考研數(shù)學(xué)知識點總結(jié).pdf

- Edited by 200510 1 . 1 1dttfy x 0 tfxf dx dy 2dttfy x x 2 1 x 1 x 2 tf xxfxxf dx dy 1122 2 0limxf0limxgl xg xf lim 10lxfxg xgxf0xgxf 20lxfxg 31lxfxg xgxf 3 0x xx sinxx tanxx arcsinxx arctan 2 2 1 cos1xxxex1xx 1ln xx11 1 2 1 1 nn xx 1 nmxnn Axn n limmA 2 nn xx 1 nMxnn Axn n limMA 2xhxfxg AxglimAxhlimAxflim 3 11 sin lim 0 x x x 2e n n n 1 1lime u u u 1 1lim evv v 1 0 1lim 4 5 0x n n x x n xx xe0 ! 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 xx01 32 1ln 1 32 12 12 1 53 0 12 1 53 arctan n n n x n xxx xx nn xx n n xxx0 ! 11 ! 2 1 11 2 6 1 0 0 10limxf0limxg 2xxfxg 3A xg xf lim A xg xf lim xg xf lim xg xf lim 21xflimxglim 2xxfxg - Edited by 200510 2 3A xg xf lim A xg xf lim 7 0 00 0 limxf x xfxxf x 8 1 0 1 1 limdxxf n k f n n k n 1 0 xxfyxf 0 x 0 xxf 2 ba,xf 1xfba, xfba, 2xf ba,M m Mm Mxf 0 ba, 0 x ba,xMxf Mxfba, m 3xfba, Mmm Mcba, cf xfba,af bfba, 0f 1 0c 0cd 1 xxdxxxd 1 xxcossin xdxxdcossin xxsincos xdxxdsincos xx 2 sectan xdxxd 2 sectan xx 2 csccot xdxxd 2 csccot xxxtansecsec xdxxxdtansecsec xxxcotcsccsc xdxxxdcotcsccsc ax x a ln 1 log1, 0 aa ax dx xd a ln log1, 0 aa x x 1 ln dx x xd 1 ln aaa xx ln1, 0 aa adxada xx ln1, 0 aa - Edited by 200510 3 xx ee dxede xx 2 1 1 arcsin x x dx x xd 2 1 1 arcsin 2 1 1 arccos x x dx x xd 2 1 1 arccos 2 1 1 arctan x x dx x xd 2 1 1 arctan 2 1 1 cot x xarc dx x xdarc 2 1 1 cot 22 22 1 ln ax axx dx ax axxd 22 22 1 ln 22 22 1 ln ax axx dx ax axxd 22 22 1 ln 2 xgxfxgxf xgxfxgxfxgxf xg xgxfxgxf xg xf 2 0xg 3 ufyxuxxuf uxfyx xxf dx du du dy dx dy dxxxfduufdy duufdyu 4 txtyxyyt t0t t t dx dy 0t 32 2 1 t tttt dt dx dt dx dy d dx dx dy d dx yd 5 xfyygx 0xf ygfxf yg 11 0xf dx dy dx xf d dy ygd yg 1 1 33 ygf ygf xf xf 0xf 6 xyy0, yxFy 0, yxFxy y y 7 y xg xfy - Edited by 200510 4 xfxg ey ln 8 xf 0 xxf 0 x 9n2n , yy n y n 1 x ey xn ey 21, 0 aaay x n xn aayln 3xysin 2 sin n xy n 4xycos 2 cos n xy n (5) xyln n n n xny!11 1 n n k knkk n n xvxuCxvxu 0 ! ! knk n C k n xuxu 0 xvxv 0 xuxvn xf 1ba, 2ba, 3bfaf ba,0f xf 1ba, 2ba, ba, f ab afbf abfafbf ba xxxfxfxxf 000 10 0 xabx 1xfba,0xfxf ba, 2xfxgba, xgxfba,cxgxfc xfxg 1,ba 2ba,0xg ba, g f agbg afbf ba xxg 1n xf 0 xn xRxx n xf xx xf xx xf xfxf n n n 0 0 2 0 0 0 0 0 ! 2! 1 - Edited by 200510 5 0 xx n n xxxR 0 0 0 xx 0lim 0 0 n n xx xx xR n xxxe x 1ln,cos,sin,x1n 2n xf 0 xba,1n ba,nbax, xRxx n xf xx xf xx xf xfxf n n n 0 0 2 0 0 0 0 0 ! 2! 1 1 0 1 !1 n n n xx n f xR 0 xx 0 xn 0 0 xn 0limxRn n 1 xfba, 0 xba, 0 x 0 xxx 0 xfxf 0 xfxf 0 xxf 0 x 0 xxx 0 xfxf 0 xfxf 0 xxf 2 xf 0 x 0 xxf 0 0 xf x0 0 xf 0 xxf 3 xf 0 x 0 0xx 0 xf0 0 xf 1 00 ,xxx 0xf 00,x xx 0xf 0 xf 0 x 2 00 ,xxx 0xf 00,x xx 0xf 0 xf 0 x 3 00 ,xx 00,x x xxf 0 xf 0 x 4 xf 0 x0 0 xf 0 0 xf 0 0 xf 0 xf 0 x 0 0 xf 0 xf 0 x - Edited by 200510 6 1xfba, xfba, k xx, 1 bfafxfxf k , 1 bfafxfxf k , 1 xfba,M xfba,m 2 1 xfI 21,x x 21 21 21 21 2 1 22 1 2 xfxf xx fxfxf xx f xfI xfy xfy xfy xfy 2 3 xfba,xf ba,x0xf xfyba, ba,x0xf xfyba, xfy xf 1 x 2 x k x 1 xf ax limxf ax lim axxfy 2 bxf x limbxf xlim byxfy 3 0lima x xf x baxxf x lim 0lima x xf x baxxf x lim baxyxfy xfyyxM, 2 3 2 1y y k0k k R 1 yxM, MD RMDDDR 1C x dxx 1 1 1 - Edited by 200510 7 2Cxdx x ln 1 3Ca a dxa xx ln 1 1, 0 aa Cedxe xx 4Cxxdxsincos 5Cxxdxcossin 6Cxdx x xdxtan cos 1 sec 2 2 7Cxdx x xdxcot sin 1 csc 2 2 8Cxxdxxsecsectan 9Cxxdxxcsccsccot 10Cxxdxcoslntan 11Cxxdxsinlncot 12Cxxxdxtanseclnsec 13Cxxxdxcotcsclncsc 14C a x xa dx arcsin 22 0a 15C a x axa dx arctan 1 22 0a 16C xa xa axa dx ln 2 1 22 0a 17Caxx ax dx 22 22 ln 0a 1 CuFduufx duuf xu xdxfdxxxf CxFCuF 1baxdbaxf a dxbaxf 1 0a 2baxdbaxf na dxxbaxf nnnn 1 1 0, 0 na 3xdxf x dx xflnlnln 4 x d x f x dx x f 111 2 5xdxf x dx xf2 6 xxxx adaf a dxaaf ln 1 1, 0 aa xxxx edefdxeef 7xdxfxdxxfsinsincossin 8xdxfxdxxfcoscossincos 9xdxfxdxxftantansectan 2 10xdxfxdxxfcotcotcsccot 2 11xdxfxdxxxfsecsectansecsec 12xdxfxdxxxfcsccsccotcsccsc 13xdxfdx x xf arcsinarcsin 1 arcsin 2 14xdxfdx x xf arccosarccos 1 arccos 2 15xdxfdx x xf arctanarctan 1 arctan 2 16xarcdxarcfdx x xarcf cotcot 1 cot 2 - Edited by 200510 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 20Cxfdx xf xf ln 0xf 2 tx0t CtGdtttf CxGCtGdtttf tx dxxf 1 xt 1 tx x n baxx n dcx bax x ebae x txg n tx tx 0 2 ACBxAx tCBxAx2tx 0A 2 2 0 lxxA0A 2 0 2 xxlA 22 xa taxsin 22 xa taxtan 22 ax taxsec 3 xuxv xduxvxvxuxdvxu dxxvxuxvxudxxvxu xu xv 1 ax n exPaxxPnsinaxxPncos xPnnan ax eaxsinaxcosxv xu 2xxPnlnxxPnarcsinxxPnarctan xPnnxPnxvxln xarcsinxarctanxu 3bxeaxsinbxeaxcos 4 - Edited by 200510 9 dx 1 1 b a a b dxxfdxxf 20 a a dxxf 3 b a b a b a dxxfkdxxfkdxxfkxfk 22112211 4 b c c a b a dxxfdxxfdxxfcba, 5baxgxfbxa b a b a dxxgdxxf 6baMxfmbxa abMdxxfabm b a 7ba b a b a dxxfdxxf 8 xfba, ba, abfdxxf b a b a dxxf ab 1 xfba, 9 0 a a dxxff aa a dxxfdxxf 0 2f 10 xfTa TTa a dxxfdxxf 0 1 xfba, x a dttfxF bax, 1xfba, x a dttfxF ba, 2xfba, x a dttfxF ba,xfxF x x dttfxF 2 1 xx 21 , xf xxfxxfxF 1122 2 xfba,xFxfba, aFbF a b xFdxxf b a xfba, xfba, 1 xfba,tx 1t, 2abt bta b a dtttfdxxf 2 xvxu,ba, dxxvxuxvxudxxvxu b a b a b a b a b a b a xduxvxvxuxdvxu - Edited by 200510 10 1 I dxxyxyS b a 121 xyxy 12 bax, II dyyxyxS d c 122 yxyx 12 dcy, 2 I drS 2 1 2 1 II drrS 2 1 2 22 2 1 3 C ty tx tabt, ,t0t Cbxax,x dtttydxS b a 1 xyybxaxy dxxyS b a 2 1 dxxydS 2 1 2 rrr, drrS 22 3 tyy txx Cttxty , CdttytxS 22 1 z czdzzdzcz zzS d c dzzSV 2 1xyy0ax bxx x - Edited by 200510 11 dxxyV b a x 2 y b a y dxxxyV2 2yxx0cy dyy y dyyxV d c y 2 x d c x dyyyxV2 AB Cxx S 1ABxyybxa dxxyxyS b a 2 12 2ABrr drrrS 22 sin2 3ABtxxtyy t dttytxtyS 22 2 1 10yQyQxP dx dy CdxxP yQ dy 2 0 2211 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 200510 12 20, 0bacbyaxf 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 , xuyv 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 dxxP Cey c 2 xQyxP dx dy dxxP exCy xC CdxexQey dxxPdxxP 3 1 , 0yxQyxP dx dy 1 yz xQzxP dx dz 11 4 xyPyQdx dy1 yQxyP dy dx yx 1 0,dyyxQdxyxP y P x Q Cyxu, yxu,dyyxQdxyxPyxdu, yxu, yxdudyyxQdxyxP, 1 2 22 yx dydyxdx 2 2 22 yx dydyxdx - Edited by 200510 13 3xydxdyydx 4xyd 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 13 222 22 1 2 1 yx d yx ydyxdx 14 222 22 1 2 1 yx d yx ydyxdx 15 22 2 22 arctan 2 1 1 yxd yx ydyxdx 16 22 2 22 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, yCyC 2 0,dyyxQdxyxP y P x Q yxR, 0,dyyxQyxRdxyxPyxR y RP x RQ yxR, yxdudyyxQyxRdxyxPyxR, Cyxu, - Edited by 200510 14 xfy n n n n xCxCdxxfy 2 1 1 yxfy, pypy pxfp, 1 ,Cxgp 1 ,Cxgy 21 ,CdxCxgy yyfy, pypy dy dp p dx dy dy dp dx dp y yy pyf pdy dp , 1 , 1 Cygp 1 ,Cyg dx dy 2 1 , Cx Cyg dy 0yxqyxpy 1 xfyxqyxpy 2 1xy1xy2 xyCxyC 22111 C 2 C xyxy 21 xy1xy2 xyCxyCy 2211 2xy1xy2 xyxy 21 3xy xy xyxy 4y xyCxyC 2211 1 C 2 C xyCxyCxyy 2211 5xy1xy2 xfyxqyxpy 1 xfyxqyxpy 2 xyxy 21 xfxfyxqyxpy 21 1 0qyypy pq 0 2 qp 104 2 qp 12 - Edited by 200510 15 xx eCeCy 21 21 204 2 qp 21 x exCCy 1 21 304 2 qp i xCxCey x sin cos 21 2n 0 1 2 2 1 1 ypypypypy nn nnn nipi, 2 , 1 0 1 2 2 1 1nn nnn pppp 1n n , 21 x n xx n eCeCeCy 21 21 2 0 knk xk k exCxCC 0 1 21 3ik nk2 xxDxDDxxCxCCe k k k k x sin cos 1 21 1 21 xfqyypy qp, xyCxyCyy 2211 xyCxyC 2211 y xfy yxfy 1xPxf n xPnn 10 nn nn n axaxaxaxRy 1 1 10 niai, 2 , 1 , 0 20xxRy n 30xRxy n 2 2 x n exPxfxPnn 1 x n exRy 2 x n exxRy 3 x n exRxy 2 3xexPxf x n sin xexPxf x n cos xPnn, 1i 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 200510 16 2i xxTxxRxey nn x sin cos 0 1 11 1 ypyxpyxpyx nn nnnn nipi, 2 , 1n t exty 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 tttt2 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 ba baba , cos 332211 bababa ba, ba, ba 0 baab ajba b Pr 0 4ba ba baba , sin baba, 321 321 bbb aaa kji ba baabbababa, 5cbacba, 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 332211 cos bbbaaa bababa ab 0ba 0 332211 bbbaba - Edited by 200510 17 ab 0ba 3 3 2 2 1 1 b a b a b a 1 npnm, 2 000 ,zyxM CBAn, 0 000 zzCyyBxxA 0 0 rrn zyxrzyxr, 0000 3 0DCzByAx CBA,zyx, CBAn, 0CzByAx 0DByAxz 0DAxyOz 0xyOz 4 111 ,zyxA 222 ,zyxB 333 ,zyxC CBA, 0 131313 121212 111 zzyyxx zzyyxx zzyyxx 5 L 0 0 2222 1111 DzCyBxA DzCyBxA L 0 2222211111 DzCyBxAkDzCyBxAk 0 , 0, 21 kk 6 0: 11111 DzCyBxA 0: 22222 DzCyBxA 12 2 2 2 2 2 2 2 1 2 1 2 1 212121 cos CBACBA CCBBAA 0 212121 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 ,zyxMM d 222 111 CBA DCzByAx d 1 SL 2 n zz m yy l xx 000 000 ,zyxnml, 3 - Edited by 200510 18 ntzz mtyy ltxx 0 0 0 tnmls, 4 111 ,zyxA 222 ,zyxB AB 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 1 L 2 L 2 2 2 2 2 2 2 1 2 1 2 1 212121 cos nmlnml nnmml l 0 212121 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 0000 00 , , , , lyxgradf yxl f 00 00 , , lyxgradflyxgradf,cos, 0000 1vufz,yxuu,yxvv, - Edited by 200510 19 x v v z x u u z x z y v v z y u u z y z 2zyxfu,yxzz, y z ff y u x z ff x u zy zx 3zyxfu,xyyxzz xzfxyff dx du zyx 4vufw,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 1yxzz, z x F F x z z y F F y z 2zyxx, 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 0 k kk yxf, 0 k 0 k kk yxf, 0, kkxx yxf kk yxf, 0, kkxx yxf kk yxf, 2n n xxfu, 1 0, 0, 1 11 nm n xx xx nm n m i iinmn xxxxfxxFF, 1 1 111 - Edited by 200510 20 0, 0, 0 0 1 11 1 1 nm n x x xxF xxF F F m n lkxx k n k , 2 , 1, 1 I xyxbxayxD 21 , x 1 x 2 ba,yxf,D DD dxdyyxfdyxf, b a x x dyyxfdx 2 1 , II yxydycyxD 21 , y 1 y 2 dc,yxf,D DD dxdyyxfdyxf, d c y y dxyxfdy 2 1 , III DID IIDD III DD D 21 ,D 12 , sin,cos,fyxfD DD ddfdyxf sin,cos, 2 1 sin,cosdfd I 21 ,20,D 21 ,2 , 0 - Edited by 200510 21 sin,cos,fyxfD DD ddfdyxf sin,cos, 2 0 sin,cos 2 1 dfd II 0 ,D , sin,cos,fyxfD DD ddfdyxf sin,cos, 0 sin,cosdfd III 0 ,20,D 2 , 0 sin,cos,fyxfD ddfdyxf DD sin,cos, 2 00 sin,cosdfd 1 dyxfyxfV D , 12 DSxy yxfz, 2 yxfz, 1 2 D d y z x z A 2 2 1 DSxyS yxzz, 1 1 Dyxyxzzyxzzyx, 21 Dxy yxzyxz, 21 Dzyxf, yxz yxz D dzzyxfdxdydvzyxf , , 2 1 , 2zDyxzzyx, zDz zD dxdyzyxfdzdvzyxf, 2 dzddzfdxdydzzyxf ,sin,cos, yx,z - Edited by 200510 22 3 cos sinsin cossin z y x 20 0 0 dxdydzzyxf, dddf sincos,sinsin,cossin 2 , L zyxf,Ln n SSS, 21 nkSk1 kkk s, n k kkkk Ssf 1 0 ,lim k Sk k nk S 1 max zyxf,L L dSzyxf, L L dSzyxf, 2 Ltxxtyy tzzt dttztytxtztytxfdSzyxf L 222 , zyxf,txtytz AB L zyxP,zyxQ,zyxR,L Ln n SSS, 21 nkSk1 111 , kkk zyx kkk zyx,L 1kkk xxx 1kkk yyy 1kkk zzz nk1 k S kkk s, n k kkkkkkkkkkkk zsRysQxsP 1 0 ,lim n zyxP,zyxQ,zyxR,L L dzzyxRdyzyxQdxzyxP, L dSF zyxRzyxQzyxPF, dzdydxdS, LL - Edited by 200510 23 2 Ltxxtyy tzzAB zyxP,zyxQ,zyxR,tx tytz AB L dzzyxRdyzyxQdxzyxP, dttztztytxRtytztytxQtxtztytxP, 1 AB L yxP,yxQ,L AB AB dsyxQyxPdyyxQdxyxPcos,cos, coscosyx, AB 2 AB L zyxP,zyxQ,zyxR,L AB dzzyxRdyzyxQdxzyxP, dszyxRzyxQzyxP AB cos,cos,cos, coscoscosAB zyx,AB 1 xyD LLD LyxP,yxQ,D L D QdyPdxdxdy y P x Q 2 xyD1n n n CCCL 100 C n CC, 1 LD yxP,yxQ,D - Edited by 200510 24 LD QdyPdxdxdy y P x Q 0 1 C n k Ck QdyPdxQdyPdx yxQyxPF,yxP,yxQ, D 1DL 0 L QdyPdx 2 AB LD AB QdyPdx AB AB L 3yxdudyyxQdxyxP, 4D y P x Q 5yxQyxPF, yxV,gradVFyxV, x V P y V Q 1 Szyxf,S Sn n SSS, 21 )1 (nkSk kkk s, k S k S k n k kkk Ssf 1 0 ,lim zyxf,S S dSzyxf, 2 SDyxyxzz,yxz, D zyxf,S DS dxdy y z x z yxzyxfdSzyxf 2 2 1, 1 S) zyxP,zyxQ,zyxR,S Sn n SSS, 21 nkSk1yz yz k S zx zxk Sxy xy k SnkSk1 kkk s, n k xykkkkzxkkkkyzkkkk SsRSsQSsP 1 0 ,lim zyxP,zyxQ,zyxR, S - Edited by 200510 25 S dxdyzyxRdzdxzyxQdydzzyxP, RQPF,dxdydzdxdydzdS, S dSF 2 Syxzz, xy Dyx, yxz, xy DzyxR,S xy DS dxdyyxzyxRdxdyzyxR, Sz xy Szyxx, yz Dzy, YZ DS dydzzyzyxPdydzzyxP, Sx S xzyy, zx Dxz, ZX DS dzdxzxzyxQdzdxzyxQ, Sy RQP,0 SS dSRQPRdxdyQdzdxPdydzcoscoscos cos,cos,cosSzyx, RQPF, cos,cos,cos 0 n SS dSnFRdxdyQdzdxPdydz 0 1 S zyxRzyxQzyxP, S RdxdyQdzdxPdydzdv z R y Q x P S dSRQPcoscoscos cos,cos,cosSzyx, 2 1n 0 S nkSk1 0 S zyxRzyxQzyxP, 0 S RdxdyQdzdxPdydzdv z R y Q x P n k SK RdxdyQdzdxPdydz 1 LSL LS zyxRzyxQzyxP,S - Edited by 200510 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 divFF S dSnFdivFdv 0 cos,cos,cos 0 n 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,cos 0 n 1 1 1n n u 1n n vba, 1n nn bvau 11n n n n vbua 2 3 4 1n n u0lim 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 n n 1 1 n n 0lim n n u 1n n u 2 - Edited by 200510 27 1 0n n ar0a 1r r a ar n n 1 0 1r 0n n ar 2p 1 1 n p n 1p 1 1 n p n 1p 1 1 n p n 1p 1 1 n p n 6 1 2 1 2 n n ), 3 , 2 , 1(0 nun 1n n u , 3 , 2 , 1 1 nSS nn n S n S 1n n u n S 1 0cNn0 nn ucv 1n n v 1n n u 1n n u 1n n v 2 0 n u0 n v, 3 , 2 , 1n A v u n n n lim 1A0 1n n u 1n n v 20A 1n n v 1n n u 3A 1n n u 1n n v 3 0 n u n n n u u 1 lim 11 1n n u 21 1n n u 31 n n n u u 1 lim 4 0 n u n n n ulim 11 1n n u - Edited by 200510 28 21 1n n u 31 1 1 0 n u 1 1 1 n n n u 2 1 1 1 n n n u 1, 3 , 2 , 1 1 nuu nn 20lim n n u 1 1 1 n n n u 1 1 1 10uu n n n 1 1n n u 1n n u 2 1n n u 1n n u 1n n u 1n n u 1n n u 3 1 2 12 1 n nn uu 12 1 n nn uu 4 1 1 1 n p n n 11p 1 1 1 n p n n 210p 1 1 1 n p n n 30p, 1 1 1 n p n n 1 xun, 3 , 2 , 1nI xu n n 1 I 2 I 0 x 0 1 xu n n0 x 1n n xu 1 0 n n xu 0 x 1n n xu 1n n xu 3 1n n xux - Edited by 200510 29 1n n xuxS x )(xS 1n n xu 1 1 0 n n n xxa)( 0 xx ), 2 , 1 , 0(nan 0 0 x 0n n nx ax 0n n nx a 0 0 n n n xxa 2 0n n nx a 1),( 0n n nx ax R 2 0n n nx a 0R 3),(RRRR,(),RR RR, R)0(R R12 ) 3( R l a a n n n 1 limla n n n lim l R 1 l0R0l R 1 )( 0 xfxa n n n1 Rx 0n n n xgxb 2 Rx 0n n nn xgxfxba 21, minRRx 0 00 00 )( n n nknkn n n n n n n xgxfxbababaxbxa 21, minRRx 2 0n n nx a0R 0n n nx axS 1xSRR, 01 1 0nn n n n n n n n xnaxaxaxS xS - Edited by 200510 30 RR, kn kn k xknnnxS11 Rx, 3 , 2 , 1k 2xSRR, 0 1 0 00 1 n 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 1 n nn R R n a dxxS 0 0 1 1 R n n n R n a dxxS iii 1 1 n n nx naRRx RSRna n n n 1 1 RS 0n n nx aRRx 1 1 n n nx naRRx 0 1 1 n nn x n a RRx 18.3 1 x x n n 1 1 0 1x 2 x n n e n x 0 ! x 3x n x n n n sin ! 12 1 12 0 x 4x n x n n n cos ! 2 1 2 0 x 5x n xn n n 1ln 1 1 1 0 11x 6xx n n n n 1 ! 11 1 1 11x 2 3 1 xf 0 x 0 xx 0 0 0 ! n n n xx n xf xf 0 x xf 0 0 x n n n x n f 0 ! 0 xf - Edited by 200510 31 2 xfRxx 0 xRxx n xf xx xf xxxfxfxf n n n 0 0 2 0 0 000 ! 2 xRnn 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 0limxRn n Rxx 0 xf 0 x 0 0 x 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 n x n n x 1x 2 0 2 ! 2 1sincos n n n n x xxx 1 1 0 2 1 1 1 1 n n n n nxx x x 1x 3 00 2. ! 1 ! 1 2 nn nntx x n t n eex 00 22 22 1 1 1 1 1 nn n n n xx xx 1x 4 x n n x dttdt t x 0 0 01 1 1ln 0 1 1 1 n n n n x 11x 0 1 1 2ln n n n x nn n n n x n x dttdt t x 0 00 12 2 0 2 12 1 1 1 arctan 11x 4 1arctan 12 1 0n n n Edited by 200510 1 1 ABBA CBACBA cBcABAcdAcAAdc AcddAc 00ccA0A s , 21 ss ccc 2211 A T AA AA T T TT T BABA T T AccA 3n nn , 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 31 40 An AA 100 010 100 0c Edited by 200510 2 AA B B iBd0 , 0 iiB n n iii B 00 Bcnn 0 Bn nn nn b b b b B 11 22 11 0 0000 *000 *00 *0 * 0 nn b iinn bb0 1 1 0 00 B n c c c 2 1 1000 000 0010 0001 , , , 22 11 nn cx cx cx n ccc, 21 nnnn n n aaa aaa aaa 21 22221 11211 AnA bcad dc ba n nnnn n n aaa aaa aaa 21 22221 11211 ! n n! n n njjj aaa 21 21 n jjj 21 n, 2 , 1 n njj aa 1 1 n jjj 21 1 n jjj 21 n, 2 , 1 n n n jjj njjj jjj aaa 21 21 21 21 1 Edited by 200510 3 n nn n bbb b b b 21 2112 1 1 * *0 *00 000 2 1 211 2 nn Cnn n ij M ij a ij ji ij MA1 D nnA aAaAaD 2222222121 1AAT 2cc AccA n 3 , 2121 321 ,A3 321 ,B BABA 332211 ,BA 332211 ,BA 3322133221 , 4 50 0 6 0 7BA B A B A0 0 8 ji ij n aa aaa )( 111 11 2 n C An mn 0A A D A D A D n , 21 i DAi n b b b 2 1 n 0A ?0A 0A rBA 00BA Edited by 200510 4 r b b b rB nn 000 *00 *0 * 22 11 0B0 ii b, i rB |n d|0 , 00 nn b0 ii b n i ii bB 1 0 ErBA 0A 1 AB ABAB AB AnmBsnABsm ABji,AiBj njinjijiij bababaC 2211 BABABAA 2121 2121 ABABBBA cBAABcBcA BCACAB TT T ABAB 0AB0A0B 0A00BAB 0ACBACAB 2n nABABn BAAB An k k AAAAEA0 lklk AAA kl l k AA kk k BAAB 01 1 1 axaxaxaxf k k k k An EaAaAaAaAf k k k k01 1 1 22 2 2BABABA BABABA 22 22 BBAABA BAAB k i iiki k k BACBA 0 A EAEAEAA332 2 3 1nm n A, 21 n T n bbb, 21 nn bbbA 2211 Edited by 200510 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 1121211