Table Of Content

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 This page intentionally left blank 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