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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)).DefinitionoftheDerivativeofaFunctionThe
2、Derivativeoffatxisgivenbyf'x=lim∆x→0fc+∆x-f(c)∆xprovidedthelimitexists.Forallxforwhichthislimitexists,f’isafunctionofx.*ThePowerRule*TheProductRule*ddxsinx=cosx*ddxcosx=-sinx*TheChainRule☺ImplicitDifferentiation(takethederivativeonbothsides;derivativeofyisy*y’)Chapter3ApplicationsofDifferentiati
3、on*Extremaandthefirstderivativetest(minimum:−→+,maximum:+→−,+&−arethesignoff’(x))*DefinitionofaCriticalNumberCOPYRIGHTSRESERVEDBYOWEN5Letfbedefinedatc.Iff’(c)=0ORIFFISNOTDIFFERENTIABLEATC,thencisacriticalnumberoff.*Rolle’sTheoremIffisdifferentiableontheopeninterval(a,b)andf(a)=f(b),thenthereisat
4、leastonenumbercin(a,b)suchthatf’(c)=0.*TheMeanValueTheoremIffiscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),thenthereexistsanumbercin(a,b)suchthatf’(c)=fb-f(a)b-a.*Increasinganddecreasingintervaloffunctions(takethefirstderivative)*Concavity(ontheintervalwhichf’’>0,co
5、ncaveup)*SecondDerivativeTestLetfbeafunctionsuchthatf’(c)=0andthesecondderivativeoffexistsonanopenintervalcontainingc.1.Iff’’(c)>0,thenf(c)isaminimum2.Iff’’(c)<0,thenf(c)isamaximum*PointsofInflection(takesecondderivativeandsetitequalto0,solvetheequationtogetxandplugxvalueinoriginalfunction)*Asym
6、ptotes(horizontalandvertical)*LimitsatInfinity*CurveSketching(takefirstandsecondderivative,makesureallthecharacteristicsofafunctionareclear)♫OptimizationProblems*Newton’sMethod(usedtoapproximatethezerosofafunction,whichistediousandstupid,DONOTHAVETOKNOWIFUDONOTWANTTOSCORE5)Chapter4&5Integration*
7、BeabletosolveadifferentialequationCOPYRIGHTSRESERVEDBYOWEN5*BasicIntegrationRules1)undu=un+1n+1+C,n≠-12)sinudu=-cosu+C3)cosudu=sinu+C4)1udu=lnu*Integralofafunctionistheareaunderthecurve*RiemannSum(divideintervalintoalotofsub
