Discovering Mathematics: A Problem-Solving Approach to Mathematical Analysis with MATHEMATICA® and Maple™

260 Pages·2011·3.17 MB·English

by  Jiří Gregor

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
[email protected] [email protected]
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