第十三次课 20环境工程1环境科学1、2班113（124）人
① 泰勒公式
② 基本初等函数的泰勒公式 ③ 罗比达法则
④ 非基础形式的未定式的转化方法
Tn?x??a0?a1(x?x0)?a2(x?x0)2?Tn?x??Tn?0??an(x?x0)nTn?2?0!?x0??Tn?x0?(x?x1!?1?0)?2!?x0?(x?x2)?0?Tn?n?n!?x0?(x?xn)?0Tn??x??a1?2a2(x?x0)?3a3(x?x0)2?Tn???x??2!a2?3!a3(x?x0)?Tn????x??3!a3?4!a4(x?x0)?Tn?k??nan(x?x0)n?1?n?n?1?an(x?x0)n?2?n?n?1??n?2?an(x?x0)n?3Tn
?n??x??n!an???ak?k!?x0????k?0,1,?f???x0?2!2!,n?
2f?x??f?x0??f??x0??x?x0??f?x??f?x0??f??x0??x?x0??f?(n?1)(?)Rn(x)?(x?x0)n?1(n?1)!Rn(x)???x?x0?
n?x?x0???f(n)(x0)?(x?x0)n?Rn(x)n!f(n)(x0)?(x?x0)n?n!f???x0??x?x0?2分别写出常见的几个基本初等函数e?,sinx,cosx,x11?,,ln?1?x?,?1?x? 在1?x1?xx?0处的带拉格朗日型余项和皮亚罗型余项麦克劳林公式
f???0?2f????0?3f?x??f?0??f??0?x?x?x?2!3!111ex?1?x?x2?x3??xn???xn?2!3!n!111sinx?x?x3?x5?x7?3!5!7!cosx?x?sin?n??f?n??0?n!xn????1?m121416x?x?x?2!4!6!?2m?1?!m??1?2m?x????x2m??2m?!x2m?1???x2m?
?0,n?2m,m?Z?????0??sin?n???m?2?????1????n?2m?1,m?Zf???0?2!x?2f?x??f?0??f??0?x?1?1?x?x2?x3?1?x1?1?x?x2?x3?1?xf????0?3!x?3?f?n??0?n!?xn???xn????1?xn???xn?n
?1?x???1??x??n??1??1?x????x?0n!?1?x?n?1x?0?n!f???x0?2!2!2f?x??f?x0??f??x0??x?x0??f?x??f?x0??f??x0??x?x0??f?(n?1)(?)Rn(x)?(x?x0)n?1(n?1)!Rn(x)???x?x0?n?x?x0???f(n)(x0)?(x?x0)n?Rn(x)n!f(n)(x0)?(x?x0)n?n!f???x0??x?x0?2
f?x??f?0??f??0?x??y?f??x0??x????x?f???0?2!x2?f????0?3!x3??f?n??0?n!?f?x??f?x0??f??x0??x?x0????x?x0?f?x??f?0??f??0?x???x?罗比达法则计算未定式的极限：
limx?af(x)f?(x) ?limx?aF(x)F?(x)limf(x)?0,limF(x)?0,?x?ax?a?f(x)f?(x)?f(x),F?x??diff?x??a???lim?limx?aF(x)x?aF?(x)?F??x??0?????????????????????????sup?f(a)?F?a??0.f(x),F?x??on?a,x??or??x,a???c.t???f(x),F?x??in?a,x??or??x,a???diff.??????a,x??or??x,a??,?F??x??0??f????f?x??f?a?f(x)f?????????F???F?x??F?a?F(x)F????f????f????f??x?f(x)lim?lim?lim?limx?aF(x)x?aF??????aF????x?aF??x?x3?3x?2???lim3??????x?1x?x2?x?1??3x2?3????lim2??? x?13x?2x?1???6x6?lim?x?16x?242??h??2hx??1???1?222x2x???x?h??x?????limx?2xlim?limh?0h?0h?0hhhx2?2hx?h2?x2?lim?lim?2x?h??2xh?0h?0 h
?x2????lim???h?0h??2?x?h??0?lim?2xh?01?x?h?2?arctanx??????1x?????x1?2x21?x?lim?lim?1x???x???1?x21?2 xx2????lim???x???1?x2??2x?lim?1x???2x?lim2
2021年高等数学课件-第十三次课 20环境工程1环境科学1、2班113(124)人
