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DiscoveringMathematics (cid:2) Jiˇrí Gregor Jaroslav Tišer Discovering Mathematics A Problem-Solving Approach to Mathematical Analysis ® with MATHEMATICA and Maple™ JirˇíGregor JaroslavTišer DepartmentofMathematics DepartmentofMathematics CzechTechnicalUniversity CzechTechnicalUniversity Techniká2 Techniká2 16627Praha6 16627Praha6 CzechRepublic CzechRepublic gregorj@math.feld.cvut.cz tiser@math.feld.cvut.cz Maple™isatrademarkofWaterlooMapleInc.,615KumpfDrive,Waterloo,Ontario,CanadaN2V1K8, http://www.maplesoft.com ‘Mathematica’andthe‘Mathematica’logoareregisteredtrademarksofWolframResearch,Inc(“WRI”, www.wolfram.com)andareusedhereinwithWRI’spermission.WRIdidnotparticipateinthecreation ofthisworkbeyondtheinclusionoftheaccompanyingsoftware,anditoffersnoendorsementbeyond theinclusionoftheaccompanyingsoftware Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com Password:[978-0-85729-054-0] ISBN978-0-85729-054-0 e-ISBN978-0-85729-064-9 DOI10.1007/978-0-85729-064-9 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2010938724 MathematicsSubjectClassification(2000): 00A07,00A35,00A05 ©Springer-VerlagLondonLimited2011 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Coverdesign:deblik Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) v ‘...mathematicshastwofaces;... Mathematics presentedintheEuclideanwayappearsasasys- tematic, deductive science; but mathematics in themakingappearsasanexperimental,inductive science.Bothaspectsareasoldasthescienceof mathematicsitself.’ G.Polya ‘Aproblem?Ifyoucansolveit,itisanexercise; otherwiseit’saresearchtopic.’ R.Bellman Prologue Thebestwaytolearnthingsisbydoingthem.Butwhymathematics? Mathematics wasalwaysoneofthe less populartopics inschool, mathematics is not easytostudy,mathematicsteachersaredemandingandanswerstotheirquestionsare oftenencouteredwithobjections,problemstheyposearedifficulttosolve,etc. Isthereawaytooverridenegativesentimentstowardsmathematics? Successisagoodstimulus.Wehavetoknowhowtosolveproblemssuccessfullyand thentheymaybecomefascinatingandrewarding.Somepeoplewanttostartwithdiffi- cultproblemsandfollowsomesuggestedpathtoasolution,othersmaywanttogradu- allydeveloptheirabilityonlesscomplicatedproblems. Anabilitytosolveproblemswillcertainlybeusefulineverydaylife. Aninterconnected netofmathematical problemsfromvarioussourcesandwithvari- ous levels of difficulty is presented together with supporting material (hints, plans of solution, definitions and theorems, answers and references) and any student, teacher, engineerorinterestedpersonmaysharpenhisskillinhisownway. Anynewpieceofknowledgecanbebuiltonlyonearlieracquiredknowledge. ThewayofapplyingCalculusinthiscollectionrequiressomebasicknowledge(afirst yearuniversitycourseissufficient).Toshowinterestingproblemswhichareusuallynot includedinCalculuscourses,theorderingoftopicsisrathernon-standard.Threemain partsareincluded:Concepts,Tools,Applications. Doweneedmathematicswhen‘allproblemscanbesolvedbycomputers’? Thiscommonstatementisknowntobemisleading.Butcomputerswithpowerfulsoft- ware can be a valuable tool when we want to get rid of routine calculation, to verify conjectures on examples or visualize results. We have chosen MATHEMATICA and Mapleasthesoftwaretoolandsomehintsaregivenonwhereandhowitcanbeused. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 PartI Concepts 1 Mappings,CompositeandInverse-Functions . . . . . . . . . . . . . . . 11 2 InfiniteSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 PartII Tools 4 FiniteSums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 CollocationandLeastSquaresMethods . . . . . . . . . . . . . . . . . . . 113 PartIII Applications 7 MaximalandMinimalValues . . . . . . . . . . . . . . . . . . . . . . . . . 131 8 ArcsandCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9 CenterofMassandMoments . . . . . . . . . . . . . . . . . . . . . . . . . 163 10 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 PartIV Appendix 11 AnswerstoProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 vii Part I Concepts Introduction WhyMathematics? Thesolutionofcomplexproblemsinengineering,ineconomicsandevenineverydaylife isoftenverydifficultandacceptingresultswhichhavenotbeenverifiedtosufficientextent maysometimesberatherrisky.Wecannothopefora‘generalsolvingmethod’,neverthe- lesssomekindofstrategiesorstructuredanticipations—‘theartofproblemsolving’—can bedescribed.Experiments,techniquesoftrial-and-error,methodsbasedonsimplification, analogy and abstraction are recognized as indispensable parts of this art. In science and technologysuchtoolsareoftensuccessfullyreplacedbymathematicalmodelsoftheprob- lemunderinvestigation.Roughlyspeaking,amathematicalmodelisacollectionofcon- ceptswithwell-definedpropertiesandinterrelationssuchthatthebehaviorofthecollection isapttoimitatethebehaviortheobjectunderinvestigation.Suchamathematicalstructure isa‘language’inwhichtheproblemhastobeexpressed.Suchexpressionisthemathe- maticalformulationiftheproblem. Theloadingcapacityofacraneorabridgecanbecalculatedbysolvingcertaindiffer- entialoralgebraicequation,asystemoflinearequationsandinequalitiesisusedinsolving optimization problems, a weather forecast can be obtained as a solution of a system of partialdifferenceequations,theRubiccubecanbesolvedusingsomeknowledgeofgroup theory,somemethodsofcodingcanbedescribedbyapplyingnumbertheory,andmany moreexamplescanbegiven. TwoStages Somestandardproblemshavestandardmathematicalmodels.Forinstance,dynamicalpro- cessesareusuallydescribedbyordinarydifferentialequationsinvarioussettings.Tosolve geometrical problems, various coordinate systems and methods of analytic or algebraic geometrycanbeused.Ontheotherhand,someproblemsleadusintoratherunexpected partsofmathematics,or—asshowninthehistoryofmathematics—theymayleadtobuild- ingnewbranchesofmathematics.Analysisofgamesleadtoprobabilitytheory,problems J.Gregor,J.Tišer,DiscoveringMathematics, 1 DOI10.1007/978-0-85729-064-9_1,©Springer-VerlagLondonLimited2011 2 Introduction likethefamous‘SevenbridgesofKönigsberg’leadstographtheory.Wehavetoconclude thatwhentryingtoestablishamathematicalmodelwecannotberestrictedtoapredefined areaofmathematics.Amathematicalmodelcanberatherdifficulttofind,oftenbecause specialistsspeakingdifferent‘languages’oftheirseparatebranchesofsciencemustunder- standeachother,findacommonlanguageandformulatetheultimategoal. Inamathematicalmodelingapproachtoproblemsolving,weneedtodiscusstwodis- tinctactivities.Firstofallasuitablemathematicalmodelandamathematicalformulation hastobefoundandsecondly,wemustbeabletosolvetheemergingmathematicalprob- lem. Solvingsomecomplicatedschemeoftrainsmovingatdifferentspeedsinoppositedi- rections,withstopsandrestrictions,weassumethatinterrelationsbetweenspeed,distance andtimeareknown.Themainproblemistoestablishasystemofequationsgoverningall therelevantquantities.Theirsolutionmightberathersimple,althoughsomesupplemen- tary steps ofverification, exclusion ofsomevirtualsolutions andotherconcluding steps might still cause difficulties. Assuming that we have a mathematical formulation of our problem,wehavetomakesurethatitssolutionexists(inmanycasesthisisfarfromobvi- ous)orperhapsthatithasmanysolutions.Thenwecantrytofinditssolution(s).Software packageslikeMATHEMATICA®1,Maple®2orotherscanbeofsubstantialhelp.Interpre- tationoftheresultanditsverificationisthefinalpartofthesolution. ProblemSolving Theonlywaytolearntheartofproblemsolvingisbydoingit.Mathematicalreasoning containsallthenecessaryconstituentsofgeneralproblemsolvingandthisisthemainrea- sonforstudyingmathematics.Acommondifficultyisthatmathematicaltextbooksoften contain(ready-made)definitionsandtheoremswithoutmotivationorjustification.Butwe havetoknowwhytheintroducedresultsarethemostadequateandusefulconcepts,and whyotherpossiblewaysarewrongormisleading.Alsoexercisesoften,bytheirlocation or context, have predefined steps of solution. These features do not support the skills of problemsolvingandamoregeneralapproachisdesirable.Aprerequisiteofsuchanap- proachisabasicknowledgeofmathematics(e.g.calculus,geometryandalgebra).Onsuch abasiswemaytrytolookbehindthecurtainsofmathematics:tolearnhowandwhyvari- ousmathematicalconceptsaredefined,toacquirebasicskillsinusingmathematicaltools anddevices,andfinallytoapplysuchknowledgetosolveproblemswhichoftenhavetheir originsoutsidemathematics. 1Mathematica® is a registered trademark of Wolfram Research, Inc., 100 Trade Center Drive, Champaign,IL61820-7237,USA,http://www.wolfram.com 2Maple® isatrademarkofWaterlooMapleInc.,615KumpfDrive,Waterloo,Ontario,Canada N2V1K8,http://www.maplesoft.com Introduction 3 HowToSolveIt Probablyoneofthebestdescriptionoftheprocessofproblemsolvingwithanemphasis on mathematics has been given by G. Polya in his book ‘How to solve it’. The author describesthefollowingfourbasicstagesofproblemsolvingindetail: 1. Understandingtheproblem. 2. Devisingaplan. 3. Carryingouttheplan. 4. Lookingback. Inallthesestagestheauthorrecommendstoanswersomequestionsandcarryoutsmall tasks.Inashortenedrewordingsomeofthemcanbeformulatedasfollows: DoIunderstandwhatiswanted,whatisknownandwhattheconditionsare? Aretheconditionssufficient,insufficientorredundant? DoIknowthesolutionofasimilar,relatedoranalogoussimplerproblem? CanIuseitsresultoritsmethod?CanIdesignsuccessivestepstowardsasolution?If not,canIformulateandsolveasimplerrelatedproblemandusetheresult? CanIcarryoutthedesignedstepstowardsasolution? CanIchecktheresultofeachofthesteps? DidIuseallthegivendataanddidIsatisfyalltheconditions? Arethereothersolutions? Cantheobtainedresultbeusedinsolvingotherrelatedproblems? Toillustratethisapproachletustrytosolvethefollowingtask: Findthepointwhichisthenearesttongivenones. Almostallthewordsinthisformulationareambiguousandwecannotevenstartthink- ingofhowtosolveit.Wehavetoanalyzeitsformulation.Themostsuspiciouspartseems tobethewording‘thenearest’.Itmightbeconnectedtodistances;onepossibleinterpreta- tioncouldbe:thepointforwhichthesumofdistancesfromthengivenonesisthesmallest. Butareallthegivenpointsmutuallydistinct?Ifnot,shouldthedistancefromtheunknown point to two coinciding ones be counted twice? Further, what is meant by ‘point’? Are theypointsinthegeometricalsense,oraretheyelementsofsomeabstractspace?Howare theydefinedinthelattercase?Lastbutnotleast:howisthedistancebetweentwopoints measured(ordefined)? Toresolvetheseambiguitieswemaywanttoreformulatethequestion.Werecallthatthe conceptofdistancehasbeenabstractlydefined.Asetsuchthatanytwoofitselements,say AandB,haveadefineddistanceρ(A,B)iscommonlycalledmetricspaceandelements ofsuchasetcanbecalledpoints.Thereforetheabovetaskcanbereformulatedasfollows: Given n mutually disti(cid:2)nct elements of a metric space, find an element x of this space such that the quantity ni=1ρ(x,xi) (i.e. the sum of the distances from x to the given x ’s) is the smallest possible. In such an abstract formulation the problem seems to be i unsolvable.Lookingbackatourstartingpointwemayfeelthatoriginallythepointsare in fact points in common ‘space’, i.e. in three-dimensional space, in which the distance meanstheEuclideandistance.Hence,forthedistanceρ(A,B)wehave ρ2(A,B)=(x −x )2+(y −y )2+(z −z )2 A B A B A B

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Discovering Mathematics: A Problem-Solving Approach to Analysis with Mathematica and Maple provides a constructive approach to mathematical discovery through innovative use of software technology. Interactive Mathematica and Maple notebooks are integral to this books’ utility as a practical tool f