichaelA.Brilleslyper,MichaelJ.Dorff,
JaneM.McDougall,JamesS.Rolf,LisbethE.Schaubroeck,RichardL.
Stankewitz,andKennethStephenson
ExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.Weller
GeometryFromAfrica:MathematicalandEducationalExplorations,Paulus
Gerdes
Historical Modules for the Teaching and Learning of Mathematics (CD),
editedbyVictorKatzandKarenDeeMichalowicz
IdentificationNumbersandCheckDigitSchemes,JosephKirtland
InterdisciplinaryLivelyApplicationProjects,editedbyChrisArney
InverseProblems:ActivitiesforUndergraduates,CharlesW.Groetsch
Keeping it R.E.A.L.: Research Experiences for All Learners, Carla D.
MartinandAnthonyTongen
LaboratoryExperiencesinGroupTheory,EllenMaycockParker
Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt
Johansson,andVictorKatz
MathMadeVisual:CreatingImagesforUnderstandingMathematics,Claudi
AlsinaandRogerB.Nelsen
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Mathematics Galore!: The First Five Years of the St. Marks Institute of
Mathematics,JamesTanton
MethodsforEuclideanGeometry, OwenByer, FelixLazebnik, Deirdre L.
Smeltzer
OrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.Sa´nchez
OvalTrackandOtherPermutationPuzzles,JohnO.Kiltinen
ParadoxesandSophismsinCalculus,SergiyKlymchukandSusanStaples
APrimerofAbstractMathematics,RobertB.Ash
ProofsWithoutWords,RogerB.Nelsen
ProofsWithoutWordsII,RogerB.Nelsen
RediscoveringMathematics:YouDotheMath,ShaiSimonson
SheDoesMath!,editedbyMarlaParker
SolveThis:MathActivitiesforStudentsandClubs,JamesS.Tanton
StudentManualforMathematicsforBusiness DecisionsPart1:Probabil-
ityandSimulation, DavidWilliamson,MarilouMendel,JulieTarr,and
DeborahYoklic
StudentManualfor Mathematicsfor Business DecisionsPart 2: Calculus
andOptimization, David Williamson,MarilouMendel, Julie Tarr, and
DeborahYoklic
TeachingStatisticsUsingBaseball,JimAlbert
VisualGroupTheory,NathanC.Carter
WhichNumbersareReal?,MichaelHenle
Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and
Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn
Rykken
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1MAA FAX:[phone]
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To our parents and families
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Contents
Introduction 1
Acknowledgments 5
I Paradoxes 7
1 FunctionsandLimits 9
1 Layingbricks . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Spiralcurves . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Aparadoxicalfractalcurve:theKochsnowflake . . . . . . 10
4 Atrickyfractalarea:theSierpinskicarpet . . . . . . . . . . 10
5 Amysteriousfractalset:theCantorternaryset . . . . . . . 11
6 Amisleadingsequence . . . . . . . . . . . . . . . . . . . . 12
7 Remarkablesymmetry . . . . . . . . . . . . . . . . . . . . 12
8 Rollingabarrel . . . . . . . . . . . . . . . . . . . . . . . . 13
9 Acatonaladder . . . . . . . . . . . . . . . . . . . . . . . 13
10 Sailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
11 EncirclingtheEarth . . . . . . . . . . . . . . . . . . . . . 14
12 Atrickyequation . . . . . . . . . . . . . . . . . . . . . . . 14
13 Asnailonarubberrope . . . . . . . . . . . . . . . . . . . 15
2 DerivativesandIntegrals 17
1 Analternativeproductrule . . . . . . . . . . . . . . . . . . 17
2 Missinginformation?. . . . . . . . . . . . . . . . . . . . . 17
3 Apaintshortage . . . . . . . . . . . . . . . . . . . . . . . 17
4 Racingmarbles . . . . . . . . . . . . . . . . . . . . . . . . 18
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