AP Calculus AB review AP微积分复习提纲

COPYRIGHTSRESERVEDBYOWEN1APCALCULUSABREVIEWChapter2DifferentiationDefinitionofTangentLinewithSlopmIffisdefinedonanopenintervalcontainingc,andifthelimitlim∆x→0∆y∆x=lim∆x→0f(c+∆x)−f(c)∆x=mexists,thenthelinepassingthrough(c,f(c))withslopemisthetangentlinetothegraphoffatthepoint(c,f(c)).DefinitionoftheDerivativeofaFunctionTheDerivativeoffatxisgivenby𝑓′(x)=lim∆x→0f(c+∆x)−f(c)∆xprovidedthelimitexists.Forallxforwhichthislimitexists,f’isafunctionofx.*ThePowerRule*TheProductRule*ddx[sinx]=cosx*ddx[cosx]=−sinx*TheChainRule☺ImplicitDifferentiation(takethederivativeonbothsides;derivativeofyisy*y’)Chapter3ApplicationsofDifferentiation*Extremaandthefirstderivativetest(minimum:−→+,maximum:+→−,+&−arethesignoff’(x))*DefinitionofaCriticalNumberLetfbedefinedatc.Iff’(c)=0ORIFFISNOTDIFFERENTIABLEATC,thencisacriticalnumberoff.COPYRIGHTSRESERVEDBYOWEN2*Rolle’sTheoremIffisdifferentiableontheopeninterval(a,b)andf(a)=f(b),thenthereisatleastonenumbercin(a,b)suchthatf’(c)=0.*TheMeanValueTheoremIffiscontinuousontheclosedinterval[a,b]anddifferentiableontheopeninterval(a,b),thenthereexistsanumbercin(a,b)suchthatf’(c)=𝑓(𝑏)−𝑓(a)b−a.*Increasinganddecreasingintervaloffunctions(takethefirstderivative)*Concavity(ontheintervalwhichf’’0,concaveup)*SecondDerivativeTestLetfbeafunctionsuchthatf’(c)=0andthesecondderivativeoffexistsonanopenintervalcontainingc.1.Iff’’(c)0,thenf(c)isaminimum2.Iff’’(c)0,thenf(c)isamaximum*PointsofInflection(takesecondderivativeandsetitequalto0,solvetheequationtogetxandplugxvalueinoriginalfunction)*Asymptotes(horizontalandvertical)*LimitsatInfinity*CurveSketching(takefirstandsecondderivative,makesureallthecharacteristicsofafunctionareclear)♫OptimizationProblems*Newton’sMethod(usedtoapproximatethezerosofafunction,whichistediousandstupid,DONOTHAVETOKNOWIFUDONOTWANTTOSCORE5)Chapter4&5Integration*Beabletosolveadifferentialequation*BasicIntegrationRulesCOPYRIGHTSRESERVEDBYOWEN31)∫𝑢𝑛𝑑𝑢=𝑢n+1n+1+C,n≠−12)∫sin𝑢𝑑𝑢=−cos𝑢+𝐶3)∫cos𝑢𝑑𝑢=sin𝑢+𝐶4)∫1u𝑑𝑢=ln𝑢*Integralofafunctionistheareaunderthecurve*RiemannSum(divideintervalintoalotofsub-intervals,calculatetheareaforeachsub-intervalandsummationistheintegral).*Definiteintegral*TheFundamentalTheoremofCalculusIfafunctionfiscontinuousontheclosedinterval[a,b]andFisananti-derivativeoffontheinterval[a,b],then∫𝑓(x)dxba=F(b)−F(a).*DefinitionoftheAverageValueofaFunctiononanIntervalIffisintegrableontheclosedinterval[a,b],thentheaveragevalueoffontheintervalis1b−a∫𝑓(x)dxba.*ThesecondfundamentaltheoremofcalculusIffiscontinuousonanopeninternalIcontaininga,then,foreveryxintheinterval,ddx[∫𝑓(t)𝑑𝑡xa]=𝑓(x).*IntegrationbySubstitution*IntegrationofEvenandOddFunctions1)Iffisanevenfunction,then∫𝑓(x)dxba=2∫𝑓(x)dxba.2)Iffisanoddfunction,then∫𝑓(x)dxba=0.*TheTrapezoidalRuleLetfbecontinuouson[a,b].ThetrapezoidalRuleforCOPYRIGHTSRESERVEDBYOWEN4approximating∫𝑓(x)dxbaisgivenby∫𝑓(x)dxba≈b−a2n[𝑓(x0)+2𝑓(x1)+2𝑓(x2)+⋯+2𝑓(xn−1)+𝑓(xn)]Moreover,an→∞,theright-handsideapproaches∫𝑓(x)dxba.*Simpson’sRule(niseven)Letfbecontinuouson[a,b].Simpson’sRuleforapproximating∫𝑓(x)dxbais∫𝑓(x)dxba≈b−a3n[𝑓(x0)+4𝑓(x1)+2𝑓(x2)+4𝑓(x3)+⋯4𝑓(xn−1)+𝑓(xn)]Moreover,asn→∞,theright-handsideapproaches∫𝑓(x)dxba*Inversefunctions(y=f(x),switchyandx,solveforx)*TheDerivativeofanInverseFunctionLetfbeafunctionthatisdifferentiableonanintervalI.Iffhasaninversefunctiong,thengisdifferentiableatanyxforwhichf’(g(x))≠0.Moreover,g′(𝑥)=1𝑓′(g(x)),f’(g(x))≠0.*TheDerivativeoftheNaturalExponentialFunctionLetubeadifferentiablefunctionofx.1.ddx[𝑒𝑥]=𝑒𝑥2.ddx[𝑒𝑢]=𝑒𝑢𝑑𝑢𝑑𝑥.*IntegrationRulesforExponentialFunctionsLetubeadifferentiablefunctionofx.∫𝑒𝑢𝑑𝑢=𝑒𝑢+𝐶.♠DerivativesforBasesotherthaneLetabeapositiverealnumber(a≠1)andletubeadifferentiablefunctionofx.1.𝑑𝑑𝑥[𝑎𝑢]=(ln𝑎)𝑎u𝑑𝑢𝑑𝑥2.𝑑𝑑𝑥[log𝑎𝑢]=1𝑢ln𝑎𝑑𝑢𝑑𝑥♠∫𝑎𝑥𝑑𝑥=(1ln𝑎)𝑎𝑥+𝐶♠limx→∞(1+1x)x=limx→∞(x+1x)x=𝑒COPYRIGHTSRESERVEDBYOWEN5*DerivativesofInverseTrigonometricFunctionsLetubeadifferentiablefunctionofx.𝑑𝑑𝑥[sin−1𝑢]=𝑢′√1−𝑢2𝑑𝑑𝑥[cos−1𝑢]=−𝑢′√1−𝑢2𝑑𝑑𝑥[tan−1𝑢]=𝑢′1+𝑢2∫𝑑𝑢√𝑎2−𝑢2=sin−1𝑢𝑎+𝐶∫𝑑𝑢𝑎2+𝑢2=1𝑎tan−1𝑢𝑎+𝐶∫𝑑𝑢𝑢√𝑢2−𝑎2=1𝑎sec−1|𝑢|𝑎+𝐶*DefinitionoftheHyperbolicFunctionssinh𝑥=𝑒𝑥−𝑒−𝑥2cosh𝑥=𝑒𝑥+𝑒−𝑥2tanh𝑥=sinh𝑥cosh𝑥csch𝑥=1sinh𝑥,𝑥≠0sech𝑥=1cosh𝑥coth𝑥=1tanh𝑥,𝑥≠0