上海财经大学英语高数课件03

Chapter3Themeanvaluetheoremandcurvesketching3.3MONOTONICFUNCTIONSANDTHEFIRSTDERIVATIVETESTInsketchingthegraphofafunctionitisveryusefultoknowwhereitrisesandwhereitfalls.ThegraphshowninFigure1risesfromAtoB,fallsfromBtoC,andrisesagainfromCtoD.Figure121xx)()(21xfxf(1)DEFINITIONAfunctionfiscalledincreasingonanintervalIifwheneverinI.AfunctionthatisincreasingordecreasingonIiscalledmonotoniconI.TESTFORMONOTONICFUNCTIONSSupposefiscontinuouson[a,b]anddifferentiableon(a,b).(a)Iff'(x)0forallxin(a,b),thenfisincreasingon[a,b].(b)Iff'(x)0forallxin(a,b),thenfisdecreasingon[a,b].PROOF(a)Letx1andx2beanytwonumbersin[a,b]withx1x2.Thenfiscontinuouson[x1,x2]anddifferentiableon(x1,x2),sobytheMeanValueTheoremthereisanumbercbetweenx1andx2suchthat(2)Nowf'(c)0byassumptionandx2-x10becausex1x2.ThustherightsideofEquation2ispositive,andsoThisshowsthatfisincreasingon[a,b].Itissimilartoprove(b).))(()()(1212xxcfxfxf).()(or21xfxf0)()(12xfxfExample1Findwherethefunctionf(x)=3x4-4x3-12x2+5isincreasingandwhereitisdecreasing.(c)Iff'doesnotchangesignatc(thatisf'ispositiveonbothsidesofcornegativeonbothsides),thenfhasnolocalextremumatc.(b)Iff'changesfromnegativetopositiveatcthenfhasalocalminimumatc.(a)Iff'changesfrompositivetonegativeatc,thenfhasalocalmaximumatc.THEFIRSTDERIVATIVETESTSupposethatcisacriticalnumberofacontinuousfunctionf.Example3Findthelocalandabsoluteextremevaluesofthefunctionf(x)=x3(x-2)2,-13.Sketchitsgraph.f(6/5)=1.20592isalocalmaximum;f(2)=0isalocalminimum;absolutemaximumvalueisf(3)=27;absoluteminimumvalueisf(-1)=-9.ThesketchedinFigure6.x3.4CONCAVITYANDPOINTSOFINFLECTIONFigure1Figure2(a)Concaveupward(b)Concavedownward(1)DEFINITIONIfthegraphoffliesaboveallofitstangentsonanintervalI,thenitiscalledconcaveupwardonI.Ifthegraphoffliesbelowallofitstangentsonf,itiscalledconcavedownwardonI.Figure3showsthegraphofafunctionthatisconcaveupward(abbreviatedCU)ontheintervals(b,c),(d,e),and(e,p)andconcavedownward(CD)ontheintervals(a,b),(c,d),and(p,q)Theequationofthistangentiswemustshowthatwhenever(SeeFigure4.)THETESTFORCONCAVITYSupposefistwicedifferentiableonanintervalI.(a)Iff(x)0forallxinI,thenthegraphoffisconcaveupwardonI.(b)Iff(x)0forallxinI,thenthegraphoffisconcavedownwardonI.))(()(axafafy))(()()(axafafxf)(axIxFigure4Proofof(a)Firstletustakethecasewherexa.ApplyingtheMeanValueTheoremtofontheinterval[a,.x],wegetanumberc,withacx,suchthat(2)f(x)-f(a)==f’(c)(x-a)Sincef0onfweknowfromtheTestforMonotonicFunctionsthatf'isincreasingonI.Thus,sinceac,wehavef'(a)f’(c)andso,multiplyingthisinequalitybythepositivenumberx-a,weget(3)Nowweaddf(a)tobothsidesofthisequality:f(a)+.f'(a)(x-a)f(a)+f'(c)(x-a)ButfromEquation2wehavef(x)=f(a)+f'(c)(x-a).Sothisinequalitybecomes(4)f(x)f(a)+f'(a)(x-a)whichiswhatwewantedtoprove..Forthecasewherexa,wehavef'(c)f'(a),butmultiplicationbythenegativenumberx-areversestheinequality,soweget(3)and(4)asbefore.))(())((axcfaxaf(2)DEFINITIONApointPonacurveiscalledapointofinflectionifthecurvechangesfromconcaveupwardtoconcavedownwardorfromconcavedownwardtoconcaveupwardatP.Example1Determinewherethecurvey=x3-3x+1isconcaveupwardandwhereitisconcavedownward.Findtheinflectionpointsandsketchthecurve.THESECONDDERIVATIVETESTSupposefiscontinuousonanopenintervalthatcontainsc.(a)Iff'(c)=0andf(c)0,thenfhasalocalminimumatc.(b)Iff'(c)=0andf(c)0,thenfhasalocalmaximumatc.Sincef'(3)=0andf(3)0,f(3)=-27isalocalminimum.Thepoint(0,0)isaninflectionpointsincethecurvechangesfromconcaveupwardtoconcavedownwardthere.Also(2,-16)sincethecurvechangesfromconcavedownwardtoconcaveupwardthere.Example2Discussthecurvey=x4-4x3withrespecttoconcavity,pointsofinflection,andlocalextrema.Usethisinformationtosketchthecurve.3.5Limitsatinfinity,horizontalasymptotesFigure1(2)DEFINITIONLetfbeafunctiondefinedonsomeinterval(,a),Thenmeansthatthevaluesoff(x)canbemadearbitrarilyclosetoLbytakingxsufficientlylargenegative.(1)DEFINITIONLetfbeafunctiondefinedonsomeinterval,Thenmeansthatthevaluesoff(x)canbemadearbitrarilyclosetoLbytakingxsufficientlylarge.Lxfx)(limLxfx)(lim(3)DEFINITIONTheliney=Liscalledahorizontalasymptoteofthecurvey=f(x)ifeitherlimf(x)=Lorlimf(x)=L(4)THEOREMIfr0isarationalnumber,thenIfr0isarationalnumbersuchthatisdefinedforallx,thenxx01limrxx01limrxx)(limxfx)(limxfx)(limxfx)(limxfxInfinitelimitsatinfinityThenotationisusedtoindicatethatthevaluesoff(x)becomelargeasxbecomeslarge.Similarmeaningsareattachedtothefollowingsymbols:Example1Example2(5)DEFINITIONLetfbeafunctiondefinedonsomeinterval.ThenmeansthatforeverythereisacorrespondingnumberNsuchthatwheneverxN.Lxfx)(limLxf)(0),(a(6)DEFINITIONLetfbeafunctiondefinedonsomeinterval.ThenmeansthatforeverythereisacorrespondingnumberNsuchthatwheneverxN.),(aLxfx)(lim0Lxf)((7)DEFINITIONLetfbeafunctiondefinedonsomeinterval.ThenmeansthatforeverypositiveMthereisacorrespondingnumberNsuchthatwheneverxN.),(a)(limxfxMxf)(3.6Curvesketching(1)Thissectionistodiscusshowtosketchthegraphofacurve.Inhighschoolwelearnttoplotpointsandjointhepoints