高中數(shù)學(xué)高考知識(shí)點(diǎn)總結(jié)附有經(jīng)典例題

數(shù)學(xué)高一數(shù)學(xué)必修1知識(shí)網(wǎng)絡(luò)集合）元素與集合的關(guān)系：屬于（）和不屬于（））集合中元素的特性：確定性、互異性、無(wú)序性集合與元素）集合的分類：按集合中元素的個(gè)數(shù)多少分為：有限集、無(wú)限集、空集）集合的表示方法：列舉法、描述法（自然語(yǔ)言描述、特征性質(zhì)描述）、圖示法、區(qū)間法子集：若，則，即是的子集。xAxBABAB、若集合中有個(gè)元素，則集合的子集有2(2-1)個(gè)。AnAnn、任何一個(gè)集合是它本身的子集，即AA注關(guān)系、對(duì)于集合,,,如果，且BC,那么AC.ABCA、空集是任何集合的（真）子集。集合真子集：若AA（即至少存在xx），則的真子集。00集合相等：且ABABABABxxAxB定義：/且集合與集合交集性質(zhì)：AAAABBAB,ABABABA定義：/或ABxxAxB并集性質(zhì)：，，，，，AAAAAABBAABAABBABABB運(yùn)算Card(AB)Card()Card(B)-Card(AB)定義：/且CAxxUxAAU補(bǔ)集性質(zhì)：CAACAAUCCAACABCACB()()，()，()()(，UUUUUUUC(AB)C)CB)UUU函數(shù)映射定義：設(shè)，B是兩個(gè)非空的集合，如果按某一個(gè)確定的對(duì)應(yīng)關(guān)系，使對(duì)于集合中的任意一個(gè)元素，在集合B中都有唯一確定的元素y與之對(duì)應(yīng)，那么就稱對(duì)應(yīng)fB為從集合A到集合B的一個(gè)映射傳統(tǒng)定義：如果在某變化中有兩個(gè)變量x,y,并且對(duì)于x在某個(gè)范圍內(nèi)的每一個(gè)確定的值，按照某個(gè)對(duì)應(yīng)關(guān)系f,y都有唯一確定的值和它對(duì)應(yīng)。那么y就是的函數(shù)。記作yf(x).定義近代定義：函數(shù)是從一個(gè)數(shù)集到另一個(gè)數(shù)集的映射。定義域函數(shù)及其表示函數(shù)的三要素值域?qū)?yīng)法則解析法列表法函數(shù)的表示方法圖象法傳統(tǒng)定義：在區(qū)間ab上，若axxb如f(x)f(x)，則f(x)在ab上遞增,ab是1212ab遞增區(qū)間；如()()，則()在,上遞減,,是的遞減區(qū)間。f1fx2fxab單調(diào)性導(dǎo)數(shù)定義：在區(qū)間ab上，若f(x)0，則f(x)在ab上遞增,ab是遞增區(qū)間；如f(x)0ab是的遞減區(qū)間。則f(x)在ab上遞減,最大值：設(shè)函數(shù)()的定義域?yàn)?，如果存在?shí)數(shù)滿足：（）對(duì)于任意的，都有()；yfxIMxIfxM函數(shù)（2）存在xI，使得f(x)M。則稱M是函數(shù)yf(x)的最大值函數(shù)的基本性質(zhì)00最值yfxINxIfxN最小值：設(shè)函數(shù)()的定義域?yàn)椋绻嬖趯?shí)數(shù)滿足：（）對(duì)于任意的，都有()；（2）存在xI，使得f(x)N。則稱N是函數(shù)yf(x)的最小值00(1)()(),定義域，則()叫做奇函數(shù)，其圖象關(guān)于原點(diǎn)對(duì)稱。fxfxxDfx奇偶性(2)f(x)f(x),定義域，則f(x)叫做偶函數(shù)，其圖象關(guān)于y軸對(duì)稱。奇偶函數(shù)的定義域關(guān)于原點(diǎn)對(duì)稱周期性：在函數(shù)f(x)的定義域上恒有f(xT)f(x)(T0的常數(shù))則f(x)叫做周期函數(shù)，T為周期；T的最小正值叫做f(x)的最小正周期，簡(jiǎn)稱周期）描點(diǎn)連線法：列表、描點(diǎn)、連線向左平移個(gè)單位：,()1y1axyfxa向右平移a個(gè)單位：yy,xaxyf(xa)11平移變換向上平移個(gè)單位：xx,ybyybf(x)11向下平移個(gè)單位：xx,ybyybf(x)11橫坐標(biāo)變換：把各點(diǎn)的橫坐標(biāo)縮短（當(dāng)時(shí)）或伸長(zhǎng)（當(dāng)時(shí)）10w11w到原來(lái)的1/倍（縱坐標(biāo)不變），即xwxyf(wx)1伸縮變換縱坐標(biāo)變換：把各點(diǎn)的縱坐標(biāo)y伸長(zhǎng)（A1)或縮短（0A1)到原來(lái)的倍1函數(shù)圖象的畫(huà)法（橫坐標(biāo)不變），即yy/Ayf(x)1（2）變換法2210xx10關(guān)于點(diǎn)(x,y)對(duì)稱：2yyf(2xx)0000212y1y0y0yx120120x關(guān)于直線對(duì)稱：(2)yf0xx0yy1y1對(duì)稱變換x11x關(guān)于直線對(duì)稱：y02()y0yfxyy021y1y0y2xx11yx關(guān)于直線對(duì)稱：yfx()y1、tanx指數(shù)函數(shù)和對(duì)數(shù)函數(shù)的底數(shù)大于零且不等于1；5、三角函數(shù)正切函數(shù)y中()cotxkkZyx中；2157法(x),g(x)f(x)g(x)f(x)f(x)fyf[g(x)]f(x)g(x)與(x)g(x)與3fyf[g(x)]50(0)0yf(x)xf(x)0ff(u)ug(x)和y(x)f(x)可以表示為5、若函數(shù)f的定義域關(guān)于原點(diǎn)對(duì)稱，則11f(x)[f(x)f(x[f(x)f(x22y,f(x)0yf(x)yf(x)[a,b]f(a)f(b)yf(x)[a,bc(a,bf(c)cf(x)0f(x)0yf(x)yf(x)x[a,bf(a)f(b)；(2)求區(qū)間(a,b)c;f(c)；f(c)cf(a)f(c)bcx0(a,b)f(c)f(b)acx0(c,b(4)a-b,a(b24。nnmna,nsamaarsra(ar,sQ)aars()(rs,Q)aaarrs()ab(abrQ)xya(a0且a1gN,axNa(MN)MN;aaaMN;Maaaa.NnnM;(aa1,MNMaloglogbab(a,c0且a,c1,bccayx(a0且aa1yxx2表yxy1a定義域值域y0,yRy(x(,0)y(1,)(,0)時(shí)，(0,1)x(0,1)y(0,)xxyx(0,y(0,1)x(0,y(1,)(1,時(shí)，(,0))xyxy性質(zhì)2y表p00111q）高中數(shù)學(xué)必修2知識(shí)點(diǎn)一、直線與方程（xx,0°（ktan即k0,900；90k0；當(dāng)90k當(dāng)k當(dāng)yy(xx)1k2xx1221xx12k與PP12（yk(xx),xyy1111yy1。lx=x。11kybbyyyxx，,yx,y（,xxxyy11yy2xx21112122211xyab1l與(a,0),y與(0,b),la,b即與。xxyC0B）12bxaaxyby（ByC0平行于已知直線Ax0（,B是不全為0的常數(shù)）的直線系：A0000AxByC0（C00yykxx,xy；k0000l:AxByC0l:AxByC0，11112222lAxByCAxByC0（2111222（l:ykxbl:ykxb當(dāng)，111222l//lkk,bb；llkk11212121212（l:AxByC0l:AxByC011112222AxByC0111AxByC0222l//ll與l；1212（(x,y（x,y1122|AB(xx)(yy)則222121Px,yl:AxByC0AxByC（d10000AB22（二、圓的方程a,bxaybr（（222；xyDxEyF022DE4F012當(dāng)22rDE,DE4F2222DE4F0DE4F0當(dāng)22當(dāng)22（D；Ca,bl:AxByC0l的距離為（1）設(shè)直線，圓C:xaybr，圓心到222AaBbCddrl與C相離drl與C相切drl與C相交；；AB22）設(shè)直線l:AxByC0C:xaybr2220l與C相切0l與C0l與C；；rx,yr20000xxyyrxy2．2220000，yr22220000．d2C:xaybr，C:xaybR22222111222ddRrdRrRrdRrdRrdRr當(dāng)當(dāng)當(dāng)當(dāng)當(dāng)d0當(dāng)三、立體幾何初步（ABCDE'AD'''''（ABCDEP'''''（ABCDEP'''''（（（（xxyy（h'（chl1ch直棱柱側(cè)面積Srhch'SSS圓錐側(cè)面積圓柱側(cè)正棱錐側(cè)面積21S(rRlS(cc)h'圓臺(tái)側(cè)面積212rrlS2rrlr2rlRlR2SS圓柱表圓錐表（1V錐31VSh柱Vr圓柱h2V圓錐r2h3111V(SSSS)h(rR)h(SSSS)h''22V臺(tái)''333圓臺(tái)V=；S=4R2（433R球（①②。③AAAAAl；點(diǎn)AlAl；llll（,Bl,A,Bl：Al（2（。IIPABABl,Pl3（（①②③④b和a和bOOA（（aαA（b（（（（（0。bab,，（。90。0（AC,,A,B,,.y軸。A（x軸yz（M(x,y,z)(x,y,z)叫MM(x,y,z)MMM（d(xx)(yy)(zz)222212121高一數(shù)學(xué)必修3公式總結(jié)以及例題§1算法初步n次多項(xiàng)式，只要作n次乘法和n次加法即可。表達(dá)式如下：axax...aaxaxax...xaxann1nn11nn1n221例題：秦九韶算法計(jì)算多項(xiàng)式3x4x5x6x7x8x1,當(dāng)x,65432?6，6即:4x5x6x7x8x1?理解算法的含義：…).3.算法含有兩大要素：flow:算法結(jié)構(gòu)：AAABpYNNppYABYNⅠ.Ⅱ.的pseudocodeBASIC語(yǔ)言編寫(xiě)的,方法。偽代碼沒(méi)有統(tǒng)一的格式，只要書(shū)寫(xiě)清楚，易于理解即可，但也要注意符號(hào)要相對(duì)統(tǒng)一，避免引起混淆。如：賦值語(yǔ)句中可以用xyxy;”Ⅰ.xyyxx”，有時(shí)在偽代碼的書(shū)寫(xiě)時(shí)也可以用“xy的“=注：=”3=a+6=aa=–1,a=+3=b=c=2,a,b,ca=3例題：將x和y的值交換pxpxxyxyyp,:yzzpⅡ.輸入語(yǔ)句（inputstatement）:Reada,b表示輸入的數(shù)一次送給a,b輸出語(yǔ)句（outstatement）：Printx,y表示一次輸出運(yùn)算結(jié)果x,y注：1.2.3.“=4.5.“；例題：當(dāng)x5=xx=5Ⅲ.條件語(yǔ)句（conditional1.行If語(yǔ)句:IfAThenB注：沒(méi)有2.塊If語(yǔ)句：注：不要忘記結(jié)束語(yǔ)句①③ABC:.a,b,caacⅣ.循環(huán)語(yǔ)句（cycleN次?.A…I…p……p2.凡是能用.3.4..例題：135....（見(jiàn)課本P）21S1S1I1S1I1ForIFrom3To99Step2SSIhileI97II2SSIEndhilePrintS?hileI99SSIII2EndhilePrintSEndForPrintSS1S1I1I1DooSSIII2II2SSILoopUntilPrintSLoopUntilI100或者I99)I99PrintSS1S1I1I1oWhileI99I100)oWhileI97或者I99)SSIII2LoopII2SSILoopPrintSPrintS4正角按逆時(shí)針?lè)较蛐D(zhuǎn)形成的角、任意角負(fù)角按順時(shí)針?lè)较蛐D(zhuǎn)形成的角零角不作任何旋轉(zhuǎn)形成的角2、角的頂點(diǎn)與原點(diǎn)重合，角的始邊與x軸的非負(fù)半軸重合，終邊落在第幾象限，則稱為第幾象限角．第一象限角的集合為k360k36090,kooo第二象限角的集合為k36090k360180,koooo第三象限角的集合為k360180k360270,koooo第四象限角的集合為k360270k360360,koooo終邊在x軸上的角的集合為k180,ko終邊在y軸上的角的集合為k18090,koo終邊在坐標(biāo)軸上的角的集合為k90,ko3、與角終邊相同的角的集合為k360,ko4、已知是第幾象限角，確定n所在象限的方法：先把各象限均分n等份，再*n從x軸的正半軸的上方起，依次將各區(qū)域標(biāo)上一、二、三、四，則原來(lái)是第幾象限對(duì)應(yīng)的標(biāo)號(hào)即為終邊所落在的區(qū)域．n5、長(zhǎng)度等于半徑長(zhǎng)的弧所對(duì)的圓心角叫做1弧度．l6、半徑為的圓的圓心角所對(duì)弧的長(zhǎng)為l，則角的弧度數(shù)的絕對(duì)值是．rr180o7、弧度制與角度制的換算公式：2，1，157.3．ooo1808、若扇形的圓心角為，半徑為，弧長(zhǎng)為l，周長(zhǎng)為C，面積為S，則r112lr，C2rl，Slr．r229、設(shè)是一個(gè)任意大小的角，的終邊上任意一點(diǎn)的坐標(biāo)是,y，它與原點(diǎn)的距離yxyx是0，則sin，cos，tanx0．rrxy22rr10、三角函數(shù)在各象限的符號(hào)：第一象限全為正，第二象限正弦為正，第三象限正切為正，第四象限余弦為正．、三角函數(shù)線：sin，cos，tan．y12、同角三角函數(shù)的基本關(guān)系：1122PTsincossin1cos,cos1sin；2tan2222xA．13、三角函數(shù)的誘導(dǎo)公式：12sin，2kcos，2tank．2sin，cos，tan．3sin，cos，tan．4sin，cos，tan．口訣：函數(shù)名稱不變，符號(hào)看象限．5，．226，．22口訣：正弦與余弦互換，符號(hào)看象限．14yxyx1yxx的圖象上所有點(diǎn)yx的圖象；再將函數(shù)yyx的圖象．1函數(shù)yx的圖象上所有點(diǎn)的橫坐標(biāo)伸長(zhǎng)（縮短）到原來(lái)的到函數(shù)ysinx的圖象；再將函數(shù)ysin的圖象上所有點(diǎn)向左（右）平移個(gè)單位長(zhǎng)度，xx的圖象上所有點(diǎn)的縱坐標(biāo)伸得到函數(shù)yx的圖象；再將函數(shù)y長(zhǎng)（縮短）到原來(lái)的yx的圖象．函數(shù)yx0,0的性質(zhì)：1振幅：；周期：；頻率：f；相位：；初相：．④x⑤①②③函數(shù)yx，當(dāng)xx時(shí)，取得最小值為y；當(dāng)xx時(shí)，取得最大值1min211，xxxx．為y，則yy，yy222max211215、正弦函數(shù)、余弦函數(shù)和正切函數(shù)的圖象與性質(zhì)：函yxxytanxy數(shù)性質(zhì)圖象定義,xxkkRR域2值域R當(dāng)x2kk當(dāng)2kk時(shí)，x2時(shí)，y1；當(dāng)y1；當(dāng)xmax2kmax最值周期既無(wú)最大值也無(wú)最小值x2kk時(shí)，y1．2mink時(shí)，y1．min性奇偶奇函數(shù)偶函數(shù)奇函數(shù)性在2k,2k22在2,2kk上k上是增函數(shù)；在是增函數(shù)；在在kk,單調(diào)222k,2k性2k,2kk上是增函數(shù)．22k上是減函數(shù)．k上是減函數(shù)．對(duì)稱中心對(duì)稱中心k,0k對(duì)稱中心對(duì)稱對(duì)性稱軸kkkk22xkk2對(duì)稱軸xkk無(wú)對(duì)稱軸01rrrrrrbababa．rrbbarrrra00aarrrrrrrrrabcabca．rrrrx,ybx,yabxx,yya，．C11221212rrrrx,yabxx,yyrbx,ya，．b11221212xxyy．,,,設(shè)、xy，xy11221212rrabCCrraa．r①ara；rarrara000ar0ra．rrrrrrrrr．a(chǎn)aaaa．a(chǎn)babrr,y,axyxy,arrrrrr0bba．a(chǎn)a與rrrbb0共rrrb0r設(shè)ara、x,ybx,yxyxy0，11221221e、e12ueerreea、a、112112221x,yx,y22、2，211122xxyy,21．121112rrrrrr⑴arrbabcosab0,0,01800oo．rrrrrrrrrrrrrabab0babab⑵性質(zhì)：設(shè)a和ba．②當(dāng)a與r與barrrrrrrrrrrrrrrbabaaaaaaaabab．2；2或rrbbarrrrrrrrrrrrrrabcacbc．rabababarrrrx,ybx,yabxxyya，．11221212r若arr,yaxyaxy22222．rrr設(shè)arx,ybx,yabxxyy0，．11221212rrrrrrx,ybx,y設(shè)a、b都是非零向量，a，，是a與b的夾角，則1122rrabxxyyrrab1212．xyxy21222221coscossinsin⑴⑵⑶⑷；；；；coscossinsinsinsincossinsinsincossintantantantantantan1tantan⑸⑹（（1tantantantantantantantan1tantan1tantan⑴sin22sin．cos2111cos⑵2222（2，21cos2sin222tantan2⑶．12tansincossin、22．5CabcCRC、、、、為abc2R．sinsinsinC2Rsinb2Rsinc2RsinC；、正弦定理的變形公式：①a，，abc②sin，sin，sinC；2R2R2R:b:csin:sin:sinC③a；abcabc④．sinsinsinCsinsinsinC111bcsinabsinCacsinS．222CCabcbcbac2ac，，222222cab2abC．222bcaacbabc222222222，，C．22CabcC、、C；a、b、c是222obcC90oabcC90o．2a22222222nnanaann12ba，，則a與bacbb為a與c2an1daaa．dn11naaaanmdaan1ddn11；nnm1naaaadnnm1④n．nmdaaaaanpqmnpq（、、、a是等差數(shù)列，且m*mnpqnnaaa2npqnpq（、、*則．npqnaann1Sn1nSnandn．221Snaan2nn、等差數(shù)列的前n項(xiàng)和的性質(zhì)：①若項(xiàng)數(shù)為，則，且*n1SSaSSnd奇偶n，．偶奇an1Sn2n1n21*SnaSnSa，nS，n奇偶12n1奇偶Sn奇Sn1a偶n22aba與bGaGbGa與bG，G為a與baqaaqa．n11n1naaqnmaaq1aan、通項(xiàng)公式的變形：①；②；③q；④n1n1nnm1aqnmn．a(chǎn)mnpq（m、n、p、qaaaaa是a是等比數(shù)列，且m*nmnpqnaaa22npqnpq（、、*則．npqnaq11anS1．a(chǎn)1qnaaqnnq11n1q1qSSn2nn*q．偶奇SSqSn②．nmnmSSS③，SS，n2nn3n2nb0abab0abab0ab、a；；．bbaab,bcacabacbc；①ab,c0ab,c0，ab,cdacbd；④a⑥a⑧a；b0,cd0ab0abn,n1nnb0abn,n1．nn2b42000axbxcy2c02bxbxx2a1,22aa012xx12c02xxxxxb12xxRa02ac02xxxxa0121和yxx,y，,y,xyC0xy．0000yC,xyC0xyC0xy，xxy000000yC,，x0000xyC0．0xyC0xyC0xyC0xyC00xyC0xyC0xyC0xyC0x，yx，yx，yx，y,y．a(chǎn)b41a、ba、ba、bab2ab0b0，2abab．若aab2aba,bR22b2a,bRa；222ababab22a0,b022a,bR．③ab222x、ys2ysxyx．4pxyxy2p．xAxCA,xCAxA.UUC(AIB)CAUCB;C(AUB)CAICB.UUUUUUAIBAAUBBABCBCAUUAICBCAUBRUU(AUB)(AIB)(AUBUC)(AIB)card(AIB)card(BIC)cardCI)card(AIBIC).a,a,L,a}222nnn12n2n(x)axbxc(a0)fffN;2(x)a(xh)k(a0)2;(x)a(xx)(xx)(a0).12f(x)MNf(x)M[f(x)Mf(x)N]0MNMNf(x)N|f(x)022Mf(x)11.f(x)NMN(x)0(k,k)f(k)f(k)08.方程f在,與不等價(jià),前者是后者的一1212bxc0(a0)(k,k).特別地,ax2內(nèi),等12bkkkkbf(k)f(k)0f(k)0且kf(k)0且2k.1212a222a121122b(x)axbxc(a0)p,qxf22abbf(p),f(q；p,qf(x)f(),f(x)x2a2abf(pf(q，f(pf(q.xp,qf(x)f(x)，2amaxmaxminminbbp,q2af(x)f(pf(q)xp,qxmin2af(x)f(pf(q)，f(x)f(pf(q).maxminf(m)f(n)0f(x)0(,n).(x)xq設(shè)f2402pqf(x)0(,)f(m)0或；pm2(m)0ff(n)0（2）方程f(x)0在區(qū)間(,n)內(nèi)為f(m)f(n)0或40或2pqpmn2(m)0(n)0f(n)0af(m)0；f或af402pqf(x)0(,n)f(m)0或.pm2(,),,,，，Lf(x,t)0t(f(x,t)0(xL).min(,)f(x,t)0t(f(x,t)0(xL).mana0a0(x)axbxc0b0或c0f42.b24ac0ｐｑ真真假真真真假真假假假假真真假假真1n1n個(gè)nx，x，x，x，p或qp且qpq或互否互為逆為逆互否否qpp是qpqqpp是qqpp是qxa,b,xxx1212f(x)f(x)(xx)f(x)f(x)00f(x在a,b21xx121212f(x)f(x)(xx)f(x)f(x)00f(x在a,b21xx121212f(x)f(x)0f(x)f(x)0，y(x)則f(x)g(x)f(x)g(x);17.如果函數(shù)f和,和函數(shù)fu)ug(x)和yf[g(xyyyf(xa)f(x)f(xa)f(xa)yf(xa)f(xa).f(x)xRf(xa)fbx)f(x)20.對(duì)于函數(shù)y(,則函數(shù)ababxf(xa)yfbx)與xy22a(x)f(xa)yf(x)(,0)f(x)f(xa)對(duì)稱;若,21.若f2yPP(x)f(x)2a(x)axaxLan1nnn1P(x)0P(x)P(x)f(x)f(x)yyxaf(ax)f(ax)f(2ax)f(x).abf(x)xf(amx)fbmx)y2f(abmx)f(mx).f(x)yf(x)x0yyyyabf(a)yfbmx)x2mf(x)yf(x)和1f(x)yf(xa)babyf(x,y)0bf(xa,yb)0f(a)bf1(b)a.a1f(b)y[f(x)b]若函數(shù)y存在反函數(shù),則其反函數(shù)為1,并不是k1y[f1(kxb)y[f1(kxb)是y[f(x)b]kf,(x)cxf(xy)f(x)f(y),f(1)c.(x)af(xy)f(x)f(y),f(1)a0.ff,x,(x)logxf(xy)f(x)f(yf(a)a0,a1).a(x)xf(xy)f(x)f(y),f(1).f,'f(x)xg(x)xg(x)()()()()()，fxyfxfygxgy，f(0)1,lim1.xx0f(x)f(xa)f(x)f(x)f(xa)0，1f(xa)(f(x)0)，或f(x)1或f(x)fx,(()0)f()1f(x)f(x)f(xa),(f(x)0,1)f(x)或221f(x)1(f(x)0)f(x)f(xa)f(x)f(x)(xx)f(a)1(f(x)f(x)1,0|xx2a)f(x)f且121f(x)f(x)12121212期f()f(x)f(x)f(x)f(x)f()f(x)f(x)f(x)f(x)()fxf(xa)f(x)f(xa)f(x)1mmnN0,,n1a（annam1mna（amnNn0,,1man(a)a.nnana；nn,a0a|a當(dāng)nan.na,a0aa(a0,r,sQ)ar.srs(a)a(a0,r,sQ).rsrs(ab)ab(a0,b0,rQ).rrr若aplogNbaN(a0,a1,Nab.NN(a0a1m0m1N0,,maamnlogblogb01,0m11,N0n(aa,mn,nmmaa若log(MN)logMlogN;aaaMNloglogMlogN;aaalogMnlogM(nR).naa(x)log(axbxc)(a0)b42f(x)f2,記.若,則Rma00f(x)Ra00a010b0x0xylog(bx)若a,,,aax11abab(0,)(,)和上y上ybxlog()axa1a1(0,)(,)bxlog()ax和，aam1p0a0a1設(shè)n，，log(np)logn））.mpmmn2logmlognlog2.aaaN(1p)pxyy.xnn1,sa(數(shù)列a}nsaaLa1ss,n2n12nnnnn1aa(nddnad(nNn)；*11nn(aa)n(n1)snad1n22n1d1n(ad)n2.221aaaqq(nN)；nn11n*1qn(1)aqn,q11s1qnna,q11aaq,q11n1或sq.nna,q11a:aqad,ab(q0)nn11nb(nd,q1abqndbq()d；n1,q1nq1nn(nd,(qs1,(q1).dqndb)n1qq11q)abb)nxa,nbb)1n(0,)sinxxtanxx.2(0,)，則1xx2若x.2|x||x1.sin1tantancot1.，，=22cosn(sin,2nsin()2n1(cos,2n(1)cos,2cos()2n1(1)sin,2)sincos;;)cosmsin).1m)sin()sinsin22.)cos()cossin22asinbcossin()(輔助角(,)所在象限由點(diǎn)ab的象限決=ab22btanasinsincos.2112222.2tantan.1tan2sin33sin4sin4sinsin()sin()3.334cos3cos4coscos()cos()3.3333)).1332yxy))xyx)xkkZ,T，2T.abc2R.sinAsinBsinCabcbcA;222bcacacosB;222cab2abC.222111ahbhch（h、h、hSS2122abcabc11absinCbcsinAcasinB.22212(|OA||OB|)OAOB)2.S2()BCCABAABC22C2(AB).2sinxaxk(1)arcsina(kZ,|a1).kcosxax2ka(kZ,|a1).xaxka(kZ,aR).sinsink(1)(kZ).kkkZ.cos2()tan(kZ).xaax(2ka,2ka),kZxaax(2ka,2ka),kZ..cosxa(|a1)x(2arccosa,2karccosa),kZ.xaax(2a,2a),kZ.tanxa(aR)x(karctana,k),kZ..2tanxa(aR)x(k,karctana),kZ2aa如果e12、λ，使得a=λe+λe．121e12212(x,y)(x,y)Pxyxy0，且ba.設(shè)a=,b=11221221a與bbbaa與b在ab(x,y)(x,y)(xx,yy),b=,b=(xx,yy)..11221212(x,y)(x,y)11221212(x,y)(x,y)(xx,yy).A11222121(x,y),R(x,y).(x,y)(x,y)(xxyy).,b=11221212xxyycos(,)(,)=xyxy1212xyxy112221222221|ABABAB=dA,B(xx)(yy)(x,y)(x,y)2221211122(x,y)(x,y)，且b設(shè)a=,b=1122xyxy0.1221aaxxyy0.1212(x,y)P(x,y)P(x,y)PP1設(shè)P，，111222212xxxy121y121y1211t（t112A(x,y)B(x,y)C(x,y)△ABC、、,則△112233xxxyyyG(,).31231233xxhxxh''OPOPPP'.'ykyyky''注:圖形FFP'(x',y)PP'''(h,k).P(x,y)a=(h,k)P(xh,yk).'yf(x)yf(xh)k.a=(h,k)yf(x),則C'CC',則C'C'(h,k)為yf(xh)k.,若CC曲線C:f(x,y)0f(xh,yk)0.按向量(h,k)平移后得到圖象C',則C'的方程為m=(x,y)a=(h,k)(x,y).設(shè)O為ABC,B,Ca,b,cO為ABC2OAOBOC.22r0O為ABCO為ABCO為ABC..r0.ABCA.O為的,bRab2aba22ababa,bR2bcabc(a0,b0,c0).a333(ab)(cd)(acbd),a,b,c,dR.22222bababa.,yxxypxyxy2p；1xysxys2.4x,yR(xy)(xy)2xy22xy是定值,則當(dāng)|xy||xy|當(dāng)|xy||xy||xy|是定值,則當(dāng)|xy|||當(dāng)|xy||xy|bxc0(或0)(a0,b4ac0)acax與222a與c2xxx(xx)(xx)0(xx)；121212xx,或xx(xx)(xx)0(xx).121212當(dāng)a>0xax2a2axa.xaxaxa或xa.22(x)0f(x)g(x)()0gxfff.f(x)g(x)(x)0f(x)0f(x)g(x)()0或gx.g(x)0f(x)[g(x2(x)0f(x)g(x)()0gx.f(x)[g(x2a1aaf(x)g(x);f(x)g(x)(x)0flogf(x)logg(x)g(x)0.aaf(x)g(x)0a1aaf(x)g(x);f(x)g(x)(x)0flogf(x)logg(x)g(x)0aaf(x)g(x)yyk（P(x,y)、P(x,y).21xx11122221yk(xx)P(x,y)k．yyl1111在y1blyyxxyyP(x,y)P(x,y)xx、((11yyxx12111222122xy1211b、b0()abC0AB:ykxbl:ykxb若l，111222||lkk,bb①l;121212lkk1.②l1212:AxByC0l:AxByC0且AABB,若l,12121111ABC2222l1ABC①l；1112222lAABB0②l；121212kk||.211kk21:ykxbl:ykxb1(l，,kk)11122212ABAB||.1221AABB1212:AxByC0l:AxByC00,AABB(l,111122221212lll與l.12212l到l12kk.211kk21:ykxbl:ykxbkk1)(l，,11122212ABAB.1221AABB1212,:AxByC0l:AxByC0AABB0(l,111122221212lll到l.12122P(x,y)yyk(xx)xx000000,其中AB是(x,y)A(xx)B(yy)0,kP00000:AxByC0l:AxByC0l,11112222(AxByC)(AxByC)0l111222kb2byAxByC0AxBy0(0(4)垂直直方程：與直線C00(A≠0，B≠0)垂直的直線系方|C|d點(diǎn)P(x,y)l：AxByC000AB2002C00或l:AxByC0AxByC00或若B0與l.若B0BC與BCAC與lAC與ll..(AxByC)(AxByC)00或1C11222:(AxByC)(AxByC)0AABB0（1112221212(AxByC)(AxByC)00或111222(AxByC)(AxByC)0111222(AxByC)(AxByC)0111222(xa)(yb)r2.22yDxEyF0DE4Fx22(22cosxar.ybrsin(xx)(xx)(yy)(yy)0A(x,y)、121211B(x,y)22(x,y)B(x,y)A,1122(xx)(xx)(yy)(yy)[(xx)(yy)(yy)(xx)]01212112112(xx)(xx)(yy)(yy)(axbyc)00,其中c1212lC0CxyDxEyF0:的交點(diǎn)的圓系方程是22過(guò)直線:與圓DxEyF(AxByC)0yDxEyF0xyDxEyF0x2y2:xC:C222211112222yDxEyF(xyDxEyF)0x2222111222(x,y)(xa)(yb)r2點(diǎn)P若d2200(ax)by)2200dr點(diǎn)Pdr點(diǎn)Pdr點(diǎn)PC0(xa)(yb)r222dr0;dr0;dr0.AaBbCd.A2B2dOr，OO121212drr;12drr;12rrdrr;1212drr條公切線;120drr.12yDxEyF0x．22(x,y)00D(xx)E(yy)xxyyF0.020200D(xx)E(yy)(x,y)xxyyF0當(dāng),00220000yk(xx)y00ykybyrx222．(x,y)xxyyr2;P00000y.kkxr1k2xacosx2y2ab.bsina2b2yx2y2aba2b2a2a2e(x)，e(x).1c2cx2yx202y2022(x,y)ab1PP..00a2b2abx2yx2y2022(x,y)ab10200a2b2abx2yxxyy2abP(x,y)01.0a2b200ab22x2y2abP(x,y)a2b200xxyy01.0a2b2x2y2ab0CAaBbc22222.a2b2x2y2aba2b2aa22|e(x)|，|e(x)|.1c2cx2yx202y2022(x,y)ab1PP..00a2b2abx2yx2y2022(x,y)ab10200a2b2abx22y22x2y2b10x.yaba2b2axy0xx22y2b.yababa2x22y22x22y22100，（ababyxy2xxyy2a0,bP(x,y)01.0a2b200ab22xy221(a0,bP(x,y)a2b200xxyy01.0a2b2xy22a0,b0CAaBbc22222.a2b2y22pxpy2px(p0)CFx.220ppxxxxpCD拋物線y2.221212y22px(,)P(2pt,2pt)或(x,y)y或P，其中上的動(dòng)點(diǎn)可設(shè)為P22pooy2o2px.obb2ayca(x)(a0)22ab4bb4b14b1222(,)(,)y為.2a4a2a4a4a(x,y)y2px(p0)y2px(p0).P2200(x,y)y2px(p0)y2px(p0).點(diǎn)P0P220(x,y)y2px(p0)2y2px(p0).200(x,y)y2px(p0)y2px(p0).點(diǎn)P0P220(x,y)x2py(p0)2x2py(p0).200(x,y)x2py(p0)x2py(p0).點(diǎn)P220點(diǎn)P0(x,y)x2py(p0)x2py(p0).2200(x,y)0x2py(p0)x2py(p0).點(diǎn)P2202pxP(x,y)yyp(xx).y20000y22pxP(x,y)yy(xx).0000y22px(p0)C0pB2AC.2(x,y)0f(x,y)0f,12f(,y)f(,y)0(12x2kby2k1kmax{a,b},其中2.當(dāng)共焦點(diǎn)的有心圓錐曲線系方程2a22kmin{a,b}當(dāng)min{a,b}kmax{a,b}222222(xx)(yy)AB2或21212ABk)(xx)|xx|1tanyy|1co|t2222（弦端點(diǎn)211212ykxb(x,yB(x,y)ax2bxc00，Ay,的ABF(x,y)01122kF(x,y)0(,)(2-,2)0PxyFxxyy.0000F(x,y)0C02B(AxByC)2(AxByC)F(x,y)0.A2B2A2B2BxyCyDxEyF0xx0代2yy代y2xAx220xyxyxxyy代xyxy代x代y0000222xxyyAxxBCyyDE0F000022200......0b=．B||APABtOP(1t)OAtOB.、||、ABCD、且ABCDpx,ypaxby．P,xyMPxMAyMB，,yOPOMxMAyMBx.xOAyOBzOCxyzk119.對(duì)空間任一點(diǎn)OOP（1Ok1OkOADxAByAC、C、D與、OD(1xy)OAxOByOC（O使設(shè)OPxOAyOBzOC.al是llAlA'Bl上B'AB|AB|cos''(a,a,a)(b,b,b)則設(shè)＝123123(ab,ab,ab)；；112233(ab,ab,ab)112233(a,a,a)＝123ababab；112233(x,y,z)(x,y,z)A111222=(xx,yy,zz).212121r設(shè)ar(x,y,z)b(x,y,z)，111222xxrrrrrrPabb12y；aby12zzrrrr12abab0xxyyzz0.121212(a,a,a)(b,b,b)設(shè)＝123123ababab.112233aaabbb212222223312(ababab)(aaa)(bbb)2212223212223112233ABCDAC與|()()|2222.2rr|a,b|rr|ab||xxyyzz|rr|a||b|=121212x21y21z21x2y2z2rr2220ba,bb（其中（ooABmarcsin(m為平面|AB||m|ABC成的角AC,BC、、,AB為ABC12sinsin(sinAsinB)sin2.22221290osinsinsin222.12,另兩邊ABCAC,BCB'為ABO、,A'12tantan(sinAsinB)tan2.222'2'1290osinsinsin222.12lrrmnrmncosrcosrarcarc或（m，n為平面，.|m||n||m||n|設(shè)與與1coscoscos.的角為與212,12sinsinsinsin2sinsincos2222;1212|180)90oo1212(x,y,z)(x,y,z)若A11|ABABAB()()()zz2.1222d=A,Bxxyy22212121Ql1ha||b|)(ab)Plla=b=PQ22|a|ur|CDn|dr|n|(lln，CDlld為ll間,、,,121212r點(diǎn)B|ABn|rd（n為平面A.AB|n|dhmnmcos.222dhmn2mncosEA,AF.222'dhmn2mncos（EAAF222''AAAEm,AFn,EFd'rrrrrrrrrrrr2abbccarrrr2|a||b|a,b2|b||c|,c2|c||a|c,a(abc)a2b2c22rrrrrrrrrrra2b2c2、l、lll123、、1231sinsinsin2l2l21l22l2222222.3123123'SS.S、'SS和Vl斜棱柱側(cè)斜棱柱是c和S11cl①S.斜棱柱側(cè)1Sl②V.斜棱柱1VFE2EEFEn1nFE；21mV2mVEE.4其體積VR,334R2．S66aa.a41VSh（Sh31VSh（Sh3NmmLm.12）nNmmLm.12nn！A=n(n(nm=*mnn，mm(nm！n1.(nm1)AAmn;m1nnA;mn1AmnnmnAAmnm1n1;AAnAnn;n1n1nnAmAAmm1.mn1nn1!233!Lnn!(n1)!1.An(n(nm！mNmnC=m==(n，m*nAm12m(nm！nmC=C;mnmnnC+C1=C.mmmn1nn1C0.nnm1C;m1CCCmmnnnC;mmnmnn1nCm;mm1n1nn2C=n;rnr0CCCCCr1n1.rrrrrr1r2nCCCC2C012n.rnnnnnnCCCCC2C135024n1.nnnnnn2CCnCn2C123n1.nnnnnCCCCCCCr0r110r.rrmnmnmnmn(C)(C)(C)(C)C0212222n.2nnnnnnAC.mmnnnm.AAAAm1Amn11m1n1n1mn1nAAA1m1m1n1.mn1k(kmn).AAkmknkknkAnk1Aknk1khkh1kAA.hkhh1mnAn當(dāng)nm1nm1C.m1Annnm1mnCn.mnm、nmn(mnCCCCC有N.nnnnn(!)mmnnmn2n2nnmnm·CCCCC(mnnnnn2nnNn.n2n!!(!)mP(P=n+n+L+n)m12mn，nnn，nn這m12m12m!!有NCCC!.nnn12m!!...!ppnnnnn11m2mP(P=n+n+L+n)m12m被分完，分別得到n，nnn，nn這m12m12mCCC!pm!!nnn12mN.ppnn1m!nnab!cn12mP(P=n+n+L+n)nnnm，12m12!mn，nn這mN.n!nn!112m2nnmP(P=n+n+L+n)12m12nmn，nn這mm12m!有N.n!nn!(!12mn+n+L+n（pp12mmnnnn，n12312n等mm!NCCC.nnpnn12m!!nnnpn1m12m信nn1111f(n)n![L(1)]n.2!3!4!n!:nnmf(n,m)nC(n1)!C(nC(n3)!C(n4)!1234mmmmL(1)C(npL(1)C(nm)!ppmmmmCCCC3CC124pmmAmnnAAA2A4L(1)pL(m].m1m2mmmApnnnnn+x+L+xmx12n+x+L+xmn,mNCx（n1.12nm1+x+L+xmn,mNCx（n1.12nnm1+x+L+xmn,mNxkkN2in1(,x（))12niCn1.(n2)(k1)m1+x+L+xmn,mNxkkN2in1(,x（12niCCCCL(1)CCCn1.1n12n1n2n2n1nm1n2mnk2n2mn2k3n2m(n2)k(ab)CaCabCabCabCb;n0n1n12n22rnrrnnnnnnnTCab(r2，n).rnrrr1nmP().nn12n12n·AA12n1k2nP(k)CP(1P).kknknnPiP0(i1,2,L);PL1.21xPxPLxPL1122nnE(b)aE)b.(,)～Bnpnp.1若Pkg(k,p)qp)E.k1p1pxEDxEpLxEpL222122nn=.；baDDa(2）若～B(n,p)Dnp(1p).qkg(k,p)qp)D若P1.k(3)p2DEE2.2x12fxe,x