{无敌必备}AP微积分公式大全

MadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------公式大全（北京新东方罗勇2012-4-27）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.FunctionandLimit1.5种基本初等函数图像性质111111;;:log:sin/cos/tan/cot/sec/cscsin/cos/tan/cot/sec/cscxaPoweryxExponentialyaLogarithmicyxTrigonometricyxxxxxxInversetrigonometricyxxxxxx2.4种表达函数的解析式()();;();()(),()()xftyfxparametricpolarrfvectorrtftgtygt3.3个重要极限0sinlim1;xxx101lim(1)lim(1)xxxxexex0010111011()()limlim()()0()mmmmmnnxxnnnanmbPxaxaxaxanmQxbxbxbxbnmMadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------()()(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,(())''(),'()'()cossinyfgxdydydudxdudxLogarithmicdifferentiationdyInversefunctiondxdxdydydydtdyddydxdtparametricdxdxdtdxdxdtdyfrcospolarvectorrtftgtdxfrMadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------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MadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------[()()]()()()()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)()dtparametricequation7.微分方程(1)();()(2):(1)lim;1:kxtdySeparationVariableMxdxNydyyKLogisticequationkyyDifferentiandyKdxKAalequatione11000(3)(4)':(')()()'()()nnnSlopefieldEluersmethodyyhyorfxfxfxxxMadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------(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,):(nnnnnnnnaRatioaaconvergencedivergencemayconvergenceordivergenceIntegralIfafnispositivecontinuousanddecreaingforxthenaandfxdxbothconvergThreetenceset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MadebyYongLuo@XDF---------------------------------------------------------------------------------------------------------(4):(1)(0).lim0,.:(||||)nnnnnnnnnnnnAlternatingseriesbbIfbbandbconvergenceErrorboundthenexttermRSSa4.幂级数和泰勒级数2301230()2()(1):.......()(2)''()()()()'()()()...()...;2!!()()()()()()'()(.0)..()!nnnnnnnnnnnnPowerseriescxccxcxcxcxTaylorseriesfafafxfafaxaxaxanLetfxPxRxfaPxfafaxaxannoLat(1)1()()()()(1)!nnnfgrangeerrorboundRxxabetweenxandan()()20233521123(3)(0)''(0)(0)()(0)'(0)......!2!!(4):1......;2!3!!sin...(1)...;3!5!(21)!11...1nnnnnnxnnMaclaurinseriesffffxxffxxxnnFourseriestorememberxxxexnxxxxxnxxxx