微积分基础知识大一课程经典讲解week

PAGEPAGE7Chapter2LimitsandContinuity(continued)1.AssumethatisahigherorderinfinitesimalofExampleareofthesameorder.5xand3xareofthesameorder.Inparticular,areequivalent(等价)infinitesimals,denotedby2.Theorem3.SomeExamplesofEquivalentInfinitesimalsProofExample1Chapter2LimitsandContinuity(continued)1.ContinuityataSingleNumberDefinition1LetIfthenwesaythefunctionfiscontinuousata.Definition2Afunctionfiscontinuousataifexists(bothexistandtheyareequal)2)f(a)isdefined.2.ThreeTypesofDiscontinuities1)Atleastoneofdoesnotexist.infinitediscontinuityBothdonotexist.2)Bothexist,butjumpdiscontinuityExamplethegreatestintegerfunctionorfisnotdefinedata.removablediscontinuity(wecanremovethediscontinuitybyredefiningfatjustthesinglenumber.)Examples3.One-SidedContinuityDefinitionAfunctionfiscontinuousfromtherightataifandfiscontinuousfromtheleftataif4.ContinuityonanIntervalfiscontinuouson(a,

b):iffiscontinuousateverynumberin(a,

b)[a,

b):iffiscontinuousateverynumberin(a,

b)and(a,

b]:iffiscontinuousateverynumberin(a,

b)and[a,

b]:iffiscontinuousateverynumberin(a,

b)and5.Theorem1)Abasicelementaryfunctioniscontinuousonitsdomain.2)Iffandgarecontinuousata,thenthefollowingfunctionsarealsocontinuousata:3)Ifgiscontinuousataandfiscontinuousatg(a),thenisalsocontinuousata.4)Anelementaryfunctioniscontinuousonitsdomain.SolutioniscontinuousonR.iscontinuouson6.Howtoshowwhetherapiecewisedefinedfunctioniscontinuousornot.Example1DeterminewhetherthefunctioniscontinuousonR.SolutionDeterminewhetherthefunctioniscontinuousatx=0,x=1.Solutionisdiscontinuousatx=0.iscontinuousatx=1.Exercise7.TheIntermediateValueTheorem(介值定理)Supposef(x)iscontinuouson[a,b]andletNbeanynumberbetweenf(a)andf(b),wherethenthereexistanumbercin(a,b)suchthatf(c)=N.ExampleShowthatthereisarootoftheequationbetween1and2.SolutionLetf(x)iscontinuouson[1,2]sinceit’sanelementaryfunction.Thereisarootoftheequationbetween1and2.Chapter3Derivatives1.Velocities(速度)Exampleaveragevelocity:instantaneous(即时的)velocity:Ingereral,theinstantaneousvelocityattimea:2.DefinitionThederivativeofafunctionfataisiftheselimitsexist.NoteExampleLetFind3.DifferentiationDefinitionAfunctionfisdifferentiable(可微的/可导的)ataifexists.4.InterpretationoftheDerivativeastheSlopeofaTangenttheslopeofthetangentlinetoy=f(x)at(a,f(a)).ExampleFindanequationofthetangentlinetoat(1,

1).SolutionAnequationofthetangentlineisy-1=2(x-1)ory=2x-1.5.One-SidedDerivativesfisdifferentiableata6.DifferentiableonanIntervalfisdifferentiableon(a,

b):iffisdifferentiableateverynumberin(a,

b).[a,

b):iffisdifferentiableateverynumberin(a,

b)andrightdifferentiableata.(a,

b]:iffisdifferentiableateverynumberin(a,

b)andleftdifferentiableatb.[a,

b]:iffisdifferentiableateverynumberin(a,

b),rightdifferentiableataandleftdifferentiableatb.7.TheoremIffisdifferentiableata,t