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1、APCALCULUSABREVIEWChapter2DifferentiationDefinitionofTangentLinewithSlopmIffisdefinedonanopenintervalcontainingc,andifthelimitexists,thenthelinepassingthrough(c,f(c))withslopemisthetangentlinetothegraphoffatthepoint(c,f(c)).DefinitionoftheDerivativeofaFunctio
2、nTheDerivativeoffatxisgivenbyprovidedthelimitexists.Forallxforwhichthislimitexists,f’isafunctionofx.*ThePowerRule*TheProductRule***TheChainRule☺ImplicitDifferentiation(takethederivativeonbothsides;derivativeofyisy*y’)Chapter3ApplicationsofDifferentiation*Extr
3、emaandthefirstderivativetest(minimum:−→+,maximum:+→−,+&−arethesignoff’(x))*DefinitionofaCriticalNumberLetfbedefinedatc.Iff’(c)=0ORIFFISNOTDIFFERENTIABLEATC,thencisacriticalnumberoff.*Rolle’sTheoremIffisdifferentiableontheopeninterval(a,b)andf(a)=f(b),thenther
4、eisatleastonenumbercin(a,b)suchthatf’(c)=0.*TheMeanValueTheoremIffiscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),thenthereexistsanumbercin(a,b)suchthatf’(c)=.*Increasinganddecreasingintervaloffunctions(takethefirstderivative)*Conc
5、avity(ontheintervalwhichf’’>0,concaveup)*SecondDerivativeTestLetfbeafunctionsuchthatf’(c)=0andthesecondderivativeoffexistsonanopenintervalcontainingc.1.Iff’’(c)>0,thenf(c)isaminimum2.Iff’’(c)<0,thenf(c)isamaximum*PointsofInflection(takesecondderivativeandseti
6、tequalto0,solvetheequationtogetxandplugxvalueinoriginalfunction)*Asymptotes(horizontalandvertical)*LimitsatInfinity*CurveSketching(takefirstandsecondderivative,makesureallthecharacteristicsofafunctionareclear)♫OptimizationProblems*Newton’sMethod(usedtoapproxi
7、matethezerosofafunction,whichistediousandstupid,DONOTHAVETOKNOWIFUDONOTWANTTOSCORE5)Chapter4&5Integration*Beabletosolveadifferentialequation*BasicIntegrationRules1)2)3)4)*Integralofafunctionistheareaunderthecurve*RiemannSum(divideintervalintoalotofsub-interva
8、ls,calculatetheareaforeachsub-intervalandsummationistheintegral).*Definiteintegral*TheFundamentalTheoremofCalculusIfafunctionfiscontinuousontheclosedinterval[a,b]andFisananti-derivativeof
