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1、高一数学必修1知识点总结全高一数学必修1各章知识点总结 第一章 集合与函数概念 一、集合有关概念 1. 集合的含义 2. 集合的中元素的三个特性: (1) 元素的确定性如:世界上最高的山 (2) 元素的互异性如:由HAPPY的字母组成的集合H,A,P,Y (3) 元素的无序性: 如:a,b,c和a,c,b是表示同一个集合 3.集合的表示: 如:我校的篮球队员，太平洋,大西洋,印度洋,北冰洋 (1) 用拉丁字母表示集合:A=我校的篮球队员,B=1,2,3,4,5 (2) 集合的表示方法:列举法与描述法。 , 注意:常用数集及其记法: 非负整数集(即自然数集) 记作:N 正整数集 N*或 N+ 整
2、数集Z 有理数集Q 实数集R 1) 列举法:a,b,c 2) 描述法:将集合中的元素的公共属性描述出来，写在大括号内表示集合的方法。x R| x-32 ,x| x-32 3) 语言描述法:例:不是直角三角形的三角形 4) Venn图: 4、集合的分类: (1) 有限集 含有有限个元素的集合 (2) 无限集 含有无限个元素的集合 2(3) 空集 不含任何元素的集合 例:x|x=,5, 二、集合间的基本关系 1.“包含”关系子集 A,B注意:有两种可能(1)A是B的一部分，;(2)A与B是同一集合。 ,反之: 集合A不包含于集合B,或集合B不包含集合A,记作AB或BA 2(“相等”关系:A=B (
3、5?5，且5?5，则5=5) 2实例:设 A=x|x-1=0 B=-1,1 “元素相同则两集合相等” 即:? 任何一个集合是它本身的子集。A A ?真子集:如果A B,且A B那就说集合A是集合B的真子集，记作AB(或BA) ?如果 A B, B C ,那么 A C ? 如果A B 同时 B A 那么A=B 3. 不含任何元素的集合叫做空集，记为 规定: 空集是任何集合的子集， 空集是任何非空集合的真子集。 nn-1, 有n个元素的集合，含有2个子集，2个真子集 - 1 - 三、集合的运算 运算交 集 并 集 补 集 类型 定 设S是一个集合，A是由所有属于A且属由所有属于集合A或义 S的一个
4、子集，由S中于B的元素所组成属于集合B的元素所所有不属于A的元素组的集合,叫做A,B的组成的集合，叫做A,B成的集合，叫做S中子:交集(记作A(读BB的并集(记作:A集A的补集(或余集) 作A交B)，即(读作A并B)，即 记作CAS:AB=,x|xA，且即AB =x|xA，,S CA= x|x,S,且x,ASxB,( 或xB)( ,A 韦 S AABB恩 A 图 图2图1示 :AA=A AA=A 性 :(CA) (CB) uu:A= A=A := C (AB) u:AB=BA AB=BA :(CA) (CB) uu:ABA AB, , := C(AB) u: ABB ABB ,质 :A (CA
5、)=U u:A (CA)= ( u1.下列四组对象，能构成集合的是 ( ) A某班所有高个子的学生 B著名的艺术家 C一切很大的书 D 倒数等于它自身的实数 2.集合a，b，c 的真子集共有 个 23.若集合M=y|y=x-2x+1,xR,N=x|x?0，则M与N的关系是 . ,4.设集合A=，B=，若AB，则的取值范围是 axxa,xx12,5.50名学生做的物理、化学两种实验，已知物理实验做得正确得有40人，化学实验做得正确得有31人，两种实验都做错得有4人，则这两种实验都做对的有 人。 6. 用描述法表示图中阴影部分的点(含边界上的点)组成的集合M= . 22227.已知集合A=x| x
6、+2x-8=0, B=x| x-5x+6=0, C=x| x-mx+m-19=0, 若B?C?，A?C=，求m的值 二、函数的有关概念 1(函数的概念:设A、B是非空的数集，如果按照某个确定的对应关系f，使对于集合A中的任意一个数x，在集合B中都有唯一确定的数f(x)和它对应，那么就称f:A?B为从集合A到集合B的一个函数(记作: y=f(x)，x?A(其中，x叫做自变量，x的取值范围A叫做函数的定义域;与x的值相对应的y值叫做函数值，函数值的集合f(x)| x?A 叫做函数的值域( 注意: 1(定义域:能使函数式有意义的实数x的集合称为函数的定义域。 求函数的定义域时列不等式组的主要依据是:
7、 (1)分式的分母不等于零; (2)偶次方根的被开方数不小于零; - 2 - (3)对数式的真数必须大于零; (4)指数、对数式的底必须大于零且不等于1. (5)如果函数是由一些基本函数通过四则运算结合而成的.那么，它的定义域是使各部分都有意义的x的值组成的集合. (6)指数为零底不可以等于零， (7)实际问题中的函数的定义域还要保证实际问题有意义. 相同函数的判断方法:?表达式相同(与表示自变量和函数值的字母无关);?定义域一致 (两点必须同时具备)(见课本21页相关例2) 2(值域 : 先考虑其定义域 (1)观察法 (2)配方法 (3)代换法 3. 函数图象知识归纳 (1)定义:在平面直角
8、坐标系中，以函数 y=f(x) , (x?A)中的x为横坐标，函数值y为纵坐标的点P(x，y)的集合C，叫做函数 y=f(x),(x ?A)的图象(C上每一点的坐标(x，y)均满足函数关系y=f(x)，反过来，以满足y=f(x)的每一组有序实数对x、y为坐标的点(x，y)，均在C上 . (2) 画法 描点法: 图象变换法; 常用变换方法有三种 1平移变换 2伸缩变换 3对称变换 4(区间的概念 (1)区间的分类:开区间、闭区间、半开半闭区间 (2)无穷区间 )区间的数轴表示( (35(映射 一般地，设A、B是两个非空的集合，如果按某一个确定的对应法则f，使对于集合A中的任意一个元素x，在集合B
9、中都有唯一确定的元素y与之对应，那么就称对应f:AB为从集合A到集合B的一个映射。记作“f(对应关系):,A(原象)B(象)” ,对于映射f:A?B来说，则应满足: (1)集合A中的每一个元素，在集合B中都有象，并且象是唯一的; (2)集合A中不同的元素，在集合B中对应的象可以是同一个; (3)不要求集合B中的每一个元素在集合A中都有原象。 6.分段函数 (1)在定义域的不同部分上有不同的解析表达式的函数。 (2)各部分的自变量的取值情况( (3)分段函数的定义域是各段定义域的交集，值域是各段值域的并集( 补充:复合函数 如果y=f(u)(u?M),u=g(x)(x?A),则 y=fg(x)=
10、F(x)(x?A) 称为f、g的复合函数。 二(函数的性质 1.函数的单调性(局部性质) (1)增函数 设函数y=f(x)的定义域为I，如果对于定义域I内的某个区间D内的任意两个自变量x，x，当xx时，都有f(x)f(x)，那么就说f(x)在区间121212D上是增函数.区间D称为y=f(x)的单调增区间. 如果对于区间D上的任意两个自变量的值x，x，当xx时，都有f(x)1212 1,f(x)，那么就说f(x)在这个区间上是减函数.区间D称为y=f(x)的单调2减区间. - 3 - 注意:函数的单调性是函数的局部性质; (2) 图象的特点 如果函数y=f(x)在某个区间是增函数或减函数，那么
11、说函数y=f(x)在这一区间上具有(严格的)单调性，在单调区间上增函数的图象从左到右是上升的，减函数的图象从左到右是下降的. (3).函数单调区间与单调性的判定方法 (A) 定义法: 1 任取x，x?D，且x1，xann*N且n?( n0,0, 负数没有偶次方根;0的任何次方根都是0，记作。 a(a,0),nnnna,|a|,a,a当n是奇数时，当n是偶数时， ,a(a,0),2(分数指数幂 正数的分数指数幂的意义，规定: m*nmn，a,a(a,0,m,n,N,n,1)m,11*na,(a,0,m,n,N,n,1) mnmana- 5 - , 0的正分数指数幂等于0，0的负分数指数幂没有意义
12、 3(实数指数幂的运算性质 rrr，sa,aa(1)? ; (a,0,r,s,R)rsrs(a),a(2) ;(a,0,r,s,R)rrs(ab),aa(3) ( (a,0,r,s,R)(二)指数函数及其性质 x1、指数函数的概念:一般地，函数叫做指数函数，y,a(a,0,且a,1)其中x是自变量，函数的定义域为R( 注意:指数函数的底数的取值范围，底数不能是负数、零和1( 2、指数函数的图象和性质 a1 0a1 0a0,a0,函数y=a与y=log(-x)的图象只能是 ( ) a2.若函数在区间上的最大值是最小值的3倍，则a= a,2af(x),logx(0,a,1)a1，x3.已知，(1)
13、求的定义域(2)求使的的取值范围 xfx()fx()0,fxaa()log(01),且a1,x第三章 函数的应用 一、方程的根与函数的零点 1、函数零点的概念:对于函数，把使成立的实数xy,f(x)(x,D)f(x),0叫做函数的零点。 y,f(x)(x,D)2、函数零点的意义:函数的零点就是方程实数根，亦即函y,f(x)f(x),0数的图象与轴交点的横坐标。 xy,f(x)即:方程有实数根函数的图象与轴有交点函数xf(x),0y,f(x),有零点( y,f(x)3、函数零点的求法: 1 (代数法)求方程的实数根; f(x),0?2 (几何法)对于不能用求根公式的方程，可以将它与函数的y,f(
14、x)?图象联系起来，并利用函数的性质找出零点( 4、二次函数的零点: 2二次函数( y,ax，bx，c(a,0)2ax，bx，c,0x(1)?,，方程有两不等实根，二次函数的图象与轴有两个交点，二次函数有两个零点( 2ax，bx，c,0x(2)?,，方程有两相等实根，二次函数的图象与轴有一个交点，二次函数有一个二重零点或二阶零点( 2ax，bx，c,0x(3)?,，方程无实根，二次函数的图象与轴无交点，二次函数无零点(- 8 - (英文版 ) easily blame, to prevent the broken window effect. Supervise the leading cad
15、res to play an exemplary role, take the lead in the strict implementation of the and , lead to safeguard the solemnity and authority 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
16、and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the masses reflect the concentration of the units and departments for supervision. - strengthening superv
17、ision, discipline inspection and supervision of cadres to set an example for compliance with the and is a man must be hexyl, blacksmith 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 stri
18、ctly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline 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 acce
19、pt the supervision of the firmness and consciousness, do comply with and . To firmly establish the discipline must first be disciplined, the supervisor will be subject to the 9 supervision of concept, and consciously safeguard and implement party compasses party, take the lead in practicing three st
20、rict real strict, so loyal, clean, play. To be good at learning, the Constitution and the as morality, politics and brought 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 id
21、eological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to 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 can
22、not control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things 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
23、precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, 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
24、 answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding 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 d
25、esire, 10 some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, 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
26、 ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is 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 attra
27、ct trouble and worry, completely depends on how we look at and deal with it. Dont always take everything back to 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,
28、 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 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
29、 will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss 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, peop
30、le are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Because the world no one can 11 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
31、 youth, Each youth are a story, the life of the world never gets easier, I want what, wish the world all 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 o
32、ften the most difficult to open ones mouth, because words will reduce its importance; to let strangers people 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. Th
33、e truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place found 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
34、may got you to comfort, but never know your heart is how wanjianchuanxin. 12, maam, like a movie, learn to 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
35、A sum of, dont lose your life and the life, respectively is: the former is a we experienced cannot escape in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things. 12 14, learn to smile, learn to strong, the world you know so man
36、y people, so many people and you are, you cannot change also cant let everyone like you, so also 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 emotio
37、ns. With the mouth is the most stupid behavior. A control negative emotions than a can take a city 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 of emotion, there is no gleam of hope, no desire, no action, no static, there is only one kind of downward