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1、高中文科数学重要公式及知识点速记 (1)高中数学公式及知识点总结 一，集合和逻辑 集合元素的特征:确定性、互异性、无序性. 空集的补集是全集 集合的性质: A,A1、?任何一个集合是它本身的子集，记为; ,A?空集是任何集合的子集，记为; ?空集是任何非空集合的真子集; A,BB,A2、如果，同时，那么A = B. A,B，B,C，那么A,C如果. ?(x，y)|xy =0，x?R，y?R坐标轴上的点集. ,?(x，y)|xy,0，x?R，y?R二、四象限的点集. ?(x，y)|xy,0，x?R，y?R 一、三象限的点集. 注:?对方程组解的集合应是点集. x，y,3,2x,3y,1,例: 解
2、的集合(2，1). ,点集与数集的交集是. (例:A =(x，y)| y =x+1 B=y|y =x2+1 则A?B =) nnn3. ?n个元素的子集有2个. ?n个元素的真子集有2 ,1个. ?n个元素的非空真子集有2,2个. ,4. ?一个命题的否命题为真，它的逆命题一定为真. 否命题逆命题. ,?一个命题为真，则它的逆否命题一定为真. 原命题逆否命题. 5、小范围推出大范围;大范围推不出小范围. x,5，,x,5或x,2例:若. 6、集合运算:交、并、补. 交:且ABxxAxB:,|,并:或ABxxAxB:,|补:且CAxUxA,U 关系: AAAAUAU,CUABBCACABAABB
3、ABAABB,;,;,.: 等价关系: ABABAABBABU,:CU 7、集合的运算律: 第1页(共93页) A:B,B:A;A:B,B:A.交换律: (A:B):C,A:(B:C);(A:B):C,A:(B:C)结合: A:(B:C),(A:B):(A:C);A:(B:C),(A:B):(A:C)分配 逻辑:1、命题:用语言、符号或式子表达的，可以判断真假的陈述句. 真命题:判断为真的语句.假命题:判断为假的语句. 2、“若，则”形式的命题中的称为命题的条件，称为命题的结论. pqpqpqqp3、原命题:“若，则” 逆命题: “若，则” ，p，q，q，p否命题:“若，则” 逆否命题:“若，
4、则” 4、四种命题的真假性之间的关系: (1)两个命题互为逆否命题，它们有相同的真假性; (2)两个命题为互逆命题或互否命题，它们的真假性没有关系( 5、若pq,，则是的充分条件，是的必要条件( pqqp若pq,，则是的充要条件(充分必要条件)( pqA,B利用集合间的包含关系: 例如:若，则A是B的充分条件或B是A的必要条件;若A=B，则A是B的充要条件; 6、逻辑联结词:?且 :命题形式;?或:命题形式; pq,pq,?非:命题形式. ，ppq, pqpq,，p真 真 真 真 假 真 假 假 真 假 真 假 真 假 真 假 真 假 假 假 ,7、?全称量词“所有的”、“任意一个”等，用“”
5、表示; 全称命题p:; 全称命题p的否定p:。 ，,x,M,p(x)，x,M,，p(x)?存在量词“存在一个”、“至少有一个”等，用“”表示; ， 特称命题p:; 特称命题p的否定p: ，x,M,p(x),x,M,，p(x)(1)二、函数、导数 log(M,N),logM，logNaaa1、函数的单调性 Mlog,logM,logN(1)设x、x,a,b,x,x那么 aaa1212Nf(x),f(x),0,f(x)在a,b上是增函数; 1212)n，logM,nlog,Maaf(x),f(x),0,f(x)在a,b上是减函数. 121n(2)设函数y,f(x)在某个区间内可导， logM,lo
6、gMaan,若f(x),0，则f(x)为增函数; logNaa,N,若f(x),0f(x)，则为减函数. 2、函数的奇偶性 logNb换底公式:logN,axf(,x),f(x)对于定义域内任意的，都有， logabf(x)则是偶函数; 推论:logb,logc,loga,1abcxf(,x),f(x)对于定义域内任意的，都有， ,loga,loga,.,loga,logaa2a3anan12n,11第2页(共93页) 是奇函数。 则f(x)奇函数的图象关于原点对称，偶函数的图象关于y轴对称。 3、函数在点处的导数的几何意义 xy,f(x)0,函数在点处的导数是曲线在处的切线的斜率，相应的切x
7、P(x,f(x)f(x)y,f(x)y,f(x)0000,线方程是. y,y,f(x)(x,x)0004、几种常见函数的导数 nn,1,0?;?; ?;?; C(x),nx(sinx),cosx(cosx),sinx11xxxx?;?; ?(logx);? ,(lnx),(a),alna(e),eaxlnax5、导数的运算法则 uuvuv,(1). (2). (3). ()(0),v()uvuv,()uvuvuv,，2vv6、会用导数求单调区间、极值、最值 ,7、求函数的极值的方法是:解方程(当时: yfx,fx,0fx,0，0,(1) 如果在附近的左侧fx,0，右侧fx,0，那么fx是极大值
8、; x，00,(2) 如果在附近的左侧fx,0，右侧fx,0，那么fx是极小值( x，00三、三角函数、三角变换、解三角形、平面向量 8、同角三角函数的基本关系式 ,sin22tan,sincos1,，,，=. cos,9、正弦、余弦的诱导公式 k,的正弦、余弦，等于的同名函数，前面加上把看成锐角时该函数的符号; ,k,，,的正弦、余弦，等于的余名函数，前面加上把看成锐角时该函数的符号。 ,210、和角与差角公式 ; sin()sincoscossin,; cos()coscossinsin,tantan,. tan(),1tantan,11、二倍角公式 sin2sincos,. 2222co
9、s2cossin2cos112sin,. 2tan,tan2,. 2,1tan,1cos2，22,2cos1cos2,cos;,，,2公式变形: 1,cos2,222sin1cos2,sin;,212、三角函数的周期 ,yx,，sin(),yx,，cos(),函数，x?R及函数，x?R(A,为常数，且A?0，,0)的第3页(共93页) 2,周期;函数，(A,为常数，且A?0，,0)的周,xkkZ,，,T,yx,，tan(),2,期. T,13、 函数的周期、最值、单调区间、图象变换 yx,，sin(),14、辅助角公式 b22 其中 ,tan,y,asinx，bcosx,a，bsin(x，,)
10、a15、正弦定理 abc,2R. sinsinsinABC16、余弦定理 222; abcbcA,，,2cos222; bcacaB,，,2cos222. cababC,，,2cos17、三角形面积公式 111. SabCbcAcaB,sinsinsin22218、三角形内角和定理 在?ABC中，有 ABCCAB，,，,()19、与的数量积(或内积) aba,b,|a|,|b|cos,20、平面向量的坐标运算 ,(1)设A，B,则. (,)xy(,)xyABOBOAxxyy,(,)11222121(2)设=(,)xy,=(,)xy，则a,b=. abxx，yy1122121222a,x，y(3
11、)设=，则 a(x,y)21、两向量的夹角公式 (,)xy(,)xy设a=,b=，且b,0，则 1122xx，yya,b1212cos, 2222abx，y,x，y112222、向量的平行与垂直 a/bb,a,xyxy0 . ,1221a,b,0,，,xxyy0 . ,a,b(a,0)1212附:三角形的五个“心”; 重心:三角形三条中线交点. 外心:三角形三边垂直平分线相交于一点. 内心:三角形三内角的平分线相交于一点. 垂心:三角形三边上的高相交于一点. 旁心:三角形一内角的平分线与另两条内角的外角平分线相交一点 第4页(共93页) 四、数列 23、数列的通项公式与前n项的和的关系 sn,
12、1,1( 数列的前n项的和为). a,asaaa,，？,nnn12nssn,2nn,1,24、等差数列的通项公式 *; aanddnadnN,，,，,(1)()n1125、等差数列其前n项和公式为 naa()，nn(1),d121n. ,，,nadn()s,，nad11n222226、等比数列的通项公式 ann,1*1; ,()aaqqnNn1q27、等比数列前n项的和公式为 n,aaq,aq(1),1n1,1q,1q,s,1,q 或 . 1,qs,n,n,naq,1,1naq,1,1五、不等式 x，y,xy28、已知x,y都是正数，则有，当x,y时等号成立。 2x，y(1)若积xy是定值p，
13、则当x,y时和有最小值; 2p12sx，yx,yxy(2)若和是定值，则当时积有最大值. s4六、解析几何 29、直线的五种方程 lkyykxx,()Pxy(,)(1)点斜式 (直线过点，且斜率为)( 11111l(2)斜截式 (b为直线在y轴上的截距). ykxb,，yyxx,11yy,Pxy(,)Pxy(,)xx,(3)两点式 ()(、 (). ,1211122212yyxx,2121xyab、ab、,0，,1(4)截距式 (分别为直线的横、纵截距，) ab(5)一般式 AxByC，,0(其中A、B不同时为0). 30、两条直线的平行和垂直 lykxb:,，lykxb:,，若， 11122
14、2?llkkbb|,; 121212?. llkk,1121231、平面两点间的距离公式 22,，,()()xxyyd(,)xy(,)xy(A，B). 2121AB,1122第5页(共93页) 32、点到直线的距离 |AxByC，00l (点,直线:). Pxy(,)AxByC，,0d,0022AB，33、 圆的三种方程 222(1)圆的标准方程 . ()()xaybr,，,2222(2)圆的一般方程 (,0). DEF，,4xyDxEyF，,0xar,，cos,(3)圆的参数方程 . ,ybr,，sin,34、直线与圆的位置关系 222直线与圆的位置关系有三种: Ax，By，C,0(x,a)
15、，(y,b),r; d,r,相离,0; d,r,相切,022. 弦长= 2r,dd,r,相交,0Aa，Bb，C其中. d,22A，B35、椭圆、双曲线、抛物线的图形、定义、标准方程、几何性质 22xa,cos,xyc222e,1，,1(0)ab椭圆:，a,c,b，离心率，参数方程是. ,22aabyb,sin,22cxyb222e,1,1c,a,b双曲线:(a0,b0)，离心率，渐近线方程是. y,x22aabapp2x,(,0)抛物线:，焦点,准线。抛物线上的点到焦点距离等于它到准线的距离. y,2px2236、双曲线的方程与渐近线方程的关系 2222xyxyb,0,1(1)若双曲线方程为渐
16、近线方程:. ,y,x2222ababa22xyxyb,0, (2)若渐近线方程为双曲线可设为. ,y,x22ababa2222xyxy,0,0,1 (3)若双曲线与有公共渐近线，可设为(，焦点在x轴上，2222abab焦点在y轴上). 237、抛物线的焦半径公式 y,2pxp2|PF,x，抛物线焦半径.(抛物线上的点到焦点距离等于它到准线的距ypxp,2(0)02离。) ppAB,x，x，,x，x，p38、过抛物线焦点的弦长. 121222七、立体几何 39、证明直线与直线平行的方法 (1)三角形中位线 (2)平行四边形(一组对边平行且相等) 40、证明直线与平面平行的方法 (1)直线与平面
17、平行的判定定理(证平面外一条直线与平面内的一条直线平行) 第6页(共93页) (2)先证面面平行 41、证明平面与平面平行的方法 平面与平面平行的判定定理(一个平面内的两条相交直线分别与另一平面平行) 42、证明直线与直线垂直的方法 转化为证明直线与平面垂直 43、证明直线与平面垂直的方法 (1)直线与平面垂直的判定定理(直线与平面内两条相交直线垂直) (2)平面与平面垂直的性质定理(两个平面垂直，一个平面内垂直交线的直线垂直另一个平面) 44、证明平面与平面垂直的方法 平面与平面垂直的判定定理(一个平面内有一条直线与另一个平面垂直) 45、柱体、椎体、球体的侧面积、表面积、体积计算公式 22
18、,rl圆柱侧面积=，表面积= 2,rl，2,r2,rl圆椎侧面积=，表面积= ,rl，,r1Sh(是柱体的底面积、是柱体的高). VSh,柱体31Sh(是锥体的底面积、是锥体的高). VSh,锥体3432R,VR球的半径是，则其体积,其表面积SR,4,( 346、异面直线所成角、直线与平面所成角、二面角的平面角的定义及计算 47、点到平面距离的计算(定义法、等体积法) 直棱柱、正棱柱、长方体、正方体的性质:侧棱平行且相等，与底面垂直。 48、正棱锥的性质:侧棱相等，顶点在底面的射影是底面正多边形的中心。 八、概率统计 49、平均数、方差、标准差的计算 xxx，？1222212nx,s,(x,x
19、)，(x,x)，？(x,x)平均数: 方差: n12nn1222s,(x,x)，(x,x)，？(x,x)标准差: n12n50、回归直线方程 nn,xxyyxynxy,，,iiii,ii,11,b,nn,2，其中. yabx,，22,xxxnx,，,ii,ii,11,aybx,2n(ac,bd)2K,51、独立性检验 (a，b)(c，d)(a，c)(b，d)2K随机变量越大，说明两个分类变量，关系越强，反之，越弱 52、古典概型的计算(必须要用列举法、列表法、树状图的方法把所有基本事件表示出来，不重复、不遗漏) 九、复数 53、复数的除法运算 第7页(共93页) a，bi(a，bi)(c,di
20、)(ac，bd)，(bc,ad)i. ,22c，di(c，di)(c,di)c，d22zabi,，54、复数的模=. ab，|abi，|z十、参数方程、极坐标化成直角坐标 222,，xy,xcos,55、 ,y,sin,ytan,(x,0),x,x,a，r,cos,22256(圆的参数方程可表示为. (,为参数)(x,a)，(y,b),r,y,b，rsin.,22,x,acos,xy 椭圆的参数方程可表示为. ，,1(,为参数)(a,b,0),22y,babsin.,2,x,2px,2 抛物线的参数方程可表示为. (t为参数)y,2px,y,2pt.,x,x，tcos,ol 经过点，倾斜角为的
21、直线的参数方程可表示为(为参数). M(x,y),t,Oooyyt,，sin.o,P(x,y) MP=t t0时P在M上方 00/PP=/t-t/12/12第8页(共93页) (英文版 ) easily blame, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the and , lead to safeguard the solemnity and author
22、ity of the party discipline, ensure that the party discipline and the laws and regulations for implementation in place. Throughout the discipline in the daily supervision and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Str
23、engthen to key areas, key departments and key projects as well as the masses reflect the concentration of the units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the and is a man must be hexyl, blac
24、ksmith needs its own hardware. Discipline inspection organs as the executor of the party discipline, and supervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of disciplin
25、e and rules consciousness, politics loyalty, sense obey. Action speak Ji Ordinance to set an example of the regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with and . To firmly establish the discipline must first be disci
26、plined, the supervisor will be subject to the 第9页(共93页) supervision of concept, and consciously safeguard and implement party compasses party, take the lead in practicing three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the as morality, politics and broug
27、ht to fruition; to implement , do not want to, dare not, not with disciplinary ruler to supervision; to discipline a ruler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to
28、have quit, the line has ended. Attached: indifferent to heart, calmly to the table in our life, there are many unpredictable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad thing
29、s finally turned out to be a good thing, but then we muddy however did not know, this is the life teach us things. 1, life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to wi
30、n, but the struggle; not to have conquered, but to have fought well. 2, the plain is the background of life. Live a plain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewar
31、ding perfunctorily, but calm attitude of life of unrestrained self-confidence. Plain living, there is no noise noisy, no earthly troubles, more did not fill in the discontent of desire, 第10页(共93页) some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie
32、, pain is a beginning, the struggle is a kind of process, death is a kind of ending. Give up this giving up is the helpless, do not give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is
33、 hell. Since the living, to live better. Sometimes we because of too narrow-minded, too care around the chores and penny wise and pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Dont always take everything back to
34、things, and dont get into a blind alley, dont want to face, dont be narrow-minded. Poke to care, is a kind of open-minded, a free and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is my intention to treat people; I am not afraid of the truth to tell the
35、 best friend, Im afraid he turned to it as a joke to tell dont 6, when we are in a positive frame of mind, you will find many good things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss
36、of sink, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over to the good Think, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a
37、hug. Because the world no one can 第11页(共93页) sympathize with you, have mercy on you. You cry, tears is your own; you pain, no one can understand. Then you only tears to smile. 8, each people have youth, Each youth are a story, the life of the world never gets easier, I want what, wish the world all
38、know, as has been the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, everything in the world is followed by birth, Rangrang, important thing is often the most difficult to open ones mouth, because words will reduce its importance; to let strangers pe
39、ople care about your life in the good things, the original is not easy 10, do not blame, do not laugh at who, also dont envy who. Like a person is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place fo
40、und a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances, how uncomfortable, and ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, maam, like a movie, learn t
41、o appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an exception is better to give than to what 13, dont envy him A sum of, dont lose your life and the life, respectively is: the former is a we experienced cannot escape
42、 in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things. 第12页(共93页) 14, learn to smile, learn to strong, the world you know so many people, so many people and you are, you cannot change also cant let everyone like you, so also
43、do not want to do. Life is too short to go crazy to love to go to waste, to chase the dream to regret. 15, when temper, a blessing to go. A wounding elegant people, the key is to control their own emotions. With the mouth is the most stupid behavior. A control negative emotions than a can take a cit
44、y more powerful water flow slow, language is expensive. People spent two years of time to learn to speak, but to spend a few years time to shut up. That is a kind of ability, that is a kind of wisdom. 16, life is not perfect, sometimes, growth is not a cry, not an eyeful of tears, there is no trace
45、of emotion, there is no gleam of hope, no desire, no action, no static, there is only one kind of downward sinking feeling, sink A murky? 6? 7? 6? 7 sink? 6? 7? 6? 7 toward the bottom of the sink. 17, in some way, do not go, you will not know the other side scenery is beautiful. To you is not good,
46、you do not mind too much, no one has an obligation to you; you learn knowledge, is you have weapons, you can start from scratch, but not unarmed; how do you treat people, does not represent how others treat you, if cannot see through this point, only inviting worry. 18, time is like a sponge in the
47、water, as long as you are willing to squeeze, the total water is still there. Every life, after the ups and downs The best test of live, to life, survival and continuation, do not stop the struggle in the joys and sorrows of life on the road, so that different soul to bear life beat, acceptance of 第13页(共93页) suffering. 19, indifferent to heart, calmly in tabl