几个初等函数的麦克劳林公式.ppt

几个初等函数的麦克劳林公式93643二、几个初等函数的麦克劳林公式第三节一、泰勒公式的建立机动目录上页下页返回结束三、泰勒公式的应用—应用用多项式近似表示函数理论分析近似计算泰勒(Taylor)公式第三章特点:)(0xf?)(0xf??一、泰勒公式的建立)(xfxy)(xfy?o))(()(000xxxfxf????以直代曲0x)(1xp在微分应用中已知近似公式:需要解决的问题如何提高精度?如何估计误差?:)(0!212xpan???,)(0xf???,?)(0)(!1xpannnn?)(0)(xfn?故?)(xpn)(0xf))((00xxxf?????!21!1nnnxxxf))((00)(??!1n200))((xxxf????!21机动目录上页下页返回结束令?)(xpn则??)(xpn???)(xpn????nan!?)()(xpnn)(00xpan?,)(0xf?)(01xpan??,)(0xf??1a)(202xxa??10)(????nnxxan?2!2a20)()1(?????nnxxann?0annxxaxxaxxa)()()(020201????????)0(之间与在nx??)()(10???nnxxxR)(2)1()(0)(xnRnnnn??????)()()(xpxfxRnn??令(称为余项),)(0xRn)(0xRn??0)(0)(???xRnn?10)()(??nnxxxRnnxnR))(1()(011??????))(1()(011nnxnR??????1022)()1()(??????nnxnnR????!)1()()1(???nRnn?则有)(0xRn?0?)(0xRn??0?)(0)(xRnn?0?x)01(之间与在xx?)102(之间与在??x机动目录上页下页返回结束)()()(xpxfxRnn??)0(之间与在xx?,0)()1(??xpnn?10)1()(!)1()()(?????nnnxxnfxR?)()()1()1(xfxRnnn????时的某邻域内当在Mxfxn??)()1(0)0(之间与在xx?10!)1()(????nnxxnMxR)())(()(00xxxxoxRnn????机动目录上页下页返回结束公式①②:阶的导数,时,有)(0xf))((00xxxf???200)(!2)(xxxf??????nnxxnxf)(!)(00)(??)(xRn?①其中10)1()(!)1()()(?????nnnxxnfxR?②则当)0(之间与在xx?泰勒目录上页下页返回结束公式③称为n阶泰勒公式的佩亚诺(Peano),泰勒公式可写为??)(0xf))((00xxxf???200)(!2)(xxxf????nnxxnxf)(!)(00)(??])[(0nxxo??])[()(0nnxxoxR??注意到③④*可以证明:④式成立机动目录上页下页返回结束特例:(1)当n=0时,泰勒公式变为?)(xf)(0xf))((0xxf????(2)当n=1时,泰勒公式变为给出拉格朗日中值定理?)(xf)(0xf))((00xxxf???20)(!2)(xxf?????可见误差?)(xf)(0xf))((00xxxf?????10)1()(!)1()(?????nnxxnf?200)(!2)(xxxf????nnxxnxf)(!)(00)(??fd)0(之间与在xx?)0(之间与在xx?)0(之间与在xx?)0(之间与在xx?机动目录上页下页返回结束称为麦克劳林(Maclaurin)公式.,)10(,00???????xx则有)0(fxf)0(????2!2)0(xf???nnxnf!)0()(?在泰勒公式中若取?)(xf)(0xf))((00xxxf?????10)1()(!)1()(?????nnxxnf?200)(!2)(xxxf????nnxxnxf)(!)(00)(??)0(之间与在xx??)(xf)0(fxf)0(????,)()1(Mxfn??则有误差估计式1!)1()(???nnxnMxR2!2)0(xf???nnxnf!)0()(?若在公式成立的区间上麦克劳林目录上页下页返回结束由此得近似公式二、几个初等函数的麦克劳林公式,)()(xkexf??),2,1(1)0()(???kfkxe?1?x?!33x???!nxn?)(xRn?!22x?其中机动目录上页下页返回结束