華東師大數(shù)學(xué)分析

1、華東師范大學(xué)2004數(shù)學(xué)分析一、（30分）計算題。1、求2120)2(coslimxxxx??解：)0(21~2sin21cos22????xxxx?1)1(120120120222)1(lim)1(lim)2(coslim????????????exxxxxxxxxx2、若求.))sin(arctan2lnxxeyx???y解：2ln11)cos(arctan)sin(arctanln22xxxxexxyx??????3、求.???d

2、xxxex2)1(解：==????dxxxex2)1(???xdxex11xxex??1????dxxxex2)1()(xxex??1dxex??cexxexx?????14、求冪級數(shù)的和函數(shù).???1nnnx)(xf解:時1||?x=?????01)(nnnx????0)1(nnxn???0nnnx???0nnx==????0nnnx01)(????nnnx???0nnx????xxx11)1(????xx11)1(122)1(xx?

3、5、為過和的曲線求L)00(O)02(?A)0(sin??axay????Ldyydxyx.)2()(3xdxaxdadyxaycossinsin???=????Ldyydxyx)2()(3?20?xdx?2033sin?xdxa?20cos2?xdxa?202cossin?xdxxa=?82??323a222aa?故????????nininignifnninifn11)1()(1))((1????nininifn1))((1?兩邊對

4、分別取極限n???10)()(?dxxgxf???ninignifn1)1()(1???10)()(?dxxgxf由夾逼性知.???????nindxxgxfnignifn110)()()1()(1lim4、若收斂，則收斂.???1nna???12nna錯誤。反例收斂但發(fā)散.?????11)1(nnn???11nn5、若在上定義的函數(shù)存在偏導(dǎo)數(shù)且在(00)上連2R)(yxf)(yxfx)(yxfy)(yxfx)(yxfy續(xù)，則在(00)

5、上可微.)(yxf正確證：)00()00(fyxfz???????=????????))00()00((yfyxf))00()00((fyf???=yyfxyxfyx?????????)00()00(21??有在(00)上連續(xù)，)(yxfx)(yxfy???????)00()00(1xfxf??????)00()00(2yfyf當(dāng)時，，)00()(???yx0????yxyfxfzyx???????????)00()00(根據(jù)定義，可知