上海财经大学英语高数课件

Chapter0ne

LimitsandRatesofChangeupdownreturn

endupdownreturnend11.4ThePreciseDefinitionofaLimit

Weknowthatitmeansf(x)ismovingclosetoLwhilexismovingclosetoaaswedesire.AnditcanreachesLasnearaswelikeonlyonconditionofthexisinaneighbor.(2)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisL,andwewrite,ifforverynumber>0thereisacorrespondingnumber>0suchthat|f(x)-L|<whenever0<|x-a|<.updownreturn

endHowtogivemathematicaldescriptionof1.4ThePreciseDefinition2

Inthedefinition,themainpartisthatforarbitrarily>0,thereexistsa>0suchthatifallxthat0<|x-a|<then|f(x)-L|<.

Anothernotationforisf(x)Lasxa.Geometricinterpretationoflimitscanbegivenintermsofthegraphofthefunctiony=L+

y=L-

y=L

a

a-a+oxy=f(x)yupdownreturn

endInthedefinition,themai3Example1ProvethatSolution

Letbeagivenpositivenumber,wewanttofindapositivenumbersuchthat|(4x-5)-7|<whenever0<|x-3|<.But|(4x-5)-7|=4|x-3|.Therefore

4|x-3|<whenever0<|x-3|<.Thatis,|x-3|</4whenever0<|x-3|<.Example2ProvethatExample3Provethatupdownreturn

endExample1ProvethatSolution4

Example4

Provethat

Similarlywecangivethedefinitionsofone-sidedlimitsprecisely.(4)DEFINITIONOFLEFT-SIDEDLIMIT

Ifforeverynumber>0thereisacorrespondingnumber>0suchthat|f(x)-L|<whenever0<a

-

x<,i.e,a

-<

x<

a.(5)DEFINITIONOFLEFT-SIDEDLIMIT

Ifforeverynumber>0thereisacorrespondingnumber>0suchthat|f(x)-L|<whenever0<x-a<,i.e,a

<

x<

a+.

Example5

Provethatupdownreturn

endExample4ProvethatSimilar5

Example6If

provethat(6)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisinfinity,andwewrite,ifforverynumberM>0thereisacorrespondingnumber>0suchthat

f(x)>Mwhenever0<|x-a|<.updownreturn

endExample6Ifprovethat(6)6ExampleProvethat

Example5

Provethat(6)DEFINITIONLetf(x)beafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.Thenwesaythatthelimitoff(x)asxapproachesaisinfinity,andwewrite,ifforverynumber

N<0thereisacorrespondingnumber>0suchthat

f(x)<Nwhenever0<|x-a|<.updownreturn

endExampleProvethatExample57

Similarly,wecangivethedefinitionsofone-sideinfinitelimits.ExampleProvethatExampleProvethatExampleProvethatupdownreturn

endSimilarly,wecangivethe81.5Continuity

Iff(x)notcontinuousata,wesayf(x)isdiscontinuousata,orf(x)hasadiscontinuityata.(1)DefinitionAfunctionf(x)iscontinuousatanumberaif.(3)Afunctionf(x)iscontinuousatanumberaifandonlyifforeverynumber>0thereisacorrespondingnumber>0suchthat|f(x)-f(a)|<whenever|x-a|<.Notethat:(1)f(a)isdefined(2)exists.updownreturn

end1.5ContinuityIff(x)notco9Exampleisdiscontinuousatx=2,sincef(2)isnotdefined.Exampleiscontinuousat

x=2..ExampleProvethatsinxiscontinuousatx=a.(2)DefinitionAfunctionf(x)iscontinuousfromtherightateverynumberaifAfunctionf(x)iscontinuousfromtheleftateverynumberaifupdownreturn

endExample10(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.)updownreturn

end(2)DefinitionAfunctionf(11(5)THEOREM(a)anypolynomialiscontinuouseverywhere,thatis,itiscontinuousonR1=().(b)anyrationalfunctioniscontinuouswhereveritisdefined,thatis,itiscontinuousonitsdomain.Example

Find(6)THEOREMIfnisapositiveeveninteger,thenf(x)=iscontinuouson[0,).Ifnisapositiveoddinteger,thenf(x)=iscontinuouson().Example

Onwhatintervalsiseachfunctioncontinuous?updownreturn

end(5)THEOREM(a)anypolynom12(8)THEOREMIfg(x)

iscontinuousataand

f(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,then

updownreturn

end(8)THEOREMIfg(x)isconti13ExampleShowthatthereisarootoftheequation4x3-6x2+3x-2=0between1and2.updownreturn

endExampleShowthatthereisa141.6Tangent,andOtherRatesofChangeA.Tangent(1)Definition

TheTangentlinetothecurvey=f(x)atpointP(a,f(a))isthelinethroughPwithslopeprovidedthatthislimitexists.ExampleFindtheequationofthetangentlinetotheparabolay=x2atthepointP(1,1).updownreturn

end1.6Tangent,andOtherRates15B.OtherratesofchangeThedifferencequotient

iscalledtheaverageratechangeofywithrespectxovertheinterval[x1,x2].(4)

instantaneousrateofchange=atpointP(x1,f(x1))withrespecttox.Supposeyisaquantitythatdependsonanotherquantityx.Thusyisafunctionofxandwewritey=f(x).Ifxchangesfromx1andx2,thenthechangeinx(alsocalledtheincrementofx)isx=x2-

x1andthecorrespondingchangeinyisx=f(x2)-f(x1).updownreturn

endB.OtherratesofchangeThedi16

(1)whatisatangenttoacircle?

Canwecopythedefinitionofthetangenttoacirclebyreplacingcirclebycurve?1.1ThetangentandvelocityproblemsThetangenttoacircleisalinewhichintersectsthecircleonceandonlyonce.Howtogivethedefinitionoftangentlinetoacurve?Forexample,updownreturn

end(1)whatisatangenttoa17

Fig.(a)InFig.(b)therearestraightlineswhichtouchthegivencurve,

buttheyseemtobedifferentfrom

thetangenttothecircle.

L2Fig.(b)L1updownreturn

endFig.(a)InFig.(b)therea18Letusseethetangenttoacircleasamovinglinetoacertainline:Sowecanthinkthetangenttoacurveisthelineapproachedbymovingsecantlines.PQupdownreturn

endQ'Letusseethetangenttoaci19

x

mPQ231.52.51.12.11.012.011.0012.001Example1:Findtheequationofthetangentlinetoaparabolay=x2atpoint(1,1).Qisapointonthecurve.Q

y=x2

Pupdownreturn

end

20ThenwecansaythattheslopemofthetangentlineisthelimitoftheslopesmQPofthesecantslines.AndweexpressthissymbolicallybywritingAndSowecanguessthatslopeofthetangenttotheparabolaat(1,1)isveryclosedto2,actuallyitis2.Thentheequationofthetangentlinetotheparabolaisy-1=2(x-2)i.ey=2x-3.updownreturn

endThenwecansaythattheslope21SupposethataballisdroppedfromtheupperobservationdeckoftheOrientalPearlTowerinShanghai,280mabovetheground.Findthevelocityoftheballafter5seconds.Fromphysicsweknowthatthedistancefallenaftertsecondsisdenotedbys(t)andmeasuredinmeters,sowehaves(t)=4.9t2.Howtofindthevelocityatt=5?(2)

Thevelocityproblem:Solutionupdownreturn

endSupposethataballisdropped22Sowecanapproximatethedesiredquantitybycomputingtheaveragevelocityoverthebrieftimeintervalofthen-thofasecondfromt=5,suchas,thetenth,twenty-thandsoon.Thenwehavethetable:TimeintervalAveragevelocity(m/s)5<t<653.95<t<5.149.495<t<5.0549.2455<t<5.0149.0495<t<5.00149.0049Sowecanapproximatethedesi23Theabovetableshowsustheresultsofsimilarcalculationsofaveragevelocityoversuccessivelysmallertimeperiods.Italsoappearsthatastimeperiodtendsto0,theaveragevelocityisbecomingcloserto49.Sotheinstantaneousvelocityatt=5isdefinedtobethelimitingvalueoftheseaveragevelocitiesovershortertimeperiodsthatstartatt=5.updownreturn

endTheabovetableshowsusthe24

1.2TheLimitofaFunction

Letusinvestigatethebehaviorofthefunctiony=f(x)=x2-x+2forvaluesofxnear2.

x

f(x)x

f(x)1.02.0000003.08.0000001.52.7500002.55.7500001.83.4400002.24.6400001.93.7100002.14.3100001.953.8525002.054.1525001.993.9701002.014.0301001.9953.9850252.0014.003001updownreturn

end1.2TheLimitofaFunction25Weseethatwhenxiscloseto2(x>2orx<2),f(x)iscloseto4.Thenwecansaythat:thelimitofthefunction

f(x)=x2-x+2asxapproaches2isequalto4.

Thenwegiveanotationforthis:Ingeneral,thefollowingnotation:Weseethatwhenxiscloseto26(1)Definition:Wewrite

Guessthevalueof.Noticethatthefunctionisnotdefinedatx=1,and

x<1f(x)x>1f(x)0.50.6666671.50.4000000.90.5263161.10.4761900.990.5025131.010.4975120.9990.5002501.0010.4997500.999.0.5000251.00010.499975Example1updownreturn

endandsay“thelimitoff(x),asxapproachesa,equalsL”.SolutionIfwecanmakethevaluesoff(x)arbitrarilyclosetoL(asclosetoLaswelike)bytakingxtobesufficientlyclosetoabutnotequaltoa.Sometimesweusenotationf(x)Lasx

a.(1)Definition:WewriteGues27

Example1

FindExample2

FindNoticethatasx

awhichmeansthatxapproachesa,xmay>aandxmay<a.Example3

Discuss,whereThefunctionH(x)approaches0asxapproaches0andx<0,anditapproaches1,asxapproaches0andx>0.SowecannotsayH(x)approachesanumberasx

a.updownreturn

endExample1FindExample2Fi28One-sideLimits:EventhoughthereisnosinglenumberthatH(x)approachesastapproaches0.thatis,doesnotexist.Butastapproaches0fromleft,t<0,H(x)approaches0.Thenwecanindicatethissituationsymbolicallybywriting:Butastapproaches0fromright,t>0,H(x)approaches1.Thenwecanindicatethissituationsymbolicallybywriting:updownreturn

endOne-sideLimits:Eventhought29WewriteAndsaytheleft-handlimitoff(x)asxapproachesa(orthelimitoff(x)asxapproachesafromleft)isequaltoL.Thatis,wecanmakethevalueoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxlessthana.Andsaytheright-handlimitoff(x)asxapproachesa(orthelimitoff(x)asxapproachesafromright)isequaltoL.Thatis,wecanmakethevalueoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxgreaterthana.WewriteHerex

a+”meansthatxapproachesaandx>a.(2)Definition:Herex

a-”meansthatxapproachesaandx<a.updownreturn

endWewriteAndsaytheleft-hand30SeefollowingFigure:Whatwillithappenasx

a

orx

b?

xOaby=f(x)yupdownreturn

endSeefollowingFigure:Whatwill31(3)Theorem:ifandonlyifExample:Find.

x1/x2±11±0.54±0.225±0.1100±0,05400±0,0110000±0.0011000000xy=1/x2Oyupdownreturn

end(3)Theorem:ifandonlyifExamp32Toindicatethekindofbehaviorexhibitedinthisexample,weusethenotation:GenerallywecangivefollowingExampleFindTheanothernotationforthisisf(x)asx

a,whichisreadas“thelimitoff(x),asxapproachesa,isinfinity”or“f(x)becomesinfinityasxapproachesa”or“f(x)increaseswithoutboundasxapproachesa”.(4)DEFINITION:Letfbeafunctiononbothsidesofa,exceptpossiblyataitself.Thenmeansthatvaluesoff(x)canbemadearbitrarilylarge(asweplease)bytakingxsufficientlyclosetoa(butnotequaltoa).updownreturn

endToindicatethekindofbehavi33y=f(x)=ln|x|yx

Obviouslyf(x)=ln|x|becomeslargenegativeasxgetscloseto0.(5)DEFINITION:Letfbeafunctiononbothsidesofa,exceptpossiblyataitself.Thenmeansthatvaluesoff(x)canbemadearbitrarilylarge(asweplease)bytakingxsufficientlyclosetoa(butnotequaltoa).Theanothernotationforthisisf(x)-asx

a,whichisreadas“thelimitoff(x),asxapproachesa,isnegativeinfinity”or“f(x)becomesnegativeinfiniteasxapproachesa”or“f(x)decreaseswithoutboundasxapproachesa”.updownreturn

endy=f(x)=ln|x|yxObviouslyf(x)=34(6)DEFINITION:Thelinex=aiscalledaverticalasymptoteofthecurvey=f(x)ifatleastoneofthefollowingstatementsistrue:Similardefinitionscanbe