考研数学知识点总结

- Edited by 2005 10 1 . 1 1 dttfy x 0 tf xf dx dy 2 dttfy x x 2 1 x1 x2 tf xxfxxf dx dy 1122 2 0lim xf 0lim xg l xg xf lim 1 0l xf xg xgxf 0 xg xf 2 0l xf xg 3 1l xf xg xgxf ~ 3 0x xx ~sin xx ~tan xx ~arcsin xx ~arctan 2 2 1 ~cos1 xx xex ~1 xx ~1ln xx ~11 1 2 1 1 nn xx 1 n mxn n Axn n lim mA 2 nn xx 1 n Mxn n Axn n lim MA 2 xhxfxg Axglim Axhlim Axflim 3 1 1 sin lim 0 x x x 2 e n n n 1 1lim e u u u 1 1lim ev v v 1 0 1lim 4 5 0x n n x x n xx xe 0 !!2 1 2 12 1253 0 !12 1 !5!3 sin n n n x n xxx xx n n n x n xxx x 2 242 0 !2 1 !4!2 1cos n n n x n xxx xx 01 32 1ln 1 32 12 12 1 53 0 12 1 53 arctan n n n x n xxx xx nn xx n n xxx 0 ! 11 !2 1 11 2 6 1 0 0 1 0lim xf 0lim xg 2 x xf xg 3 A xg xf lim A xg xf lim xg xf lim xg xf lim 2 1 xflim xglim 2 x xf xg - Edited by 2005 10 2 3 A xg xf lim A xg xf lim 7 0 00 0 lim xf x xfxxf x [ ] 8 1 0 1 1 lim dxxf n k f n n k n [ ] 1 0x xfy xf 0x 0x xf 2 ba, xf 1 xf ba, xf ba, 2 xf ba, M m M m Mxf 0 ba, 0x ba, x Mxf M xf ba, m 3 xf ba, M m m M c ba, cf xf ba, af bf ba, 0f 1 0c 0cd 1 xx dxxxd 1 xx cossin xdxxd cossin xx sincos xdxxd sincos xx 2sectan xdxxd 2sectan xx 2csccot xdxxd 2csccot xxx tansecsec xdxxxd tansecsec xxx cotcsccsc xdxxxd cotcsccsc ax xa ln 1 log 1,0 aa ax dx xd a ln log 1,0 aa x x 1 ln dx x xd 1 ln aaa xx ln 1,0 aa adxada xx ln 1,0 aa - Edited by 2005 10 3 xx ee dxede xx 21 1 arcsin x x dx x xd 21 1 arcsin 21 1 arccos x x dx x xd 21 1 arccos 21 1 arctan x x dx x xd 21 1 arctan 21 1 cot x xarc dx x xdarc 21 1 cot 22 22 1ln ax axx dx ax axxd 22 22 1ln 22 22 1ln ax axx dx ax axxd 22 22 1ln 2 xgxfxgxf xgxfxgxfxgxf xg xgxfxgxf xg xf 2 0xg 3 ufy xu x x uf u xfy x xxf dx du du dy dx dy dxxxfduufdy duufdy u 4 tx ty xyy t t 0t t t dx dy 0t 32 2 1 t tttt dt dxdt dx dy d dx dx dy d dx yd 5 xfy ygx 0xf ygfxf yg 11 0xf dx dydx xf d dy ygd yg 1 1 33 ygf ygf xf xf 0xf 6 xyy 0, yxF y 0, yxF x y y y 7 y xg xfy - Edited by 2005 10 4 xfxgey ln 8 xf 0x xf 0x 9 n 2n ,,, yy ny n 1 xey xn ey 2 1,0 aaay x nxn aay ln 3 xy sin 2 sin n xy n 4 xy cos 2 cos n xy n (5) xy ln n nn xny !11 1 n n k knkk n n xvxuCxvxu 0 !! ! knk n C kn xuxu 0 xvxv 0 xu xv n xf 1 ba, 2 ba, 3 bfaf ba, 0f xf 1 ba, 2 ba, ba, f ab afbf abfafbf ba xxxfxfxxf 000 10 0x a b x 1 xf ba, 0xf xf ba, 2 xf xg ba, xgxf ba, cxgxf c xf xg 1 ],[ ba 2 ba, 0xg ba, g f agbg afbf ba xxg 1 n xf 0x n xRxx n xf xx xf xx xf xfxf n n n 0 02 0 0 0 0 0 !!2!1 - Edited by 2005 10 5 0xx n n xxxR 00 0xx 0lim 0 0 n n xx xx xR n xxxe x 1ln,cos,sin, x1 n 2 n xf 0x ba, 1n ba, n bax , xRxx n xf xx xf xx xf xfxf n n n 0 02 0 0 0 0 0 !!2!1 1 0 1 !1 n n n xx n f xR 0x x 0x n 00x n 0lim xRn n 1 xf ba, 0x ba, 0x 0xxx 0xfxf 0xf xf 0x xf 0x 0xxx 0xfxf 0xf xf 0x xf 2 xf 0x 0x xf 00xf x 00xf 0x xf 3 xf 0x 00 xx 0xf 00xf 1 00 , xx x 0xf 00 , xx x 0xf 0xf 0x 2 00 , xx x 0xf 00 , xx x 0xf 0xf 0x 3 00 , xx 00 , xx x xf 0xf 0x 4 xf 0x 00xf 00xf 00xf 0xf 0x 00xf 0xf 0x - Edited by 2005 10 6 1 xf ba, xf ba, kxx ,,1 bfafxfxf k ,,,,1 bfafxfxf k ,,,,1 xf ba, M xf ba, m 2 1 xf I 21, xx 21 21 21 21 2 1 22 1 2 xfxf xx fxfxf xx f xf I xfy xfy xfy xfy 2 3 xf ba, xf ba, x 0xf xfy ba, ba, x 0xf xfy ba, xfy xf 1x 2x kx 1 xf ax lim xf ax lim ax xfy 2 bxf x lim bxf x lim by xfy 3 0lim a x xf x baxxf x lim 0lim a x xf x baxxf x lim baxy xfy xfy yxM , 2 3 2 1 y y k 0k k R 1 yxM , M D RMD D D R 1 C x dxx 1 1 1 - Edited by 2005 10 7 2 Cxdx x ln 1 3 Ca a dxa xx ln 1 1,0 aa Cedxe xx 4 Cxxdx sincos 5 Cxxdx cossin 6 Cxdx x xdx tan cos 1 sec 2 2 7 Cxdx x xdx cot sin 1 csc 2 2 8 Cxxdxx secsectan 9 Cxxdxx csccsccot 10 Cxxdx coslntan 11 Cxxdx sinlncot 12 Cxxxdx tanseclnsec 13 Cxxxdx cotcsclncsc 14 C a x xa dx arcsin 22 0a 15 C a x axa dx arctan 1 22 0a 16 C xa xa axa dx ln 2 1 22 0a 17 Caxx ax dx 22 22 ln 0a 1 CuFduuf x duuf xu xdxfdxxxf CxFCuF 1 baxdbaxf a dxbaxf 1 0a 2 baxdbaxf na dxxbaxf nnnn 11 0,0 na 3 xdxf x dx xf lnlnln 4 x d x f x dx x f 111 2 5 xdxf x dx xf 2 6 xxxx adaf a dxaaf ln 1 1,0 aa xxxx edefdxeef 7 xdxfxdxxf sinsincossin 8 xdxfxdxxf coscossincos 9 xdxfxdxxf tantansectan 2 10 xdxfxdxxf cotcotcsccot 2 11 xdxfxdxxxf secsectansecsec 12 xdxfxdxxxf csccsccotcsccsc 13 xdxfdx x xf arcsinarcsin 1 arcsin 2 14 xdxfdx x xf arccosarccos 1 arccos 2 15 xdxfdx x xf arctanarctan 1 arctan 2 16 xarcdxarcfdx x xarcf cotcot 1 cot 2 - Edited by 2005 10 8 17 x d x fdx x x f 1 arctan 1 arctan 1 1 arctan 2 18 2222 22 22 lnln ln axxdaxxfdx ax axxf 0a 19 2222 22 22 lnln ln axxdaxxfdx ax axxf 0a 20 Cxfdx xf xf ln 0xf 2 tx 0t CtGdtttf CxGCtGdtttf tx dxxf 1 xt 1 tx x n bax x n dcx bax xe bae x txgn tx tx 0 2 ACBxAx tCBxAx2 tx 0A 22 0 lxxA 0A 2 0 2 xxlA 22 xa tax sin 22 xa tax tan 22 ax tax sec 3 xu xv xduxvxvxuxdvxu dxxvxuxvxudxxvxu xu xv 1 ax n exP axxPn sin axxPn cos xPn n a n axe axsin axcos xv xu 2 xxPn ln xxPn arcsin xxPn arctan xPn n xPn xv xln xarcsin xarctan xu 3 bxeax sin bxeax cos 4 - Edited by 2005 10 9 dx 1 1 b a a b dxxfdxxf 2 0 a a dxxf 3 b a b a b a dxxfkdxxfkdxxfkxfk 22112211 4 b c c a b a dxxfdxxfdxxf c ba, 5 ba xgxf bxa b a b a dxxgdxxf 6 ba Mxfm bxa abMdxxfabm b a 7 ba b a b a dxxfdxxf 8 xf ba, ba, abfdxxf b a b a dxxf ab 1 xf ba, 9 0 a a dxxf f aa a dxxfdxxf 0 2 f 10 xf T a TTa a dxxfdxxf 0 1 xf ba, x a dttfxF bax , 1 xf ba, x a dttfxF ba, 2 xf ba, x a dttfxF ba, xfxF x x dttfxF 2 1 xx 21 , xf xxfxxfxF 1122 2 xf ba, xF xf ba, aFbF a b xFdxxf b a xf ba, xf ba, 1 xf ba, tx 1 t , , 2 a b t bta b a dtttfdxxf 2 xvxu , ba, dxxvxuxvxudxxvxu b a b a b a b a b a b a xduxvxvxuxdvxu - Edited by 2005 10 10 1 I dxxyxyS b a 121 xyxy 12 bax , II dyyxyxS d c 122 yxyx 12 dcy , 2 I drS 21 2 1 II drrS 21 2 22 2 1 3 C ty tx t a b t , , t 0t C bxax , x dtttydxS b a 1 xyy bxa [ xy ] dxxyS b a 2 1 dxxydS 2 1 2 rr [ r , ] drrS 22 3 tyy txx C t [ tx ty , ] C dttytxS 22 1 z cz dz z dzcz z zS d c dzzSV 2 1 xyy 0 ax bx x x - Edited by 2005 10 11 dxxyV b a x 2 y b ay dxxxyV 2 2 yxx 0 cy dy y y dyyxV d c y 2 x d c x dyyyxV 2 AB C x x S 1 AB xyy bxa dxxyxyS b a 2 12 2 AB rr drrrS 22 sin2 3 AB txx tyy t dttytxtyS 22 2 1 1 0yQyQxP dx dy CdxxP yQ dy 2 02211 dyyNxMdxyNxM Cdy yN yN dx xM xM 1 2 2 1 0,0 12 yNxM 2 1 x y f dx dy u x y uf dx du xu dx dy cxc x dx uuf du ||ln - Edited by 2005 10 12 2 0,0 bacbyaxf dx dy ucbyax ubfa dx du cxdx ubfa du 3 222 111 cybxa cybxa f dx dy 0 22 11 ba ba 0 0 222 111 cybxa cybxa , xu yv u v ba u v ba f vbua vbua f du dv 22 11 22 11 0 22 11 ba ba 1 2 1 2 b b a a 211 111 cybxa cybxa f dx dy ybxau 11 2 1 1111 cu cu fba dx dy ba dx du 1 0yxP dx dy dxxPCey c 2 xQyxP dx dy dxxPexCy xC CdxexQey dxxPdxxP 3 1,0yxQyxP dx dy 1yz xQzxP dx dz 11 4 xyPyQdx dy 1 yQxyP dy dx y x 1 0,, dyyxQdxyxP y P x Q Cyxu , yxu , dyyxQdxyxPyxdu ,,, yxu , yxdudyyxQdxyxP ,,, 1 2 22 yx dydyxdx 2 2 22 yx dydyxdx - Edited by 2005 10 13 3 xydxdyydx 4 xyd xy xdyydx ln 5 22 22 ln 2 1 yxd yx ydyxdx 6 22 22 ln 2 1 yxd yx ydyxdx 7 x y d x ydxxdy 2 8 y x d y xdyydx 2 9 y x d yx xdyydx arctan 22 10 x y d yx ydxxdy arctan 22 11 yx yx d yx xdyydx ln 2 1 22 12 yx yx d yx ydxxdy ln 2 1 22 13 22222 1 2 1 yx d yx ydyxdx 14 22222 1 2 1 yx d yx ydyxdx 15 22 222 arctan 2 1 1 yxd yx ydyxdx 16 22 222 arctan 2 1 1 yxd yx ydyxdx yx yx dyyxQdxyxPyxuyxu , ,00 00 ,,,, y y x x dyyxQdxyxPyxu 00 ,,, 000 yxP x u , yCdxyxPyxu ,, y yCdxyxP yy u yxQ ,, yC yC 2 0,, dyyxQdxyxP y P x Q yxR , 0,,,, dyyxQyxRdxyxPyxR y RP x RQ yxR , yxdudyyxQyxRdxyxPyxR ,,,,, Cyxu , - Edited by 2005 10 14 xfy n nn n xCxCdxxfy 2 1 1 yxfy , py py pxfp , 1,Cxgp 1,Cxgy 21, CdxCxgy yyfy , py p y dy dp p dx dy dy dp dx dp y y y pyf pdy dp , 1 ,, 1Cygp 1,Cyg dx dy 2 1, Cx Cyg dy 0yxqyxpy 1 xfyxqyxpy 2 1 xy1 xy2 xyCxyC 2211 1C 2C xyxy 21 xy1 xy2 xyCxyCy 2211 2 xy1 xy2 xyxy 21 3 xy xy xyxy 4 y xyCxyC 2211 1C 2C xyCxyCxyy 2211 5 xy1 xy2 xfyxqyxpy 1 xfyxqyxpy 2 xyxy 21 xfxfyxqyxpy 21 1 0qyypy p q 02 qp 1 042 qp 1 2 - Edited by 2005 10 15 xx eCeCy 21 21 2 042 qp 21 xexCCy 121 3 042 qp i xCxCey x sin cos 21 2 n 01 2 2 1 1 ypypypypy nn nnn nipi ,,2,1 0 1 2 2 1 1 nn nnn pppp 1 n n,,, 21 x n xx neCeCeCy 21 21 2 0 k nk xk k exCxCC 01 21 3 i k nk2 xxDxDDxxCxCCe kk k k x sin cos 121 1 21 xfqyypy qp, xyCxyCyy 2211 xyCxyC 2211 y xf y y xf y 1 xPxf n xPn n 1 0 nn nn n axaxaxaxRy 1 1 10 niai ,,2,1,0 2 0 xxRy n 3 0 xRxy n 2 2 x n exPxf xPn n 1 x n exRy 2 x n exxRy 3 x n exRxy 2 3 xexPxf xn sin xexPxf xn cos xPn n , 1 i xxTxxRey nn x sin cos nn nn n axaxaxaxR 1 1 10 niai ,,1,0 nn nn n bxbxbxbxT 1 1 10 nibi ,,1,0 - Edited by 2005 10 16 2 i xxTxxRxey nn x sin cos 01 11 1 ypyxpyxpyx nn nnnn nipi ,,2,1 n tex t y dt dy xdt dy e dx dt dt dy dx dy t 1 dt dy dx dy x dt dy e dt yd e dt dy e dt d e dx dy dt d dx dt dx yd tttt 2 2 2 2 2 2 dt dy dt yd x 2 2 2 1 dt dy dt yd dx yd x 2 2 2 2 2 321321 ,, aaakajaiaa 321321 ,, bbbkbjbibb 321321 ,, ccckcjcicc 1 332211 ,, babababa 332211 ,, babababa 2 321 ,, aaa 3 bababa ,cos 332211 bababa ba, ba, ba 0ba a b ajba bPr 0 4 ba bababa ,sin ba ba, 321 321 bbb aaa kji ba ba abba ba ba, 5 cbacba ,, 321 321 321 ,, ccc bbb aaa cba cba ,, cba ,, 321321 ,,,,, bbbbaaaa ba, ba ba cos 2 3 2 2 2 1 2 3 2 2 2 1 332211cos bbbaaa bababa a b 0ba 0332211 bbbaba - Edited by 2005 10 17 a b 0ba 3 3 2 2 1 1 b a b a b a 1 n pnm ,, 2 000 ,, zyxM CBAn ,, 0000 zzCyyBxxA 00rrn zyxrzyxr ,,,,, 0000 3 0DCzByAx CBA ,, zyx ,, CBAn ,, 0CzByAx 0DByAx z 0DAx yOz 0x yOz 4 111 ,, zyxA 222 ,, zyxB 333 ,, zyxC CBA ,, 0 131313 121212 111 zzyyxx zzyyxx zzyyxx 5 L 0 0 2222 1111 DzCyBxA DzCyBxA L 02222211111 DzCyBxAkDzCyBxAk 0,0, 21 kk 6 0: 11111 DzCyBxA 0: 22222 DzCyBxA 1 2 2 2 2 2 2 2 2 1 2 1 2 1 212121cos CBACBA CCBBAA 0212121 CCBBAA 2 1 2 1 2 1 2 1 D D C C B B A A 2 1 2 1 2 1 2 1 D D C C B B A A 0DCzByAx 111 ,, zyxM M d 222 111 CBA DCzByAx d 1 S L 2 n zz m yy l xx 000 000 ,, zyx nml ,, 3 - Edited by 2005 10 18 ntzz mtyy ltxx 0 0 0 tnmls ,,, 4 111 ,, zyxA 222 ,, zyxB A B 12 1 12 1 12 1 zz zz yy yy xx xx 5 0 0 2222 1111 DzCyBxA DzCyBxA 222111 ,,,, CBACBAS 6 1 1 1 1 1 1 1 : n zz m yy l xx L 2 2 2 2 2 2 2 : n zz m yy l xx L 1L 2L 2 2 2 2 2 2 2 1 2 1 2 1 212121cos nmlnml nnmmll 0212121 nnmmll 2 1 2 1 2 1 n n m m l l 0DCzByAx L n zz m yy l xx 000 L 222222 sin nmlCBA CnBmAl L C n B m A l L 0CnBmAl L 0CnBmAl L 1 yxz , 000 , yxP cos,cosl t yxftytxf yxl f t 0000 0 00 ,cos,cos lim , yxfz , 000 , yxP y yxf x yxf yxgradf 000000 , , , , lyxgradf yxl f 00 00 , , lyxgradflyxgradf ,,cos, 0000 1 vufz , yxuu , yxvv , - Edited by 2005 10 19 x v v z x u u z x z y v v z y u u z y z 2 zyxfu ,, yxzz , y z ff y u x z ff x u zy zx 3 zyxfu ,, xyy xzz xzfxyff dx du zyx 4 vufw , zyxuu ,, zyxvv ,, z v f z u f z w y v f y u f y w x v f x u f x w vu vu vu 0,, zyxF 1 yxzz , z x F F x z z y F F y z 2 zyxx , x y F F y x x z F F z x 3 xzyy , y z F F z y y x F F x y yxfz , 0, 0, yxf yxf y x kk yx , lk ,,2,1 2 ,,, kkxykkyykkxxk yxfyxfyxf 0k kk yxf , 0k 0k kk yxf , 0, kkxx yxf kk yxf , 0, kkxx yxf kk yxf , 2n nxxfu ,,1 0,, 0,, 1 11 nm n xx xx nm n m i iinmn xxxxfxxFF ,,,,,,,,, 1 1 111 - Edited by 2005 10 20 0,, 0,, 0 0 1 111 1 nm n x x xxF xxF F F m n lkxx kn k ,,2,1,,1 I xyxbxayxD 21,, x1 x2 ba, yxf , D DD dxdyyxfdyxf ,, b a x x dyyxfdx 2 1 , II yxydycyxD 21,, y1 y2 dc, yxf , D D D dxdyyxfdyxf ,, d c y y dxyxfdy 2 1 , I II D I D II D D I II D D D 21,,D 1 2 , sin,cos, fyxf D D D ddfdyxf sin,cos, 2 1 sin,cos dfd I 21,20,D 21 , 2,0 - Edited by 2005 10 21 sin,cos, fyxf D DD ddfdyxf sin,cos, 2 0 sin,cos 2 1 dfd II 0,,D , sin,cos, fyxf D DD ddfdyxf sin,cos, 0 sin,cos dfd III 0,20,D 2,0 sin,cos, fyxf D ddfdyxf D D sin,cos, 2 0 0 sin,cos dfd 1 dyxfyxfV D ,, 12 D S xy yxfz ,2 yxfz ,1 2 D d y z x z A 22 1 D S xy S yxzz , 1 1 Dyxyxzzyxzzyx ,,,,,, 21 D xy yxzyxz ,,, 21 D zyxf ,, yxz yxz D dzzyxfdxdydvzyxf , , 2 1 ,,,, 2 zDyxzzyx ,,,, zD z zD dxdyzyxfdzdvzyxf ,,,, 2 dzddzfdxdydzzyxf ,sin,cos,, yx, , z - Edited by 2005 10 22 3 cos sinsin cossin z y x 20 0 0 dxdydzzyxf ,, dddf sincos,sinsin,cossin 2 ,, L zyxf ,, L n nSSS ,,, 21 nkSk 1 kkk s,, n k kkkk Ssf 1 0 ,,lim kS k k nk S 1 max zyxf ,, L L dSzyxf ,, L L dSzyxf ,, 2 L txx tyy tzz t dttztytxtztytxfdSzyxf L 222 ,,,, zyxf ,, tx ty tz ABL zyxP ,, zyxQ ,, zyxR ,, L L n nSSS ,,, 21 nkSk 1 111 ,, kkk zyx kkk zyx ,, L 1kkk xxx 1kkk yyy 1kkk zzz nk1 kS kkk s,, n k kkkkkkkkkkkk zsRysQxsP 1 0 ,,,,,,lim n zyxP ,, zyxQ ,, zyxR ,, L L dzzyxRdyzyxQdxzyxP ,,,,,, L dSF zyxRzyxQzyxPF ,,,,,,,, dzdydxdS ,, L L - Edited by 2005 10 23 2 L txx tyy tzz A B zyxP ,, zyxQ ,, zyxR ,, tx ty tz AB L dzzyxRdyzyxQdxzyxP ,,,,,, dttztztytxRtytztytxQtxtztytxP ,,,,,, 1 ABL yxP , yxQ , L ABAB dsyxQyxPdyyxQdxyxP cos,cos,,, cos cos yx, A B 2 ABL zyxP ,, zyxQ ,, zyxR ,, L AB dzzyxRdyzyxQdxzyxP ,,,,,, dszyxRzyxQzyxP AB cos,,cos,,cos,, cos cos cos AB zyx ,, A B 1 xy D L L D L yxP , yxQ , D L D QdyPdxdxdy y P x Q 2 xy D 1n n nCCCL 10 0C nCC ,,1 L D yxP , yxQ , D - Edited by 2005 10 24 LD QdyPdxdxdy y P x Q 0 1 C n k Ck QdyPdxQdyPdx yxQyxPF ,,, yxP , yxQ , D 1 D L 0 L QdyPdx 2 ABL D AB QdyPdx A B ABL 3 yxdudyyxQdxyxP ,,, 4 D y P x Q 5 yxQyxPF ,,, yxV , gradVF yxV , x V P y V Q 1 S zyxf ,, S S n nSSS ,,, 21 )1( nkSk kkk s,, kS kS k n k kkk Ssf 1 0 ,,lim zyxf ,, S S dSzyxf ,, 2 S Dyxyxzz ,,, yxz , D zyxf ,, S DS dxdy y z x z yxzyxfdSzyxf 22 1,,,,, 1 S ) zyxP ,, zyxQ ,, zyxR ,, S S n nSSS ,,, 21 nkSk 1 yz yzkS zx zxk S xy xyk S nkSk 1 kkk s,, n k xykkkkzxkkkkyzkkkk SsRSsQSsP 1 0 ,,,,,,lim zyxP ,, zyxQ ,, zyxR ,, S - Edited by 2005 10 25 S dxdyzyxRdzdxzyxQdydzzyxP ,,,,,, RQPF ,, dxdydzdxdydzdS ,, S dSF 2 S yxzz , xyDyx, yxz , xyD zyxR ,, S xyDS dxdyyxzyxRdxdyzyxR ,,,,, S z xy S zyxx , yzDzy, YZDS dydzzyzyxPdydzzyxP ,,,,, S x S xzyy , zxDxz, ZXDS dzdxzxzyxQdzdxzyxQ ,,,,, S y RQP ,, 0 SS dSRQPRdxdyQdzdxPdydz coscoscos cos,cos,cos S zyx ,, RQPF ,, cos,cos,cos0n SS dSnFRdxdyQdzdxPdydz 0 1 S zyxRzyxQzyxP ,,,,,,,, S RdxdyQdzdxPdydzdv z R y Q x P S dSRQP coscoscos cos,cos,cos S zyx ,, 2 1n 0S nkSk 1 0S zyxRzyxQzyxP ,,,,,,,, 0S RdxdyQdzdxPdydzdv z R y Q x P n k SK RdxdyQdzdxPdydz 1 L S L L S zyxRzyxQzyxP ,,,,,,,, S - Edited by 2005 10 26 S L RQP zyx dxdydzdxdydz RdzQdyPdx S dxdy y P x Q dzdx x R z P dydz z Q y R S L dS RQP zyx RdzQdyPdx coscoscos zyxuu ,, zyx ,, u z u y u x u gradu ,, u 1 zyxRzyxQzyxPF ,,,,,,,, F z R y Q x P divF F S dSnFdivFdv 0 cos,cos,cos0n 2 zyxRzyxQzyxPF ,,,,,,,, RQP zyx kji FrotF k y P x Q j x R z P i z Q y R F S L dSnrotFdrF 0 dzdydxdr ,, , cos,cos,cos0n 1 1 1n nu 1n nv ba, 1n nn bvau 11 n n n n vbua 2 3 4 1n nu 0lim n n u 1 1 1 n n 1)1(lim n n 1 1 1 n n 1 1 n n 0 1 lim nn 1 1 n n 0lim n n u 1n nu 2 - Edited by 2005 10 27 1 0n nar 0a 1r r a ar n n 10 1r 0n nar 2 p 1 1 n pn 1p 1 1 n pn 1p 1 1 n pn 1p 1 1 n pn 6 1 2 1 2 n n ),3,2,1(0 nun 1n nu ,3,2,11 nSS nn nS nS 1n nu nS 1 0c Nn 0nn ucv 1n nv 1n nu 1n nu 1n nv 2 0nu 0nv ,3,2,1n A v u n n n lim 1 A0 1n nu 1n nv 2 0A 1n nv 1n nu 3 A 1n nu 1n nv 3 0nu n n n u u 1lim 1 1 1n nu 2 1 1n nu 3 1 n n n u u 1lim 4 0nu n n n ulim 1 1 1n nu - Edited by 2005 10 28 2 1 1n nu 3 1 1 1 0nu 1 1 1 n n n u 2 1 1 1 n n n u 1 ,3,2,11 nuu nn 2 0lim n n u 1 1 1 n n n u 1 1 1 10 uu n n n 1 1n nu 1n nu 2 1n nu 1n nu 1n nu 1n nu 1n nu 3 1 2 1 2 1 n nn uu 1 2 1 n nn uu 4 1 1 1 n p n n 1 1p 1 1 1 n p n n 2 10 p 1 1 1 n p n n 3 0p , 1 1 1 n p n n 1 xun ,3,2,1n I xu n n 1 I 2 I 0x 0 1 xu n n 0x 1n n xu 1 0 n n xu 0x 1n n xu 1n n xu 3 1n n xu x - Edited by 2005 10 29 1n n xuxS x )(xS 1n n xu 1 1 0 n n n xxa )( 0xx ),2,1,0(nan 00x 0n n n xa x 0n n n xa 0 0 n n n xxa 2 0n n n xa 1 ),( 0n n n xa x R 2 0n n n xa 0R 3 ),( RR RR,( ), RR RR, R )0( R R 1 2 )3( R l a a n n n 1lim lan n n lim l R 1 l 0R 0l R 1 )( 0 xfxa n n n 1Rx 0n n n xgxb 2Rx 0n n nn xgxfxba 21 ,min RRx 0 00 00 )( n n nknkn n n n n n n xgxfxbababaxbxa 21 ,min RRx 2 0n n n xa 0R 0n n n xaxS 1 xS RR, 0 1 1 0 n n n n n n n n n xnaxaxaxS xS - Edited by 2005 10 30 RR, kn kn k xknnnxS 11 Rx ,3,2,1k 2 xS RR, 0 1 0 00 1n nn n x n n x x n a dttadttS 3 0n n n xSxa RRx i 0 lim n n n Rx RaxS 0 lim n n n Rx RaxS ii 0 1 0 1n nn R R n a dxxS 0 0 1 1R n nn R n a dxxS iii 1 1 n n n xna RRx RSRna n n n 1 1 RS 0n n n xa RRx 1 1 n n n xna RRx 0 1 1n nn x n a RRx 1 8.3 1 x x n n 1 1 0 1x 2 x n n e n x 0 ! x 3 x n x n n n sin ! 12 1 12 0 x 4 x n x n n n cos ! 2 1 2 0 x 5 x n xn n n 1ln 1 1 1 0 11 x 6 xx n n n n 1 ! 11 1 1 11 x 2 3 1 xf 0x 0xx 0 0 0 !n n n xx n xf xf 0x xf 00x n n n x n f 0 ! 0 xf - Edited by 2005 10 31 2 xf Rxx 0 xRxx n xf xx xf xxxfxfxf n n n 0 02 0 0 000 !!2 xRn n 10 ! 1 1 0 00 1 n n n xx n xxxf xR 0 0 0 !n n n xx n xf xf Rxx 0 0lim xRn n Rxx 0 xf 0x 00x 1 0 0 n n n xxaxf Rxx 0 ! 0 n xf a n n ,2,1,0n 0 ! 1 n nx x n e x 0 12 ! 12 1sin n n n n x x x 1 ! 11 11 n nx n n x 1x 2 0 2 ! 2 1sincos n n n n x xx x 1 1 0 2 1 1 1 1 n n n n nxx xx 1x 3 0 0 2. ! 1 ! 12 n n nntx x n t n ee x 0 0 22 22 1 1 1 1 1 n n nnn xx xx 1x 4 x n nx dttdt t x 0 0 0 1 1 1ln 0 1 1 1 n nn n x 11 x 0 1 1 2ln n n n x n n nn nx n x dttdt t x 0 0 0 12 2 0 2 12 1 1 1 arctan 11 x 4 1arctan 12 1 0n n n Edited by 2005 10 1 1 ABBA CBACBA cBcABAc dAcAAdc AcddAc 00 ccA 0A s,,, 21 ssccc 2211 A TA A AA TT TTT BABA TT AccA 3 n n n ,, 2211 aa *00 0*0 00* E3 300 030 003 IE 100 010 001 *00 **0 *** AAT AAT BA c 34 12 01 00000 32000 15210 02014 0 0 0 0 3 1 4 0 A n A A 100 010 100 0 c Edited by 2005 10 2 A A B B i B d0,,0 ii B n n iii B 00 B cnn 0B n nn nn b b b b B 11 22 11 0 0000 *000 **00 ***0 **** 0 nnb iinn bb 01 1 0 00B nc c c 2 1 1000 000 0010 0001 , , , 22 11 nn cx cx cx nccc ,,, 21 nnnn n n aaa aaa aaa 21 22221 11211 A n A bcad dc ba n nnnn n n aaa aaa aaa 21 22221 11211 !n n !n nnjjj aaa 21 21 njjj 21 n,,2,1 nnjj aa 11 n jjj 211 njjj 21 n,,2,1 n n n jjj njjj jjj aaa 21 21 21 211 Edited by 2005 10 3 n nn n bbb b b b 21 2112 1 1 *** **0 *00 000 2 1 211 2 nn Cnn n ijM ija ij ji ij MA 1 D nn AaAaAaD 2222222121 1 AAT 2 c c AccA n 3 ,,,,,, 2121 321 ,,A 3 321 ,,B BABA 332211 ,,BA 332211 ,,BA 3322133221 ,,,, 4 5 0 0 6 0 7 BA B A B A 0 0 8 ji ij n aa aaa )( 111 11 2 nC A n m n 0A A D A D A D n,,, 21 iD A i nb b b 2 1 n 0A ? 0A 0A rBA 00 BA Edited by 2005 10 4 r b b b rB nn000 *00 **0 *** 22 11 0B 0iib , i rB | n d|0,,0 0nnb 0iib n i iibB 1 0 ErBA 0A 1 A B A B A B AB A nm B sn AB sm AB ji, A i B j njinjijiij bababaC 2211 BABABAA 2121 2121 ABABBBA cBAABcBcA BCACAB TTT ABAB 0AB 0A 0B 0A 00 BAB 0A CBACAB 2 n n A B AB n BAAB A n k k AAAA EA0 lklk AAA kllk AA kkk BAAB 01 1 1 axaxaxaxf k k k k A n EaAaAaAaAf kk k k 01 1 1 222 2 BABABA BABABA 22 22 BBAABA BAAB k i iiki k k BACBA 0 A EAEAEAA 3322 3 1 nm nA ,,, 21 n T nbbb ,,, 21 nnbbbA 2211 Edited by 2005 10 5 333322311 233222211 133122111 3 2 1 333231 232221 131211 321 ababab ababab ababab b b b aaa aaa aaa 332211 33 23 13 3 32 22 12 2 31 21 11 1 bbb a a a b a a a b a a a b mnmnm m nn nn bxaxa xa bxaxaxa bxaxaxa 22 11 22222121 11212111 A nA ,,, 21 T nxxx ,,1 nmnmm nn nn xaxaxa xaxaxa xaxaxa Ax 2211 2222121 1212111 Ax T mbbb ,,, 21 nnxxx 2211 2 CAB sB ,,, 21 srrrC ,,, 21 siAr ii ,,2,1, sAAAAB ,,, 21 , , , , 1 21 11 22 22221 11211 21 22221 11211 mnsnn s s mnmm n n c c c bbb bbb bbb aaa aaa aaa nniiiii bbbAr 2211 AB i ir A n,,, 21 B i AB i B A i TTT CAB A A A A AAE AEA kAkEA kAAkE k k 4 3 1 jiE , E ji, E ji, 5n 10000 00010 00100 01000 00001 4,2E 2 )(ciE 0c E i i 5n 10000 01000 00100 0000 00001 )(2 c cE 3 )(, cjiE E j c i E i c j Edited by 2005 10 6 5n 10000 01000 00100 00010 0001 )(4,1 c cE A A )(4,1,,,, 54321 cE 10000 01000 00100 00010 0001 ,,,, 54321 c 541321 ,,,, c 5 6 A B A B A B 1 BA, 4 2221 1211 AA AA A 2221 1211 BB BB B 1iA jB1 2iA jB2 2222122121221121 2212121121121111 BABABAAA BABABABA AB 2 kkA A A 00 00 00 22 11 kkkkkkkk BA BA BA B B B A A A 00 00 00 00 00 00 00 00 00 2222 1111 22 11 22 11 n A Atr A A TkTkT 1 TkTtr 1 1 CAB C A B BAxI BxAII A 0A I xEBA II TTT BxA TTT xEBA 2 0a a a 11 acab 1a cb A n n H EAH EHA A H A 1A Edited by 2005 10 7 A A CBACAB CBCABA n A 0A EAA 1 11 EAA A 0 A A 11 H EAx ExA 0A EAx B ExA C EAB ECA CABCBCAB A 1A EAx 1A 1AEEA A B n EBAEAB i A TA TT AA 1 1 kA kk AA 1 1 0c cA 1 1 1 A c cA 0AccA n EAA c cA c cA 11 11 ii A B n AB 111 ABAB A B n A B AB jiEjiE ,, 1 c iEciE 11 cjiEcjiE ,, 1 1, cjiE kkA A A A 000 000 000 000 22 11 iiA 1 1 22 1 11 1 000 000 000 000 kkA A A A 3 n A *A T ij nnnn n n A AAA AAA AAA A 21 22212 12111 * EAAAAA ** Edited by 2005 10 8 A A A AAA AAA AAA aaa aaa aaa nnnn n n nnnn n n 000 000 000 000 21 22212 12111 21 22221 11211 A E A A A * A A A *1 2n dc ba A ac bd A* bcad ac bd A 1 1* AAA EA A A * 11* A A A A A A AAA 1111 * 1 * n AA , ,** TT AA ** 1 AccA n , *,** ABAB kk AA ** AAA n 2 ** 2n dc ba A A dc ba dc ba A *** T k 1 * i) TT AA ** ** 1 1 AA ii) 111 , ABABABAB TT T *** ABAB kkk ABAB 1 2 3 4 5 BAx BAx 1 1 s,,, 21 s,,, 21 sccc ,,, 21 ssccc 2211 s,,, 21 s,,,0 21 si ,,, 21 sss xxx 221121 ,,, xs,,, 21 Edited by 2005 10 9 T sxxx ,,1 Ax A 2 st ,,,,,, 2121 si ,,, 21 srrrCAB ,,, 21 nA ,,, 21 nsrrr ,,,,,, 2121 st ,,,,,, 2121 C Cst ,,,,,, 2121 3211 322 2 323 32 311 221 001 ,,,, 321321 pst rrr ,,,,,,,,, 212121 pt rrr ,,,,,, 2121 3 s,,, 21 t,,, 21 ts ,,,,,, 2121 ts ,,,,,, 2121 1 s,,, 21 0 0 1 1 0 1 0 2 1 0 0 3 0 1 1 4 i sii ,,,,, 111 i 0 sccc ,,, 21 02211 ssccc s,,, 21 s,,, 21 01c ssccc 2211 s s c c c c 1 2 1 2 1 s,,, 21 011 sscc 01 scc s,,, 21 011 ssxx 0,,, 21 xs 1s 0x 0 2s 21, naaa ,,, 211 nbbb ,,, 212 21 , nn bababa ::: 2211 2 s n n1 ,, 01 n Edited by 2005 10 10 nA ,,, 21 0Ax 0A ns s,,, 21 0Ax n s s,,, 21 54321 ,,,, 421 ,, s,,, 21 ,,,, 21 s s,,, 21 ccc s ,,,1 0 011 ccc ss 0c scc ,,1 0 011 sscc s,,1 s s c c c c 1 1 s,,1 s1 s1 st ,,,, 11 st t,,1 sA ,,1 tB ,,1 ts C ACB 0Cx s t ts 0C 0ACB 0Bx t,,1 s,,, 21 ns s,,, 21 s1 s,,1 s,,1 st 11 t1 st s1 t1 ts s,,, 21 0 0 1 1 0 0 0 2 0 0 1 3 0 0 1 1 0 0 2 2 0 1 0 3 0 1 1 4 1 s,,, 21 I Edited by 2005 10 11 i I ii I I s,,, 21 III s1 s,,, 21 Is #,,, 21 s,,, 21 0 sns ,min,, 0 1 3,, 1 s 6421 ,,, 21, I I# I6421 ,,, I 2 s,,, 21 ss,,, 21 sss ,, ,,,, ,,, 12121 s,,, 21 I I ,,,, 21 s ,I ,,,1 IIs ss ss s ,,/,1,, ,,, ,,, 11 11 1 s,,1 sss ,, ,,, 11 stsst ,, ,,,,, ,,,, 11111 st ,, ,, 11 ts ,,,, 11 ttss ,, , ,, 1111 s,,, 21 s,,, 21 B Bs,, 1 3 ss ,,,,,,, 2121 0, 2211 ssxxx 02211 ssxxx 421 ,, 421 ,, 421 ,, 0 421 ,, ccc 0442211 ccc 0,,0,,0,, 421 ccc 02211 ssxxx 02211 ssxxx 0442211 ccc 321 ,, 42134213 2323 Edited by 2005 10 12 sA ,,, 21 sB ,,, 21 02211 ssxxx 0Ax 02211 ssxxx 0Bx s,,, 21 s,,, 21 0Ax 0Bx 0Ax 0Bx A B A nm A Ar CA 00000 01000 10110 20301 1001424 52712 12031 21301 A C || A C Ar A B B Ar AAr nmAr ,min0 00 AAr A mAr A nAr n A nAr A A 0A A 0Ax Ax 3 ArAr T 0c ArcAr BrArBAr BrArABr ,min A BrABr B ArABr BrABr ABAB 1 ABrBr 0AB nBrAr A B A BrABr B ArABr BrArnABr Edited by 2005 10 13 1 mnmnmm nn nn bxaxaxa bxaxaxa bxaxaxa 2211 22222121 11212111 2 Ax A 3 nnxxx 2211 n,,, 21 1 0Ax e,,, 21 eeccc 2211 00, 2211 eeii cccAA 2 0Ax e,,, 21 Ax eeccc 2211 Ax 121 eccc eeccc 2211 0Ax 021 eccc iA i eeee AcAcAccccA 22112211 eccc 21 21 , Ax 21 0Ax 0 Ax n Ax 0 0Ax Ax nnxxx 2211 n,,, 21 nn ,,,,,,, 2121 AA | AA | nAA | nAA | m mAmA ,| mA mA | nm nA 0Ax nA A nA A A 00 BAB CBACAB sB ,,, 21 sAAAB ,,1 0AB i 0iA i 0Ax 0Ax 0i 0CBA 0CB A BAB Edited by 2005 10 14 0ABx 0Bx 0ABx 0AB 0B 0Bx 1 0Ax J 0Ax J 0Ax e,,, 21 0Ax i 0Ax e,,, 21 0Ax e,,, 21 AnJ nAJ A B BA 0Bx A 00J An e,,, 21 0Ax Anl 0AB nBA sB ,,, 21 0AB i 0Ax AnJB s,,, 21 nBA e,,, 21 0Ax 0Ax eeccc 2211 ic 0 0Ax e,,, 21 0Ax Ax eeccc 22110 ic A n n A 20 11 A 0 1 1 0 1 1 1 2 0 1 0 1 A 2 1 1 0 A 2 2 1 1 A 2 3 1 2 A 1 0 A A A A E n A E A cccAcA Edited by 2005 10 15 221122112211 22 11 ccAcAcccA A A 2 A n A 0,A 0,0AE 0xAE A 0 AE 0 xAE AxE A A A AxE AxE n A n n ,, , 21 AxE AxE i 0 xAEi i 1 A 3 2 1 00 0 ** A 321 3 2 1 00 0 ** xxx x x x AxE 2 1Ar A Atr,0,,0,0 3 n A AErn A n ,, , 21 An 21 Atrn 21 4321 44434241 34333231 24232221 14131211 xxxx axaaa aaxaa aaaxa aaaax 3x 4321 3x 44332211 axaxaxax Atraaaa 44332211 4 A A A A Af xfAf 3 AAAA Edited by 2005 10 16 1222 AEA 1 2 4 24 235235 EAAA A 11A || * A A 111 AAA || **|| A AAA A n ,,, 2 1 Af nfff ,,, 2 1 A 1A n 1 ,, 1 , 1 2 1 *A n AAA 2 1 || ,, || , || TA n ,, , 21 AxEAxEAxE TT || 5 nA ,,, || 2 1 A EAAE 0 EA A 0Af 0cf cEA A f Af 0f ccf 0 A AcE n A B n n U BAUU 1 A B BA ~ UAAU AB UAAU AB i ABBA ~~ BAUU 1 1UBUA ii BA ~ CB ~ CA ~ BAUU 1 CBVV 1 CBVVAUVUVUVAUV 111 1 BA ~ A B BA AUAUAUUB 11 BA A B AxEUAxEUAUUxEBxE 11 A B A 1U B 11111 11 UAUUUUAU UUBA n A 10 11 A 2 11 0 0 AUU 1 21 EA n A U AUU 1 A A n nU ,,, 21 n AUU 000 000 000 000 2 1 1 Edited by 2005 10 17 nn n n UA ,,, 000 000 000 000 ,,, 2211 2 1 21 iiiA ni ,,2,1 A A AEn i AErn i 1 1 A n A U i 0xAEi n n,,1 nU ,,, 21 n AUU 000 000 000 000 2 1 1 i i 1 323121 2 3 2 2 2 1321 56423,, xxxxxxxxxxxxf n ji ji ij n i iiin xxaxaxxxf 2,,, 1 2 21 22 1 22 2 2 1 qppp xxxxx n n A T nxxxx ,,, 21 AxxT 3n 333231 232221 131211 aaa aaa aaa A 3 2 1 333231 232221 131211 321 ,, x x x aaa aaa aaa xxxAxxT 333232131 323222121 313212111 321 ,, xaxaxa xaxaxa xaxaxa xxx 3 1, ji jiij xxa A ji xx jiij aa 323121 2 3 2 2 2 1321 56423,, xxxxxxxxxxxxf AxxT A 311a 222a 133a 42112 aa 63113 aa 53223 aa A A Axxxxxf Tn,,, 21 A A 2 1 2 2 2 2 b y a x n nxxxf ,,, 21 Edited by 2005 10 18 nyyy ,,, 21 nxxx ,,, 21 nnnnnn nn nn ycycycx ycycycx ycycycx 2211 22221212 12121111 nnnn n n ccc ccc ccc C 21 22221 11211 nxxxf ,,, 21 nyy ,,1 nyyg ,,1 nxxf 1 T nyyyY ,,, 21 CYx n TTT n yygACYCYAxxxxf ,,11 nyyg ,1 ACC T ACCCACACC TTTT TT 3 n A B n C BACC T A B BA~ Axxxxf Tn1 BABYYyyg Tn ~1 1 A Q AQQD 1 DAQQAQQT 1 DA ~ DA~ 2 3 0 0 A 1 nxxxf ,,, 21 nxx ,,1 0 0,,, 21 nxxxf 22 22 2 1121 ,,, nnn xdxdxdxxxf 0id ni ,,1 11x 02 xxx 00,,0,1 1df 0id AxxT 0x 0AxxT Edited by 2005 10 19 n 000 000 00 0 000 2 1 0 i ni ,,1 2 nxxxf ,,, 21 nyyyg ,,, 21 BA~ A B ACCB T A 0x 0ACxCxACxCxBxx TTTT C 0x 0Cx A EA~ C CCA T A n A 0 A 0 A n rA A r rA A r r A 1 n , , nn b b b a a a 2 1 2 1 , nnbababa 2211, T 2 ,, ,,, 2121 2121 ,,, ccc ,,, 0, 00, n i ia 1 2, n i ia 1 2, 00 cc 1 0 0 1 0 1 0 2 2 0 2 2 0 1 1 , 0, Edited by 2005 10 20 n s,,, 21 n A EAAT 1AAT A A A naA ,,, 21 2 1 2 212 121 2 1 , , ,, nn n T AA n T EAA ,,, 21 c12 321 ,, 11 1 11 21 22 , , 122 k 111212 ,,, k 11 12 , , k 12 , 2 22 32 1 11 31 33 , , , , 1 1 1 2 2 2 3 3 3 321 ,, 321 ,, A A AErn A Q AQQ 1 0xAE 1 n nR n n V nR 1 21, V 21 V 2 V c c V V nR n 0AX nR 0AX AX nR nR s,,, 21 isss ccccL 221121 ,,, Edited by 2005 10 21 nR 2 V nR 0 0 V Vdim V V 0AX Arn ss rL ,,,,,,dim 2121 s,,, 21 k,,, 21 V V k,,, 21 kkccc 2211 kccc ,,, 21 k,,, 21 k 1 k,,, 21 kccc ,,, 21 kddd ,,, 21 k,,, 21 kkkk dddcccdcdcdc ,,,,,,,,, 21212211 2 k,,, 21 kccc ,,, 21 c k,,, 21 kk cccccccccc ,,,,,, 2121 V t,,, 21 k,,, 21 t,,, 21 t,,, 21 t,,, 21 3 k,,, 21 k,,, 21 V 1 k,,, 21 kiii ccc ,,, 21 kkkk k k ccc ccc ccc C 21 22221 11211 C k,,, 21 k,,, 21 Ckk ,,,,,, 2121 V k,,, 21 k,,, 21 T kxxxx ,,, 21 T kyyyy ,,, 21 xk,,, 21 k,,, 21 Cyy k,,, 21 Cyx 4 V k,,, 21 kccc ,,, 21 kddd ,,, 21 kk dcdcdc 2211, - Edited by 2005 10 1 . A A 1A 2A 11 )( i i i i APAP A n21, n PPP n 1 )()()( 21 A m21 , )()()( 21 m )()()( 21 mPPP n m A B )( )( AP ABP )/( ABP )( )( AP ABP nBBB ,,, 21 nBBB ,,, 21 ),,2,1(0)( niBP i n i iBA 1 )|()()|()()|()()( 2211 nn BAPBPBAPBPBAPBPAP 1B 2B nB A 1B 2B nB )(BiP i n n i iBA 1 0)(AP n j jj ii i BAPBP BAPBP ABP 1 )/()( )/()( )/( )( iBP 1i 2 n )/( ABP i 1i 2 n A 1B 2B nB A B )()()( BPAPABP A B A B 0)(AP )( )( )()( )( )( )|( BP AP BPAP AP ABP ABP A B A B A B A B 本人花万元报名参加北京一内部考研辅导班，该辅导班考前会发布押题，押题命中率百分之90左右， 去年该培训班考生全部高分过线。如果需要发布的押题可以联系我QQ673351717免费索取 来者不拒 一一发布 希望大家都能顺利高分通过2012研考 - Edited by 2005 10 2 n A A n A A A n p A A qp1 )(kPn n A )0( nkk X X ,,,, ,,,, | )( 21 21 k k k ppp xxx xXP X 0kp ,2,1k 1 1 k kp 0)( xXP X 0)( 0xXP X ],( ba X x )()( xXPxF )()()( aFbFbXaP ],( ba )(xF )(xF ,, 21 xx X kx )(xF kx ,1)(0 xF x )(xF 21 xx )( 1xF )( 2xF 0)(lim)( xFF x 1)(lim)( xFF x )()0( xFxF )(xF )0()()( xFxFxXP )(xF X )(xf x x dxxfxF )()( X )(xf X )(xf )(xF )()()()()()( 1221212121 xFxFxXxPxXxPxXxPxXxP 4 1° 0)(xf 2° 1)( dxxf - Edited by 2005 10 3 1)()( dxxfF 1 )(xf 1° 2° 3° )( 21 xXxP )()( 12 xFxF 2 1 )( x x dxxf 4° )(xf x )()( xfxF dxxfdxxXxP )()( kk pxXP )( )(),(, APAE )()()()( xXPxFxXX X 0)( xXP )( xX Ø hx x dxxfhxXxPxXP )()()( 0h 0)( xXP 0)( xXP Ø n A p A X X n,,2,1,0 knkk n n qpkPkXP C)()( nkppq ,,2,1,0,10,1 X n p ),(~ pnBX nknkk n n n nn pqpqpnpqqkXP X CC ,,,,,, | )( 2221 1n kk qpkXP 1)( 1.0k X e k kXP k ! )( 0 2,1,0k X )(~X ),min( ,2,1,0 ,)( nMl lk C CC kXP n N kn MN k M ,3,2,1,)( 1 kpqkXP k X )(xf ,0 , )( k xf ab 1 X 0 xb )(xf ,xe 0x , 0, 0x , )(xF ,1 xe 0x , ,0 x<0 - Edited by 2005 10 5 )( ixg iP )( ixg X ),( YX ),)(,( yxyxf D dxdyyxfDYXP ,),(}),{( .1),( dxdyyxf .),()(),()( dxyxfyfdyyxfxf YX ),,2,1,()},(),{( jipyxYXP ijji ,)|( i ij ij p p xXyYP )( ),( )|( yf yxf yxf Y )( ),( )|( xf yxf xyf X 0)(,0)( yfxf YX ,0 ),( 1 ),( Dyx S yxf D , 12 1 ),( 2 2 2 21 21 2 1 1 2 21 ))((2 )1(2 1 2 yyxx eyxf 1||,0,0,, 2121 ).,,, 22 2 1,21 ).(~),, 22,2 2 11 NY },{),( yYxXPyxF })(,)(|),{( 2121 yYxX ;1),(0 yxF - Edited by 2005 10 6 );0,(),(),,0(),( yxFyxFyxFyxF .1),(,0),(),(),( FxFyFF , 12 1 ),( 2 2 2 21 21 2 1 1 2 21 ))((2 )1(2 1 2 yyxx eyxf X P( kxX ) pk k=1,2, ,n n k kk pxXE 1 )( X f(x) dxxxfXE )()( 1 E(C)=C 2 E(CX)=CE(X) 3 E(X+Y)=E(X)+E(Y) n i n i iiii XECXCE 1 1 )()( 4 E(XY)=E(X) E(Y) X Y X Y 5 Y=g(X) n i kk pxgYE 1 )()( dxxxfXE )()( dxxfxgYE )()()( D(X)=E[X-E(X)]2 )()( XDX k kk pXExXD 2)]([)( dxxfXExXD )()]([)( 2 1 D(C)=0 E(C)=C 2 D(aX)=a2D(X) E(aX)=aE(X) 3 D(aX+b)= a2D(X) E(aX+b)=aE(X)+b 4 D(X)=E(X2)-E2(X) 5 D(X+Y)=D(X)+D(Y) X Y X Y D(X Y)=D(X)+D(Y) 2E[(X-E(X))(Y-E(Y))] E(X+Y)=E(X)+E(Y) 0-1 ),1( pB p )1( pp ),( pnB np )1( pnp )(P )( pG p 1 2 1 p p - Edited by 2005 10 7 ),,( NMnH N nM 1 1 N nN N M N nM ),( baU 2 ba 12 )( 2ab )(e 1 2 1 ),( 2N 2 ~B(n,p) knkk nn qpCkP )( !k e xk n N kn MN k M C CC kXP )( N nM 1)( kpqkXP p 1 2p q ~U[a,b] f(x)= ab 1 [a, b ] 2 ba 12 )( 2ab xe 1 2 1 ~N( ) 2 2 2 )( 2 1 )( x exf E(X)= 11 ),cov( YXXY ))].())(([(11 YEYXEXEXY XY X Y D X D Y XX YY (i) cov (X, Y)=cov (Y, X); )()( YDXD XY XY 1 1 0 0XY ,,,, 22 2 121 0 0XY - Edited by 2005 10 8 ),(),(),( ),(),( )],([ YXdxdyyxfyxG YXpyxG YXGE i j ijji . ,)( Xdxxpx Xpx u k i i k i k k .,2,1,))(( kXEXE kk . ,)())(( ))(( XdxxpXEx XpXEx u k i i k i k )( lkYXE klu ))].(())([( YEYXEXEu kkl 2 2 )( XP )( XP .1)( 11 lim 11 n i i n i i n XE n X n P .1 1 lim 1 n i i n X n P .1lim XP n .1lim p n P n .0lim p n P n .1 1 lim 1 n i i n X n P - Edited by 2005 10 9 ),2,1(0)(,)( 2 kXDXE kk n nX Y n k k n 1 x t n k k n n n dtex n nX PxF . 2 1 lim)(lim 21 2 )1,0( / N n X n x t n n dtex pnp npX P . 2 1 )1( lim 2 2 ),(, knp N M N knkk nN N kn MN K M pPC C CC )1( ).(N 0, npn e k pPC k knkk n ! )1( ).(n nxxx ,,, 21 nxxx ,,, 21 nxxx ,,, 21 . 1 1 n i ix n x n i i xx n S 1 22 .)( 1 1 .)( 1 1 1 2 n i i xx n S k n i k ik kx n M 1 .,2,1, 1 n i k ik kxx n M 1 .,3,2,)( 1 n i i XX n S 1 22 )( 1 * )(XE n XD 2 )( - Edited by 2005 10 10 22 )(SE 22 1 )*( n n SE n i i XX n S 1 22 )( 1 * nXXX ,,, 21 n i iXW 1 2 .0,0 ,0 2 2 1 )( 2 1 2 2 u ueu nuf un n 2 2 . 2 0 1 2 dxex n x n 2 ),(2 ii nY ).(~ 21 1 2 k k i i nnnYZ ),(~),1,0(~ 2 nYNX nY X T / 2 1 2 1 2 2 1 )( n n t n n n tf ).( t 2n n )(~),(~ 2 2 1 2 nYnX 2 1 / / nY nX F ,0 ,1 22 2 )( 2 2 1 1 2 2 2 1 21 21 21 1 1 y yy n n y n n nn nn yf nn n n 1 )()(1 ntnt ),( 1 ),( 12 211 nnF nnF X 2S nxxx ,,, 21 ),( 2N ).1,0(~ / N n x u def nxxx ,,, 21 - Edited by 2005 10 11 ),( 2N ),1(~ / nt nS x t def 2 nxxx ,,, 21 ),( 2N ),1(~ )1( 2 2 2 n Sn w def )1(2 n 2 nxxx ,,, 21 ),( 2N nyyy ,,, 21 ),( 22N ),1,1(~ / / 212 2 2 2 2 1 2 1 nnF S S F def ,)( 1 1 2 11 2 1 1n i i xx n S ;)( 1 1 2 12 2 2 2n i i yy n S )1,1( 21 nnF 11n 12n m,,, 21 ).,,,;( 21 mxF ),,2,1)(( mkXEv kk m,,, 21 ),,,( 21 mkk vv nxxx ,,, 21 X n k n i k ik x n v 1 1 ).,,2,1( mk n i m imm n i im n i im x n v x n v x n v 1 21 1 2 212 1 211 . 1 ),,,( , 1 ),,,( , 1 ),,,( ),,,( 21 m m,,, 21 X ),,,;( 21 mxf m,,, 21 nxxx ,,, 21 ),,,;(),,,( 1 11 22 n i mimn xfL Ln. X ),,,;(}{ 21 mxpxXP ),,,;(),,,;,,,( 1 111 222 n i mimn xpxxxL ),,,;,,,( 22 11 mnxxxL m,,, 21 m,,, 21 - Edited by 2005 10 12 m,,, 21 Ln m,,, 21 m,,, 21 lnx Ln lnLn mi L ii i n ,,2,1,0 ln ),,2,1)(,,,( 21 mixxx nii i Ln i ),,,,( 21 nxxx E = X E X D X x S2 E X D X E x =E X E S2 =D X ),,,,( 2111 nxxx ),,,,( 2122 nxxx 21 )( DD 21 n ,0)|(|lim n n P n nxxx ,,,, 21 ),,,,( 2111 nxxx ),,,,( 2122 nxxx )( 21 ],[ 21 )10(1 ,1}{ 21P ],[ 21 1 nxxx ,,,, 21 ),(~ 2NX 1 2 ],[ 21 i ii 1 iii ],[ 21 2 0 2 2 0 n i ix n x 1 1 ).1,0(~ /0 N n x u (ii) 1 )|(| uP 1 - Edited by 2005 10 13 .1 /2 n x P iii 2 )( nx ,00 n x n x n x n x 00 , 1 i nxxx ,,,, 21 ),( 2N 2 u n i i xx n S 1 22 )( 1 1 2 ).1(~ / nt nS x t (ii) 1 t )|(| uP 1 .1 / nS x P iii 2 )( nx , n S x n S x n S x n S x , 1 i nxxx ,,,, 21 ),( 2N n i i xx n S 1 22 )( 1 1 2 2 ).1(~ )1( 2 2 2 n Sn ii 1 2 21 2 .1)( 21P .1 )1( 22 2 1 Sn P iii 22 2 1 )1( Sn 1 2 2 2 2 )1()1( SnSn 1 2 S n S n 12 1 , 1 1