高中數(shù)學(xué)知識點匯總

目錄概率極限導(dǎo)數(shù)復(fù)數(shù)0v1.0可編輯可修改§01.集合與簡易邏輯知識要點AA；A；ABBAA=B.，.ABBCAC=ZS中AA，N則CA=sCA=B=CDB=BASAAxyxyRxy}xy32x3y1.yyx則B=）2n2n2nnn集有2－2個.nab，則a或b3a=2且b=a+b=②x且y，.3xyx+yx=1或y=xy3xy3是12xy2xy2v1.0可編輯可修改xx或x2.補(bǔ)：CA{xU,且x}A,,AU,AUABABAABBCABUUABB;ABB.(AB)CA(BC);(AB)CA(BC)UUUUUUUUv1.0可編輯可修改12m++-xxxxxxxm3nn1n2012n0yax2bxc（a02bx,x(xx)1212120b(a的解集212R2axbxc0(a0)的解集12f(x)g(x)f(x)f(x)f(x)g(x)g(x)g(x)f(x)0f(x)g(x)f(x)0()()0fxgxg(x)0g(x)g(x)bcbc(c或且)。互逆原命題逆命題若則q若則p互F否為逆逆為否互逆否命題若則┐p否命題若則┐q互逆5且P與q或p與qP則q；q則pq是q是p若pq且qp是qp6和§02.函數(shù)知識要點定義F:AB反函數(shù)一般研究具體函數(shù)圖像映射性質(zhì)函數(shù)二次函數(shù)指數(shù)對數(shù)指數(shù)函數(shù)對數(shù)函數(shù)7v1.0可編輯可修改y在C中的任何一個值，通過x=在Ay是自變量yyf(x)(x),習(xí)慣上改寫成y11Ix1212121正確理解奇、偶函數(shù)的定義。必須把握好兩個問題：f(x)為奇（1）定義域在數(shù)軸上關(guān)于原點對稱是函數(shù)函數(shù)或偶函數(shù)的必要不充分條件；fx（2）()()或fxf(x)f(x)是定義域上的恒等式。2．奇函數(shù)的圖象關(guān)于原點成中心對稱圖形，偶函數(shù)的圖象關(guān)于軸成軸對稱圖形。反之亦真，因此，也y可以利用函數(shù)圖象的對稱性去判斷函數(shù)的奇偶性。3.奇函數(shù)在對稱區(qū)間同增同減；偶函數(shù)在對稱區(qū)間增減性相反.4fxfx是偶函數(shù)，則()f(x)若奇函數(shù)在x0時有意義，則f(0)0。()fxfx(),,ababy21在yxf(x)()()()()0()0fx1.fxfxfxfxf(x)()fxfx(),,ababyx3在f(x)f(x)f(x)y=）f(x)f(x)0()01.fx()fxyf（）y軸對稱yf（x）x軸對稱y）y）稱yf（）（xx(xx)f(x)f(x)xb2222xb121212122222xbxbx19xBA1x.()(())，()BRA|1.BAfxffxBfxRxx①f(xy)f(x)f(y)f(xy)f(x).f(y)f(y)f(xy)f(x)fxy)y]f(xy)f(y)f(x)xf()f(x)f(y)f(xy)f(x)f(y)②yxxf(x)f(y)f()f(y)yy1|x1|x|1|xyyyy2|x|→||→→xy222▲y(0,1)xxxyy|2x2x1|2→||yxx2x17y2{|}，xxxRx3x3{|}xyyyR2x310ya(a且ax圖象y=1y=1性質(zhì)RRlogxalog(MN)logMlogNaaaMNloglogMlogNaaalogMnlogM12)naa1logMlogMnaanNNaalogN換底公式：logNblogaab推論：logblogcloga1abclogaloga...logalogaa2a3anan12n1111MN0,a0,a1,b0,b1,c0,c1,a,a0且1）12n12yOxx(時)yx時,0abab)a)b).0MnM00Mn0Mlogx22logx(2logxx2中中＞0而aaaayx⑵yax（aa1a當(dāng)a1log的01yxxaaa13log(MN)logMlogNaaaMNloglogMlogNaaalogMnlogM12)naa1logMlogMnaanlogNaNalogN換底公式：logNblogaab推論：logblogcloga1abclogaloga...logalogaa2a3anan12n11M0,N0,a0,a1,b0,b1,c0,c1,a,a...a0且1）12n,0ablog(ab)log(a)log(b).M0M00nn0MMlogx22logx(2logxx2中而中aaaaylogx⑵yax（1aaa當(dāng)a1ylog的01xxaaaxx)11212與214nn數(shù)列項15nnn1nanaaaqnm；；1nn1nnnmaamdnmnaa(nd（,0aaqn1aq）n1n11AankGa0)2（n,kN,nk0）（n,kN,nk0）**n前n項aa121nnnaqn和n11n(nnd12aaa(m,n,p,qN*,aaaa(m,n,p,qN*,mnpq)amnpqmnpqmnpq)aaaaadn1q常數(shù)）nn1nnn16n1aqnka=a+（）a+（）aaqn1kn1k1n(aa)n(n(q求和公式sd1n1212naqnddq(n11n2212aba=an22nmnmn若aamaa。若則mnpqnpq若k}kNa}也若k}kN2nnknnnkn34sss2ns2nnnnnaqd，nm1mnnn1mnam5①aad(n)(n2)nn1aann1an1③(,aknbnkn①aaq(n,且nn1②a2an1an1(n2，0①aaann1n1b成等比的雙非條件，即b17bbb且0acac③(,ancqnlog1xaanxnsa(n11aaanSs(n2)nsnnnnn1（[注]：①1aandndaddn11dd2nddn→2aSAnBn2na212nndd①等差數(shù)列依次每k項的和仍成等差數(shù)列，其公差為原公差的2倍S,SS,SS...；k2kk3k2kS奇aSSnd，n2nnN；偶奇S偶an1na21S21，S奇nnnNSSan2n1n奇偶S偶n1.代入n2nnn1n=2nn12n1②123222n262nn1③123333n32591；101.aannnnn18ra1r.)arn1nn[ar)n]1r)aar)ar)2...ar)n1.ra)rananarr)12]1r)ar)ar)ar)...ar)=.am為marma1rmx1rm1m11rxr11m2......11mxxrxrxarr11rm⑴apaqan2n1nx2（2qxxa,xxxxan2n11212x1x2(),,.accnxnaaccacxncxnn.11212112122n⑵naParnn1aa,accPnccPaqa1n2n1nnn1212由,aa12r().axPaxaxxn1nn1nP1rr()()(ax)P1xaParPParraaP1nnnn1n2n1P1P11P1aP2rPrrnn.1arn1aaPaPaa（P）aPa.narn1n1n1n1nnnnn1rrrrc，ca，acP1c（a）P1.nn11P21P1n211P11P0nSdSnnn19ddaan0nSn2(a)nnn1212nn11例如：1,...(21,...2nnn24dd12a)為同一常數(shù)。通項公式法。中項公式法:驗證驗證aa(nnn1an12aaa(aaa)nN2n1n1nn2nn2a0aSa0mann1m1a0msamsmm01amm1ca0aannn1abab0nnnnnn(n=220=n21212nn(n33321123nn(nn22226111111(n(nnn1n(n2)2nn21)1111()(pq)qppq考試內(nèi)容：考試要求：21§04.三角函數(shù)知識要點①與（＜360°）終邊相同的角的集合（角與角的終邊重合）：▲y|360,kZk23x|180,kZk41xy|18090,kZk14|90,kZk32SIN\COS三角函數(shù)值大小關(guān)系圖x|18045,kZk1、2、3、4表示第一、二、三、四象限一半所在區(qū)域|18045,kZyxkx360ky360180k180k的關(guān)系：36090k1801801||1l||r.sr222ya的終邊y；sin（rroyrcscr；；；.xcotxxrtancosxyyx22yy-+yy-+++TPo-+o--xo+-xxOMAx余弦、正割正切、余切正弦、余割16.幾個重要結(jié)論:(1)(2)yy|sinx|>|cosx|sinx>cosxOOxxcosx>sinx|sinx|>|cosx|(3)若o<x<sinx<x<tanx21|且,xxRxk2|且,xxRxkkZ1|且,xxRxk2x|且,xxRxkkZf(x)sintancossincotcostancot1cscsin1seccos1sincos1sectan11222222把k223sin(2x)sinxsin(x)sinxx=xxxx22cos(2x)cosxcos(x)cosxxtan(2x)tanxcot(2x)cotxtan(x)tanxcot(x)cotxx=22xx22xx)sinxx)sinxx)sinxx)cosxx)cosxx)cosxx)tanxx)cotxx)tanxx)tanxx)cotxx)cotx)coscossinsinsin2sincos)coscossinsin)sincoscossincoscos2sin22cos2112sin2221sin21cos)sincoscossin2)tantan1tantan1coscos22)tantan1tantan2111cos11sincossinsin211cos()sin2tancossinsinsin22sincostan2111tan2sin()coscoscoscoscos22211sinsincoscostan()cot1tan21tan222122)222221)222tan2221tan2221sin()cos22222,62,,.62tan15cot7523tan75cot1523sin75cos154424yAsinxycosx1RRR,kZxxRxkkZ|且2[RR當(dāng)當(dāng)[2k1,；[2k,222）2k]上為增函上為增函數(shù)212數(shù)()A[2,數(shù)；2數(shù)2]322）()上為減函數(shù)（kZsincos與cos））ysin與xyxyxyxyf(x)在[,]ab()在[,]yfxaby②yx與ycosx.x③y)或cos()（0xT.Oyxx2（TTytan2④y)x（kZcos()xkkyx2x)x（1ykZ,0k225（,02y2xy原點對稱2x)2xtan·tank(kZ)；tan·tan(kZ).22⑥cos與y(x)yxsin2yxk21.y(x)))xkx2ytanx在Rytanxf(x)f(x)f(x)f(x)f(x)）ytan(x1)ytanx30xf(x)f(0)00x⑨yxy為周期函數(shù)（Txx1/2y為周期函數(shù)（Tyxxxy=cos|x|圖象1的周期為y=|cos2+1/2圖象x|ycos2x2yf(x)5f(xkkR.⑩yacosbsina2b2)cosb有.bya22a11、三角函數(shù)圖象的作法：26Tf1||x;||T由y由xx1||由x由y由x，x，221，1］，值域是．－，22，，x，22．，22，27πyarcsinx是奇函數(shù)，故arcsin()xxarcsinx，sinxx與yyx,sin(arcsin)，，.xxx,x22arccos()arccos()2，.xxkxyarccosx.0,x)，，xxx②ycosxarccos和arcsinyxyxysinx,(,)yarctanxyx22，x(,).arctan(x)x，xtan(arctan)xx.(,)yarccotx,yarccot(,)x222，x(,).cot(x)x)kcot()，x(,).xx②yarcsin與)arctanarccos與yyarccotxxyxxyxx)x2,x[xx)2,x[.⑵aa①sinxa②cosxaaaaaakZ|2arccos,x|x2k,akZxxkaxxk|,akZax|xk1karcsin,akZ28|xkarctana,kZ③tanxxa③cotxxa|xk,kZsin21n3sin43sin2sin2sinsincoscos2cos4...cos2n2sinn13343coscos22sinncoscoscoscoscos2k2482n2nsink12nsin((nd)x)dnx)xxd)x)x)xxd)x)k0sin((nd)x)dnk0tantantantantantan1tantantantantantantan()f(x)sin在)xsinx＜x＜tanx,x(0,)2x若ABC2A2B2Cx2y2z229§05.平面向量知識要點AB；aOOｘｙｘｙ）121122v1.0可編輯可修改abbaab(xx,yy)1212aba(b)1212,OBOAABa是一個向量,滿a數(shù)乘向量|a|||a|與a(x,y)aa與aaa//bab0.a向量的數(shù)量積a?bb?aa?b)?ba?)a?)(ab)?ca?cb?ca|a||=xya?bxxyy12122222ab|a||b|cos(a,b)ee12λ，1λλeλe.21112v1.0可編輯可修改bPPPPPλ1212121211OP＝112xx,121yy11xxxy1122yy12.212則OP＝OPa或abcabcbc，222bca，222cab.222ABCh，h，h，abc.Sah=bh=chS=Pr△S=abc/R△△abcSAS=PPaPbPc△△Sb+c-arb+a-cr=1/2a+c-br△acb43AAAFEcAcbbOCcDNaFbBCBEDCaBIFCraBaEraI圖234圖1中的I為SS=Pr△ABC△圖2中的I為SS=1/2b+c-ar△aabcOBCACABcsABCAE=sab+c-a）]2BN=sba+c-b）FC=sca+b-c）abcabRt=2abctanAtanBtanCtanAtanBtanC.33AtantanBC,tantanCtan，ABCAB1tanAtanBACBDABBC22ABCD是BCAD2ABCDAD2.BDDCBC22ABBDcos①B2ABBD2ABBCAC22cosB2ABBCACBDABBC22AAD2BDDC圖5BC1AD是BCADA；22amb22c2a2ppatbcBpCDabc2aAD是BCphppapbpcac2a2b2ABC+=2ABC＜a2b2+cc222ABC＞a2b2+2bc2cosCa22ABCcos022Cabc222abc22ababab2(ab)222234OBOAABabBAOAOBab()aRbba(ab)ca(bc)a()bab3abab．a(chǎn)、babababbaalA0bbaPltta．a(chǎn)laaOAa平行于平面a//．如果兩個向量a不共線，與向量,bpa,b,共面的充要條件是存在實數(shù)xy使pxaybP,yMABxMPxMAyMBOOPOMxMAyMB①M(fèi)AB357a,b,cp,y,zpxaybzcO,,B,CP,y,zOPxOAyOBzOC8,baOAaOBbO,AOBa與的夾角，記作a,b；且規(guī)定0a,b，顯然有a,bb,a；若ba,baab.與b2設(shè)aOAaa.||ab||||cos,abab．a(chǎn)l，e是llA在lA，B在lBABle||ABAB|cos,||aeae．ABe|a|a,e0|a|2aaababa．(a)b(ab)a(b)abba(bc)abacxy36za,aa(,,)abbbb123123ab(ab,ab,ab)(,,)()aaaaRabababab123112233112233a1a2a3(R)∥,,0abababababababab112233112233b1b2b3)aaaa2a2a2a2aaaaa213ababababcosa,b|a||b|112233222222bbaaab123123()2(y)2(z)2.dxxy1z12122，aaa叫做平面an|ABn|B.A|n|,分別是二面角l,nn12則,,,nnnnnn121212a，,ABaCD則使AB求ABCDCECDCE解,若,,ABB▲nCD▲n2EAC37不等式0;0;0.abababababababbaab,bcacabacbc,abcdacbdabcdacbd,.,0abcab,c0ab0,cdacbd0ababcdcd3811(10)ab,ab0abab0anbn(nZ,且nab0ab(nZ,且nnn2若aR,則|a0,a02|ab2ab)bR,ab2abab2222bab.2,,xyRxySP,,1P,x=y2Syabc(4)若、、cR,則abc33ba若則2ab(6)a|xaxaxa或x;|xaxaaxa2222若、,則|||||||||abRabababb2aba2b2.ab1122abab)bab)a2ba2222ab(a=b(ab）22222a2b2c2abc2(a,,cR,ab)331a21a...a(aa...a)2222n12nn(acbd)(ab)(cd)22222.1111111(n2)nn1n(n1)n2(1)nnn1n39111②n1nnn1(nnn12nnn1若a,a,a,,aR,b,b,b,bR;則123n123n（abababab)(aaaabbbb)22222222232n112233nn1123n當(dāng)且僅當(dāng)1aa時取等號aa223nbbbb13n有x,x(xx1212xxf(x)f(x)xxf(1f(x)f(x)f()或).21212212222axbaxbxf(x)g(x)0f(x)0f(x)g(x)0;f(x)0g(x)g(x)g(x)0f(x)0f(x)g(x)g(x)0定義域1○f(x)g(x)()0fx()0fx2○f(x)0f(x)()()0gxgxf(x)g(x)g(x)0或2g(x)0()[()]2gx()[()]fxfxgxaafx()a()(a1)f(x)g(x);af(x)a()(0a1)f(x)g(x)gxgxf(x)b(a0,b0)f(x)lgalgb40f(x)0f(x)0logf(x)logg(x)(a1)g(x)0;logf(x)logg(x)(0a1)g(x)0aaaaf(x)g(x)f(x)g(x)g(x)0|f(x)g(x)g(x)f(x)g(x)g(x)0f(x)g(xf(x)g(x)|f(x)g(x)g(x)0(f(xg(x或x1124①x(1x)2x(1x)(1x)()23223272x(1x)(1x)12423222②(1)()yyxx2y23223279111ysinxcosxsinx(1sinx)22|x|x|||(與同號，故取等)2xxx41§07.直線和圓的方程知識要點xx角，其中直線與軸平行或重合時，其傾斜角為0，故直線傾斜角的范圍是0(0).注：①當(dāng)90或lxx2x1x(,0),),(0)abababxy42xy1.ab22注：若2是一直線的方程，則這條直線的方程是2，但若yxyx332yx2(x0)3ykxb,kb,kbbkbkb∥和和lllllkkl12121212,,∥llbb，yllkk12121212且或,b1b2ll）ABBACC12121212,的傾斜角為則∥ll.ll12121212和和k1這llkllkk12121212,②ll00lllklk212121210ABAB1221到到lllll12121k2k1.與)，當(dāng)90時tanl21kk12與與與llllll121212和90ll122k2k.11kk1243l:AxByC0過兩直線1111的交點的直線系方程l:AxByC02222(AxByCAxByC)0(0AxByC111222222(,):到PxyldlAxByCP00Cd00.AB22P2.2|PP(xx)(yy)211122122121O|xy22定比分點坐標(biāo)分式。若點分有向線段,其中PP所成的比為PP即PP1212xxyyP1xy,12121122211ktanyy(xx)(,),(,的直線的斜率公式：PxyPxy.1k2111222x12x21xx,yyx＝90當(dāng)1212:)，lAxByClAxByCCC:0(112212CCd12.dAB22注；0?0?xyxy1111llA?12111222l.244yxby換x換y.xxx–,–(,)0fxyC(,)(,)0Mxyfxy(,)(,)0(,)0xyfxyfxyCPC0000(,)為半徑的圓的標(biāo)準(zhǔn)方程是(Cab)().222rxaybrr.222xyr()()a2yb2b2rbabab[,圓心(,或(,)]xx()(yb)yxa222a[raabab,(,(,軸(xy)()a2ya2a2raaa[,(,)]x0.22xyF45D2E24FDE當(dāng)40C,r.DE22F222DE當(dāng)當(dāng)24F0,.2DE2240D2E2Fxarcos（ybrsin②方程Ax20表示圓的充要條件是：0且0且BAC2F.D2E2AF40(,)(,))0AxyBxyxxxxyyyy()()()(11221212(,):(MxyCxa)().222ybr00①內(nèi)C()2()2r2ybMxa00②上（xa)2(yb)2r2MC00③外C()2()2r2ybMxa00：(xa)(yb)r(r0)；：l0(0)；22AB222CCCAB2(,)Cab.dl2①dr與lC220xyDxEyF111220xyDxEyF222②與drlCC:xyDxEyF0221111C:xyDxEyF0222222()()()0.DDxEEyFF121212③與drlC46220xyDxEyF111OO12220xyDxEyF222()2()22xaybryx0AxBxC0l與C0l0l與C與C0，022xyDxEyF22xyDxEyF111222圓的切線方程：圓的斜率為的切線方程是k12過圓y222xyrkrx2y2DxEyF0xxyy(,)Pxy0.0FyyD0Ex0x00022xyx–x–y–y–R.20000A圓(,)Pxy.222xyrxxyyr20000yyk(xx)1010xybyk(ax)kB.C00R11D(a,b)R21圓.已知的方程0又以為圓為方程為22xyFO(xx)(xa)(yy)(xb)k2AA(xa)2(yb)2R2AA4CCCC47C48§08.PFPF2aFF2方程為橢圓,12PFPF2aFF2無軌跡,12PFPF2aFFF,F為端點的線段121212xx2y2ab0).ya2b2.y22xab0)a2b2x22y12A2Ba2b2xacos（一象限0bsin2y()或)(,0)x22ababyaba2c焦點：(,0)(或).④焦距：ccFF2c,ca2b2.⑤準(zhǔn)線：x或cc122yaec(0eca22y設(shè)xab0),FFPFaexPF,aex0P(x,y)1210200a2b249x22yab0),PFaey,PFaey(,)PxyFFb2a200121020a2a2pFe(x)aex(xpF(x)exa(xe10c002c000(cos,sin)Nabxdb2(c,)和(c,b2b2)a2aa22xy0)abc)ca2b2(ea2b2a2y2程x(00)ttabcea2b2a22Pxy1.,FFPF1F22FPF121a2b2b2tan2a2bcot.2122)(,)(,NxPFPF22aFF2方程為雙曲線1PFPF22aFF2無軌跡N的軌跡是橢圓1PFPF22aFF以F,F的一個端點的一條射線11212x22y22yab,0),x雙曲線標(biāo)準(zhǔn)方程：,0).一般方程：aba2b2a2b20).22xa2c(,0),(,0)(,0),(,0)acxxy0或acabx22y0a2b2a2),).),(0,)..yaaccycxasecxbtanyasec22yx0或yx0或.btanaba2b2y.2a2c,2,2xyeac502b..222ccabe,aa22xy1（,FF12a2b2MFexaMFexa1010MFMF2aMFexaMFexa122020▲▲yyMFeyaFM10MMFeyaxx20FFMFey0a1MFey0aF2yxe2.222xya22x2ya2b22與x2ya2b22xy0.a2b22222xy(0)xy0a2b2a2b2▲y22xy0xy(0).ab4a2b2211)1yxp22x12y1.2x2()得xy2428223425122Pxy1m=Pa2b2PF1d=m.e1d2PF2ne設(shè)p022y2yxxxxOOOOpppF(0,)2222軸xe1pPFx2pPFx2pPFy2pPFy21111524acb24ab2a2cx().②y22(0);x22py(p0).PyPpx22x2x22④y222y2tx2y22Fle當(dāng)01e當(dāng)1e當(dāng)1ec當(dāng)0cabeea與FF1212212跡ee53x222xy21(ab方ab2ab2))cosxasecyb參數(shù)為離心角）xtan參數(shù)為離心角）Rx0,xxx軸pFF2F11a22））cc(0(eeeeaapa2a22ccprar2b2aaa2a2Pcc54y與x22考試內(nèi)容考試要求5556§09.立體幾何知識要點3或41或30或18a平行于平面b與⑦,,ababab57）00,90）1122）0）0[0]),P,,lllllll,l12121212（或與LLLL1212∥.aa與平面aaaa58v1.0可編輯可修改與平面、∥（×）（、lPaOA若PA⊥，⊥⊥aa⊥.⊥aPMAOBθO作,，ll12,,,則PM,.PMOAPMOBOAPMOBlm2n2d22mncos（為）212coscoscos（121θ2（SChCh60（lSClC11四棱柱底面是平行六面體底面直平行六面體矩形長方體側(cè)棱垂直底面是底面是側(cè)面與正方體正方形正四棱柱底面邊長相等平行四邊形推論一：長方體一條對角線與同一個頂點的三條棱所成的角為,,，則.cos2cos2cos21推論二：長方體一條對角線與同一個頂點的三各側(cè)面所成的角為,,，則.cos2cos2cos22．61V3V.棱柱棱柱1S''）Ch2S底S側(cè)）⊥，cos，.albcclabal1b1Salcosablb則①，S③1222SS底得.側(cè)620IAbacBCD令A(yù)BaADcACb,,Dacbbac0FE得BCACABba,ADcBCADbcacACO'acbc0則.BCAD0HGB'ooACBOACAC,OBEFFGEFGH2.SROr4VR3.3PP63,AB點AB2（Vrhrh②圓錐體積：V12（rhrh31（ShVShRO3633h，，Sa2aSa2344底側(cè)36a313R342432a6得a/33.aa2a2a2RR43444411VBSR3SRSh33側(cè)底底與與與0babbcac,,abc∥ab0bab00(0)bbaa,(0)，∥的充要條件是存在實數(shù)abbab.ab或在與∥.aaaa,,abPaby使Pxayb.64O(是PABCOPxOAyOBzOCxyzOPyzOAyOBzOCAPyABzAC)四,,abcPpxaybzc.,Az使OPxOAyOBzOCD,,,其ABbACcADdBGM1C中Q()用AQAMMQAQabc3xyza,aa(,,)bbbba123123ab(ab,ab,ab)(,,)()aaaaRabababab123112233112233a1a2a3b1b2b3∥b,,()0abababRababababa112233112233)aaaa2a2a2a2aaaaa213ababababcosa,b|a||b|112233222222bbaaab123123()2(y)2(z)2.dxxy1z12122，aaa叫做平面an65|ABn||n|B.A,分別是二面角,lnn12則,,,nnnnnn121212a，,ABaCD則使AB求ABCDCECDCE解,若,,ABB▲nCD▲n2EAC222266=====BOD1bcacababc222222222；V3222C2Rabc；22242m；2a2b2c3=6768OA,∠ABF=12與coscos;12和1A1成；23123B1DCScos射原69，則Scos；側(cè)底,,,2222,,,22SS=S+S底=cS側(cè)1=S344R23SR3弧70§10.排列組合二項定理知識要點從mnnmmnm=mmnnm(mnm71從nm≤nnm從nmAmn!An(n(nm(mn,n,mN)(nm)!n!(n!!=1AAACAmAAnAC0Cn1mmmm1mm1mm1n1n1nmnnnnnn有kan12!!!...!nnn數(shù)為n，且n=n,則S.n12k12k12kn3n1.nm≤nn出mAmnAmn(n(nm!!CmnCm!(nm)!nmCmCnm;CmCmCm1②nnnnn1nmnnm二類，一類是含紅球選法有CCC一類是不含紅球的選法有C）m111m1mnnnm72nmCm1nmCnnCmCC1mmn1.nnnmCCC2012nnnnnn2CCCCCC024135n1nnnnnnCCCCCmmmm2mm1mn1nm1mnkCnCkk1n1n11CkCk1k1n1nn1123n1n1!111）(n(n(n!CCCCC33343534n1.CCCmm1mn1nnnC)C)C)C0212n2nnnn2n())xxxxnnn2nC()()()CCCCn1CCn2CCC02C12Cn2n0n12n02nnnnnnnnnnnnm(mn)個.其中Anm1AmmAnm1nm1nm173Ammn2.AA1n1A22nnAn1.A22n1nA2An1.nn1nA22的2mAAnmnmmnm1n–即m≤n12nAnn()mmnmAmmnmAnnAmmm=An/A.mnmCCCnnnkn.(k1)nnAkkC2422374（PC81810/20C2）2CmAA/Am，當(dāng)n–即m≤n1nmmnmnm1m2xxxx12123434,,,,,,xxxxxxxx12xxxx123412341234(y,y,y,y)12341234.C3注意：若為非負(fù)數(shù)解的x個數(shù)，即用中a等于，有a,aax1ii12n，axxxxAa1a1a1A123n12nCn1An.nkrAArkrnrr.rnAAAm11或AmAAmmn11m1m1n1n1nn1Amn1從nkrC后A.CCAkrnrCCrkrnrrkkrr從nkr75C后AC.CAkkknrknr從nkCCsCCAks.nrsksnrkrsC后Arkrnmr/AKAArr.AkkCCC2/A44421082CCCCCC2/AA1122222441098642nAAmm2、3、5，去參加不同的勞動，其安排方法為：C2C38C55A339C2C3C45A33種108mr.mA/AArrmC2C4C44A338A2276nmACmCm1…Cm2knn-mn-(mm...m)112k-1C25202C38C55出6C.126001C29C37(a).n0nCabbC0n01n1nCabnrrnrabCabnnn①1n②0,1,2,,r,,nC;CCCCnnnnn③naCab(0rn,rZ).rnrrn（ab)nr1Tr1n當(dāng)nn1項，它的二項式系數(shù)C22nn1n1當(dāng)nn1n11C2C222nn012nnnCCCnn02421n3nn1CCCCCnnn()n(,ab當(dāng)a或b1AA,AA(A為T或k1k1kk1k1kAAAAk1kkk77⑷如何來求(abc)n展開式中含apbqcr的系數(shù)呢其中p,q,rN,且pqrn把(abc)[(ab)c]CC(ab)C，在nnrrnnrr(ab)nr中含有b的項為，故在(abc)中含abc的項為pqrqCnanrqbCnqapbqqn!(nr)!!!!!rqpCrCqapbqcrCCCCrC.rqpnqnpnnr!()!!()!nrrnrqnrqnr當(dāng)a1n不大時，常用近似公式)1anC2a)1aCaCa233nnnnnna考試內(nèi)容：考試要求：n§11.概率知識要點A781AmnAP(A)m.n(即A、B)A、B.AA))))12n12nP(A)P(A)P(A1.積，即P(A·B)=P(A)·P(B).ABA與B很有可能不是獨立事件，但P(A)41,P(B)2611,P(A)P(B)522521326K21K.P(A)P(B)P(AA,A,,AP(AAA)P(A)P(A)P(A).12n12n12nA與BA與B,A與B，與ABnnn79k.P(k)CP(1P)knknkn()PABPAPBPAB)()()(§12.知識要點ξabξ())fxfξ,,,,xxx12iξxi)ξξPxp)1ii…………xxx12iPppp12i；②pi11.ppp12i.例如：80即nkP(ξCpq,n,qp]1kknknkξξ～.Cpqb(k;nknknknnkkAAAA,)qξk)P(AAAA)kkk2k1k1()ξP(A)P(A)P(A)qpkk112k1kξ1q23……k……P2qpqk1pξqpqpk1.k1Nn(1nN)則其中的次品數(shù)ξ是一離散型隨機(jī)變量，分布列為CCkMnkNM(0kM,0nkNM)MkP(ξk)CnN＜時C0kmrrmanCCkankξP(ξk).k0,1,,n.bCnab81anξabn次共有(ab)n個可能結(jié)果，等可能：(k)含Cab個結(jié)果，故ηknknkaCabaaknknk，即～B(n)nk,k0,1,2,,nk)P(ηk)C()k(1knab(ab)nababab可CknP(k)P(k)ξηξ…………xxx12iPppp12i為ξExpxpxp1122nn()abEEabaEb0()Ebba1)ξξaEbEbb0E()q=1c.EcPcξP0q1p+01ppEq!Ek其分布列為～(,)Bnppqknkk!(nk1E～(,)qkppξP()()xpkkk82為ξ0D.為ξD()()()2pxExExED2p2p1122nnξξ)D(aba)abD0DPpξP0q1p+q=DpqDnpqqDp2和)EEEEEξ和),)EEEDDD⑷E)E)E()()2DE2E0.EEx[,)內(nèi)abxxaxb▲yy=f(x)ξ()ξfx(,)”xxabx1()2xξf(x).e2（xR,,0ξ,～()()fxN2()N2～()ξ,.EDN2283xxx時曲線處于最高點，當(dāng)xx＜＞xx以xx21x(x)ξe2ξ的概率函數(shù)為(x)即～(有())，()1()xxNxPxξ(Pa)()().bba()的X取0()0.5當(dāng)()的X0xxx0.5(x)0.5()0.07930.5則yS～()則ξNxxμa()P(x)F(x)().ξFxσS陰設(shè)里的變量服從正態(tài)分布N(確定一次試驗中的取值是否落入范圍a2)做出判斷：如果(,)，接受統(tǒng)計假設(shè).如果a(,)a(,)ξ()則ξ(,)內(nèi)N(,)ξ84極限考試內(nèi)容：教學(xué)歸納法．?dāng)?shù)學(xué)歸納法應(yīng)用．?dāng)?shù)列的極限．函數(shù)的極限．根限的四則運(yùn)算．函數(shù)的連續(xù)性．考試要求：（1）理解數(shù)學(xué)歸納法的原理，能用數(shù)學(xué)歸納法證明一些簡單的數(shù)學(xué)命題．（2）了解數(shù)列極限和函數(shù)極限的概念．（3）掌握極限的四則運(yùn)算法則；會求某些數(shù)列與函數(shù)的極限．（4）了解函數(shù)連續(xù)的意義，了解閉區(qū)間上連續(xù)函數(shù)有最大值和最小值的性質(zhì)．§13.極限知識要點n（nkkN,）knn00nk1()Pnnnn（()Pnn0N0nk（,()Pn1()nkPnkNkn0nn()Pn0①aannn.aan①（CCCn②lim10(kN,是常數(shù))nkn85當(dāng)|1lim0anan當(dāng)a1a=liman11limlim(annannn當(dāng)1limaann,aabbnbnn①()ababnnn②()ababnnn③limaabb0)nnbnC(Ca)Ca.nnnnna11q1(q.qS()，fxxxxa00()lim()fxfxaxx().xxafxa00xx0xx()()在xx0fxfxx00()在()在()在fxfxfxxx0xxx000lim()fxxx0x1x1如()Px在1lim()1Pxxxx1x1x1lim()fxa,limg(x)bxx0xx086①lim(()fxgxab())xx0②lim(()())fxgxabxx0f(x)a③limbg(x)bxx0ClimCf(x))Climf(x).xx0xx0lim[f(x)][limf(x)]（nN）nnxx0xx0①lim10xn②limax0lim0（aaxaxx③limsinxx1lim1xsinxx0x01④lim1)x，lim)x（2.71828183）eeexxxx0f(x)xx()(),()(),fxgxfxgx(g(x)0()gxxx0xx0xxlim()xx0fx0xx0lim()().fxfx0xx0xx0xxx00xx()lim()lim()fxfxfx00xx0xx0但lim()().fxfx0xx087f（[,]()()0(,)ababfafb()（＜＜)0.fxabf()[,]fxabf(a),fb)B得),(,)ABCab（＜＜fCab0|xx()≤()≤()lim()lim()hxAgxfxhxgx0xx0xx0lim()fxA.xx0|xx|||xx0x00①limqnq1n②liman0(0)a!n③limnk0(a0kann④limlnnnn(lnn)k⑤limknn88導(dǎo)數(shù)§14.導(dǎo)數(shù)知識要點常見函數(shù)的導(dǎo)數(shù)導(dǎo)數(shù)的運(yùn)算法則函數(shù)的單調(diào)性函數(shù)的最值x()在處yfxxx00有增量，則函數(shù)值y也引起相應(yīng)的增量()()；比值yfxxfxx0089yx()()fxxfx()到x00yfxxx0x0limyx0limf(xx)f(x)()00yfxxxx0x0yf(xx)f(x)(x)在x()或|()=limlim.0yf'fxy''fx000xx00xxx0x0xxyf(x)，()與.ABAyfx'BAB()yfxxx00()()yfxyfxxx00yf(x)()點yfxxx00xx0