微积分第二章课件

Chapter2

LimitsandDerivatives

2.1Thetangentandvelocityproblems

2.1.1Thetangentproblem

Example1FindanequationofthetangentlinetotheparabolaatthepointP(1,1).

SOLUTIONWewillbeabletofindanequationofthetangentlinetassoonasweknowitsslopem.Thedifficultyisthatweknowonlyonepoint,P,ont,whereasweneedtwopointstocomputetheslope.ButobservethatwecancomputeanapproximationtombychoosinganearbyQ(x,x2)ontheparabolaandcomputingtheslopemPQofthesecantlinePQ.1Chapter2LimitsandDeriva

Wechoose,then.Forinstance,forthepointQ(1.5,2.25)wehaveThecloserQistoP,thecloserxisto1andtheclosermPQisto2.Thissuggeststhattheslopeofthetangentlinetshouldbem=2.

Figure1PQ2Wechoose,then2.2LimitsofFunctions2.2.1LimitofaFunctionf(x)asxApproachesaDefinition1Letfbeafunctiondefinedonsomeopenintervalcontainingaexceptpossiblyataitself,andletLbearealnumber.Wesaythatthelimitoff(x)asxapproachesaisL,andwriteandsay“thelimitoff(x),asxapproachesa,equalsL”

ifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoa(oneithersideofa)butnotequaltoa.32.2LimitsofFunctions3InordertounderstandtheprecisemeaningofafunctioninDefinition,letusbegintoconsiderthebehaviorofafunction

asxapproaches1.FromthegraphoffshowninFigure2,wecanintuitivelyseethatasxgetscloserto1frombothsidesbut

x≠1,f(x)

getscloserto3/2.Inthiscase,weusethenotationandsaythatthelimitoff(x),asxapproaches1,is3/2,orthatf(x)

Figure2

/2asxapproaches1.

approaches3。.4InordertounderstandExample2Guessthevalueof.SOLUTIONThefunctionf(x)=sinx/xisnotdefinedatx=0.FromthetableandthegraphinFigure3weguessthatThisguessisinfactcorrect,aswillbeprovedinChapter3.Figure35Example2GuessthevalueofExample3TheHeavisidefunctionHisdefinedbyAstapproaches0fromtheleft,H(t)approaches0.Astapproaches0fromtheright,H(t)approaches1.ThereisnosinglenumberthatH(t)approachesastapproaches0.Figure46Example3TheHeavisidefunct2.2.2One-SidedLimitsDefinition2Letfbeafunctiondefinedonanopenintervaloftheform(a,c)forsomerealnumberc,andletLbearealnumber.Wesaythattheright-handlimitoff(x)asx

approachesafromtherightisL,andwrite

ifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxgreaterthana.72.2.2One-SidedLimits7Similarly,wegetdefinitionoftheright-handlimitoff(x)asx

approachesa.Letfbeafunctiondefinedonanopenintervaloftheform(c,a)forsomerealnumberc,andletLbearealnumber.Wesaythattheleft-handlimitoff(x)asxapproachesafromtheleftisL,andwriteifwecanmakethevaluesoff(x)arbitrarilyclosetoLbytakingxtobesufficientlyclosetoaandxlessthana.88Forexample,Example4Usethegraphofy=g(x)tofindthefollowinglimits,iftheyexist.SolutionThisgraphshowsthatFigure59Forexample,SolutionThisgraForinstance,since,thereforedoesnotexist.Example5Supposethat

(1)Findand(2)DiscussSolution(1),.(2)Because

≠,sodoesnotexist.10Forinstance,sinceExample6ShowthatSolutionRecallthatWehaveTherefore,11Example6Showthat112.2.3InfiniteLimitsDefinition3Letfbeafunctiondefinedonbothsidesofa,exceptpossiblyataitself.Thenmeansthatthevalueoff(x)canbemadearbitrarilylargebytakingxsufficientlyclosetoa,butnotequaltoa.Example6Findifitexists.SOLUTIONAsxbecomescloseto0,x2alsobecomescloseto0,and1/x2becomesverylarge.(Seethetableonthenextpage.)122.2.3InfiniteLimits12Itappearsfromthegraphofthefunctionf(x)showninFigurethatthevalueofthef(x)canbemadearbitrarilyxlargebytakingxcloseenoughto0.ThusFigure613ItappearsfromthegraphoftDefinition4Letfbeafunctiondefinedonbothsidesofa,exceptpossiblyataitself.Thenmeansthatthevalueoff(x)canbemadearbitrarilylargenegativebytakingxsufficientlyclosetoa,butnotequaltoa.AsanexamplewehaveSimilardefinitionscanbegivenfortheone-sidedinfinitelimits14Definition4LetfbeafunctExamplesofthesefourcasesaregiveninFigure7.Figure715ExamplesofthesefourcasesaDefinition5Thelinex=aiscalledaverticalasymptoteofthecurvey=f(x)ifatleastoneofthefollowingstatementsistrue:Forinstance,they-axisisaverticalasymptoteofthecurvey=1/x2because16Definition5Thelinex=aisExample7FindSolutionIfxiscloseto3butlargerthan3,thenthedenominatorx-3isapositivenumberand2xiscloseto6.Sothequotient2x/(x-3)isalargepositivenumber.Thus,weseethatLikewise,ifxiscloseto3butsmallerthan3,thenx-3isasmallnegativenumberbut2xisstillapositivenumber(closeto6).So2x/(x-3)isanumericallylargenegativenumber.Thus17Example7Find17Thelinex=3isaverticalasymptote.Example8Findtheverticalasymptotesoff(x)=tanx.SolutionBecausetanx=sinx/cosx,therearepotentialverticalasymptoteswherecosx=0.Infact,wehaveThisshowsthatthelineisaverticalasymptote.Similarreasoningshowsthatthelines,wherenisaninteger,areallverticalasymptotesoff(x)=tanx.18Thelinex=3isaverticalas2.3CalculatingLimitsUsingtheLimitLawsLimitLawsIflimf(x)andlimg(x)bothexist,then1lim[f(x)+g(x)]=limf(x)+limg(x),lim[f(x)-g(x)]=limf(x)-limg(x),2lim[f(x)·g(x)]=limf(x)·limg(x),3limc·f(x)=c·limf(x),foranynumberc,4lim[f(x)/g(x)]=limf(x)/limg(x),providedlimg(x)≠0.Example1FindSolution192.3CalculatingLimitsUsinExample2Iffisapolynomialfunctionandaisarealnumber,thenSolutionSincefisapolynomialfunction,wemayassumethatforrealnumbersbn,bn-1,…b0andsomepositiveintegern.Applyinglimitlawsforanumberoftimesyieldsthat20Example2IffisapolynoArationalfunction

fisaratiooftwopolynomials:wherePandQarepolynomials.Thedomainconsistsofallvaluesofxsuchthat.DirectSubstitutionPropertyIffisarationalfunctionandaisinthedomainoff,then21ArationalfunctionfisaratExample3Findeachlimit.Solution(1)(2)Since

the

limitlawscannotbeappliedtothequotientHowever,22Example3Findeachlimit.2Theorem1Ifwhenxisneara(exceptpossiblyata)andthelimitsoffandgbothexistasxapproachesa,thenTheorem2(TheSqueezeTheorem)

Supposef(x)≤h(x)≤g(x)foreveryxinanopenintervalcontaininga,exceptpossiblyata.(SandwichTheoremorPinchingTheorem)

23Theorem1IfExample4ShowthatSolutionFirstnotethatwecannotusebecausedoesnotexist.However,sinceWehaveandBytheSqueezeTheorem,weknowthatFigure124Example4ShowthatFigure1242.4ThePreciseDefinitionofaLimitDefinition1Letfbeafunctiondefinedonsomeopenintervalcontainingaexceptpossiblyataitself,andletLbearealnumber.Wesaythatthelimitoff(x)asxapproachesaisL,andwrite

ifforeverynumberε>0,thereisanumberδ>0suchthat

252.4ThePreciseDefinitionofGeometricinterpretationofthelimitFigure126GeometricinterpretationofthDefinition1canbestatedasfollows:1meansthatthedistancebetweenf(x)andLcanbemadearbitrarilysmallbytakingthedistancefromxtoasufficientlysmall(butnot0).2meansthatthevaluesoff(x)canbemadeascloseaswepleasetoandLbytakingxcloseenoughtoa(butnotequaltoa).3meansthatforevery(nomatterhowsmallis)wecanfindsuchthatifxliesintheopenintervaland,thenf(x)liesintheopeninterval27Definition1canbestatedasExample1UseDefinition1toprovethat(1)Wherecisarealnumber.(2)Solution(1)Letf(x)=c.Since

andsince0islessthananyε>0,itfollowsfromDefinition1thatf(x)hasthelimitcasxapproachesa.Thus(2)Letf(x)=x.Sinceitisobviousthatifchooseεasδ，thenweobtainwheneverItfollowsthat28Example1UseDefinition1toExample2ProvethatSolutionForeverynumberε>0,sincetomakeitissufficienttomakeHenceletδ=ε/2，thenweobtainthat Thisimpliesthat29Example2Provethat29Definition2Letfbeafunctiondefinedonanopenintervaloftheform(a,c)forsomerealnumberc,andletLbearealnumber.Wesaythattheright-handlimitoff(x)asx

approachesafromtherightisL,andwriteifforeverynumberε>0,thereisanumberδ>0suchthat3030Definition3Letfbeafunctiondefinedonanopenintervaloftheform(c,a)forsomerealnumberc,andletLbearealnumber.Wesaythattheleft-handlimitoff(x)asx

approachesafromtheleftisL,andwriteifforeverynumberε>0,thereisanumberδ>0suchthat

31Definition3LetfbeafunctExample3ProvethatSolution1.Guessingavaluefor

δ.Letεbeagivenpositivenumber.Wewanttofindanumberδsuchthatthatis,Thissuggeststhatweshouldchoose2.Showingthatthis

δworks.Givenε>0,letIf,thenThisshowsthat32Example3Provethat32Example4ProvethatSolution1.Guessingavaluefor

δ.Letε>0

begiven.Wehavetofindanumberδ>0suchthatthatis,NoticethatifwecanfindapositiveconstantCsuchthatthenandwecanmakebytaking33Example4Provethat33WecanfindsuchanumberCifwerestrictxtolieinsomeintervalcenteredat3.Infact,sinceweareinterestedonlyinvaluesofxthatarecloseto3,itisreasonabletoassumethatx

iswithinadistance1from3,thatis,ThensoThus,wehaveandsoC=7isasuitablechoicefortheconstant.Butnowtherearetworestrictionsonnamely34WecanfindsuchanumberCifTomakesurethatbothoftheseinequalitiesaresatisfied,wetakeδtobethesmallerofthetwonumbers1andε/7.Thenotationforthisis2.Showingthatthis

δworks.Givenε>0,letIf,thenWealsohavesoThisshowsthat35TomakesurethatbothofthesInfiniteLimitsDefinition4Letfbeafunctiondefinedonsomeopenintervalthatcontainsthenumbera,exceptpossiblyataitself.ThenmeansthatforeverypositivenumberMthereisapositivenumberδsuchthat36InfiniteLimits36Example5ProvethatSolution1.Guessingavalueforδ.Given

M>0,wewanttofindδ>0suchthatThatis,Thissuggeststhatweshouldtake2.Showingthatthisδworks.IfM>0isgiven,letIf,thenTherefore,byDefinition4,37Example5Provethat372.5ContinuityDefinition1Afunctionfiscontinuousatanumberaif382.5ContinuityDefinition1AThekindofdiscontinuityillustratedinpart(3)ofFigure1iscalledaremovablediscontinuitybecausewecouldremovethediscontinuitybyredefiningatc.Thediscontinuityinpart(2)iscalledajumpdiscontinuitybecausethefunction“jumps”fromonevaluetoanother.Iffapproaches+∞or-∞asxapproachescfromeitherside,as,forexample,inpart(1),wesaythatfhasan

infinitediscontinuityatc.Figure139ThekindofdiscontinuityIngeneral,ifafunctionfisnotcontinuousatc,thenithasaremovablediscontinuityatanumberciftheright-handandtheleft-handlimitsexistatcandareequal;ajumpdiscontinuityatcifthetwoone-sidedlimitsarenotequal.Example1Whereareeachofthefollowingfunctionsdiscontinuous?40Ingeneral,ifafunctiSolution(a)Noticethatf(2)isnotdefined,sofisdiscontinuousat2.(b)WehaveSofisdiscontinuousat0.(c)Heref(2)=1isdefinedandBut,sofisnotcontinuousat2.(d)Thegreatestintegerfunctionhasdiscontinuitiesatalloftheintegersbecausedoesnotexistifnisaninteger.?41Solution(a)Noticethatf(2)Figure2showsthegraphsofthefunctionsinExample1.Figure2(a)(c)(d)(b)。。．．。。。。。42Figure2showsthegraphsoftDefinition2Afunctionfiscontinuousfromtherightatanumberaifandfiscontinuousfromtheleftat

aifExample2Ateachintegern,thefunctioniscontinuousfromtherightbutdiscontinuousfromtheleftbecause

43Definition2AfunctionfisDefinition3Afunctionfiscontinuousonanintervalifitiscontinuousateverypointintheinterval.(Iffisdefinedonlyononesideofanendpointoftheinterval,weunderstandcontinuousattheendpointtomeancontinuousfromtherightorcontinuousfromtheleft.)Example3Showthatthefunctioniscontinuousontheinterval[-1,1].Solution44Definition3AfunctionfThus,byDefinition1,fiscontinuousataif-1<a<1.Similarcalculationsshowthatsofiscontinuousfromtherightat-1andcontinuousfromtheleftat1.Therefore,accordingtoDefinition3,fiscontinuouson[-1,1].ThegraphoffissketchedinFigure3.Figure345Thus,byDefinition1,fiscoTheorem4Iffandgarecontinuousataandcisaconstant,thenthefollowingfunctionsarealsocontinuousata:ItfollowsfromTheorem4andDefinition3thatiffandg

arecontinuousonaninterval,thensoarethefunctionsf+g,f-g,cf,fg,and(ifgisnever0)f/g.ThefollowingtheoremwasstatedastheDirectSubstitutionProperty.46Theorem4IffandgarecontTheorem5(a)Anypolynomialiscontinuouseverywhere;thatis,itiscontinuouson(b)Anyrationalfunctioniscontinuouswhereveritisdefined;thatis,itiscontinuousonitsdomain.Seepages19-20.Theorem6Thefollowingtypesoffunctionsarecontinuousateverypointintheirdomains:

polynomialsrationalfunctionsrootfunctionstrigonometricfunctionsinversetrigonometricfunctionsexponentialfunctionslogarithmicfunctions47Theorem5(a)AnypolynomialExample5Whereisthefunctioncontinuous?SolutionWeknowfromTheorem6thatthefunctiony=lnxiscontinuousforx>0andy=arctanxiscontinuousonR.Thus,y=lnx+arctanxiscontinuouson(0,∞).Thedenominator,y=x2-1,isapolynomial,soitiscontinuouseverywhere.Therefore,fiscontinuousatallpositivenumbersxexceptwherex2-1=0.Sofiscontinuousontheintervals(0,1)and(1,∞).48Example5WhereisthefunctioTheorem7IffiscontinuousatbandthenInotherwords,Example6EvaluateSolutionSincearcsinisacontinuousfunction,wehave49Theorem7IffiscontinuousTheorem8Ifgiscontinuousataandfiscontinuousatg(a),thenthecompositefunctionf(g(x))iscontinuousata.Example7Whereisthefunctioncontinuous?SolutionWeknowfromtheTheorem6thatf(x)=lnxiscontinuousandg(x)=1+cosxiscontinuous(becausebothy=1andy=cosxarecontinuous).Therefore,byTheorem8,F(x)=f(g(x))iscontinuouswhereveritisdefined.Nowln(1+cosx)isdefinedwhen1+cosx>0.Soitisundefinedwhenx=±π

,±3π

,….Thus,Fhasdiscontinuitieswhenxisanoddmultipleofπandiscontinuousontheintervalsbetweenthesevalues(seeFigure4).50Theorem8IfgiscontinuousFigure4Theorem9Iff(x)isone-to-onecontinuousfunctiondefinedoninterval,thenitsinversefunctionf-1isalsocontinuous.51Figure4Theorem9Iff(x)isTheIntermediateValueTheorem

10Supposethatfiscontinuousontheclosedinterval[a,b]andletNbeanynumberbetweenf(a)andf(b),wheref(a)≠f(b).Thenthereisatleastonenumbercin(a,b)suchthatf(c)=N.SomepropertiesofcontinuousfunctionsonaclosedintervalTheorem11Ifafunctionfiscontinuousonaclosedinterval[a,b],thenthereexistsapositivenumberMsuchthat

│f(x)│<M

foreveryxin[a,b].Theorem12Iffiscontinuousonaclosedinterval[a,b],thenfattainsamaximumvalue

f(ξ)andaminimumvalue

f(η)atsomenumbersξandηin[a,b].52TheIntermediateValueTheoremTheIntermediateValueTheoremstatesthatacontinuousfunctiontakesoneveryintermediatevaluebetweenthefunctionvaluesf(a)andf(b).ItisillustratedbyFigure5.NotethatthevalueNcanbetakenononceormorethanonce.Figure553TheIntermediateValueTheoremExample8Showthatthereisarootoftheequationbetween1and2.SolutionLetWearelookingforasolutionofthegivenequation,thatis,anumbercbetween1and2suchthatf(c)=0.Therefore,wetakea=1,b=2,andN=0inTheorem10.Wehave

f(1)=4-6+3-2=-1<0andf(2)=32-24+6-2=12>0Thus,f(1)<0<f(2);thatis,N=0isanumberbetweenf(1)andf(2).54Example8ShowthatthereisNowfiscontinuoussinceitisapolynomial,sotheIVTsaysthereisanumbercbetween1and2suchthatf(c)=0.Inotherwords,theequationhasatleastonerootcintheinterval(1,2).WecanusecomputersoftwareorgraphingcalculatortoillustratetheuseoftheIntermediateValueTheoreminExample8(seeFigure6).Figure655Nowfiscontinuoussinceiti2.6LimitsatInfinity;HorizontalAsymptotesLet’sbeginbyinvestigatingthebehaviorofthefunctionfdefinedbyasxbecomeslarge.Thetableattheleftgivesvaluesofthisfunctioncorrecttosixdecimalplaces,andthegraphoffhasdrawninFigure1.

x

f(x)Figure1562.6LimitsatInfinity;HorizDefinition1LetfbeafunctiondefinedonsomeintervalThenmeansthatthevaluesoff(x)canbemadearbitrarilyclosetoLbytakingxsufficientlylarge.Anothernotationforisisreadas“asx

approachesinfinity,thelimitoff(x)isL”

becomesinfinity

increaseswithoutbound

57Definition1LetfbeafunctiGeometricillustrationsofDefinition1areshowninFigure2.Noticethattherearemanywaysforthegraphofftoapproachtheliney=L(whichiscalledahorizontalasymptote)aswelooktothefarrightofeachgraph.Figure258GeometricillustrationsofDefinition2LetfbeafunctiondefinedonsomeintervalThenmeansthatthevaluesoff(x)canbemadearbitrarilyclosetoLbytakingxsufficientlylargenegative.Definition3Theliney=Liscalledahorizontalasymptoteofthecurvey=f(x)ifeitherForinstance,thecurveillustratedinFigure1hastheliney=1asahorizontalasymptotebecause59Definition2LetfbeafunctiAnotherexampleofacurvewithtwohorizontalasymptotesisy=arctanx.(SeeFigure3.)Sincesobothofthelinesandarehorizontalasymptotes.Figure360AnotherexampleofacurvewiExample1ComputeSolutionObservethatwhenxislarge,1/xissmall.Bytakingxlargeenough,wecanmake1/xascloseto0.Therefore,accordingtoDefinition1,wehaveSimilarreasoningshowsthatwhenxislargenegative,1/xissmallnegative,sowealsohaveIfr>0isarationalnumber,thenIfr>0isarationalnumbersuchthatxrisdefinedforallx,then61Example1ComputeIfr>0isExample2FindSolutionAsimilarcalculationshowsthatthelimitasisalso3/5.62Example2FindAsimilarcalcExample3FindthehorizontalandverticalasymptotesofthegraphofthefunctionSolutionDividingbothnumeratoranddenominatorbyxandusingthepropertiesoflimits,wehaveTherefore,thelineisahorizontalasymptoteofthegraphoff.63Example3FindthehorizontalIncomputingthelimitas,wemustrememberthatforx<0,wehaveThus,thelineisalsoahorizontalasymptote.Averticalasymptoteislikelytooccurwhenthedenominator,3x-5,is0,thatis,whenx=5/3.64IncomputingthelimitasSinceWeknowthattheverticalasymptoteisx=5/3.AllthreeasymptotesaredrawninFigure4.

Figure465SinceFigure465Example4FindSolutionAsxincreases,thevaluesofsinxoscillatebetween1and-1infinitelyoftenandsotheydon’tapproachanydefinitenumber.Thus,doesnotexist.66Example4FindAsxincreInfiniteLimitsatInfinityThenotationisusedtoindicatethatthevaluesoff(x)becomelargeasxbecomeslarge.Similarmeaningsareattachedtothefollowingsymbols:Forinstance,67InfiniteLimitsatInfinityForPreciseDefinitionsDefinition4Letfbeafunctiondefinedonsomeinterval.ThenmeansthatforeverythereisacorrespondingnumberMsuchthatDefinition5Letfbeafunctiondefinedonaninfinite

interval(-∞,c）forarealnumberc,andletLbeareal

number.Wesaythatthelimitoff(x)asxapproaches-∞isL,andwrite

ifforeveryε>0,thereisanumberm<0suchthat

68PreciseDefinitions68Example5Provey

y=L+εSolutionLet.Ly=L-ε

0Mx

Foreveryε>0,wemayassumethat0<ε<1,since

69Example5ProveDefinition1Thederivativeofafunctionfatanumbera,denotedby,isifthislimitexists.2.8DerivativesLetx=a+h,thenh=x–aandhapproaches0ifandonlyifxapproachesa.Therefore,anequivalentwayofstatingthedefinitionofthederivativeis

Thisistheslopeofthetangenttoacurvewithequationy=f(x)atthepointx=a.70Definition1ThederivativeofExample1Findthederivativeofthefunctionatthenumbera.SolutionFromtheDefinition1,wehave71Example1FindthederivativeInterpretationoftheDerivativeastheSlopeofaTangentThetangentlinetoy=f(x)atP(a,f(a))isthelinethrough(a,f(a))whoseslopeequalto,thederivativeoffata.Ifweusethepoint-slopeformoftheequationofaline,wecanwriteanequationofthetangentlinetothecurvey=f(x)atpoint(a,f(a)):

Figure172InterpretationoftheDerivatiExample2Findanequationofthetangentlinetotheparabolaatthepoint(3,-6).SolutionFromExample1weknowthattheslopeofthetangentlineat(3,-6)isThus,anequationofthetangentline,showninFigure2,isFigure273Example2Findanequationof2.9TheDerivativeasaFunctionIntheprecedingsectionweconsideredthederivativeofafunctionfatafixednumbera:Herewechangeourpointofviewandletthenumberavary.Ifwereplaceabyavariablex,weobtainGivenanynumberxforwhichthislimitexists,weassigntoxthenumberSowecanregardf’asanewfunction,calledthederivativeoff.742.9TheDerivativeasaFunctExample1(a)Iffindaformulafor(b)Illustratebycomparingthegraphsoffandf’.Solution(a)75Example1(a)If(b)Weuseacomputersoftwaretographfandf’inFigure1.Figure176(b)Weuseacomputer