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1、高中数学必修5,必修2公式大全71.常用不等式: 22(1)(当且仅当a,b时取“=”号)( abab，,2abR,ab，(2)(当且仅当a,b时取“=”号)( ,ababR,2333(3) abcabcabc，,3(0,0,0).(4)柯西不等式 22222 ()()(),.abcdacbdabcdR，,，,(5). a,b,a，b,a，b72.极值定理 已知都是正数，则有 x,y(1)若积是定值，则当时和有最小值; x，yxypx,y2p12s(2)若和x，y是定值，则当时积有最大值. x,yxys422推广 已知，则有 x,y,R(x，y),(x,y)，2xy(1)若积是定值,则当最大时
2、,最大; xy|x,y|x，y|当最小时,最小. |x,y|x，y|(2)若和是定值,则当最大时, 最小; |x，y|x,y|xy|当最小时, 最大. |x,y|xy|2273.一元二次不等式，如果与aaxbxc，,0(0)或(0,40)abac,22axbxc，axbxc，同号，则其解集在两根之外;如果与异号，则其解集在两根之a间.简言之:同号两根之外，异号两根之间. ; xxxxxxxxx,()()0()121212. xxxxxxxxxx,()()0()或12121274.含有绝对值的不等式 当a 0时，有 22xaxaaxa,. 22xaxaxa,或. xa,75.无理不等式 fx()
3、0,(1) . fxgx()(),gx()0,fxgx()(),fx()0,fx()0,(2). fxgx()(),或gx()0,gx()0,2,fxgx()(),fx()0,(3). fxgx()(),gx()0,2,fxgx()(),76.指数不等式与对数不等式 a,1(1)当时, fxgx()(); aafxgx,()()of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction
4、and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG . Pressure gauges, using an installed spark arrestor for acetylene cy
5、linders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construct
6、ion person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 12 million. Direct responsibility for the accident
7、and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident
8、and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury accident occurs, the direct punishment 500-1000, who is directly responsible for the accident responsibility, give notice of criticis
9、m and 50-100 economic sanctions against them. (4) to conceal the accident, reported without undue delay or false, to inform the administrative leadership of the criticism, resulting in serious fx()0,log()log()()0fxgxgx,. ,aa,fxgx()(),01,a(2)当时, fxgx()(); aafxgx,()()fx()0,log()log()()0fxgxgx, ,aa,fxg
10、x()(),77.斜率公式 yy,21(、). Pxy(,)Pxy(,)k,111222xx,2178.直线的五种方程 lk(1)点斜式 (直线过点，且斜率为)( yykxx,()Pxy(,)11111l(2)斜截式 (b为直线在y轴上的截距). ykxb,，yyxx,11(3)两点式 ()(、 (). yy,Pxy(,)Pxy(,)xx,1211122212yyxx,2121xyab、ab、,0，,1(4)截距式 (分别为直线的横、纵截距，) ab(5)一般式 (其中A、B不同时为0). AxByC，,079.两条直线的平行和垂直 (1)若， lykxb:,，lykxb:,，111222?;
11、 llkkbb|,121212?. llkk,11212(2)若,且A、A、B、B都不为零, lAxByC:0，,lAxByC:0，,121211112222ABC111?; ll|,12ABC222?; llAABB,，,012121280.夹角公式 kk,21(1)tan|,. ,1kk，21lykxb:,，lykxb:,，(，,) kk,111122212ABAB,1221(2)tan|,. ,AABB，1212lAxByC:0，,lAxByC:0，,(,). AABB，,0111122221212,ll,直线时，直线l与l的夹角是. 12122ll81. 到的角公式 12kk,21ta
12、n,(1),. 1kk，21lykxb:,，lykxb:,，(，,) kk,111122212of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and accep
13、tance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG . Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure m
14、easures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the jo
15、b unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 12 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brough
16、t to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personne
17、l Department of the company. (3) personal injury accident occurs, the direct punishment 500-1000, who is directly responsible for the accident responsibility, give notice of criticism and 50-100 economic sanctions against them. (4) to conceal the accident, reported without undue delay or false, to i
18、nform the administrative leadership of the criticism, resulting in serious ABAB,1221(2). tan,AABB，1212(,). lAxByC:0，,lAxByC:0，,AABB，,0111122221212,直线时，直线l到l的角是. ll,1212282(四种常用直线系方程 (1)定点直线系方程:经过定点的直线系方程为(除直线Pxy(,)yykxx,()00000k),其中是待定的系数; 经过定点的直线系方程为xx,Pxy(,)0000,其中是待定的系数( AxxByy()()0,，,AB,00(2)共点直
19、线系方程:经过两直线,的交点lAxByC:0，,lAxByC:0，,11112222的直线系方程为(除)，其中是待定的系数( ()()0AxByCAxByC，,l1112222(3)平行直线系方程:直线中当斜率k一定而b变动时，表示平行直线ykxb,，,0系方程(与直线平行的直线系方程是()，是AxByC，,0AxBy，,0参变量( (4)垂直直线系方程:与直线 (A?0，B?0)垂直的直线系方程是AxByC，,0,是参变量( BxAy,，,083.点到直线的距离 |AxByC，00l(点,直线:). Pxy(,)AxByC，,0d,0022AB，,084. 或所表示的平面区域 AxByC，,
20、0,0设直线，则或所表示的平面区域是: lAxByC:0，,AxByC，,0B,0lBB若，当与同号时，表示直线的上方的区域;当与AxByC，AxByC，l异号时，表示直线的下方的区域.简言之,同号在上,异号在下. B,0lAA若，当与同号时，表示直线的右方的区域;当与AxByC，AxByC，l异号时，表示直线的左方的区域. 简言之,同号在右,异号在左. ,085. 或所表示的平面区域 ()()0AxByCAxByC，,111222设曲线CAxByCAxByC:()()0，,(AABB,0)，则 1112221212,0()()0AxByCAxByC，,或所表示的平面区域是: 111222()
21、()0AxByCAxByC，,所表示的平面区域上下两部分; 111222()()0AxByCAxByC，,所表示的平面区域上下两部分. 11122286. 圆的四种方程 222(1)圆的标准方程 . ()()xaybr,，,2222DEF，,4(2)圆的一般方程 (,0). xyDxEyF，,0xar,，cos,(3)圆的参数方程 . ,ybr,，sin,()()()()0xxxxyyyy,，,(4)圆的直径式方程 (圆的直径的端点是1212Axy(,)Bxy(,)、). 112287. 圆系方程 Axy(,)Bxy(,)(1)过点,的圆系方程是 1122()()()()()()()()0xx
22、xxyyyyxxyyyyxx,，,，, 1212112112,，,，,()()()()()0xxxxyyyyaxbyc,axbyc，,0,其中是直线1212AB的方程,是待定的系数( of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump in
23、stallation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG . Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yu
24、an. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job
25、 in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 12 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerni
26、ng administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative s
27、anctions should be dealt with by the personnel Department of the company. (3) personal injury accident occurs, the direct punishment 500-1000, who is directly responsible for the accident responsibility, give notice of criticism and 50-100 economic sanctions against them. (4) to conceal the accident
28、, reported without undue delay or false, to inform the administrative leadership of the criticism, resulting in serious 22lC(2)过直线:与圆:的交点的圆系方程AxByC，,0xyDxEyF，,022是,是待定的系数( xyDxEyFAxByC，,()02222(3) 过圆:与圆:的交CCxyDxEyF，,0xyDxEyF，,0212221112222点的圆系方程是,是待定的xyDxEyFxyDxEyF，,()0111222系数( 88.点与圆的位置关系 222点与圆的位
29、置关系有三种 Pxy(,)(x,a)，(y,b),r0022daxby,，,()()若，则 00dr,dr,dr,点在圆外;点在圆上;点在圆内. PPP89.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)，(y,b),r; d,r,相离,0; d,r,相切,0. d,r,相交,0Aa，Bb，C其中. d,22A，B90.两圆位置关系的判定方法 设两圆圆心分别为O，O，半径分别为r，r，OO,d 121212; d,r，r,外离,4条公切线12; d,r，r,外切,3条公切线12r,r,d,r，r,相交,2条公切线; 1212d,r,r,内切,1条公切线; 1
30、20,d,r,r,内含,无公切线. 1291.圆的切线方程 22(1)已知圆( xyDxEyF，,0?若已知切点(,)xy在圆上，则切线只有一条，其方程是 00DxxEyy()()，00 xxyyF，,0. 0022DxxEyy()()，00(,)xyxxyyF，,0当圆外时, 表示过两个切点000022的切点弦方程( yykxx,()?过圆外一点的切线方程可设为，再利用相切条件求k，这时必00有两条切线，注意不要漏掉平行于y轴的切线( ?斜率为k的切线方程可设为，再利用相切条件求b，必有两条切线( ykxb,，222(2)已知圆( xyr，,2Pxy(,)?过圆上的点的切线方程为xxyyr，
31、,; 000002k?斜率为的圆的切线方程为. ykxrk,，1109(证明直线与直线的平行的思考途径 (1)转化为判定共面二直线无交点; (2)转化为二直线同与第三条直线平行; (3)转化为线面平行; of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan a
32、nd pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG . Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator f
33、ined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work a
34、nd the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 12 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such a
35、s concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning admini
36、strative sanctions should be dealt with by the personnel Department of the company. (3) personal injury accident occurs, the direct punishment 500-1000, who is directly responsible for the accident responsibility, give notice of criticism and 50-100 economic sanctions against them. (4) to conceal th
37、e accident, reported without undue delay or false, to inform the administrative leadership of the criticism, resulting in serious (4)转化为线面垂直; (5)转化为面面平行. 110(证明直线与平面的平行的思考途径 (1)转化为直线与平面无公共点; (2)转化为线线平行; (3)转化为面面平行. 111(证明平面与平面平行的思考途径 (1)转化为判定二平面无公共点; (2)转化为线面平行; (3)转化为线面垂直. 112(证明直线与直线的垂直的思考途径 (1)转化
38、为相交垂直; (2)转化为线面垂直; (3)转化为线与另一线的射影垂直; (4)转化为线与形成射影的斜线垂直. 113(证明直线与平面垂直的思考途径 (1)转化为该直线与平面内任一直线垂直; (2)转化为该直线与平面内相交二直线垂直; (3)转化为该直线与平面的一条垂线平行; (4)转化为该直线垂直于另一个平行平面; (5)转化为该直线与两个垂直平面的交线垂直. 114(证明平面与平面的垂直的思考途径 (1)转化为判断二面角是直二面角; (2)转化为线面垂直. 115.空间向量的加法与数乘向量运算的运算律 (1)加法交换律:a，b=b，a( (2)加法结合律:(a，b)，c=a，(b，c)(
39、(3)数乘分配律:(a，b)=a，b( 116.平面向量加法的平行四边形法则向空间的推广 始点相同且不在同一个平面内的三个向量之和，等于以这三个向量为棱的平行六面体的以公共始点为始点的对角线所表示的向量. 117.共线向量定理 对空间任意两个向量a、b(b?0 )，a?b存在实数使a=b( ,PAB、APtAB,三点共线. APAB|,OPtOAtOB,，(1),ABCD、ABCD、CDABtCD,AB、共线且不共线且不共线. ABCD|,118.共面向量定理 xy,向量p与两个不共线的向量a、b共面的存在实数对,使( paxby,，,xy,推论 空间一点P位于平面MAB内的存在有序实数对,使
40、， ,MPxMAyMB,，,xy,或对空间任一定点O，有序实数对，使. OPOMxMAyMB,，,O119.对空间任一点和不共线的三点A、B、C，满足OPxOAyOBzOC,，k,1Ok,1()，则当时，对于空间任一点，总有P、A、B、C四点共面;当xyzk，,O,O,时，若平面ABC，则P、A、B、C四点共面;若平面ABC，则P、A、B、C四点不共面( ,AB、 C DACABAD四点共面与、共面 ,ADxAByAC,，,O,(平面ABC). ODxyOAxOByOC,，(1)120.空间向量基本定理 如果三个向量a、b、c不共面，那么对空间任一向量p，存在一个唯一的有序实数组x，y，z，使
41、p,xa，yb，zc( of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 H
42、G20201-2000 construction installation engineering construction standard 7, HG . Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a
43、 fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit wil
44、l charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 12 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant De
45、partment. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) pe
46、rsonal injury accident occurs, the direct punishment 500-1000, who is directly responsible for the accident responsibility, give notice of criticism and 50-100 economic sanctions against them. (4) to conceal the accident, reported without undue delay or false, to inform the administrative leadership
47、 of the criticism, resulting in serious 推论 设O、A、B、C是不共面的四点，则对空间任一点P，都存在唯一的三个有序实,数x，y，z，使. OPxOAyOBzOC,，121.射影公式 ,llll已知向量=a和轴，e是上与同方向的单位向量.作A点在上的射影，作BAABl点在上的射影，则 B,，e=?e aaABAB,|cos122.向量的直角坐标运算 设a,，b,则 (,)aaa(,)bbb123123(1)a，b,; (,)ababab，112233(2)a,b,; (,)ababab,112233(3)a, (?R); (,),aaa123(4)a?b,; ababab，112233123.设A，B，则 (,)xyz(,)xyz111222,= . ABOBOA,