上海财经大学英语高数课件

Chapter0neLimitsandRatesofChangeupdownreturnend1.4ThePreciseDefinitionofaLimit?)(limLxfaxWeknowthatitmeansf(x)ismovingclosetoLwhilexismovingclosetoaaswedesire.AnditcanreachesLasnearaswelikeonlyonconditionofthexisinaneighbor.(2)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisL,andwewrite,ifforverynumber0thereisacorrespondingnumber0suchthat|f(x)-L|whenever0|x-a|.Lxfax)(limupdownreturnendHowtogivemathematicaldescriptionofInthedefinition,themainpartisthatforarbitrarily0,thereexistsa0suchthatifallxthat0|x-a|then|f(x)-L|.Anothernotationforisf(x)Lasxa.Lxfax)(limGeometricinterpretationoflimitscanbegivenintermsofthegraphofthefunctiony=L+y=L-y=Laa-a+oxy=f(x)yupdownreturnendExample1Provethat.7)54(lim3xxSolutionLetbeagivenpositivenumber,wewanttofindapositivenumbersuchthat|(4x-5)-7|whenever0|x-3|.But|(4x-5)-7|=4|x-3|.Therefore4|x-3|whenever0|x-3|.Thatis,|x-3|/4whenever0|x-3|.Example2Provethat.8lim32xxExample3Provethat.sinsinlimaxaxupdownreturnendExample4Provethat101459lim24tttSimilarlywecangivethedefinitionsofone-sidedlimitsprecisely.(4)DEFINITIONOFLEFT-SIDEDLIMITIfforeverynumber0thereisacorrespondingnumber0suchthat|f(x)-L|whenever0a-x,i.e,a-xa.Lxfax)(lim(5)DEFINITIONOFLEFT-SIDEDLIMITIfforeverynumber0thereisacorrespondingnumber0suchthat|f(x)-L|whenever0x-a,i.e,axa+.Lxfax)(lim.0lim0xxExample5ProvethatupdownreturnendExample6IfLxfax)(lim,)(lim,Mxgaxprovethat,)]()([limMLxgxfax,)]()([limLMxgxfax).0(,)()(limMMLxgxfax(6)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisinfinity,andwewrite,ifforverynumberM0thereisacorrespondingnumber0suchthatf(x)Mwhenever0|x-a|.)(limxfaxupdownreturnend.)1(1lim21xxExampleProvethatExample5Provethat(6)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisinfinity,andwewrite,ifforverynumberN0thereisacorrespondingnumber0suchthatf(x)Nwhenever0|x-a|.)(limxfax||lnlim0xxupdownreturnendSimilarly,wecangivethedefinitionsofone-sideinfinitelimits.)(limxfax)(limxfax)(limxfax)(limxfaxxx102limExampleProvethatExampleProvethat02lim10xx11lim1xxExampleProvethatupdownreturnend1.5ContinuityIff(x)notcontinuousata,wesayf(x)isdiscontinuousata,orf(x)hasadiscontinuityata.(1)DefinitionAfunctionf(x)iscontinuousatanumberaif)()(limafxfax.(3))()(limafxfaxAfunctionf(x)iscontinuousatanumberaifandonlyifforeverynumber0thereisacorrespondingnumber0suchthat|f(x)-f(a)|whenever|x-a|.Notethat:(1)f(a)isdefined)(limxfax(2)exists.updownreturnendExampleisdiscontinuousatx=2,sincef(2)isnotdefined.22)(2xxxxfExampleiscontinuousatx=2..22322)(2xxxxxxfExampleProvethatsinxiscontinuousatx=a.(2)DefinitionAfunctionf(x)iscontinuousfromtherightateverynumberaifAfunctionf(x)iscontinuousfromtheleftateverynumberaif)()(limafxfax)()(limafxfaxupdownreturnend(2)DefinitionAfunctionf(x)iscontinuousonanintervalifitiscontinuousateverynumberintheinterval.(atanendpointoftheintervalweunderstandcontinuoustomeancontinuousfromtherightorcontinuousfromtheleft)ExampleAteachintegern,thefunctionf(x)=[x]iscontinuousfromtherightanddiscontinuousfromtheleft.ExampleShowthatthefunctionf(x)=1-(1-x2)1/2iscontinuousontheinterval[-1,1].(4)TheoremIffunctionsf(x),g(x)iscontinuousataandcisaconstant,thenthefollowingfunctionsarecontinuousata:1.f(x)+g(x)2.f(x)-g(x)3.f(x)g(x)4.f(x)[g(x)]-1(g(a)isn’t0.)updownreturnend(5)THEOREM(a)anypolynomialiscontinuouseverywhere,thatis,itiscontinuousonR1=().(b)anyrationalfunctioniscontinuouswhereveritisdefined,thatis,itiscontinuousonitsdomain.ExampleFind.2553lim22xxxx(6)THEOREMIfnisapositiveeveninteger,thenf(x)=iscontinuouson[0,).Ifnisapositiveoddinteger,thenf(x)=iscontinuouson().nxnxExampleOnwhatintervalsiseachfunctioncontinuous?,453)(22xxxa.11111)(22xxxxxbupdownreturnend(8)THEOREMIfg(x)iscontinuousataandf(x)iscontinuousatg(a)then(fog)(x))=f(g(x))iscontinuousata.(7)THEINTERMEDIATEVALUETHEOREMSupposethatf(x)iscontinuousontheclosedinterval[a,b].LetNbeanynumberstrictlybetweenf(a)andf(b).Thenthereexistsanumbercin(a,b)suchthatf(c)=Nyxby=Na(7)THEOREMIff(x)iscontinuousatband,thenbg(x)axlim).lim()()(limg(x)fbf)xf(gaxaxupdownreturnendExampleShowthatthereisarootoftheequation4x3-6x2+3x-2=0between1and2.updownreturnend1.6Tangent,andOtherRatesofChangeA.Tangent(1)DefinitionTheTangentlinetothecurvey=f(x)atpointP(a,f(a))isthelinethroughPwithslopeprovidedthatthislimitexists.axafxfmax)()(limExampleFindtheequationofthetangentlinetotheparabolay=x2atthepointP(1,1).updownreturnendB.OtherratesofchangeThedifferencequotientiscalledtheaverageratechangeofywithrespectxovertheinterval[x1,x2].1212)()(xxxfxfxy(4)instantaneousrateofchange=atpointP(x1,f(x1))withrespecttox.12120)()(limlim1