微积分英文版电子档

DifferentialCalculusNewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.Thecentralideaofdifferentialcalculusisthenotionofderivative.Liketheintegral,thederivativeoriginatedfromaproblemingeometry—theproblemoffindingthetangentlineatapointofacurve.Unliletheintegral,however,thederivativeevolvedverylateinthehistoryofmathematics.Theconceptwasnotformulateduntilearlyinthe17thcenturywhentheFrenchmathematicianPierredeFermat,attemptedtodeterminethemaximaandminimaofcertainspecialfunctions.Fermat’sidea,basicallyverysimple,canbeunderstoodifwerefertoacurveandassumethatateachofitspointsthiscurvehasadefinitedirectionthatcanbedescribedbyatangentline.Fermatnoticedthatatcertainpointswherethecurvehasamaximumorminimum,thetangentlinemustbehorizontal.Thustheproblemoflocatingsuchextremevaluesisseentodependonthesolutionofanotherproblem,thatoflocatingthehorizontaltangents.Thisraisesthemoregeneralquestionofdeterminingthedirectionofthetangentlineatanarbitrarypointofthecurve.ItwastheattempttosolvethisgeneralproblemthatledFermattodiscoversomeoftherudimentaryideasunderlyingthenotionofderivative.Atfirstsightthereseemstobenoconnectionwhateverbetweentheproblemoffindingtheareaofaregionlyingunderacurveandtheproblemoffindingthetangentlineatapointofacurve.Thefirstpersontorealizethatthesetwoseeminglyremoteideasare,infact,ratherintimatelyrelatedappearstohavebeenNewton’steacher,IsaacBarrow(1630-1677).However,NewtonandLeibnizwerethefirsttounderstandtherealimportanceofthisrelationandtheyexploitedittothefullest,thusinauguratinganunprecedentederainthedevelopmentofmathematics.Althoughthederivativewasoriginallyformulatedtostudytheproblemoftangents,itwassoonfoundthatitalsoprovidesawaytocalculatevelocityand,moregenerally,therateofchangeofafunction.Inthenextsectionweshallconsideraspecialprobleminvolvingthecalculationofavelocity.Thesolutionofthisproblemcontainsalltheessentialfcaturesofthederivativeconceptandmayhelptomotivatethegeneraldefinitionofderivativewhichisgivenbelow.Supposeaprojectileisfiredstraightupfromthegroundwithinitialvelocityof144feetpersecond.Neglectfriction,andassumetheprojectileisinfluencedonlybygravitysothatitmovesupandbackalongastraightline.Letf(t)denotetheheightinfeetthattheprojectileattainstsecondsafterfiring.Iftheforceofgravitywerenotactingonit,theprojectilewouldcontinuetomoveupwardwithaconstantvelocity,travelingadistanceof144feeteverysecond,andattimetwewoulehavef(t)=144t.Inactualpractice,gravitycausestheprojectiletoslowdownuntilitsvelocitydecreasestozeroandthenitdropsbacktoearth.Physicalexperimentssuggestthatastheprojectileisaloft,itsheightf(t)isgivenbytheformula.Theterm–16t2isduetotheinfluenceofgravity.Notethatf(t)=0whent=0andwhent=9.Thismeansthattheprojectilereturnstoearthafter9secondsanditistobeunderstoodthatformula(1)isvalidonlyfor0t9.Theproblemwewishtoconsideristhis:Todeterminethevelocityoftheprojectileateachinstantofitsmotion.Beforewecanunderstandthisproblem,wemustdecideonwhatismeantbythevelocityateachinstant.Todothis,weintroducefirstthenotionofaveragevelocityduringatimeinterval,sayfromtimettotimet+h.Thisisdefinedtobethequotient.Changeindistanceduringtimeinterval=f(t+h)-f(t)/h.engthoftimeintervalThisquotient,calledadifferencequotient,isanumberwhichmaybecalculatedwheneverbothtandt+hareintheinterval[0,9].Thenumberhmaybepositiveornegative,butnotzero.Weshallkeeptfixedandseewhathappenstothedifferencequotientaswetakevaluesofhwithsmallerandsmallerabsolutevalue.Thelimitprocessbywhichv(t)isobtainedfromthedifferencequotientiswrittensymbolicallyasfollows:Theequationisusedtodefinevelocitynotonlyforthisparticularexamplebut,moregenerally,foranyparticlemovingalongastraightline,providedthepositionfunctionfissuchthatthedifferercequotienttendstoadefinitelimitashapproacheszero.Theexampledescribeintheforegoingsectionpointsthewaytotheintroductionoftheconceptofderivative.Webeginwithafunctionfdefinedatleastonsomeopeninterval(a,b)onthexaxis.Thenwechooseafixedpointinthisintervalandintroducethedifferencequotient[f(x+h)-f(x)]/h.Wherethenumberh,whichmaybepositiveornegative(butnotzero),issuchthatx+halsoliesin(a,b).Thenumeratorofthisquotientmeasuresthechangeinthefunctionwhenxchangesfromxtox+h.Thequotientitselfisreferredtoastheaveragerateofchangeoffintheintervaljoiningxtox+h.Nowwelethapproachzeroandseewhathappenstothisquotient.Ifthequotient.Ifthequotientapproachessomedefinitevaluesasalimit(whichimpliesthatthelimitisthesamewhetherhapproacheszerothroughpositivevaluesorthroughnegativevalues),thenthislimitiscalledthederivativeoffatxandisdenotedbythesymbolf’(x)(readas“fprimeofx”).Thustheformaldefinitionoff’(x)maybestatedasfollowsDefinitionofderivative.Thederivativef’(x)isdefinedbytheequation