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1、APCALCULUSABREVIEWChapter2DifferentiationDefinitionofTangentLinewithSlopmIffisdefinedonanopenintervalcontainingc,andifthelimitlim∆x→0∆y∆x=lim∆x→0fc+∆x-f(c)∆x=mexists,thenthelinepassingthrough(c,f(c))withslopemisthetangentlinetothegraphoffatthepoint(c,f(c)).DefinitionoftheDeri
2、vativeofaFunctionTheDerivativeoffatxisgivenbyf'x=lim∆x→0fc+∆x-f(c)∆xprovidedthelimitexists.Forallxforwhichthislimitexists,f’isafunctionofx.*ThePowerRule*TheProductRule*ddxsinx=cosx*ddxcosx=-sinx*TheChainRule☺ImplicitDifferentiation(takethederivativeonbothsides;derivativeofyis
3、y*y’)Chapter3ApplicationsofDifferentiation*Extremaandthefirstderivativetest(minimum:−→+,maximum:+→−,+&−arethesignoff’(x))*DefinitionofaCriticalNumberCOPYRIGHTSRESERVEDBYOWEN5Letfbedefinedatc.Iff’(c)=0ORIFFISNOTDIFFERENTIABLEATC,thencisacriticalnumberoff.*Rolle’sTheoremIffisdi
4、fferentiableontheopeninterval(a,b)andf(a)=f(b),thenthereisatleastonenumbercin(a,b)suchthatf’(c)=0.*TheMeanValueTheoremIffiscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),thenthereexistsanumbercin(a,b)suchthatf’(c)=fb-f(a)b-a.*Increasinganddecreasing
5、intervaloffunctions(takethefirstderivative)*Concavity(ontheintervalwhichf’’>0,concaveup)*SecondDerivativeTestLetfbeafunctionsuchthatf’(c)=0andthesecondderivativeoffexistsonanopenintervalcontainingc.1.Iff’’(c)>0,thenf(c)isaminimum2.Iff’’(c)<0,thenf(c)isamaximum*PointsofInflect
6、ion(takesecondderivativeandsetitequalto0,solvetheequationtogetxandplugxvalueinoriginalfunction)*Asymptotes(horizontalandvertical)*LimitsatInfinity*CurveSketching(takefirstandsecondderivative,makesureallthecharacteristicsofafunctionareclear)♫OptimizationProblems*Newton’sMethod
7、(usedtoapproximatethezerosofafunction,whichistediousandstupid,DONOTHAVETOKNOWIFUDONOTWANTTOSCORE5)Chapter4&5Integration*BeabletosolveadifferentialequationCOPYRIGHTSRESERVEDBYOWEN5*BasicIntegrationRules1)undu=un+1n+1+C,n≠-12)sinudu=-cosu+C3)cosudu=sinu+C4)1udu=lnu*Integralofaf
8、unctionistheareaunderthecurve*RiemannSum(divideintervalintoalotofsub
