APCalculusABreviewAP微積分復習提綱（共5頁）本頁僅作為內頁可以根據(jù)需求調整合適字體及大小檔封面，使用時請直接刪除即可APCALCULUSABREVIEWChapter2DifferentiationDefinitionofTangentLinewithSlopmIffisdefinedonanopenintervalcontainingc,andifthelimitf(c+Ax).ffc)Ax=mexists,thenthelinepassingthrough(c,/(c))withslopemisthetangentlinetothegraphof/atthepoint(c,/(c)).DefinitionoftheDerivativeofaFunctionTheDerivativeof/atxisgivenbyf(x)=limAx一0f(c+Ax)-f(c>Akprovidedthelimitexists.Forallxforwhichthislimitexists,fisafunctionofx.*ThePowerRule*TheProductRule*日I本Isinx*壯I8sX-sinx*TheChainRule?ImplicitDifferentiation(takethederivativeonbothsides;derivativeofyisy*y')Chapter3ApplicationsofDifferentiation*Extremaandthefirstderivativetest(minimum:一—+,maximum:+-一，+&-arethesignoff(x))*DefinitionofaCriticalNumberLet/bedefinedatc.If/(c)=0ORIFFISNOTDIFFERENTIABLEATC,thencisacriticalnumberoff.*Rolle,sTheoremIf/isdifferentiableontheopeninterval(a,b)and/(a)=/(b),thenthereisatleastonenumbercin(a,b)suchthat/(c)=0.*TheMeanValueTheoremIf/iscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),thenthereexistsanumbercin(a,b)suchthat/(c)=f(b)-f⑸.b-a"Increasinganddecreasingintervaloffunctions(takethefirstderivative)*Concavity(ontheintervalwhichf>0,concaveup)*SecondDerivativeTestLetfbeafunctionsuchthatf’(c)=0andthesecondderivativeoffexistsonanopenintervalcontainingc.Iff’’(c)>0,thenf(c)isaminimumIff’’(c)<0,thenf(c)isamaximum*PointsofInflection(takesecondderivativeandsetitequalto0,solvetheequationtogetxandplugxvalueinoriginalfunction)Asymptotes(horizontalandvertical)LimitsatInfinityCurveSketching(takefirstandsecondderivative,makesureallthecharacteristicsofafunctionareclear)「OptimizationProblemsChapter4&5Integration*Beabletosolveadifferentialequation*BasicIntegrationRules此+11)/典11=i十C,nA12)/&出udu=-cosu+C3)Joosudu=sinu+C4)J，du=Inu*lntegralofafunctionistheareaunderthecurve*RiemannSum(divideintervalintoalotofsub-intervals,calculatetheareaforeachsub-intervalandsummationistheintegral).*Definiteintegral*TheFundamentalTheoremofCalculusIfafunction/iscontinuousontheclosedinterval[a,b]andFisanantiderivativeof/ontheinterval[a,b],thenJ：f(x)dx=F(b)-F(a)-*DefinitionoftheAverageValueofaFunctiononanIntervalIf/isintegrableontheclosedinterval[a,b],thentheaveragevalueof/ontheintervalis*ThesecondfundamentaltheoremofcalculusIf/iscontinuousonanopeninternal/containinga,then,foreveryxintheinterval,i睜如dt]=f(x).*lntegrationbySubstitution*lntegrationofEvenandOddFunctionsIf/isanevenfunction,then%x)dx=2J^f(x)dx-If/isanoddfunction,thenfbf(x)dx=0.*Inversefunctions"=f(x),switchyandx,solveforx)*TheDerivativeofanInverseFunctionLet/beafunctionthatisdifferentiableonaninterval/.If/hasaninversefunctiong,thengisdifferentiableatanyxforwhich/(g(x))^0.Moreover,ga)=上，r(g(x))R.*TheDerivativeoftheNaturalExponentialFunctionLetubeadifferentiablefunctionofx.1.96]=爐2.*lntegrationRulesforExponentialFunctionsLetubeadifferentiablefunctionofx.Jcudu=+C,^DerivativesforBasesotherthaneLetabeapositiverealnumber(awl)andletubeadifferentiablefunctionofx.2?白嘀u]=1duulnadx1都U]=Qn助a喂x—*001Kx+1+-)=lim(—)*DerivativesofInverseTrigonometricFunctionsLetubeadifferentiablefunctionofx.d「——iiit d「_]-|-nr丁si