高数总结笔记.pdf

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2、v101lim450xnnxxnxxxe 0!2121212530!121!5!3sinnnnxnxxxxxnnnxnxxxx22420!21!4!21cosnnnxnxxxxx 01321ln1321212153012153arctannnnxnxxxxxnnxxnnxxx 0!11!2111261001 0lim xf 0lim xg2 x xf xg3 AxgxflimAxgxflimxgxflimxgxflim2 1 xflim xglim2 x xf xg23 AxgxflimAxgxflim70000lim xfxxfxxfx 81011lim dxxfnkfnnkn 10x xf

3、y xf0x0x xf2ba, xf1 xf ba,xf ba,2 xfba, MmM mMxf0ba,0xba, x MxfM xf ba,m3 xf ba,M m mM c ba,cfxf ba, afbf ba,0f10c 0cd1xx dxxxd1xx cossin xdxxd cossinxx sincos xdxxd sincosxx2sectan xdxxd2sectanxx2csccot xdxxd2csccotxxx tansecsec xdxxxd tansecsecxxx cotcsccsc xdxxxd cotcsccscaxxaln1log 1,0 aaaxdxxda

4、lnlog 1,0 aaxx1ln dxxxd1lnaaaxxln 1,0 aaadxadaxxln 1,0 aa3xxee dxedexx211arcsinxx dxxxd211arcsin211arccosxx dxxxd211arccos211arctanxx dxxxd211arctan211cotxxarc dxxxdarc211cot22221lnaxaxxdxaxaxxd22221ln22221lnaxaxxdxaxaxxd22221ln2xgxfxgxfxgxfxgxfxgxfxgxgxfxgxfxgxf20xg3ufy xu x x ufu xfy xxxfdxdududyd

5、xdydxxxfduufdyduufdy u4tx ty xyy tt 0tttdxdy0t3221tttttdtdxdtdxdyddxdxdyddxyd5xfy ygx0xfygfxfyg110xfdxdydxxfddyygdyg1133ygfygfxfxf0xf6xyy 0,yxF y0,yxF x yyy7yxgxfy4xfxgeyln8xf0x xf0x9 n 2n,yynyn1xeyxney2 1,0 aaayx nxnaay ln3 xy sin2sinnxyn4 xy cos2cosnxyn(5) xy lnnnnxny !111nnkknkknnxvxuCxvxu0!knknC

6、knxuxu0xvxv0xu xv nxf1 ba,2 ba,3 bfafba, 0fxf1 ba,2 ba,ba,fabafbfabfafbf baxxxfxfxxf000100x a b x1 xf ba, 0xf xfba,2 xf xg ba,xgxf ba, cxgxf cxf xg1 , ba2 ba, 0xgba,gfagbgafbfbaxxg1 nxf0x nxRxxnxfxxxfxxxfxfxfnnn00200000!2!150xxnnxxxR000xx0lim00nnxxxxxRnxxxex1ln,cos,sin, x1 n2 nxf0x ba, 1nba, n bax ,

7、xRxxnxfxxxfxxxfxfxfnnn00200000!2!1101!1nnnxxnfxR0x x0x n00x n0lim xRnn1xf ba,0x ba,0x0xxx0xfxf0xf xf0x xf0x0xxx0xfxf0xf xf0x xf2xf0x0x xf00xfx 00xf0x xf3xf0x00 xx0xf 00xf100,xx x0xf00,xx x0xf0xf0x200,xx x0xf00,xx x0xf0xf0x300,xx00,xxx xf0xf0x4xf0x 00xf00xf00xf0xf0x00xf0xf0x61 xf ba,xf ba,kxx ,1bfafx

8、fxfk,1bfafxfxfk,1xf ba, Mxf ba, m21xf I21,xx21212121212212xfxfxxfxfxfxxfxf Ixfyxfyxfyxfy23xf ba, xfba, x 0xfxfy ba,ba, x 0xfxfy ba,xfyxf1x2xkx1xfaxlim xfaxlimax xfy2bxfxlim bxfxlimby xfy30lim axxfxbaxxfxlim0lim axxfxbaxxfxlimbaxy xfyxfy yxM ,2321 yyk 0kkR1yxM ,M DRMD D D R1 Cxdxx11172 Cxdxxln13 Caad

9、xaxxln11,0 aaCedxexx4 Cxxdx sincos5 Cxxdx cossin6 Cxdxxxdx tancos1sec227 Cxdxxxdx cotsin1csc228 Cxxdxx secsectan9 Cxxdxx csccsccot10 Cxxdx coslntan11 Cxxdx sinlncot12 Cxxxdx tanseclnsec13 Cxxxdx cotcsclncsc14 Caxxadxarcsin220a15 Caxaxadxarctan1220a16 Cxaxaaxadxln21220a17 Caxxaxdx2222ln 0a1CuFduuf xd

10、uufxuxdxfdxxxfCxFCuF1 baxdbaxfadxbaxf10a2 baxdbaxfnadxxbaxfnnnn110,0 na3 xdxfxdxxf lnlnln4xdxfxdxxf11125 xdxfxdxxf 26xxxxadafadxaafln11,0 aaxxxxedefdxeef7 xdxfxdxxf sinsincossin8 xdxfxdxxf coscossincos9 xdxfxdxxf tantansectan210 xdxfxdxxf cotcotcsccot211 xdxfxdxxxf secsectansecsec12 xdxfxdxxxf csccs

11、ccotcsccsc13 xdxfdxxxfarcsinarcsin1arcsin214 xdxfdxxxfarccosarccos1arccos215 xdxfdxxxfarctanarctan1arctan216 xarcdxarcfdxxxarcfcotcot1cot2817xdxfdxxxf1arctan1arctan11arctan21822222222lnlnlnaxxdaxxfdxaxaxxf0a1922222222lnlnlnaxxdaxxfdxaxaxxf0a20 Cxfdxxfxfln 0xf2tx 0tCtGdtttfCxGCtGdtttftxdxxf1xt1txxnba

12、x xndcxbaxxe baextxgntxtx02ACBxAxtCBxAx2tx0A220lxxA 0A202xxlA22xatax sin22xatax tan22axtax sec3xu xvxduxvxvxuxdvxudxxvxuxvxudxxvxuxuxv1axnexP axxPnsin axxPncosxPnn a naxe axsin axcos xvxu2 xxPnln xxPnarcsin xxPnarctanxPnn xPnxv xlnxarcsin xarctan xu3 bxeaxsin bxeaxcos49dx11baabdxxfdxxf2 0aadxxf3baba

13、badxxfkdxxfkdxxfkxfk221122114bccabadxxfdxxfdxxf c ba,5 ba xgxf bxababadxxgdxxf6 ba Mxfm bxaabMdxxfabmba7 bababadxxfdxxf8 xf ba,ba,abfdxxfbabadxxfab1xf ba,90aadxxf faaadxxfdxxf02 f10xf T aTTaadxxfdxxf01xf ba,xadttfxFbax ,1 xf ba,xadttfxFba,2 xf ba,xadttfxFba, xfxFxxdttfxF21xx21,xfxxfxxfxF11222xf ba,

14、xF xf ba,aFbFabxFdxxfbaxf ba,xf ba,1xf ba, tx1 t , ,2 a b tbtabadtttfdxxf2xvxu , ba,dxxvxuxvxudxxvxubabababababaxduxvxvxuxdvxu101I dxxyxySba121xyxy12bax ,II dyyxyxSdc122yxyx12dcy ,2I drS2121II drrS21222213Ctytxt a b t , t 0tC bxax , xdtttydxSba1xyy bxa xy dxxySba21dxxydS212rr r , drrS223tyytxxC t tx

15、 ty, C dttytxS221zcz dz z dzczz zSdcdzzSV21 xyy 0 axbx xx11dxxyVbax2ybaydxxxyV 22 yxx 0 cydy yydyyxVdcy2xdcxdyyyxV 2ABC x xS1ABxyy bxadxxyxySba2122 AB rrdrrrS22sin23 AB txx tyytdttytxtyS22211 0yQyQxPdxdyCdxxPyQdy202211dyyNxMdxyNxMCdyyNyNdxxMxM12210,012yNxM21xyfdxdyuxyufdxduxudxdycxcxdxuufdu|ln122 0,

16、0 bacbyaxfdxdyucbyaxubfadxducxdxubfadu3222111cybxacybxafdxdy02211baba00222111cybxacybxa,xu yvuvbauvbafvbuavbuafdudv2211221102211baba1212bbaa211111cybxacybxafdxdyybxau11211111cucufbadxdybadxdu10yxPdxdydxxPCeyc2xQyxPdxdydxxPexCyxCCdxexQeydxxPdxxP31,0yxQyxPdxdy1yzxQzxPdxdz114xyPyQdxdy 1yQxyPdydxy x10,

17、dyyxQdxyxPyPxQCyxu ,yxu , dyyxQdxyxPyxdu ,yxu ,yxdudyyxQdxyxP ,1222yxdydyxdx2222yxdydyxdx133 xydxdyydx4 xydxyxdyydxln52222ln21yxdyxydyxdx62222ln21yxdyxydyxdx7xydxydxxdy28yxdyxdyydx29yxdyxxdyydxarctan2210xydyxydxxdyarctan2211yxyxdyxxdyydxln212212yxyxdyxydxxdyln21221322222121yxdyxydyxdx1422222121yxdyx

18、ydyxdx1522222arctan211yxdyxydyxdx1622222arctan211yxdyxydyxdxyxyxdyyxQdxyxPyxuyxu,0000,yyxxdyyxQdxyxPyxu00,000yxPxu,yCdxyxPyxu ,yyCdxyxPyyuyxQ ,yC yC20, dyyxQdxyxPyPxQyxR ,0, dyyxQyxRdxyxPyxRyRPxRQyxR ,yxdudyyxQyxRdxyxPyxR ,Cyxu ,14xfynnnnxCxCdxxfy211yxfy ,py pypxfp ,1,Cxgp1,Cxgy21, CdxCxgyyyfy ,py p

19、 ydydppdxdydydpdxdpyy ypyfpdydp,1,1Cygp1,Cygdxdy21,CxCygdy0yxqyxpy 1xfyxqyxpy 21 xy1xy2xyCxyC2211 1C2Cxyxy21xy1xy2xyCxyCy22112 xy1xy2xyxy213 xyxyxyxy4 yxyCxyC22111C2CxyCxyCxyy22115 xy1xy2xfyxqyxpy1xfyxqyxpy2xyxy21xfxfyxqyxpy2110qyypyp q02qp1 042qp1 215xxeCeCy21212 042qp21xexCCy1213 042qpixCxCeyxsinc

20、os212 n012211ypypypypynnnnnnipi,2,1012211 nnnnnpppp1 nn,21xnxxneCeCeCy212120k nkxkkexCxCC01213 i knk2xxDxDDxxCxCCekkkkxsincos121121xfqyypy qp,xyCxyCyy2211xyCxyC2211yxf yy xf y1 xPxfnxPnn1 0nnnnnaxaxaxaxRy1110niai,2,1,02 0 xxRyn3 0 xRxyn22xnexPxf xPnn1xnexRy2xnexxRy3xnexRxy23 xexPxfxnsinxexPxfxncosxP

21、nn ,1 ixxTxxReynnxsincosnnnnnaxaxaxaxR1110niai,1,0nnnnnbxbxbxbxT1110nibi,1,0162 ixxTxxRxeynnxsincos01111ypyxpyxpyxnnnnnnnipi,2,1 ntex t ydtdyxdtdyedxdtdtdydxdyt1dtdydxdyxdtdyedtydedtdyedtdedxdydtddxdtdxydtttt 222222dtdydtydx2221dtdydtyddxydx22222321321, aaakajaiaa321321, bbbkbjbibb321321, ccckcjcicc

22、1332211, babababa332211, babababa2321, aaa3bababa,cos332211babababa,ba,ba0ba a bajbabPr04 babababa,sinba ba,321321bbbaaakjibaba abba ba ba,5 cbacba ,321321321,cccbbbaaacbacba , cba ,321321, bbbbaaaaba,babacos232221232221332211cosbbbaaabababaa b0ba0332211bbbaba17a b0ba332211bababa1n pnm ,2000, zyxMCB

23、An ,0000zzCyyBxxA00rrnzyxrzyxr ,000030DCzByAxCBA , zyx ,CBAn ,0CzByAx0DByAx z0DAx yOz0x yOz4111, zyxA222, zyxB333, zyxCCBA ,0131313121212111zzyyxxzzyyxxzzyyxx5L0022221111DzCyBxADzCyBxAL02222211111DzCyBxAkDzCyBxAk0,0,21kk60:11111DzCyBxA0:22222DzCyBxA1 2222222212121212121cosCBACBACCBBAA0212121CCBBAA21

24、212121DDCCBBAA21212121DDCCBBAA0DCzByAx111, zyxM Md222111CBADCzByAxd1S L2nzzmyylxx000000, zyx nml ,318ntzzmtyyltxx000tnmls ,4111, zyxA222, zyxBA B121121121zzzzyyyyxxxx50022221111DzCyBxADzCyBxA222111, CBACBAS61111111:nzzmyylxxL2222222:nzzmyylxxL1L2L222222212121212121cosnmlnmlnnmmll0212121nnmmll212121n

25、nmmll0DCzByAxLnzzmyylxx000L222222sinnmlCBACnBmAlLCnBmAlL 0CnBmAlL0CnBmAlL1yxz ,000,yxPcos,cosltyxftytxfyxlft0000000,cos,coslim,yxfz ,000,yxPyyxfxyxfyxgradf000000,lyxgradfyxlf0000,lyxgradflyxgradf ,cos,00001 vufz , yxuu , yxvv ,19xvvzxuuzxzyvvzyuuzyz2 zyxfu , yxzz ,yzffyuxzffxuzyzx3 zyxfu , xyy xzzxz

26、fxyffdxduzyx4 vufw , zyxuu , zyxvv ,zvfzufzwyvfyufywxvfxufxwvuvuvu0, zyxF1 yxzz ,zxFFxzzyFFyz2 zyxx ,xyFFyxxzFFzx3xzyy ,yzFFzyyxFFxyyxfz ,0,0,yxfyxfyxkkyx , lk ,2,12,kkxykkyykkxxkyxfyxfyxf0k kkyxf ,0k0k kkyxf ,0,kkxxyxfkkyxf ,0,kkxxyxfkkyxf ,2nnxxfu ,10,0,111nmnxxxxnmnmiiinmnxxxxfxxFF ,11111200,0,00

27、11111nmnxxxxFxxFFFmnlkxxknk,2,1,1IxyxbxayxD21,x1x2ba, yxf , DDDdxdyyxfdyxf ,baxxdyyxfdx21,IIyxydycyxD21,y1y2dc, yxf , DD Ddxdyyxfdyxf ,dcyydxyxfdy21,I IID I DII D DI IID DD21,D1 2,sin,cos, fyxf DD Dddfdyxf sin,cos,21sin,cos dfdI21,20,D21, 2,021sin,cos, fyxf DDDddfdyxf sin,cos,20sin,cos21dfdII0,D,sin

28、,cos, fyxf DDDddfdyxf sin,cos,0sin,cos dfdIII0,20,D2,0sin,cos, fyxf DddfdyxfD Dsin,cos,20 0sin,cos dfd1dyxfyxfVD,12D S xyyxfz ,2yxfz ,12DdyzxzA221D S xy Syxzz ,11Dyxyxzzyxzzyx ,21D xyyxzyxz ,21D zyxf ,yxzyxzDdzzyxfdxdydvzyxf,21,2 zDyxzzyx ,zD zzDdxdyzyxfdzdvzyxf ,2dzddzfdxdydzzyxf ,sin,cos,yx, , z22

29、3cossinsincossinzyx2000dxdydzzyxf ,dddf sincos,sinsin,cossin2,Lzyxf , L nnSSS ,21nkSk1kkks,nkkkkkSsf10,limkS kknkS1maxzyxf , LLdSzyxf ,LLdSzyxf ,2L txx tyytzz tdttztytxtztytxfdSzyxfL222,zyxf , tx ty tzABLzyxP , zyxQ , zyxR , LL nnSSS ,21nkSk1111,kkkzyxkkkzyx , L1kkkxxx1kkkyyy1kkkzzznk1kSkkks,nkkkkkk

30、kkkkkkkzsRysQxsP10,limnzyxP , zyxQ , zyxR , LLdzzyxRdyzyxQdxzyxP ,LdSFzyxRzyxQzyxPF ,dzdydxdS ,L L232L txx tyytzz A BzyxP , zyxQ , zyxR , txty tzABLdzzyxRdyzyxQdxzyxP ,dttztztytxRtytztytxQtxtztytxP ,1ABLyxP , yxQ , LABABdsyxQyxPdyyxQdxyxP cos,cos,cos cos yx,A B2ABLzyxP , zyxQ , zyxR , LABdzzyxRdyzyx

31、QdxzyxP ,dszyxRzyxQzyxPABcos,cos,cos,cos cos cos ABzyx , A B1xy DL L DL yxP , yxQ , DLDQdyPdxdxdyyPxQ2xy D 1nnnCCCL10 0CnCC ,1L DyxP , yxQ , D24LDQdyPdxdxdyyPxQ01CnkCkQdyPdxQdyPdxyxQyxPF , yxP , yxQ ,D1 D L0LQdyPdx2ABL DABQdyPdxA BABL3 yxdudyyxQdxyxP ,4 DyPxQ5 yxQyxPF ,yxV , gradVF yxV ,xVPyVQ1S zyx

32、f , SS nnSSS ,21)1( nkSk kkks,kSkSknkkkkSsf10,limzyxf , SSdSzyxf ,2S Dyxyxzz , yxz ,Dzyxf , SDSdxdyyzxzyxzyxfdSzyxf221,1S )zyxP , zyxQ , zyxR , SS nnSSS ,21nkSk1 yzyzkSzxzxkS xyxykS nkSk1kkks,nkxykkkkzxkkkkyzkkkkSsRSsQSsP10,limzyxP , zyxQ , zyxR ,S25SdxdyzyxRdzdxzyxQdydzzyxP ,RQPF , dxdydzdxdydzdS ,

33、SdSF2S yxzz ,xyDyx,yxz ,xyD zyxR , SxyDSdxdyyxzyxRdxdyzyxR ,S zxyS zyxx ,yzDzy,YZDSdydzzyzyxPdydzzyxP ,S xSxzyy ,zxDxz,ZXDSdzdxzxzyxQdzdxzyxQ ,S yRQP , 0SSdSRQPRdxdyQdzdxPdydz coscoscoscos,cos,cos S zyx ,RQPF , cos,cos,cos0nSSdSnFRdxdyQdzdxPdydz01SzyxRzyxQzyxP ,SRdxdyQdzdxPdydzdvzRyQxPSdSRQP coscosc

34、oscos,cos,cos S zyx ,21n0SnkSk10SzyxRzyxQzyxP ,0SRdxdyQdzdxPdydzdvzRyQxPnkSKRdxdyQdzdxPdydz1L S LL SzyxRzyxQzyxP , S26SLRQPzyxdxdydzdxdydzRdzQdyPdxSdxdyyPxQdzdxxRzPdydzzQyRSLdSRQPzyxRdzQdyPdxcoscoscoszyxuu ,zyx,uzuyuxugradu , u1zyxRzyxQzyxPF ,FzRyQxPdivF FSdSnFdivFdv0cos,cos,cos0n2zyxRzyxQzyxPF ,RQP

35、zyxkjiFrotFkyPxQjxRzPizQyRFSLdSnrotFdrF0dzdydxdr , , cos,cos,cos0n111nnu1nnv ba,1nnnbvau11 nnnnvbua2341nnu 0limnnu111nn1)1(limnn111nn11nn01limnn11nn0limnnu1nnu22710nnar 0a1rraarnn101r0nnar2 p11npn1p11npn1p11npn1p11npn61212nn),3,2,1(0 nun1nnu,3,2,11nSSnnnSnS1nnunS10c Nn 0nnucv1nnv1nnu1nnu1nnv20nu 0nv

36、 ,3,2,1nAvunnnlim1 A01nnu1nnv2 0A1nnv1nnu3 A1nnu1nnv30nunnnuu1lim1 11nnu2 11nnu3 1nnnuu1lim40nunnnulim1 11nnu282 11nnu3 1110nu111nnnu2111nnnu1 ,3,2,11nuunn2 0limnnu111nnnu11110 uunnn11nnu1nnu21nnu1nnu1nnu1nnu1nnu312121nnnuu121nnnuu4111npnn1 1p111npnn2 10 p111npnn3 0p , 111npnn1xun,3,2,1n Ixunn1I2I0x

37、01xunn 0x1nnxu10nnxu0x1nnxu1nnxu31nnxu x291nnxuxS x)(xS1nnxu110nnnxxa )(0xx),2,1,0(nan00x0nnnxa x0nnnxa00nnnxxa20nnnxa1 ),(0nnnxa xR20nnnxa0R3 ),( RR RR,( ),RRRR,R )0( RR 1 2)3(Rlaannn1lim lannnlimlR1l 0R 0lR1)(0xfxannn 1Rx0nnnxgxb2Rx0nnnnxgxfxba21,min RRx00000)(nnnknknnnnnnnxgxfxbababaxbxa21,min RR

38、x20nnnxa 0R0nnnxaxS1 xS RR,0 110 n nnnnnnnnxnaxaxaxSxS30RR,knknkxknnnxS 11Rx ,3,2,1k2 xS RR,010001nnnnxnnxxnadttadttS30nnnxSxa RRxi0limnnnRxRaxS0limnnnRxRaxSii0101nnnRRnadxxS0011RnnnRnadxxSiii11nnnxna RRxRSRnannn11RS0nnnxa RRx11nnnxna RRx011nnnxnaRRx1 8.31xxnn1101x2xnnenx0!x3 xnxnnnsin!121120x4 xnxnnncos!2120x5 xnxnnn1ln111011 x6 xxnnnn1!111111 x231xf0x0xx000!nnnxxnxfxf0xxf00xnnnxnf0!0xf