Paradoxes and Sophisms in Calculus PDF
Preview Paradoxes and Sophisms in Calculus
s s e m s u s o i l h u d p c a l o a r k S u a C h c m s P n d n giy KlyStaple a i Serusan S Classroom Resource Materials i i “master” — 2013/4/15 — 11:01 — page i — #1 i i Paradoxes and Sophisms in Calculus i i i i i i “master” — 2013/4/15 — 11:01 — page ii — #2 i i (cid:13)c 2013bytheMathematicalAssociationofAmerica,Inc. LibraryofCongressCatalogCardNumber2013934605 PrinteditionISBN978-0-88385-781-6 ElectroniceditionISBN978-1-61444-110-6 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10987654321 i i i i i i “master” — 2013/4/15 — 11:01 — page iii — #3 i i Paradoxes and Sophisms in Calculus Sergiy Klymchuk AucklandUniversity of Technology and Susan Staples Texas Christian University PublishedandDistributedby TheMathematicalAssociationofAmerica i i i i i i “master” — 2013/4/15 — 11:01 — page iv — #4 i i 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 i i i i i i “master” — 2013/4/15 — 11:01 — page v — #5 i i 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. 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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:1-301-206-9789 i i i i i i “master” — 2013/4/15 — 11:01 — page vii — #7 i i To our parents and families i i i i i i “master” — 2013/4/15 — 11:01 — page viii — #8 i i i i i i i i “master” — 2013/4/15 — 11:01 — page ix — #9 i i 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 ix i i i i
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