高中數(shù)學(xué)知識點歸納(理科)

高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納必修一第一章集合與函數(shù)的概念,B,C,a,b,c,(aN,N,Z,Q,RA),(aA),BABABABABBABAABBAABABABABABxB,x,且ABAAABBBABAn2n21n22nABABABxx,x“A并BABABA與集合BAxx,且x作：“A交BAB1高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納AUAAU=xxU,xAAAUUA,,若A則CAUUAB,BCAC;(AB),(AB)B;A(AB),B(AB).A(BC)(AB)(ACA(BC)(AB)(ACA,,.1AA,.,)CAACAUCUCUCCAAUUUUUU(CA)(CB)C(AB);(CA)(CB)C(AB).UUUUUU設(shè)、BxfAB(x)f:ABAByf(xxA，f其中，xxxyf(x)xAB.(a,b,b;,b,,b;,a,,a;,,,,,,,2高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納映射：設(shè)、BAfx:AB合ByfA到集合B）①(x)IIx,xfD12xxf(x)f(x)f(x)在區(qū)間DDf(x)1212②(x)IDx,xfI12xxf(x)f(x)f(x)在區(qū)間DDf(x)1212①f(x)IMxIf(x)My滿足有；；If(x)MMyf(x)）存在x00②f(x)IM滿足xIf(x)MyIf(x)MMyf(x)）存在x00①(x)(x)xf(x)f(x)fy②f(x)f(x)fx3高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納第二章基本初等函數(shù)axn，那么x叫做a的n當(dāng)nnna(a0)當(dāng)nnn。(a)aanannnnmaa00nmnaaa,(a)a,(ab)ab,其中a0,b；r,sQrsrsrsrsrrra(a0,a1)yxR。xyxa1圖象R(0,)x在R在R性質(zhì)x0y1時；x0時0y1；x0時y1；4高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納N(a0,a1)xaxlogNNaxalgNlogNlnNlogN10eNxlogN,(a0,a1)axalog10,loga1aa(MN)MNaaaMNMNaaaMnM(a0,aM0,N0)naab(a0,ab0)alogbloglogbb(a0,ac0,cb0)clogaac1logb,logblogclogdlogdlogaaabcabylogx(a0,a1)x0,。aya0a1圖象0,Rx在(0,)在性質(zhì)x1時y0；x1時y0；5高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納xyaxa12yx1yxRRRRR無y,0,0及f(x)yf(xyf(x)yyf(x)yf(xyf(xyf(x)第三章函數(shù)的應(yīng)用yf(x)，我們把使f(x)0yf(x)的實數(shù)x(x)0yf(x)xyf(x)ff(x)a,bf(a)fb)0如果函數(shù)yyf(x)a,bca,bf(c)0c也就是方程f(x)06高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納,bf(a)fb)0yf(x)f(x)af(x),bf(a)fb)0；a,)ca(c)f(c)0fff，則c(a)f(c)0bc(c)fb)0acx(a,c)0x(c,b)0abab。必修二第一章空間幾何體7高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納45①xoy（xoy0）②③，xyz22r2S2Srlr(RrRlrl)S22SR2ShVVVV底1Sh底31（SSSS)h3上上下下4R33第二章點、直線、平面之間的位置關(guān)系DCαAB①，28高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納②且lα·1LB·α·C·使2β3αL44,bOaa,bbab與a與ab,①a與babOO②兩條異面直線所成的角(0,；2③④⑤9aaabα,b,a//ba,b,abP,a,b,b,ab10高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納llllll⊥PPa,lb,a,b,abPll//b,abaaPOAPOOAO,等或PllPBl和QAN,OM和ONMON,aa,ba//ba11l與x取xxlll與xl與xkk=l與xk=l與xklkyyPPkP2111122212121l12P(x,y)k00000lky與（bl在yykxb11222212121112112高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納l(a,0)y(0,b)B與xAxyab01abx,yAxByC0B）P(x,x),P(x,y)()()xxyyPP12221122222121AxByC(x,y)l:C0dP0000A2B2C0l已知兩條平行線直線l和ll：1ByC，：Ax0，則l與122112CCld212A2B2第四章圓與方程(x)(y)r(a,b)xy1222r22(x,y)(xa)(yb)r2M2200(xa)(yb)r>22200(xa)(yb)r=22200(xa)(yb)r<22200x2yDxEyF0222EF0和yDx2213高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納,rd當(dāng)drlCdrlCdrlC）000當(dāng)lClClCllrrCC1212lrrCC1212|lrrrrCC121212||lrrCC1212||lrrCC1212ROMQyPM'M(x,y,z)x，、xy、zR在x、y、z(x,y,z)M(x,y,z)，M的zMM(x,y,z)xP2PyMzM1OHNMNy22MM1N(x,y,z)P(x,y,z)P111112222x14高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納PP(xx)(yy)(zz)22212121212必修三第一章算法初步15高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納ABABBPABPABABAAPAAPAPAPAAAPP成立不成立16高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納INPUTDispPRINT表達式變量變量＝表達式17高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納語句2圖1圖2對語句（圖3）（圖4）循環(huán)體滿足條件？WEND否與18高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納LOOPUNTIL條件否是在mnSR；0000n,nR0RS1R1R為n；000R,nR0RRR為S211101R；2R0Rnn1與19高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納0f(x)=ax+ax+…+ax+ann-1nn-11f(x)=ax+axx+a=(ax+ax)x+a=((ax+ax)x+a)x+a0nn-1n-1n-2n-2n-3nn-110nn-11=......=(...(ax+a)x+a)x+...+a)x+a0nn-1210nn-1n-210v=ax+an1n-1v=vx+a1v=vx+a2......v=vx+ann-12n-23n-30nnnnn為k1kkaaaa(0ak,0a,...,a,ak)，nn110(k)nn1105（121202021212543210(2)783848283210(8)2k(k1,kN)k第二章統(tǒng)計,xx,x,x,,x123n20高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納k=N/nk21高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納xxxxn12nxxxx12nn(xx)(xx)(xx)222ss212nnk倍(x3s,x3s)nxynxyiii1bya，nxnx22ii1ybxaYYx的2222高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納第三章概率SSSSnAnAn()nAAfAAAnnAAf(A)nP()AnnnAnAABBAAB，ABABABABABAB)ABABAB)ABABA與BA與BA與B23高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納A與BABABABA與BABA發(fā)生BBAAP()；構(gòu)成事件P()；必修四第一章三角函數(shù)24高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納120~0225001lr1800rad0rad1800~030,45,60,90,270,360000000000001lR2lSryxryxsincostanrsin(2k)2k),2k),(kZ)sinsincos1,tan22cos3k與sin(2k)cos(2k)tan(2k)kZ)25與sin()與與222與222y,RR2Rk22xk226高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納kk2k,2k在222,2kk222k,2k22上是減函數(shù)．k,0kk222xkk對稱軸無對稱軸xAxA0,0及yAA0,0y（x0,0y1fx；⑤初相：．xxxyxxyy1min2max11，，yyyyxxxx．2222112xy1yx數(shù)x倍yxyxy1xy27高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納ysinxysinxx倍yxyxy第二章平面向量或aAB3．向量的模：向量的長度(a或AB)0）eababababaa0a0a的aaaa0a0()());()ababaaaaaa(a0)b與baeea，12、eee、ae1112221228高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納0aba(x,y)(x,y),b(x,yR4．平面向量的坐標(biāo)運算：設(shè)a1122b(xx,yy)ab(xx,yy)a(x,y)a；12121212(x,y),B(x,y)(xx,yy)A11222121(x,y),b(x,yabxyxya則與11221221abcos（其中abcosabaababab=abababaa=a2=a或a2=aaabbacababab()()()(ab)cacbc(x,y),b(x,y)a1122,axy2abxxyyaxy222212121111abxxyyabab0xxyy0cos12121212xabx1yy2222122第三章三角恒等變換sin()sincoscossincos()coscossinsin(S)sin()sincoscossin(S)()()(C))C)()()29高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納tan()T)tan()T)11()()sinxcosx2sin(x)sinxcosx2sin(x)44sinx3cosx2sin(x)sinx3cosx2sin(x)33baasinxbcosxabsin(x)tan221tan1tan1tan1tantan()4tan()4sin22sincosS2cos2cossin222cos1212sinC222tantan2T1tan221cos21cos21cos2sin2,cos2,tan2222221cos2必修五第一章解三角形abc2RRABC(為)sinAsinBsinC30高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納abcbccos,bac2accosB,cab2abcosC222222222bcaacbabc222222222ABC22111absinCbcsinAacsinBS222bCcB,bcAaC,caBbA．三角形中的射影定理：a11212bc)a,m2(ac)b,m2(ab)cm2222222222abc1(abc)p）S2p(pa)(pb)(pc)(pa)(pb)(pc)）rp222p(pa)(p)(pchp(pa)(p)(pchp(pa)(p)(pc)）habcabc第二章數(shù)列an項nanna2nn1。2:…31高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納:a2有窮數(shù)列n2na2n按項數(shù):a無窮數(shù)列nn:an12n:a(2nnn1aaann1n2ad(nN*,d為常數(shù)）da。n1naba,,bab與A2Aab或。A2an1dan1nnS,(n1)Saaaaa，1SS,(n2)n123nnnn1naann11nS1dn22naananmd,aanmd,md）anmnmnm）,b,2bac為a與cbanpqmnp（、、、aaaa，qmmnpq2aaa(、kN,kn)*nnknkSSS）S，S，n2nn3n2nn(aa)aa）S中n121nn32高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納2aq0)qq(）。q1ana,G,b組成等比數(shù)列，此時G與bG2ab。aqann1n1(q(q1S(1qn)na1n1qaaq,qnm）anmnanmm）,b,b2為a與cbanpqmnaaaa，qm（、、、pmnpqaa(、nN,kn)a2*nnknkSSS）S，S，n2nn3n2nS求annaaf(n)f(n)a（n1n1nan1n,Sn(nna26nn122an,Sn(n1)）3nn33高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納第三章不等式b0abab0abab0ab．1．大小關(guān)系與不等關(guān)系：a；；bba①a②a；,bcac；bacbc③a④a；b,c0ab,c0，；b,cdacbd⑤a⑥a⑦a⑧a；b0,cd0；；b0abn,n1nnb0abn,n1．nnax2bxc0,(a0)；①②③④①z34高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納(xa)(yb)②z22ab22ab0；ab0b0，2ab若aab．a(chǎn)b2aba、ba、bab2x、ys2ysxyx．4pxyxy2p．選修—1第一章常用邏輯用語pqpqp稱為命題的條件，qpqqppqqp真真真真假真假假假真真假假假35高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納pq是是pqqp是q．ppqpqpqpq、pqpq當(dāng)、pqq．ppq當(dāng)、pqpqqpq、pp．pppp若p，pxxxpx，pxxxpxpxpx：，xpx，：pxpx，pxpx：p：，36第二章圓錐曲線與方程F，F(xiàn)FF2112222a2b2a2b212121212b2aab1212FFccabc2212xyc2e1aa2a2xyc00右00設(shè)MM到FdM到Fd112237e12d1d2F，F(xiàn)FF2112222a2b2a2b21212b2a12122212xcb2e1e1aa2a2xycbayyxab左0下0右0上038設(shè)MM到FdM到Fd1122e12d1d2FlFlA、By2y2x2x22y軸軸xp,0,0F0,F0,2222ppppxyy2222e10y0x0x0ypppp22220000M39第三章空間向量與立體幾何．ABAB010aaa．a(chǎn)、b，aC與ba，bab．a(chǎn)與a0a0．a(chǎn)a當(dāng)與40高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納aa。，a，baaabab．,yAPxAByACPABCxOAxAByACOOPO,B,CP,B,C,y,zOPxOAyOBzOC(其中xyz1),bab。a0aba，bbab．，ab，AOBa，b的a和bOa,ba,b0,．a(chǎn)和ba,b，bab．a(chǎn)2cosa,babab的數(shù)量積，記作．即已知兩個非零向量a和b，則ab稱為，ababcosa,b0．babbcosa,baa與在a若a，be41高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納aaeaa,e①e②abab0ab2，aaa，abaa③aabbabcosa,b④abbab⑤abbaababababcacbc．①ai，j，kp,y,zpxiyjzkxi，yj，p在i，j，ka，b，cp,y,zpxaybzc．xayb,,,za，b，cpp,b,ca，b，caa，b，c設(shè)eeeOeee123123Oe，e，exyz123OpOOPp,y,zpx，y，zp123x,y,zpOP底e，e，ep123,y,zx．x,y,zbx,y,z設(shè)a，11122242高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納bxx,yy,zzabxx,yy,zza，121212121212,,xyza111bxxyyzza121212bab0xxyyzz0a、ba．1212120ababx,,bxyyzz．212121aaxyza212121abxxyyzzcosa,b．121212abxyzxyz212222222211,y,zBx,y,zAx，，111222．ABxxyyzz則d222AB212121的位置可以用向量稱OPP空間中任意一條直線llAAlallPAallOab．Px,yOPxaybO與a，bllaa，ba，baa//ba//b0abR，ababab．43高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納////aanaaaanan0aaanan//，．////ab，a，bab，abab．0abcoscosa，ba，b．a(chǎn)blln，l與，l與nlnsincos．ln，，lnn設(shè)n，n1212nncosl12．nn12點ABABAB在直線l上找一點P，過定點且垂直于直線l的向量為n，則定點A到直線l的距離為andcos,n．n點PAnPncos,n為d．n44高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納選修2—2第一章導(dǎo)數(shù)及其應(yīng)用ff(x)f(x),ff(x)f(xxxx12xxx212121f(xx)f(x)ff(x)yf(x)y在xxlimlim000xxx0x0f(xx)f(x)xf(x或y,f(x)lim在x.00x000x0x0f(x)yf(x)x(,())在Pxfxfx，()y0000()()()當(dāng)xfx是xfxfx或yf(xx)f(x)()limfxyxx0c0（c(x)(),)x；xxxnxnN；n1n*11(e)e,(a)alna(a0,a1)(lnx,(logxloge(a0,a1)；xxxxxxaa(x)g(x)f(x)g(x)f.(x)g(x)f(x)g(x)f(x)g(x)f.()fx()()()()fxgxfxgx(g(x)0).(x)g2g(x)fu和ug(x)yxyu，45高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納yfu和ug(x)yf(g(xyuyxux(a,b)f(x)0yf(x)()0fx()yfx()0,得xxfx0()0xfx0()0()fxfx0()0xfx0()0()fxfx0f(x)(a,b)在yyf(x)f(afb)1nnf()xf()Sniix0ni1i11nnv()tv()Sniix0ni1i1(x)a,ba,bxb等faxxxi1x01inbannxn,x(i1,2,,n)f()xf()，i1iniiii1i146高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納當(dāng)nf(x)a,bbanf().bbf(x)dx,即f(x)niaani1與bxf(x)(x)b,(0xaxbabyyfx()fa(x)dxkf(x)dx(kbbkfaa(x)dxf(xdxf(xdx；f(x)f1bbb212aaa(x)f(x)f(x)dx(acb)bcbfaacf(x)a,bF(x)f(x)(x)Fb)F(a).bfaFb)F(a)F(x)()()()()bbfxFxbFbFaaaaxx當(dāng)xx47高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納第二章推理與證明nnn0k(kn,kN)nk1n0*nn0第三章數(shù)系的擴充與復(fù)數(shù)的引入48高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納bi(a,bR)i(i1)Caz2abi(a,bR)bazabicdiabdzabi復(fù)平面內(nèi)的點Z(a,b)復(fù)平面內(nèi)的向量OZ一一對應(yīng)一一對應(yīng)bi(a,bR)abi(a,bR)與aabi,zcdizz(abi)(cdi)(ac)(bd)i若z1212abi,zcdiz，12z(abi)(cdi)(ac)(bd)i則z12abi,zcdizz(abi)(cdi)(acbd)(adbc)i若z1212abi,zcdi若z，12za(abi)(cdi)i則1zc(cdi)(cdi)cdcd2222249高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納選修—3第一章計數(shù)原理1m2nNmn②1m2有nmnN②2nn個從nm(mn)個nm從nm(mn)nmAmnn(nn2)(nm1)AmnAn(nn2)321n!（nn0!1）nnAn(nn(nmmnn(nn(nmnm)321(nm)321n!(nmn!即Amn(nm50高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納從nm(mn)nm從nm(mn)nm個CmnACAAAn(n1)(n(nm1)mmmmCmnnnmm!nmmn!mnmCmn性質(zhì)1CCC1)nm0mnnn性質(zhì)2CCCmmm1n1nn(ab)CaCabCabCb(nN)n0n1n11knkknn*nnnn(k0,1,2,,n)CknTCab(k0,1,2,,n)knkkk1n若a1,bx(1)CCxCxCx(nN)x01*nkknnnnnn(ab11(ab)221(ab)3331(ab)44641(ab)5551(ab)6661(ab)777151(ab)8881高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納123CC1n12n136CC2n13n14CC3n14nCCCCCr1rrrr二項式121rrrnnCCmnmnnn1C當(dāng)nn2nn1n1當(dāng)nnCC1,22nn第二章隨機變量及其分布列XX,Y,XX,x,,x,x,XxX12inx(i1,2,,n)P(Xx)pXXiiixx…………xXP12np1ppp2inn0p1,i1,2,,np1．①iii1XXpP(X1)Xp011pp52高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納MNn(nN)X(XM)CCkkNMnN(X),k,mMm01CC0CC1CCmn0n1pMNMMNMMCnNCnNCnNM,nnN,mN,n,M,NN*.m，XX,BP()0AABP(B)ABP(AB)0P(B)1P(B),P()若B和CP(BC)P(B)PC)ABBA()()()A與B與B,與B,與BnnnnXAAp，(Xk)Cp(1p),k0,1,2,,n則PkknknXXp01……k……nC0p0(1)pCp(1)pCpkk(1)pCpnn(1)0pn11n1nnnnXXB(n,p)p53高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納XxxxXP…………12np1ppp2in(X)xpxpxpxpEX1122iinnaXba,bYE(b)aE(X)b若YPYaxb)P(Xx),i1,2,,n,ii則YaxbaxbaxbaxbYP…………12inp1ppip2nEY)aE(X)bE(b)aE(X)bXxxxXP…………12np1ppp2inE(X)則(xE(X))2x(i1,2,,n)iin(X)(xE(Xp()XEX而D2iii1D(X)XD(X)XYaXba,bD(aXb)aD(X)254ME(X)nNN11e概率密度曲線就是或近似地是函數(shù)、（babPaXbX,aa,xbxXa,b，xX()PaXbbxX()a,aN(,)2XXN(,)2。xx1x；x55高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納xP(X)0.6826P(X)0.9544P(X)0.9974(,)(,)以第三章統(tǒng)計案例??bxy??aybxnn(xx)(yy)xynxy?iiiibi1i1nn(xx)2xnx22iii1i11n1nnnxx,yiy，(x,y)ii1i1,xy,y和x和XY1212yy12abxcd2acbdabcd56高中數(shù)學(xué)必備(必須理解與記憶)知識點歸納HX與Y1較精確地給出這種判斷的可靠