集合与函数思维导图
2024-04-12 10:17:38 1 举报AI智能生成
集合与函数是数学领域中的重要概念。集合是一种特殊的数学对象，用于表示一组具有特定性质的元素。它包括空集、子集、真子集、并集、交集、补集等概念。函数则是一种特殊的映射，它以输入值x和输出值y的关系为基础，将一种集合转换为另一种集合。 集合与函数思维导图展示了集合与函数之间的关联和属性。首先，从空集开始，我们定义了基本的集合概念，如空集、子集、真子集、并集、交集、补集等。接着，我们详细介绍了不同集合之间的包含关系和运算关系。 在函数的部分，思维导图定义了函数的基本概念，包括函数、对应法则、定义域、值域等。同时，还介绍了一些常见的函数类型，如线性函数、二次函数、指数函数、对数函数等。最后，我们还探讨了函数与集合之间的联系，特别是集合作为函数定义域和值域的应用。 总的来说，集合与函数思维导图以可视化的形式展示了集合与函数之间的联系和属性，有助于理解和掌握这些重要的数学概念。
作者其他创作
大纲/内容
集合相关知识点
集合基本概念
集合中不存在重复的元素
空集是任何一个集合的子集
<span class="equation-text" contenteditable="false" data-index="0" data-equation="一个集合中，所有的子集合数量有2^n个，其中n为元素的数量；真子集有2^n-1个；非空子集的数量有2^n-1"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="个（真子集不包括全集本身"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="2" data-equation="）"><span></span><span></span></span>
公式
两种元素
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A\cup B=A+B-A\cap B"><span></span><span></span></span><br>eg:一共80人，50个男人，60个女人，那么有30个既是男的又是女的<br>
三种元素
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A \bigcup B \bigcup C =A+B+C-A\bigcap C-A\bigcap B-B\bigcap C+A\bigcap B\bigcap C"><span></span><span></span></span><br> = 1x1次+2x1次+3x1次<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A \bigcup B \bigcup C = A+B+C-1\times2次-2\times3次"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A+B+C = 1\times1次+2\times2次+3\times3次"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A \bigcup B \bigcup C 代表总数量（不重复）；A+B+C代表所有数（重复）"><span></span><span></span></span><br><br>
题目
eg:共50道题，小明会40道，小华会45道，小李会50道，那么他们三人至少会多少道题
解：40+45+50-（3-1）x50=35道
解题思路：根据公式<span class="equation-text" contenteditable="false" data-index="0" data-equation="A \bigcup B \bigcup C =A+B+C-A\bigcap C-A\bigcap B-B\bigcap C+A\bigcap B\bigcap C"><span></span><span></span></span><br>63+89+47-（46+24+24+24）+24+15=120人
整式的因式和余式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)含有(ax-b)因式↔f(x)能被(ax-b)整除↔f({b \over a})=0;"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)含有(x-a)因式↔f(x)能被(x-a)整除↔f(a)=0"><span></span><span></span></span>
整式及运算（常考公式）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x\pm y)^2 = x^2 \pm 2xy+ y^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2 + y^2 = (x+y)^2-2xy"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2 + y^2 = (x-y)^2 + 2xy"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2 + y^2 + z^2 \pm xy \pm xz \pm yz = {1 \over 2}[(x\pm y)^2 + (x\pm z)^2 + (y\pm z)^2]"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2 + {1 \over x}^2 = (x+{1 \over x})^2-2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^2-y^2=(x+y)(x-y)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="2(xy+xz+yz)=(x+y+z)^2 - (x^2+y^2 +z^2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{1 \over x} + {1 \over y} + {1 \over z}=0=> (x+y+z)^2 = x^2 + y^2 + z^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^3 + y^3 = (x+y)(x^2-xy+y^2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^3+y^3 = (x+y)^3-3xy(x+y)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^3+({1 \over x})^3 = (x+{1 \over x})^3-3(x+{1 \over x})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^3-y^3 = (x-y)(x^2+xy+y^2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x+y)^3 = x^3 + 3x^2y+3xy^2+y^3"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x-y)^3=x^3-3x^2y+2xy^2+y^3"><span></span><span></span></span>
函数
一元二次函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=ax^2 + bx + c(c\neq0)"><span></span><span></span></span>
图像：抛物线
开口方向：a>0向上；a<0向下
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对称轴：-{b \over 2a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="顶点坐标：\left\{ - {b \over 2a},{4ac-b^2 \over 4a}\right\}"><span></span><span></span></span>
图像与坐标轴交点坐标：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac>0 ;与x轴有两个不同交点;存在两个不相等的根"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac=0 ;与x轴有一个交点;存在两个相等的根"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac<0; 与x轴无交点，无根"><span></span><span></span></span>
最大最小值：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>0时有最小值,为 {4ac-b^2 \over 4a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a<0时有最大值,为{4ac-b^2 \over 4a}"><span></span><span></span></span>
恒正，恒负：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="恒正需满足 a>0且 \triangle <0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="恒负需满足 a<0 且\triangle <0"><span></span><span></span></span>
指数函数和对数函数
运算公式
图像及性质
方程
定义：<span class="equation-text" contenteditable="false" data-index="0" data-equation="ax^2+bx+c=0(a≠0)x_1,x_2为方程的两根"><span></span><span></span></span>
解法：
十字相乘法
求根公式法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="{ -b\pm\sqrt{b^2-4ac}\over 2a}"><span></span><span></span></span>
根的判断：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle>0 有两个不相等的实根"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=0 有两个相等的实根"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle<0 无实根"><span></span><span></span></span>
根与系数关系：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="两根之和:x_1+x_2 = -{b \over a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="两根之积:x_1 \cdot x_2 = {c \over a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="两根倒数之和:{1 \over x_1}+{1 \over x_2}=-{b \over c}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="两根距离:|x_1-x_2|={\sqrt{b^2-4ac} \over |a|}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x_1-x_2)^2=(x_1+x_2)^2-4x_1x_2"><span></span><span></span></span>
根的分布：
有两正根
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle = b^2-4ac \geq 0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1+x_2 = -{b \over a} >0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1\cdot x_2 = {c \over a} >0"><span></span><span></span></span>
有两负根
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac \geq0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1+x_2=-{b \over a}<0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1\cdot x_2={c \over a}>0"><span></span><span></span></span>
一正一负
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\triangle=b^2-4ac>0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1\cdot x_2={c \over a}<0 (a,c异号)"><span></span><span></span></span>
两根绝对值大小关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|正|>|负|"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1+x_2 = -{b \over a}>0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1\cdot x_2 ={c \over a} <0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|正|<|负|"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1+x_2 = -{b \over a}<0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1\cdot x_2 ={c \over a} <0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="有两个根，一个根比m大，一个根比m小;可知a\cdot f(m)<0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="情况1：因为M在A，B两个根中间，所以f(M)<0；所以a\cdot f(M)<0(a>0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="情况2：因为M在A,B两个根之间，所以f(M)>0;所以a\cdot f(M)<0 (a<0)"><span></span><span></span></span>
均值不等式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="算数平均值:设n个数x_1,x_2,...,x_n,称\overline{x}={x_1+x_2+...+x_n \over n}为这n个数的算数平均值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="几何平均根:设n个数x_1,x_2,...,x_n,称a_g=\sqrt[n]{x_1\cdot x_2 ... x_n}为这n个数的几何平均值(几何平均值是对于正数而言)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="均值不等式:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{x_1+x_2+...+x_n \over n}\geq\sqrt[n]{x_1\cdot x_2...x_n}(当且这些数都相等时等式成立)"><span></span><span></span></span>
一正数、二定值、三相等
其他不等式
不等式的基本性质：
传递性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b,b>c => a>c"><span></span><span></span></span>
同向相加性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b,c>d=>a+c>b+d"><span></span><span></span></span>
同向皆正相乘性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b>0,c>d>0=>ac>bd"><span></span><span></span></span>
皆正倒数性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b>0=>{1 \over b}>{1 \over a}"><span></span><span></span></span>
皆正乘方性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b>0=>a^n>b^n(n\in z^+)"><span></span><span></span></span>
一元二次不等式：
简单绝对值不等式（绝对值不等式的核心思想是先脱掉绝对值符号转化为一般不等式再去解题）：
零点分类讨论法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)|=\begin{Bmatrix}f(x) & f(x)>0 \\-f(x)& f(x)<0\end{Bmatrix}"><span></span><span></span></span>
公式法：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)|<a(a>0)=>-a<f(x)<a"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)|>0(a>0)=>f(x)>a 或 f(x)<-a"><span></span><span></span></span>
平方法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)|>|g(x)|=>|f(x)|^2>|g(x)|^2 => (f(x)+g(x))(f(x)-g(x))>0"><span></span><span></span></span>
简单分式不等式（简单的分式不等式主要考察的是其解法，当分母不确定的情况下，不要轻易去分母，<br>一定要先移项之后再通分合并求解，它的标准型分为如下几种情况:）：<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{f(x) \over g(x)}\geq p(x)=>{f(x)-g(x)p(x) \over g(x)}\geq 0=>[f(x)-g(x)p(x)]\cdot g(x)\geq 0 (g(x)\neq 0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{f(x) \over g(x)}\leq p(x)=>{f(x)-g(x)p(x) \over g(x)}\leq 0=>[f(x)-g(x)p(x)]\cdot g(x)\leq0 (g(x)\neq0)"><span></span><span></span></span>
数列基本定义及概念
<span class="equation-text" contenteditable="false" data-index="0" data-equation="数列的定义:按一定次序排列的一列数叫做数列，一般形式：a_1,a_2...a_n,记作\lbrace a_n \rbrace"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="数列的前n项和:S_n=a_1+a_2+...+a_n=\begin{matrix} \sum_{i=1}^n a_i \end{matrix}"><span></span><span></span></span>
等差数列
<span class="equation-text" contenteditable="false" data-index="0" data-equation="定义：如果在数列\lbrace a_n\rbrace中，a_{n+1}-a_n = d (常数),则称数列\lbrace a_n\rbrace为等差数列，d为公差"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="通项公式:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n=a_1+(n-1)d=a_m+(n-m)d=dn+(a_1-d)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当d\neq0 时,a_n=dn+(a_1-d)有以下特点:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n是关于d的一次函数,斜率为公差d,所有数字和d+(a_1-d)=a_1"><span></span><span></span></span>
图像为一条直线
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d>0时为单调递增函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d<0时为单调递减函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=0时为一条横线"><span></span><span></span></span>
最值问题
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d>0,a_n=0时,S_n有最小值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d<0,a_n=0时,S_n有最大值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="等差中项:a,A,b为等差数列，则2A=a+b"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="前n项和公式(重点):"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_n={(a_1+a_n )n\over 2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_n={d \over 2}n^2+(a_1-{d \over 2})n (当公差d不为0时有以下性质)"><span></span><span></span></span>
常数项为0，那么过零点
开口方向由d的符号决定
二次项系数为半公差<span class="equation-text" contenteditable="false" data-index="0" data-equation="{d \over 2}"><span></span><span></span></span>
对称轴<span class="equation-text" contenteditable="false" data-index="0" data-equation="x={1 \over 2}-{a_1 \over d} 求最值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="等差数列平均值\overline{x}={x_1+x_1+...+x_n \over n}={x_1+x_n \over 2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="等差数列的性质:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若m,n,l,k\in Z^+,m+n=l+k则a_m+a_n=a_l+a_k"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\left\{a_n \right\}为等差数列，S_n为前n项的和，则S_n,S_{2n}-S_n,S_{3n}-S_{2n},...也为等差数列，其公差为n^2d"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\left \{ \frac{\lbrace a_n\rbrace为等差数列其前n项和为S_N}{\lbrace b_n\rbrace为等差数列其前n项和为T_N} \right. 则{ a_k\over b_k}={S_{2k-1} \over T_{2k-1}}"><span></span><span></span></span>
等比数列
<span class="equation-text" contenteditable="false" data-index="0" data-equation="定义:如果在数列\lbrace a_n\rbrace中,{a_{n+1} \over a_n}=q(常量)(q\neq0),则称\lbrace a_n\rbrace为等比数列,q为公比"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="通项公式:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n=a_1 q^{n-1}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n={a_1 \over q}q^n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="等比中项:a,G,b成等比数列,则G^2=ab"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="前n项和公式:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_n=na_1(q=1)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_n={a_1(1-q^n) \over 1-q}={a_1 \over 1-q}-{a_1q^n \over 1-q}(q\neq0且q\neq1)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="无穷递减等比数列前n项和:对于无穷递缩等比数列(|q|<1,q\neq0),当n\to 正无穷时,q^n->0,从而存在所有项之和为S_n={a_1 \over 1-q}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="等比数列的性质:"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若m,n,l,k\in Z^+,m+n=l+k则a_m\ast a_n=a_l \ast a_k"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\left\{a_n \right\}为等比数列，S_n为前n项的和，则S_n,S_{2n}-S_n,S_{3n}-S_{2n},...也为等比数列，其公差为q^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="{S_n \over S_m}={1-q^n \over 1-q^m}"><span></span><span></span></span>
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