微积分英文版电子档.doc

《微积分英文版电子档.doc》由会员上传分享，免费在线阅读，更多相关内容在应用文档-天天文库。

1、DifferentialCalculus/D/x*p2R6r$]$NewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthesemenwereprimarilyduetothefactt

2、hattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.+V0E$f!k%w6l-{(Thecentralideaofdifferentialcalculusisthenotionofderivative.Liketheintegral,thederivativeoriginatedfromaproblemingeometry—theproblemoffindingthetangentlineatapointofacurve.U

3、nliletheintegral,however,thederivativeevolvedverylateinthehistoryofmathematics.Theconceptwasnotformulateduntilearlyinthe17thcenturywhentheFrenchmathematicianPierredeFermat,attemptedtodeterminethemaximaandminimaofcertainspecialfunctions.Fermat’sidea,basicallyverysimple,canbeunderstoodifweref

4、ertoacurveandassumethatateachofitspointsthiscurvehasadefinitedirectionthatcanbedescribedbyatangentline.Fermatnoticedthatatcertainpointswherethecurvehasamaximumorminimum,thetangentlinemustbehorizontal.Thustheproblemoflocatingsuchextremevaluesisseentodependonthesolutionofanotherproblem,thatof

5、locatingthehorizontaltangents.Thisraisesthemoregeneralquestionofdeterminingthedirectionofthetangentlineatanarbitrarypointofthecurve.ItwastheattempttosolvethisgeneralproblemthatledFermattodiscoversomeoftherudimentaryideasunderlyingthenotionofderivative.Atfirstsightthereseemstobenoconnectionw

6、hateverbetweentheproblemoffindingtheareaofaregionlyingunderacurveandtheproblemoffindingthetangentlineatapointofacurve.Thefirstpersontorealizethatthesetwoseeminglyremoteideasare,infact,ratherintimatelyrelatedappearstohavebeenNewton’steacher,IsaacBarrow(1630-1677).However,NewtonandLeibnizwere

7、thefirsttounderstandtherealimportanceofthisrelationandtheyexploitedittothefullest,thusinauguratinganunprecedentederainthedevelopmentofmathematics.Althoughthederivativewasoriginallyformulatedtostudytheproblemoftangents,itwassoonfoundthatitalsoprovidesaway