TheBadZhang 发布的文章

概率

排列组合

排列 组合
$A_n^k = \frac {n!}{(n-k)!}$ $C_n^k = \frac {n!}{(n-k)!k!}$

均值与方差

期望 方差
$E(X) = \sum_{i=1}^{n} (x_i p_i)$ $D(X) = \sum_{i=1}^{n} (x_i - E(X))^2 p_i$
  1. 均值方差的性质
    • $E(aX+b) = aE(X) + b$
    • $D(aX+b) = a^2D(X)$
  2. 两点分布与二项分布的均值、方差
    • 若$X$服从两点分布,则$E(X) = p$,$D(X) = p(1-p)$
    • 若$X \sim B(n, p)$,则$E(X) = np$,$D(X) = np(1-p)$

不等式

(1)$\frac{a+b}{2}\geq\sqrt{ab}$

(2)$a^2+b^2\geq2ab$

(3)${a+b+c}{3}\geq{(abc)}^\frac{1}{3}$

(4)$a^3+b^3+c^3\geq 3abc$

(5)$\frac{a_1+a_2+\dots+a_n}{n} \geq {(a_1a_2…a_n)}^\frac{1}{n}$

(6)$\frac{2}{\frac{1}{a}+\frac{1}{b}}\leq\sqrt{ab}\leq\frac{a+b}{2}\leq\sqrt{\frac{a^2+b^2}{2}}$

  1. 均值不等式:
    • 两数均值不等式:$\frac{a+b}{2} \geq \sqrt {ab}$
    • n 数均值不等式:$$\frac {a_1+a_2+ \cdots + a_n}{n} \geq \sqrt [n]{a_1a_2 \cdots a_n}$$
    • 调和平均数、几何平均数、算术平均数、平方平均数
    • $\frac {2}{\frac{1}{a}+\frac{1}{b}} \leq \sqrt{ab} \leq \frac {a+b}{2} \leq \sqrt {\frac{a^2 + b^2}{2}}$
  2. 柯西不等式:$(a^2 + b^2)(c^2 + d^2) \geq (ab + cd)^2$
  3. 糖水不等式:$\frac {b}{a} < \frac {b+c}{a+c}$

不等式拓展阅读

三角函数

特殊值

正弦 余弦 正切
$\sin (0) = 0$ $\cos (0) = 1$ $\tan (0) = 0$
$\sin (\frac {\pi}{6}) = \frac {1}{2}$ $\cos (\frac {\pi}{6}) = \frac {\sqrt {3}}{2}$ $\tan (\frac {\pi}{6}) = \frac {\sqrt{3}}{3}$
$\sin (\frac {\pi}{4}) = \frac {\sqrt {2}}{2}$ $\cos (\frac {\pi}{4}) = \frac {\sqrt {2}}{2}$ $\tan (\frac {\pi}{4}) = 1$
$\sin (\frac {\pi}{3}) = \frac {\sqrt {3}}{2}$ $\cos (\frac {\pi}{3}) = \frac {1}{2}$ $\tan (\frac {\pi}{3}) = \sqrt {3}$
$\sin (\frac {\pi}{2}) = 1$ $\cos (\frac {\pi}{2}) = 0$ $\tan (\frac {\pi}{2}) = +\infty$

诱导公式

和差角公式

  • $\cos (a+b) = \cos a \cos b - \sin a \sin b$
  • $\cos (a-b) = \cos a \cos b + \sin a \sin b$
  • $\sin (a \pm b) = \sin a \cos b \pm \cos a \sin b$
  • $\tan (a+b)=\frac{\tan a+ \tan b}{1 - \tan a \cdot \tan b}$
  • $\tan (a-b)=\frac{\tan a- \tan b}{1+ \tan a \cdot \tan b}$

和差化积

  • $\sin a+ \sin b=2 \sin \frac{a+b}{2} \cos \frac{a-b}{2}$
  • $\sin a- \sin b = 2 \cos \frac{a+b}{2} \sin \frac{a-b}{2}$
  • $\cos a+ \cos b = 2 \cos \frac{a+b}{2} \cos \frac{a-b}{2}$
  • $\cos a- \cos b = -2 \sin \frac{a+b}{2} \sin \frac{a-b}{2}$
  • $\tan a \pm \tan b = \frac {\sin (a \pm b)}{\cos a \cdot \cos b}$
  • $\cot a \pm \cot b = \pm \frac {\sin (a \pm b)}{\sin a \cdot \sin b}$

积化和差

  • $\sin \alpha \cos \beta = \frac{1}{2}[\sin (\alpha + \beta) + \sin(\alpha - \beta)]$
  • $\cos \alpha \sin \beta = \frac{1}{2}[\sin (\alpha - \beta) + \sin(\alpha - \beta)]$
  • $\cos \alpha \cos \beta = \frac{1}{2}[\cos (\alpha + \beta) + \cos(\alpha - \beta)]$
  • $\sin \alpha \sin \beta = - \frac{1}{2}[\cos (\alpha - \beta) + \cos(\alpha - \beta)]$

二倍角

  • $\sin (2x) = 2 \sin(x) \cos(x)$
  • $\cos (2x) = \cos^2(x) - \sin^2(x) = 2 \cos^2(x) - 1 = 1 - 2 \sin^2(x)$
  • $\tan (2x) = \frac{2 \tan a}{1 - \tan^2 a}$

正弦定理

$$\frac{a}{\sin(A)}=\frac{c}{sin(C)}=\frac{c}{sin(C)}=2R$$

余弦定理

$$cos(C) = \frac {a^2+b^2-c^2}{2ab}$$

升幂降角

升角降幂

$$S_{\triangle ABC} = \frac {1}{2}AB \sin C$$

向量

数列

$a_n = a_1 + (n-1)d$

若$a_n$是等差数列,则有

$$S_n = \frac {(a_1 + a_n) * n}{2}$$

$a_n = a_1 q^{n-1}$

若$a_n$是等比数列,则有

$$S_n = \frac {a_1 * (1 - q^n)}{1-q}$$

$a_n = Sn - S{n-1}$

  1. 裂项相消
    1. $a_n=\frac{1}{n(n+1)}$

$$\begin{aligned}S_n&=a_1+a_2+a_3+\cdots+a_n \\ &=\frac{1}{1\times 2}+\frac{1}{2\times 3}+\cdots+\frac{1}{n(n+1} \\ &=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdots+\frac{1}{n}-\frac{1}{n+1} \\ &=1-\frac{1}{n+1}\end{aligned}$$

  1. 错位相减

    函数

奇函数:$f(x) = -f(x)$

偶函数:$f(x)=f(-x)$
$a^r \times a^s = a^{r+s}$
${(ab)}^r = a^rb^r$
${(a^r)}^s=a^{rs}$

二次函数

  1. $\Delta=b^2-4ac$
  2. $x = \frac{-b \pm \sqrt{\Delta}}{2ab}$
  3. 韦达定理

$\left\{\begin{aligned}x&=1\\y&=2+x\end{aligned}\right.$

复数

$a+bi$

空间几何

$S{圆柱体}=2\pi r(r+l)$
$V
{柱体}=Sh$
$S{圆锥}$
$V
{锥}$
$S{圆台}$
$V
{台}$
$S{球}$
$V
{球}$

解析几何

直线$y = kx + b$$Ax + By + C = 0$
$y = \frac {1}{x}$
$y = ax^2$

圆锥曲线

圆$x^2 + y^2 = r^2$
椭圆$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$
与椭圆相交的直线,交点线段长:$|AB| = \sqrt{1+k^2}|x_1 - x_2| = \sqrt{1+\frac{1}{k^2}}|y_1 - y_2|$
抛物线$x = 2py$
抛物线焦点弦长:$|AB| = x_1 + x_2 + p = \frac{2p}{\sin^2 \theta} \geq 2p$

$\frac{1}{|AF|}+\frac{1}{|BF|} = \frac{2}{p}$

$x_1x_2 = \frac{p^2}{4}$ $y_1y_2 = -p^2$

$|AF| = \frac{p}{1- \cos \theta}$$|BF| = \frac{p}{1+ \cos \theta}$

$S_{\triangle ABC} = \frac{p^2}{2 \sin \theta}$

双曲线$\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1$

渐近线$y = \pm \frac {b}{a} x$

斜率公式:$k_{p_1p_2} = \frac{y_1 - y_2}{x_1 - x_2}$

倾斜角$\alpha$:$k = \tan \alpha (\alpha \neq \frac{\pi}{2})$

点到直线距离公式:$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$

中点公式:$x = \frac{x_1+x_2}{2}$$x = \frac{y_1+y_2}{2}$

重心公式:$x = \frac{x_1+x_2+x_3}{3}$$y = \frac{y_1+y_2+y_3}{3}$

线段长度:$s=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

对数

$(N>0,a>0,a\neq 1)$
$\log_a{MN}=\log_aM+\log_aN$
$\log^a{\frac{M}{N}}=\log_aM-\log_aN$
$\log_a{N^n}=n\log_aN$
$a^{\log_a{N}} = N$
$\log_a{a} = 1$
$\log_a{1} = 0$
$(a>0且a \neq 1, c>0 且 c \neq 1)$
$\log_a{b} = \frac{\log_c{b}}{\log_c{a}}$

导数

原函数 导函数
$kx + b$ $k$
$x^a$ $ax^{a-1}$
$\frac{1}{x}$ $- \frac{1}{x^2}$
$\ln{x}$ $\frac{1}{x}$
$a^x$ $a^x \ln{a}$
$log_a{x}$ $\frac{1}{x \ln{a}}$
$\sin x$ $\cos x$
$\cos x$ $-\sin x$
$uv$ $uv'+u'v$
$u+v$ $u'+v'$

https://zhuanlan.zhihu.com/p/41855459
https://www.mohu.org/info/symbols/symbols.htm
https://texwiki.texjp.org/?LaTeX%E5%85%A5%E9%96%80%2F%E7%B0%A1%E5%8D%98%E3%81%AA%E6%95%B0%E5%BC%8F%282%29#ma22efee

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