On the shoulders of giants: A course in single variable calculus

291 Pages·2002·2.057 MB·English

by  Gordon Mclelland

TLFeBOOK
CONTENTS
Preface v
1 Terror,tragedyandbadvibrations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheTowerofTerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Intothinair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Musicandthebridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Rulesofcalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Functions 11
2.1 Rulesofcalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Intervalsontherealline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Graphsoffunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Examplesoffunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Continuityandsmoothness 27
3.1 Smoothfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Differentiation 41
4.1 Thederivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Rulesfordifferentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Velocity,accelerationandratesofchange . . . . . . . . . . . . . . . . . . . . . . . 53
5 Fallingbodies 57
5.1 TheTowerofTerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Solvingdifferentialequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Generalremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Increasinganddecreasingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5 Extremevalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 Seriesandtheexponentialfunction 75
6.1 Theairpressureproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Infiniteseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Convergenceofseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4 Radiusofconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
ii CONTENTS
6.5 Differentiationofpowerseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6 Thechainrule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.7 Propertiesoftheexponentialfunction . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.8 Solutionoftheairpressureproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Trigonometricfunctions 109
7.1 Vibratingstringsandcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Trigonometricfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3 Moreonthesineandcosinefunctions . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4 Triangles,circlesandthenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
(cid:1)
7.5 Exactvaluesofthesineandcosinefunctions. . . . . . . . . . . . . . . . . . . . . . 122
7.6 Othertrigonometricfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8 Oscillationproblems 127
8.1 Secondorderlineardifferentialequations . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Complexnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3 Complexseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4 Complexrootsoftheauxiliaryequation . . . . . . . . . . . . . . . . . . . . . . . . 143
8.5 Simpleharmonicmotionanddamping . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.6 Forcedoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9 Integration 167
9.1 AnotherproblemontheTowerofTerror . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2 Moreonairpressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.3 Integralsandprimitivefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.4 Areasundercurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.5 Areafunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9.6 Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.7 Evaluationofintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.8 Thefundamentaltheoremofthecalculus. . . . . . . . . . . . . . . . . . . . . . . . 187
9.9 Thelogarithmfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10 Inversefunctions 197
10.1 Theexistenceofinverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.2 Calculatingfunctionvaluesforinverses . . . . . . . . . . . . . . . . . . . . . . . . 205
10.3 Theoscillationproblemagain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
10.4 Inversetrigonometricfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.5 Otherinversetrigonometricfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 221
11 Hyperbolicfunctions 225
11.1 Hyperbolicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
11.2 Propertiesofthehyperbolicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.3 Inversehyperbolicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
CONTENTS iii
12 Methodsofintegration 235
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.2 Calculationofdefiniteintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
12.3 Integrationbysubstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.4 Integrationbyparts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
12.5 Themethodofpartialfractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
12.6 Integralswithaquadraticdenominator . . . . . . . . . . . . . . . . . . . . . . . . . 247
12.7 Concludingremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
13 Anonlineardifferentialequation 251
13.1 Theenergyequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
13.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Answers 261
Index 281
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PREFACE
IfIhaveseenfurtheritisbystandingontheshouldersofGiants.
SirIsaacNewton,1675.
This book presents an innovative treatment of single variable calculus designed as an introductory
mathematics textbook for engineering and science students. The subject material is developed by
modellingphysicalproblems,someofwhichwouldnormallybeencounteredbystudentsasexperi-
mentsinafirstyearphysicscourse. Thesolutionsoftheseproblemsprovideameansofintroducing
mathematicalconceptsastheyareneeded.Thebookpresentsallofthematerialfromatraditionalfirst
yearcalculuscourse, butit willappearfordifferentpurposesandin a differentorderfromstandard
treatments.
Therationaleofthebookisthatthemathematicsshouldbeintroducedinacontexttailoredtothe
needs of the audience. Each mathematical concept is introduced only when it is needed to solve a
particularpracticalproblem,soatallstages, thestudentshouldbeabletoconnectthemathematical
concept with a particular physical idea or problem. For various reasons, notions such asrelevance
or justintimemathematicsare common catchcries. We have responded to these in a way which
maintainstheprofessionalintegrityofthecoursesweteach.
Thebookbeginswithacollectionofproblems. Adiscussionoftheseproblemsleadstotheidea
ofafunction,whichinthefirstinstancewillberegardedasarulefornumericalcalculation. Insome
cases, real or hypotheticalresults will be presented, from which the function can be deduced. Part
of the purpose of the book is to assist students in learning how to define the rules for calculating
functionsandtounderstandwhysuchrulesareneeded.Themostcommonwayofexpressingaruleis
bymeansofanalgebraicformulaandthisisthewayinwhichmoststudentsfirstencounterfunctions.
Unfortunately,manyofthemareunabletoprogressbeyondthefunctionsasformulasconcept. Our
stance in this book is that functions are rules for numerical calculation and so must be presented
in a form which allows function values to be calculated in decimal form to an arbitrary degree of
accuracy.Forthisreason,trigonometricfunctionsfirstappearaspowerseriessolutionstodifferential
equations, rather than through the common definitions in terms of triangles. The latter definitions
may be intuitively simpler, but they are of little use in calculating function values or preparing the
studentforlaterwork.Webeginwithsimplefunctionsdefinedbyalgebraicformulasandmoveonto
functionsdefinedbypowerseriesandintegrals. Asweprogressthroughthebook,differentphysical
problemsgiverisetovariousfunctionsandifthecalculationoffunctionvaluesrequiresthenumerical
evaluation of an integral, then this simply has to be accepted as an inconvenient but unavoidable
propertyofthe problem. We wouldlike studentstoappreciatethe factthatsomeproblems,suchas
the nonlinearpendulum, requiresophisticated mathematicalmethodsfor their analysis and difficult
mathematicsisunavoidableifwewishtosolvetheproblem.Itisnotintroducedsimplytoprovidean
vi PREFACE
intellectualchallengeortofilterouttheweakerstudents.
Ourattitudetoproofsandrigouristhatwebelievethatallresultsshouldbecorrectlystated,but
notallofthem needformalproof. Mostof all, we do notbelievethatstudentsshouldbepresented
with handwaving argumentsmasquerading as proofs. If we feel that a proof is accessible and that
there is somethingusefulto be learnedfrom the proof, then we provideit. Otherwise, we state the
resultandmoveon. Studentsarequitecapableofusingtheresultsonterm-by-termdifferentiationof
apowerseriesforinstance,eveniftheyhavenotseentheproof.However,wethinkthatitisimportant
toemphasisethatapowerseriescanbedifferentiatedinthiswayonlywithintheinteriorofitsinterval
ofconvergence.Bythismeanswecantaketheapplicationsinthisbookbeyondtheartificialexamples
oftenseeninstandardtexts.
Wediscusscontinuityanddifferentiationintermsofconvergenceofsequences.Wethinkthatthis
isintuitivelymoreaccessiblethantheusualapproachofconsideringlimitsoffunctions. Iflimitsare
(cid:2)
treatedwiththefullrigourofthe - approach,thentheyare toodifficultfortheaveragebeginning
(cid:1)
student,whileanon-rigoroustreatmentsimplyleadstoconfusion.
Theremainderofthisprefacesummarisesthecontentofthisbook. Ourlistofphysicalproblems
includestheverticalmotionofaprojectile,thevariationofatmosphericpressurewithheight,themo-
tionofabodyinsimpleharmonicmotion,underdampedandoverdampedoscillations,forceddamped
oscillationsandthenonlinearpendulum.Ineachcasethesolutionisafunctionwhichrelatestwovari-
ables. Anappealtothestudent’sphysicalintuitionsuggeststhatthegraphsofthesefunctionsshould
havecertainproperties.Closeranalysisoftheseintuitiveideasleadstotheconceptsofcontinuityand
differentiability. Modellingtheproblemsleadstodifferentialequationsforthedesiredfunctionsand
insolvingtheseequationswediscusspowerseries,radiusofconvergenceandterm-by-termdifferen-
tiation. Indiscussingoscillationwehavetoconsiderthecasewheretheauxiliaryequationmayhave
non-realrootsanditisatthispointthatweintroducecomplexnumbers.Notalldifferentialequations
areamenabletoasolutionbypowerseriesandintegrationisdevelopedasamethodtodealwiththese
cases. Along the way it is necessary to use the chain rule, to define functions by integrals and to
defineinversefunctions.Methodsofintegrationareintroducedasapracticalalternativetonumerical
methodsforevaluatingintegralsifaprimitivefunctioncanbefound. Wealsoneedtoknowwhether
afunctiondefinedbyanintegralisneworwhetheritisaknownelementaryfunctioninanotherform.
Wedonotgoverydeeplyintothistopic.Withtheadventofsymbolicmanipulationpackagessuchas
Mathematica,thereseemstobelittleneedforscienceandengineeringstudentstospendtimeevalu-
atinganythingbutthesimplestofintegralsbyhand. Thebookconcludeswithacapstonediscussion
ofthenonlinearequationofmotionofthesimplependulum. Ourpurposehereistodemonstratethe
factthattherearephysicalproblemswhichabsolutelyneedthemathematicsdevelopedinthisbook.
Variousadhocprocedureswhichmighthavesufficedforsomeoftheearlierproblemsarenolonger
useful. TheuseofMathematicamakesplottingofellipticfunctionsandfindingtheirvaluesnomore
difficultthanisthecasewithanyofthecommonfunctions.
WewouldliketothankTimLangtryforhelpwithLATEX.TimLangtryandGraemeCohenreadthe
textofthepreliminaryeditionofthisbookwithmeticulousattentionandmadenumeroussuggestions,
commentsandcorrections. Otherusefulsuggestions,contributionsandcorrectionscamefromMary
CouplandandLeighWood.
1
CHAPTER
TERROR, TRAGEDY AND BAD VIBRATIONS
1.1 INTRODUCTION
Mathematics is almost universally regarded as a useful subject, but the truth of the matter is that
mathematics beyond the middle levels of high school is almost never used by the ordinary person.
Certainly, simple arithmetic is needed to live a normallife in developedsocieties, but when would
weeverusealgebraorcalculus? Inmathematics,asinmanyotherareasofknowledge,wecanoften
get by with a less than complete understandingof the processes. People do nothave to understand
how a car, a computer or a mobile phone works in order to make use of them. However, some
people do have to understand the underlying principles of such devices in order to invent them in
the first place, to improvetheir design or to repairthem. Most people do notneed to knowhowto
organisetheOlympicGames,schedulebaggagehandlersforaninternationalairlineoranalysetraffic
flowin acommunicationsnetwork,butonceagain,someonemustdesignthe systemswhichenable
these activities to be carried out. The complex technical, social and financial systems used by our
modernsocietyallrelyonmathematicstoagreaterorlesserextentandweneedskilledpeoplesuch
as engineers, scientists and economists to manage them. Mathematics is widely used, but this use
is not always evident. Part of the purposeof this bookis to demonstratethe way that mathematics
pervades many aspects of our lives. To do this, we shall make use of three easily understood and
obviouslyrelevantproblems.Byexploringeachoftheseinincreasingdetailwewillfinditnecessary
tointroducealargenumberofmathematicaltechniquesinordertoobtainsolutionstotheproblems.
Aswebecomemorefamiliarwiththemathematicswedevelop,weshallfindthatitisnotlimitedto
theoriginalproblems,butisapplicabletomanyothersituations.
Inthischapter,wewillconsiderthreeproblems: anamusementparkrideknownastheTowerof
Terror, the disastrous consequencesthat occurredwhen an aircraft cargodoor flew open in mid-air
andanunexpectednoisepollutionproblemonanewbridge.Theseproblemswillbeusedasthebasis
forintroducingnewmathematicalideasandinlaterchapterswewillapplytheseideastothesolution
ofotherproblems.
1.2 THE TOWER OF TERROR
Sixteenpeoplearestrappedintoseatsinasixtonnecarriageatrestonahorizontalmetaltrack. The
power is switched on and in six seconds they are travelling at 160km/hr. The carriage traverses a
shortcurvedtrackandthenhurtlesverticallyupwardstoreachtheheightofa38storeybuilding. It
comesmomentarilytorestandthenfreefallsforaboutfivesecondstoagainreachaspeedofalmost
2 TERROR,TRAGEDYANDBADVIBRATIONS
Figure1.1:TheTowerofTerror
THETOWEROFTERROR 3
(cid:4)
(cid:3)
(cid:2) (cid:1)
Figure1.2:TheTowerofTerror(Schematic)
160km/hr. Ithurtlesbackaroundthecurvetothehorizontaltrackwherepowerfulbrakesbringitto
restbackatthestart. Thewholeeventtakesabout25seconds(Figure1.1).
Thishair-raisingjourneytakesplaceeveryfewminutesatDreamworld,alargeamusementpark
ontheGoldCoastinQueensland,Australia. Parkslikethishavebecomecommonaroundtheworld
withthebestknownbeingDisneylandintheUnitedStates. Oneofthemainfeaturesoftheparksare
therideswhichareofferedandasaresultofcompetitionbetweenparksandtheneedtocontinually
changetherides,theyhavebecomelarger,fasterandmoreexciting. Theridejustdescribedisaptly
namedtheTowerofTerror.
These trends have resulted in the developmentof a specialised industry to develop and test the
rideswhichtheparksoffer. Therearetwoaspectstothis. Firsttheconstructionmustensurethatthe
equipmentwillnotcollapseunderthestrainsimposedonit. Suchfailure,withtheresultingshowerof
fast-movingdebrisoverthepark,wouldbedisastrous. Second,andequallyimportant,istheneedto
ensurethatpatronswillbeabletophysicallywithstandtheforcestowhichtheywillbesubjected. In
fact,manyrideshaverestrictionsonwhocantaketherideandthereareoftenwarningnoticesabout
thedangeroftakingtherideforpeoplewithvariousmedicalproblems.
Let’slookatsomeaspectsoftherideintheTowerofTerrorillustratedintheschematicdiagram
in Figure1.2. Thecarriageis acceleratedalonga horizontaltrackfromthe startingpoint (cid:1) . When
itreaches (cid:2) afteraboutsix seconds,itistravellingat160km/hranditthentravelsaroundacurved
portion of track until its motion has become vertical by the point (cid:3) . From (cid:3) the speed decreases
undertheinfluenceofgravityuntilitcomesmomentarilytorestat (cid:4) , 115metresabovetheground
or the heightof a 38 storey building. The motion is then reversed as the carriage free falls back to
(cid:3) . Duringthisportionoftheride,theridersexperiencethesensationofweightlessnessforfiveorsix
seconds. Thecarriagethengoesroundthecurvedsectionofthetracktoreachthehorizontalportion
ofthetrack,thebrakesareappliedat(cid:2) andthecarriagecomestoastopat(cid:1) .
Themostimportantfeatureoftherideisperhapsthetimetakenforthecarriagetotravelfrom(cid:4)
backto (cid:3) . Thisisthetimeduringwhichtheridersexperienceweightlessnessduringfreefall. Ifthe
timeistooshortthentheridewouldbepointless. Thelongerthetimehowever,thehigherthetower
must be, with the consequent increase in cost and difficulty of construction. The time depends on
thespeedatwhichthecarriageistravellingwhenitreaches(cid:3) ontheoutwardjourneyandthehigher
this speed the longer the horizontalportion of the track must be and the more power is required to