AP微积分BC公式大全

1APAPAPAP微积分公式大全AP微积分考试为闭卷考试，考试时也不给任何数学公式。因此，熟练掌握考试大纲要求的微积分公式十分必要。为了帮助考生更快的掌握AP微积分的相关公式，本文总结了AP微积分考试当中常用的重要公式，供大家参考学习。Chapter1.Chapter1.Chapter1.Chapter1.FunctionFunctionFunctionFunctionandandandandLLLLimitimitimitimit1.5种基本初等函数图像性质111111;;:log:sin/cos/tan/cot/sec/cscsin/cos/tan/cot/sec/cscxaPoweryxExponentialyaLogarithmicyxTrigonometricyxxxxxxInversetrigonometricyxxxxxxµ−−−−−−=====2.4种表达函数的解析式()();;();()(),()()xftyfxparametricpolarrfvectorrtftgtygtθ=⎧===⎨=⎩����3.3个重要极限0sinlim1;xxx→=101lim(1)lim(1)→∞→+=⇒+=xxxxexex0010111011()()limlim()()0()mmmmmnnxxnnnanmbPxaxaxaxanmQxbxbxbxbnm−−−→∞→∞−⎧=⎪⎪++++⎪==∞⎨++++⎪⎪⎪⎩⋯⋯Chapter2.Chapter2.Chapter2.Chapter2.DerivativesDerivativesDerivativesDerivatives21.导数定义式0000000000000000000()()(1)'()limlim()()()()(2):()limlim;()()()()()limlim;xxxxxxxxfxxfxyfxxxfxfxfxxfxOnesidefxxxxfxfxfxxfxfxxxx−−++∆→∆→−→∆→+→∆→+∆−∆==∆∆−+∆−′−==−∆−+∆−′==−∆2.求导公式和法则1222(1):111()',()',()';211()'(ln),()';(log)',(ln)';ln(sin)cos,(cos)'sin,(tan)'sec,(cot)'csc,(sec)'sectnnxxxxaDerivativeformulasoffivebasicelementryfunctionsxnxxxxxaaaeexxxaxxxxxxxxxxx−==−=====′==−==−=2222222an,(csc)'csccot;111(arcsin),(arccos),(arctan),111111(cot),(sec),(cot)1||1||1''(2):()''',()''',()';(3)xxxxxxxxxxarcxarcxarcxxxxxxuuvuvOperationuvuvuvuvuvvvChainrule=−′′′==−=+−−′′′=−==−+−−−±=±=+=22[()](4)1(5)(),'()sin,(())''(),'()'()cossinyfgxdydydudxdudxLogarithmicdifferentiationdyInversefunctiondxdxdydydydtdyddydxdtparametricdxdxdtdxdxdtdyfrcospolarvectorrtftgtdxfrθθθθθθ=⇒=⋅===+==−����Chapter3.Chapter3.Chapter3.Chapter3.IntegralIntegralIntegralIntegral31.不定积分定义式'()()()();definitionfxdxfxCmethoddfxfxC=+=+∫∫2.求不定积分的四种方法1121221(1):,1lnsincos,cossin,tanln|cos|cotln|sin|,secln|sectan|,cscln|csccot|sin(1),tan,11nnuuxFormulasxdxCaduaCnaxdxxCxdxxCxdxxCxdxxCxdxxxCxdxxxCdxdxdxxCxxCxxx+−−=+=++=−+=+=−+=+=+=−+=+=++−∫∫∫∫∫∫∫∫∫∫122sec(1)1(2)(3)(4):(ln,,sin)xxCxxUsubstitutionPartialfractionsIntegrationbypartsudvuvvdustoryaboutxxxande−=+−−=−∫∫∫3.反常积分的两种形式(1)()lim()()lim()()()()(2)()lim()()limbaabbbaabbaccaacbIntegralonanInfiniteIntervalfxdxfxdxfxdxfxdxfxdxfxdxfxdxIntegrandwithInfinitediscontinuitiesfxdxfxdxfxdx+−+∞→+∞−∞→−∞+∞+∞+∞−∞−∞−∞→→===+==∫∫∫∫∫∫∫∫∫()()()()bcabcbaacfxdxfxdxfxdxfxdx=+∫∫∫∫∫4.定积分定义和运算法则(1)(,,,)(2):()limlim()(3):biiannRiemannsumleftrightmidtrapezoidaldefinitionfxdxAfxxpropertiesofIntegral→∞→∞==∆∑∑∫4[()()]()()()()bbbaaabbaafxgxdxfxdxgxdxkfxdxkfxdx±=±=∫∫∫∫∫0()()()()0()()()0(())()2()(())bcbaacaababaaaaaafxdxfxdxfxdxfxdxfxdxfxdxfxdxfxoddfxdxfxdxfxeven−−=+==−==∫∫∫∫∫∫∫∫∫5.微积分两条基础理论'()()()()()'()()()(),[,];bbaaxafxdxfbfaorfbfafxdxdAxdftdtfxxabdxdx=−=+==∈∫∫∫6.定积分应用22112222()(1):()()(2)(//),(3)(///sec),(4):1()1()()(baxyxyfxdxMeanvaluetheoremfcbaAreaverticalslicehorizontalslicepolarVolumediskwashershellknowncrosstiondydxLengthofcurveLdxdydxdyorLxtyt=−−=+=+′′=+∫∫∫)()dtparametricequationβα∫7.微分方程(1)();()(2):(1)lim;1:kxtdySeparationVariableMxdxNydyyKLogisticequationkyyDifferentiandyKdxKAalequatione−→∞==−⇒==+11000(3)(4)':(')()()'()()nnnSlopefieldEluersmethodyyhyorfxfxfxxx−−=+=+−5Chapter4.Chapter4.Chapter4.Chapter4.SeriesSeriesSeriesSeries1.级数的定义与收敛性1231123111:(1)..........;(2):.....;(3)lim,;lim,.nnnnnnnnnnnnnnnnaaaaapartialsumSaaaaaIfSexitsthenaconvergesIfSdoesnotexitthenadivergesDefinition∞==∞→∞=∞→∞==+++++==++++∑∑∑∑2.判定级数收敛性的三大审敛法1111(1):lim(1,1,1);(2):(),1,):(nnnnnnnnaRatioaaconvergencedivergencemayconvergenceordivergenceIntegralIfafnispositivecontinuousanddecreaingforxthenaandfxdxbothconvergThreetencesetρρρρ∞+→∞=∞∞====≥∑∑∫1111;(3):0,,;,.nnnnnnnnnnordivergerenceComparisonLetabforallnIfbconvergesthenaconvergesIfadivergesthenbdiverges∞∞==∞∞==≤≤∑∑∑∑3.四种重要的级数12311111:11111(1):............();123(2):.....||1,;||1,;11(3):.nnnnnpnHarmonicseriesdivergencennGeometricseriesaararararaIfrconvergencearIfrdivergencerFourserPseriesInies∞=∞−=∞−=∞==+++++++++=⇒=≥−−∑∑∑∑1,;1,;fpconvergenceIfpdivergence≤611111(4):(1)(0).lim0,.:(||||)nnnnnnnnnnnnAlternatingseriesbbIfbbandbconvergenceErrorboundthenexttermRSSa∞+=+→∞++−=≤−=∑4.幂级数和泰勒级数2301230()2()(1):.......()(2)''()()()()'()()()...()...;2!!()()()()()()'()(.0)..()!nnnnnnnnnnnnPowerseriescxccxcxcxcxTaylorseriesfafafxfafaxaxaxanLetfxPxRxfaPxfafaxaxannoLat∞==++++++=+−+−++−+=+=+−++−=∞∑(1)1()()()()(1)!nnnfgrangeerrorboundRxxabetweenxandanξξ++=−+()()20233521123(3)(0)''(0)(0)()(0)'(0)......!2!!(4):1......;2!3!!sin...(1)...;3!5!(21)!11...1nnnnnnxnnMaclaurinseriesffffxxffxxxnnFourseriestorememberxxxexnxxxxxnxxxx∞=−+==+++++=++++++=−+−+−+−=+++++−∑24221...;cos1...(1)...;2!4!(22)!nnnxxxxxn−++=−+−+−+−