考研数学知识点总结.pdf

1、- Edited by 2005 10 1. 1 1 dttfyx0tf xfdxdy2 dttfyxx21x1x2tfxxfxxfdxdy11222 0lim xf 0lim xg lxgxflim 1 0l xf xgxgxf 0 xg xf2 0l xf xg 3 1l xf xgxgxf 3 0x xxsin xxtan xxarcsin xxarctan 221cos1 xx xex1 xx 1lnxx 111 2 1 1nnxx1n mxnnAxnnlim mA 2nnxx1n MxnnAxnnlim MA 2 xhxfxg Axglim Axhlim Axflim 3 1 1si

2、nlim0xxx2 ennn11lim euuu11limev vv101lim 4 50xnnxxnxxxe 0!2121212530!121!5!3sinnnnxnxxxxx nnnxnxxxx22420!21!4!21cos nnnxnxxxxx 01321ln1321212153012153arctannnnxnxxxxx nnxxnnxxx 0!11!211126 1001 0lim xf 0lim xg 2 x xf xg 3 Axgxflim Axgxflim xgxflimxgxflim 2 1 xflim xglim 2 x xf xg - Edited by 2005 10

3、 23 Axgxflim Axgxflim 7 0000lim xfxxfxxfx 8 1011lim dxxfnkfnnkn 1 0x xfy xf0x0x xf2 ba, xf1 xf ba,xf ba, 2 xfba, Mm M m Mxf0ba,0xba, x MxfM xf ba,m 3 xf ba,M m mM c ba,cf xf ba, afbf ba, 0f 1 0c 0cd 1xx dxxxd1xx cossin xdxxd cossin xx sincos xdxxd sincos xx2sectan xdxxd2sectan xx2csccot xdxxd2csccot

4、 xxx tansecsec xdxxxd tansecsec xxx cotcsccsc xdxxxd cotcsccsc axxaln1log 1,0 aa axdxxdalnlog 1,0 aa xx1ln dxxxd1ln aaaxxln 1,0 aa adxadaxxln 1,0 aa - Edited by 2005 10 3xxee dxedexx211arcsinxx dxxxd211arcsin 211arccosxx dxxxd211arccos 211arctanxx dxxxd211arctan 211cotxxarc dxxxdarc211cot 22221lnaxa

5、xx dxaxaxxd22221ln22221lnaxaxx dxaxaxxd22221ln 2 xgxfxgxf xgxfxgxfxgxf xgxgxfxgxfxgxf20xg 3 ufy xu x x ufu xfy xxxfdxdududydxdydxxxfduufdy duufdy u4 tx ty xyy tt 0t ttdxdy0t 3221tttttdtdxdtdxdyddxdxdyddxyd5 xfy ygx0xf ygfxfyg110xf dxdydxxfddyygdyg1133ygfygfxfxf0xf 6 xyy 0,yxF y0,yxF x yyy 7 y xgxfy-

6、 Edited by 2005 10 4xfxgeyln8 xf0x xf0x 9 n 2n ,yynyn 1xey xney 2 1,0 aaayxnxnaay ln 3 xy sin 2sinnxyn4 xy cos 2cosnxyn(5) xy ln nnnxny !111n nkknkknnxvxuCxvxu0!knknCknxuxu0xvxv0xu xv n xf 1 ba, 2 ba, 3 bfaf ba, 0f xf 1 ba, 2 ba, ba, fabafbfabfafbf ba xxxfxfxxf00010 0x a b x 1 xf ba, 0xf xfba, 2 xf

7、xg ba,xgxf ba, cxgxf cxf xg 1 , ba 2 ba, 0xg ba, gfagbgafbfba xxg1 n xf0x n xRxxnxfxxxfxxxfxfxfnnn00200000!2!1- Edited by 2005 10 50xx nnxxxR00 0xx0lim00nnxxxxxRnxxxex1ln,cos,sin, x1 n2 n xf0x ba, 1nba, n bax , xRxxnxfxxxfxxxfxfxfnnn00200000!2!1101!1nnnxxnfxR0x x0x n00x n 0lim xRnn1 xf ba,0x ba,0x0x

8、xx0xfxf0xf xf0x xf 0x0xxx0xfxf0xf xf0x xf 2 xf0x0x xf00xf x 00xf0x xf3 xf0x00 xx0xf 00xf 1 00,xx x0xf00,xx x0xf0xf0x 2 00,xx x0xf00,xx x0xf0xf0x 3 00,xx00,xxx xf0xf0x4 xf0x 00xf00xf 00xf0xf0x 00xf0xf0x - Edited by 2005 10 61 xf ba, xf ba,kxx ,1bfafxfxfk,1bfafxfxfk,1xf ba, Mxf ba, m 2 1 xf I21,xx2121

9、2121212212xfxfxxfxfxfxxfxf I xfyxfy xfyxfy 2 3 xf ba, xf ba, x 0xfxfy ba, ba, x 0xfxfy ba, xfy xf 1x2xkx 1 xfaxlim xfaxlim ax xfy 2 bxfxlim bxfxlim by xfy 3 0lim axxfxbaxxfxlim 0lim axxfxbaxxfxlim baxy xfy xfy yxM ,2321 yyk 0kkR1yxM ,M DRMD D D R1 Cxdxx111 - Edited by 2005 10 72 Cxdxxln13 Caadxaxxln

10、11,0 aa Cedxexx4 Cxxdx sincos 5 Cxxdx cossin 6 Cxdxxxdx tancos1sec227 Cxdxxxdx cotsin1csc228 Cxxdxx secsectan 9 Cxxdxx csccsccot 10 Cxxdx coslntan 11 Cxxdx sinlncot 12 Cxxxdx tanseclnsec 13 Cxxxdx cotcsclncsc 14 Caxxadxarcsin220a 15 Caxaxadxarctan1220a 16 Cxaxaaxadxln21220a 17 Caxxaxdx2222ln 0a 1 Cu

11、Fduuf x duufxuxdxfdxxxf CxFCuF 1 baxdbaxfadxbaxf10a 2 baxdbaxfnadxxbaxfnnnn110,0 na 3 xdxfxdxxf lnlnln 4xdxfxdxxf11125 xdxfxdxxf 2 6xxxxadafadxaafln11,0 aa xxxxedefdxeef 7 xdxfxdxxf sinsincossin 8 xdxfxdxxf coscossincos 9 xdxfxdxxf tantansectan210 xdxfxdxxf cotcotcsccot211 xdxfxdxxxf secsectansecsec

12、 12 xdxfxdxxxf csccsccotcsccsc 13 xdxfdxxxfarcsinarcsin1arcsin214 xdxfdxxxfarccosarccos1arccos215 xdxfdxxxfarctanarctan1arctan216 xarcdxarcfdxxxarcfcotcot1cot2- Edited by 2005 10 817xdxfdxxxf1arctan1arctan11arctan21822222222lnlnlnaxxdaxxfdxaxaxxf0a 1922222222lnlnlnaxxdaxxfdxaxaxxf0a 20 Cxfdxxfxfln 0

13、xf 2 tx 0tCtGdtttf CxGCtGdtttftxdxxf1xt1tx xnbax xndcxbaxxe baextxgntxtx 0 2ACBxAxtCBxAx2tx0A220lxxA 0A202xxlA 22xa tax sin 22xa tax tan 22ax tax sec 3 xu xv xduxvxvxuxdvxu dxxvxuxvxudxxvxu xuxv 1axnexP axxPnsin axxPncosxPnn a naxe axsin axcos xvxu 2 xxPnln xxPnarcsin xxPnarctanxPnn xPnxv xlnxarcsin

14、 xarctan xu3 bxeaxsin bxeaxcos4- Edited by 2005 10 9dx 1 1baabdxxfdxxf 2 0aadxxf 3bababadxxfkdxxfkdxxfkxfk221122114bccabadxxfdxxfdxxf c ba,5 ba xgxf bxa babadxxgdxxf 6 ba Mxfm bxa abMdxxfabmba7 bababadxxfdxxf 8 xf ba,ba, abfdxxfbabadxxfab1xf ba,9 0aadxxf f aaadxxfdxxf02 f 10 xf T a TTaadxxfdxxf01 xf

15、 ba,xadttfxFbax , 1 xf ba,xadttfxFba, 2 xf ba,xadttfxFba, xfxF xxdttfxF21xx21,xf xxfxxfxF11222 xf ba, xF xf ba,aFbFabxFdxxfbaxf ba,xf ba,1 xf ba, tx 1 t , , 2 a b tbta badtttfdxxf 2 xvxu , ba, dxxvxuxvxudxxvxubabababababaxduxvxvxuxdvxu - Edited by 2005 10 101 I dxxyxySba121xyxy12bax , II dyyxyxSdc12

16、2yxyx12dcy , 2 I drS2121II drrS21222213 Ctytxt a b t , t 0tC bxax , xdtttydxSba1 xyy bxa xy dxxySba21 dxxydS21 2 rr r , drrS223 tyytxxC t tx ty, C dttytxS221 zcz dz z dzczz zSdcdzzSV 2 1 xyy 0 axbx x x - Edited by 2005 10 11dxxyVbax2y baydxxxyV 2 2 yxx 0 cydy y y dyyxVdcy2x dcxdyyyxV 2 ABC x xS 1ABx

17、yy bxa dxxyxySba212 2ABrr drrrS22sin2 3ABtxx tyyt dttytxtyS222 1 1 0yQyQxPdxdyCdxxPyQdy202211dyyNxMdxyNxM CdyyNyNdxxMxM12210,012yNxM 2 1xyfdxdyuxyufdxduxudxdycxcxdxuufdu|ln - Edited by 2005 10 122 0,0 bacbyaxfdxdyucbyax ubfadxducxdxubfadu3222111cybxacybxafdxdy02211baba00222111cybxacybxa, xu yv uvbau

18、vbafvbuavbuafdudv2211221102211baba1212bbaa211111cybxacybxafdxdyybxau11211111cucufbadxdybadxdu1 0yxPdxdydxxPCeyc 2 xQyxPdxdydxxPexCy xC CdxexQeydxxPdxxP3 1,0yxQyxPdxdy1yz xQzxPdxdz11 4xyPyQdxdy 1yQxyPdydxy x 1 0, dyyxQdxyxPyPxQCyxu , yxu , dyyxQdxyxPyxdu , yxu , yxdudyyxQdxyxP , 1222yxdydyxdx 2222yxd

19、ydyxdx - Edited by 2005 10 133 xydxdyydx 4 xydxyxdyydxln 52222ln21yxdyxydyxdx62222ln21yxdyxydyxdx7xydxydxxdy28yxdyxdyydx29yxdyxxdyydxarctan2210xydyxydxxdyarctan2211yxyxdyxxdyydxln212212yxyxdyxydxxdyln21221322222121yxdyxydyxdx1422222121yxdyxydyxdx1522222arctan211yxdyxydyxdx1622222arctan211yxdyxydyxdx

20、yxyxdyyxQdxyxPyxuyxu,0000, yyxxdyyxQdxyxPyxu00,000yxPxu, yCdxyxPyxu , y yCdxyxPyyuyxQ , yC yC 2 0, dyyxQdxyxP yPxQyxR , 0, dyyxQyxRdxyxPyxRyRPxRQyxR , yxdudyyxQyxRdxyxPyxR , Cyxu , - Edited by 2005 10 14xfynnnnxCxCdxxfy211yxfy , py py pxfp ,1,Cxgp 1,Cxgy21, CdxCxgy yyfy , py p ydydppdxdydydpdxdpy y

21、ypyfpdydp,1,1Cygp1,Cygdxdy21,CxCygdy0yxqyxpy 1 xfyxqyxpy 2 1 xy1xy2xyCxyC2211 1C2Cxyxy21xy1xy2xyCxyCy22112 xy1xy2xyxy213 xyxyxyxy 4 yxyCxyC22111C2CxyCxyCxyy22115 xy1xy2xfyxqyxpy1xfyxqyxpy2xyxy21xfxfyxqyxpy211 0qyypy p q 02qp 1 042qp1 2- Edited by 2005 10 15xxeCeCy21212 042qp21xexCCy1213 042qpi xCxCe

22、yxsin cos212 n 012211ypypypypynnnnnnipi,2,1 0 12211 nnnnnpppp 1 nn, 21xnxxneCeCeCy212120k nk xkkexCxCC01213 i knk2 xxDxDDxxCxCCekkkkxsin cos121121xfqyypy qp, xyCxyCyy2211xyCxyC2211y xf yy xf y 1 xPxfnxPnn 1 0nnnnnaxaxaxaxRy1110niai,2,1,0 2 0 xxRyn3 0 xRxyn22xnexPxf xPnn1xnexRy 2xnexxRy 3xnexRxy23 xe

23、xPxfxnsin xexPxfxncos xPnn , 1 ixxTxxReynnxsin cos nnnnnaxaxaxaxR1110niai,1,0 nnnnnbxbxbxbxT1110nibi,1,0 - Edited by 2005 10 162 ixxTxxRxeynnxsin cos 01111ypyxpyxpyxnnnnnnnipi,2,1 ntex t ydtdyxdtdyedxdtdtdydxdyt1dtdydxdyx dtdyedtydedtdyedtdedxdydtddxdtdxydtttt 222222dtdydtydx2221dtdydtyddxydx2222232

24、1321, aaakajaiaa 321321, bbbkbjbibb 321321, ccckcjcicc 1332211, babababa 332211, babababa 2321, aaa 3bababa,cos 332211bababa ba,ba, ba 0ba a b ajbabPr04 ba bababa,sin ba ba, 321321bbbaaakjiba ba abba ba ba,5 cbacba ,321321321,cccbbbaaacba cba , cba ,321321, bbbbaaaa ba,babacos 232221232221332211cosb

25、bbaaabababaa b0ba 0332211bbbaba- Edited by 2005 10 17a b0ba 332211bababa1 n pnm ,2 000, zyxMCBAn , 0000zzCyyBxxA 00rrn zyxrzyxr ,00003 0DCzByAx CBA , zyx ,CBAn , 0CzByAx 0DByAx z 0DAx yOz 0x yOz 4 111, zyxA222, zyxB333, zyxCCBA , 0131313121212111zzyyxxzzyyxxzzyyxx5 L0022221111DzCyBxADzCyBxAL02222211

26、111DzCyBxAkDzCyBxAk0,0,21kk 6 0:11111DzCyBxA 0:22222DzCyBxA 1 2222222212121212121cosCBACBACCBBAA0212121CCBBAA 21212121DDCCBBAA21212121DDCCBBAA0DCzByAx111, zyxM Md 222111CBADCzByAxd 1 S L2 nzzmyylxx000000, zyx nml ,3 - Edited by 2005 10 18ntzzmtyyltxx000tnmls , 4 111, zyxA222, zyxBA B 121121121zzzzyy

27、yyxxxx5 0022221111DzCyBxADzCyBxA222111, CBACBAS 6 1111111:nzzmyylxxL 2222222:nzzmyylxxL 1L2L222222212121212121cosnmlnmlnnmmll0212121nnmmll 212121nnmmll0DCzByAx L nzzmyylxx000L222222sinnmlCBACnBmAlL CnBmAlL 0CnBmAl L 0CnBmAl L 1 yxz ,000,yxPcos,cosl tyxftytxfyxlft0000000,cos,coslim,yxfz ,000,yxP yyxf

28、xyxfyxgradf000000, lyxgradfyxlf0000,lyxgradflyxgradf ,cos,00001 vufz , yxuu , yxvv , - Edited by 2005 10 19xvvzxuuzxzyvvzyuuzyz2 zyxfu , yxzz , yzffyuxzffxuzyzx3 zyxfu , xyy xzz xzfxyffdxduzyx4 vufw , zyxuu , zyxvv , zvfzufzwyvfyufywxvfxufxwvuvuvu0, zyxF 1 yxzz ,zxFFxzzyFFyz2 zyxx ,xyFFyxxzFFzx3 xzy

29、y ,yzFFzyyxFFxyyxfz , 0,0,yxfyxfyxkkyx , lk ,2,1 2,kkxykkyykkxxkyxfyxfyxf 0kkkyxf , 0k0kkkyxf , 0,kkxxyxf kkyxf , 0,kkxxyxf kkyxf , 2n nxxfu ,10,0,111nmnxxxxnm nmiiinmnxxxxfxxFF ,11111- Edited by 2005 10 200,0,0011111nmnxxxxFxxFFFmnlkxxknk,2,1,1IxyxbxayxD21, x1x2ba, yxf , DDDdxdyyxfdyxf , baxxdyyxfd

30、x21, IIyxydycyxD21, y1y2dc, yxf , DD Ddxdyyxfdyxf , dcyydxyxfdy21, I IID I DII D DI IID DD 21,D 1 2,sin,cos, fyxf D D Dddfdyxf sin,cos, 21sin,cos dfd I21,20,D 21, 2,0- Edited by 2005 10 21sin,cos, fyxf D DDddfdyxf sin,cos, 20sin,cos21dfd II0,D ,sin,cos, fyxf D DDddfdyxf sin,cos, 0sin,cos dfd III0,20

31、,D 2,0sin,cos, fyxf D ddfdyxfD Dsin,cos, 20 0sin,cos dfd 1 dyxfyxfVD,12D S xyyxfz ,2yxfz ,12 DdyzxzA221 D S xy Syxzz , 1 1 Dyxyxzzyxzzyx ,21D xyyxzyxz ,21D zyxf ,yxzyxzDdzzyxfdxdydvzyxf,21, 2 zDyxzzyx , zD z zDdxdyzyxfdzdvzyxf , 2 dzddzfdxdydzzyxf ,sin,cos,yx, , z - Edited by 2005 10 223 cossinsinco

32、ssinzyx2000dxdydzzyxf , dddf sincos,sinsin,cossin2,Lzyxf , L nnSSS ,21nkSk1 kkks,nkkkkkSsf10,lim kS kknkS1max zyxf , LLdSzyxf , LLdSzyxf , 2 L txx tyytzz t dttztytxtztytxfdSzyxfL222,zyxf , tx ty tzABLzyxP , zyxQ , zyxR , LL nnSSS ,21nkSk1111,kkkzyxkkkzyx , L1kkkxxx1kkkyyy1kkkzzznk1kSkkks, nkkkkkkkkk

33、kkkkzsRysQxsP10,limnzyxP , zyxQ , zyxR , LLdzzyxRdyzyxQdxzyxP , LdSF zyxRzyxQzyxPF , dzdydxdS , L L- Edited by 2005 10 232 L txx tyytzz A BzyxP , zyxQ , zyxR , txty tz ABLdzzyxRdyzyxQdxzyxP , dttztztytxRtytztytxQtxtztytxP ,1 ABLyxP , yxQ , L ABABdsyxQyxPdyyxQdxyxP cos,cos,cos cos yx,A B 2 ABLzyxP ,

34、zyxQ , zyxR , L ABdzzyxRdyzyxQdxzyxP ,dszyxRzyxQzyxPABcos,cos,cos,cos cos cos ABzyx , A B 1 xy DL L DL yxP , yxQ , DLDQdyPdxdxdyyPxQ2 xy D 1nnnCCCL10 0CnCC ,1L DyxP , yxQ , D- Edited by 2005 10 24LDQdyPdxdxdyyPxQ01CnkCkQdyPdxQdyPdx yxQyxPF , yxP , yxQ ,D1 D L0LQdyPdx 2ABL DABQdyPdxA BABL3 yxdudyyxQd

35、xyxP , 4 DyPxQ5 yxQyxPF ,yxV , gradVF yxV ,xVPyVQ 1 S zyxf , SS nnSSS ,21)1( nkSk kkks,kSkSknkkkkSsf10,lim zyxf , SSdSzyxf , 2 S Dyxyxzz , yxz ,D zyxf , S DSdxdyyzxzyxzyxfdSzyxf221,1 S )zyxP , zyxQ , zyxR , SS nnSSS ,21nkSk1 yzyzkSzxzxkS xyxykS nkSk1kkks,nkxykkkkzxkkkkyzkkkkSsRSsQSsP10,limzyxP , zyx

36、Q , zyxR ,S- Edited by 2005 10 25SdxdyzyxRdzdxzyxQdydzzyxP , RQPF , dxdydzdxdydzdS , SdSF 2 S yxzz ,xyDyx, yxz ,xyD zyxR , S xyDSdxdyyxzyxRdxdyzyxR , S zxyS zyxx ,yzDzy, YZDSdydzzyzyxPdydzzyxP , S xSxzyy ,zxDxz, ZXDSdzdxzxzyxQdzdxzyxQ , S yRQP , 0SSdSRQPRdxdyQdzdxPdydz coscoscoscos,cos,cos S zyx ,RQ

37、PF , cos,cos,cos0n SSdSnFRdxdyQdzdxPdydz01 SzyxRzyxQzyxP ,SRdxdyQdzdxPdydzdvzRyQxPSdSRQP coscoscos cos,cos,cos S zyx ,2 1n0SnkSk10SzyxRzyxQzyxP ,0SRdxdyQdzdxPdydzdvzRyQxPnkSKRdxdyQdzdxPdydz1L S LL SzyxRzyxQzyxP , S- Edited by 2005 10 26SLRQPzyxdxdydzdxdydzRdzQdyPdx SdxdyyPxQdzdxxRzPdydzzQyRSLdSRQPzy

38、xRdzQdyPdxcoscoscoszyxuu ,zyx, uzuyuxugradu , u 1 zyxRzyxQzyxPF , FzRyQxPdivF FSdSnFdivFdv0cos,cos,cos0n 2 zyxRzyxQzyxPF , RQPzyxkjiFrotF kyPxQjxRzPizQyRF SLdSnrotFdrF0dzdydxdr , , cos,cos,cos0n 1 11nnu1nnv ba, 1nnnbvau11 nnnnvbua 2341nnu 0limnnu 111nn1)1(limnn111nn11nn01limnn11nn0limnnu1nnu 2 - Edi

39、ted by 2005 10 271 0nnar 0a 1rraarnn101r0nnar 2 p 11npn1p 11npn1p 11npn1p 11npn61212nn),3,2,1(0 nun1nnu,3,2,11nSSnnnSnS 1nnunS1 0c Nn 0nnucv 1nnv1nnu 1nnu1nnv 2 0nu 0nv ,3,2,1n Avunnnlim 1 A01nnu1nnv2 0A1nnv1nnu 3 A1nnu1nnv 3 0nunnnuu1lim 1 11nnu 2 11nnu 3 1 nnnuu1lim 4 0nunnnulim 1 11nnu - Edited b

40、y 2005 10 282 11nnu 3 1 11 0nu111nnnu 2 111nnnu 1 ,3,2,11nuunn2 0limnnu 111nnnu 11110 uunnn1 1nnu1nnu 2 1nnu1nnu 1nnu1nnu1nnu3 12121nnnuu121nnnuu 4 111npnn1 1p111npnn2 10 p111npnn3 0p , 111npnn1 xun,3,2,1n Ixunn1I 2 I 0x01xunn 0x1nnxu 10nnxu0x1nnxu 1nnxu3 1nnxu x- Edited by 2005 10 291nnxuxS x )(xS1

41、nnxu1 10nnnxxa )(0xx),2,1,0(nan00x0nnnxa x 0nnnxa00nnnxxa 2 0nnnxa 1 ),(0nnnxa xR 20nnnxa0R 3 ),( RR RR,( ),RRRR,R )0( R R 1 2)3(R laannn1lim lannnlimlR1l 0R 0lR 1 )(0xfxannn 1Rx0nnnxgxb2Rx 0nnnnxgxfxba21,min RRx 00000)(nnnknknnnnnnnxgxfxbababaxbxa21,min RRx 2 0nnnxa 0R0nnnxaxS 1 xS RR, 0 110 n nnnn

42、nnnnxnaxaxaxS xS- Edited by 2005 10 30RR, knknkxknnnxS 11Rx ,3,2,1k 2 xS RR, 010001nnnnxnnxxnadttadttS 30nnnxSxa RRxi0limnnnRxRaxS0limnnnRxRaxS ii0101nnnRRnadxxS0011RnnnRnadxxS iii11nnnxna RRx RSRnannn11RS 0nnnxa RRx11nnnxna RRx011nnnxnaRRx 1 8.31xxnn1101x 2xnnenx0!x 3 xnxnnnsin! 121120x 4 xnxnnncos! 2120x 5 xnxnnn1ln111011 x 6 xxnnnn1!111111 x 231 xf0x0xx000!nnnxxnxfxf0x xf00x nnnxnf0!0xf