ap前兩次課及講義

Chapter3TheDefinitionoftheDerivativeContentsDerivingtheFormula1TheSlopeofACurve2DefinitionofDerivative

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Differentiability4Slopeoftheline:determinedbytwopoints§3.1DerivingtheFormulaASimplestContinuousFunction:AlineMstandsforslope（斜率）AlsoHavenothingwiththesubtractionorderSlopeoftheline:determinedbytwopoints§3.1DerivingtheFormulaASimplestContinuousFunction:AlineLet:DifferenceQuotient(差商)SlopeoftheSecantline（割線）§3.2TheSlopeofACurveSlopeofthetangentLine（切線）Theaveragerateofchangeofthefunctiony:（函數(shù)平均變化率）§3.2TheSlopeofACurveDenotedby:Thederivativeofafunctionf(x)isanotherfunctionwhosevalueatanynumberis:Def:Ifthislimitdoesexist,wesaythatf(x)isdifferentiable(可微)at.Findingaderivativeiscalleddifferentiation(微分).providedthatthislimitexistsandisnot.§3.3DefinitionofDerivativeⅠDefinitionofDerivative（導(dǎo)數(shù)定義）

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DefinitionofDerivativeLetThatis:Someotherforms:LetThen§3.3DefinitionofDerivativeAverageandInstantaneousVelocity(平均速度和瞬時(shí)速度)Fallingbodyproblem(自由落體)AnExample:ApplicationinPhysics§3.3DefinitionofDerivativeS=gt2/21.Ifthelimitdoesnotexist,thenwesaythatf(x)isnot

differentiable

at.Notes:2.Ifthelimitis,thenwesaythatthederivativeoff(x)atis(butnotdifferentiable).expressestheaveragechangerateofafunction,andexpressestheinstantaneouschangerateat.§3.3DefinitionofDerivative4.One-Sidedderivative(單側(cè)導(dǎo)數(shù))left-handderivativeright-handderivative§3.3DefinitionofDerivativeTheoremA:f(x)isdifferentiableatifandonlyif:(當(dāng)且僅當(dāng))IIIDifferentiationbyDefinitionSteps:1.Findtheincrements:2.Findthequotient:3.Takethelimit:§3.3DefinitionofDerivativeExamples:5.LetFind2.Findthederivativeofatat.Findthederivativeof3.Findand4.Findthederivativeof§3.3DefinitionofDerivative1.Whatisaveragerateofthefunctionfgivenbyf(x)=x4-5xontheclosedinterval[0,3]?A)8.5B)8.7C)22D)33E)66Solution:6.DeterminewhetherisdifferentiableatThatis,is

not

differentiableatSo,§3.3DefinitionofDerivativeIVTheGeometricInterpretationofDerivative（導(dǎo)數(shù)的幾何意義）expressestheslopeofthetangentlineofatthatis:isobliquity).So,theequationofthetangentlineis:Andtheequationofthenormallineis:

(法線)§3.3DefinitionofDerivativeWhenis90degree,thederivativeexists?7.FindthetangentlineandthenormallineofhyperbolaExample:atpointThetangentlineis:Solution:Thenormallineis:§3.3DefinitionofDerivativeOneoftheimportantrequirementsforthedifferentiabilityofafunctionisthatthefunctionbecontinuous.§3.4DifferentiabilityButifafunctioniscontinuousatapoint,thefunctionis

differentiable?AsharpcornerItcandrawtwotangentlineat

。Theslopesoftwotangentlinesisnotequal!!Sothefunctionisnotdifferentiablehere.

TheoremB:

DifferentiabilityImpliesContinuityProof:TheRelationbetweenDifferentiability(可微性)andContinuity(連續(xù)性)isdifferentiableatiscontinuousatNotice:

Theconverseofthistheoremisfalse.§3.4DifferentiabilityNotice:Theconverseofthistheoremisfalse.Sometypicalinstances:01★Thederivativeofatis011/π－1/π★isnotdifferentiableatThederivativeofatdoesnotexist.★§3.4DifferentiabilityExample:Solution:isbounded,iscontinuousatbutdoesnotexist,8.LetdiscussitscontinuityanddifferentiabilityatisnotdifferentiableatSo,§3.4DifferentiabilityExample:Solution:9.Letfindwhenwhenwhen§3.4DifferentiabilitySummery

1.Essentialofderivative:thelimitofthequotientofincrement,ortheinstantaneouschangerate.3.Thegeometricinterpretationofderivative:

theslopeofthetangentline.4.Adifferentiablefunctionisacontinuousone,butacontinuousfunctionisnotalwaysadifferentiableone.

（可導(dǎo)一定連續(xù)，連續(xù)不一定可導(dǎo)）§3.4DifferentiabilityIDerivingtheformulaIIDefinitionof