重要公式

第一类重要极限

$$\lim\limits_{x\rightarrow{0}} \frac{\sin{x}}{x} = 1$$

第二类重要极限

$$\lim\limits_{x\rightarrow\infty}(1+\frac{1}{x})^{x} = \lim\limits_{x\rightarrow\infty}{e}^{x\cdot\ln{(1+\frac{1}{x})}}=\lim\limits_{x\rightarrow\infty}e^{{x}\cdot{\frac{1}{x}}}=e$$

$\frac{0}{0}$ 型等价无穷小量

等价公式	广东普通专升本	重要程度
$\sin x \sim x$	√
$\arcsin x \sim x$	√
$\tan x \sim x$	√
$\arctan x \sim x$	√
$e^x-1 \sim x$	√	$\bigstar$
$ln(1+x) \sim x$	√	$\bigstar$
$1-\cos x \sim \frac{1}{2}x^2$	√	$\bigstar$
$\sqrt[n]{1+x}-1 \sim \frac{x}{n} \iff (1+ax)^b-1 \sim abx (特点是 -1)$	√	$\bigstar$
$x-\sin x \sim \frac{1}{6}x^3$
$\tan x-x \sim \frac{1}{3}x^3$
$\tan x-\sin x \sim \frac{1}{2}x^3$
$a^x-1 \sim x\ln a$
$\ln(1+x)-x \sim -\frac{1}{2}x^2$

导数公式

导数公式	广东普通专升本	重要程度
$C^{'} = 0\ (C为常数)$	√	$\bigstar$
$(x^a) = ax^{a-1}$	√	$\bigstar$
$(\frac{1}{x})^{'} = -\frac{1}{x^2}$	√	$\bigstar$
$(\sqrt{x})^{'} = \frac{1}{2\sqrt{x}}$	√	$\bigstar$
$(a^x)^{'} = a^x\ln a\ (a > 0 且 a \ne 1)$	√	$\bigstar$
$(e^x)^{'} = e^x$	√	$\bigstar$
$(log_ax)^{'} = \frac{1}{xlna}\ (a > 0 且 a \ne 1)$	√	$\bigstar$
$(lnx)^{'} = \frac{1}{x}$	√	$\bigstar$
$(\sin x)^{'} = \cos x$	√	$\bigstar$
$(\cos x)^{'} = -\sin x$	√	$\bigstar$
$(\tan x)^{'} = \sec^2{x}$	√	$\bigstar$
$(\cot x)^{'} = -\csc^2{x}$	√	$\bigstar$
$(\sec x)^{'} = \sec x \cdot \tan x$	√	$\bigstar$
$(\csc x)^{'} = -\csc x \cdot \cot x$	√	$\bigstar$
$(\arcsin x)^{'} = \frac{1}{\sqrt{1-x^2}}$	√	$\bigstar$
$(\arccos x)^{'} = -\frac{1}{\sqrt{1-x^2}}$	√	$\bigstar$
$(\arctan x)^{'} = \frac{1}{1+x^2}$	√	$\bigstar$
$(arccot\ {x})^{'} = -\frac{1}{1+x^2}$	√	$\bigstar$

不定积分

积分公式	广东普通专升本	重要程度
$\int k \text{d}x = kx+C\ (k为常数)$	√	$\bigstar$
$\int x^a \text{d}x = \frac{1}{a+1}x^{a+1} + C$	√	$\bigstar$
$\int \frac{1}{\sqrt{x}} \text{d}x = 2\sqrt{x}+C$	√	$\bigstar$
$\int \frac{1}{x^2} \text{d}x = -\frac{1}{x}+C$	√	$\bigstar$
$\int \frac{1}{x} \text{d}x = \ln |x|+C$	√	$\bigstar$
$\int e^x \text{d}x = e^x + C$	√	$\bigstar$
$\int a^x \text{d}x = \frac{a^x}{\ln a}+C$	√	$\bigstar$
$\int \cos{x}\text{d}x=\sin{x}+C$	√	$\bigstar$
$\int \sin{x}\frac{x-1}{x+a}=-\cos{x}+C$	√	$\bigstar$
$\int \tan{x}\text{d}x=-\ln{|\cos{x}|}+C$	√	$\bigstar$
$\int \cot{x}\text{d}x=\ln|\sin{x}|+C$	√	$\bigstar$
$\int \sec{x}\text{d}x=\ln|\sec{x}+\tan{x}|+C$	√	$\bigstar$
$\int \csc{x}\text{d}x=\ln|\csc{x}-\cot{x}|+C$	√	$\bigstar$
$\int \sec^2{x}\text{d}x=\tan{x}+C$	√	$\bigstar$
$\int \csc^2{x}\text{d}x=-\cot{x}+C$	√	$\bigstar$
$\int \sec{x}\tan{x}=\sec{x}+C$	√	$\bigstar$
$\int \csc{x}\cot{x}\text{d}x=-\csc{x}+C$	√	$\bigstar$
$\int \frac{1}{\sqrt{1-x^2}}\text{d}x=\arcsin{x}+C$	√	$\bigstar$
$-\int\frac{1}{\sqrt{1-x^2}}\text{d}x=\arccos{x}+C$	√	$\bigstar$
$\int\frac{1}{1+x^2}\text{d}x=\arctan{x}+C$	√	$\bigstar$
$-\int\frac{1}{1+x^2}\text{d}x=arccot\ {x}+C$	√	$\bigstar$
$\int \frac{1}{x^2-a^2} \text{d}x = \frac{1}{2a}\ln|\frac{x-1}{x+a}|+C$
$\int\ln{x}\text{d}{x}=x\ln{x}-x+C$
$\int{x}e^x\text{d}{x}=xe^x-e^x+C$

直角三角形

特殊三角函数值表

角度	$0^{\circ}$	$30^{\circ}$	$45^{\circ}$	$60^{\circ}$	$90^{\circ}$	$120^{\circ}$	$135^{\circ}$	$150^{\circ}$	$180^{\circ}$
$\alpha的弧度$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$\sin\alpha$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$0$
$\cos\alpha$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$
$\tan\alpha$	$0$	$\frac{\sqrt{3}}{3}$	1	$\sqrt{3}$	不存在	$-\sqrt{3}$	$-1$	$-\frac{\sqrt{3}}{3}$	0
$\cot\alpha$	$\infty$	$\sqrt{3}$	1	$\frac{\sqrt{3}}{3}$	0	$-\frac{\sqrt{3}}{3}$	$-1$	$-\sqrt{3}$	$\infty$

勾股定理

$$a^2+ b^2 = c^2$$

三角关系

	三角关系公式	广东普通专升本	重要程度
正弦	$\sin \alpha = \frac{a}{c} = \frac{对边}{斜边}$
余弦	$\cos \alpha = \frac{b}{c} = \frac{邻边}{斜边}$
正切	$\tan \alpha = \frac{a}{b} = \frac{对边}{邻边}$
余切	$\cot \alpha = \frac{b}{a} = \frac{邻边}{对边}$
正割	$\sec\alpha=\frac{c}{b}=\frac{斜边}{邻边}$
余割	$\csc\alpha=\frac{c}{a}=\frac{斜边}{对边}$

三角函数转换

1. $\sec \alpha = \frac{1}{\cos \alpha}$
2. $\csc \alpha = \frac{1}{\sin \alpha}$
3. $\frac{1}{\cos^2{x}}=\sec^2{x}$

常见三角函数化简公式

1. 平方和
   1. $\sin^{2}x + \cos^{2}x = 1$
   2. $1+\tan^{2}x = \sec^{2}x$
   3. $1+\cot^{2}x = csc^{2}x$
2. 二倍角
   1. $\sin{2x} = 2\sin{x}cos{x}$
   2. $\cos{2x} = \cos^{2}x - \sin^{2}x = 1 - 2\sin^{2}x = 2\cos^{2}x - 1$
3. 降次
   1. $\sin^{2}x = \frac{1-\cos{2x}}{2}$
   2. $\cos^{2}x = \frac{1+\cos{2x}}{2}$