https://github.com/TangNPC/Math
https://github.com/TangNPC/Math
Last synced: 8 months ago
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- Host: GitHub
- URL: https://github.com/TangNPC/Math
- Owner: HeyTang233
- Created: 2021-05-24T10:40:26.000Z (about 5 years ago)
- Default Branch: main
- Last Pushed: 2021-05-24T10:45:58.000Z (about 5 years ago)
- Last Synced: 2024-11-24T03:10:24.340Z (over 1 year ago)
- Size: 3.17 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# 函数
## 1. 函数的定义
- 万有引力公式：
$$
S = \frac{1}{2}gt^2
$$
- 可以写成
$$
S= 5t^2 \longrightarrow y=f(x)
$$
$$
x\longrightarrow 自变量\\
y\longrightarrow 因变量\\
f\longrightarrow 对应法则(表达式)
$$
### 两函数相同的条件
$$
f(x)=
\begin{cases}
定义域& (不化简)\\
对应法则& (化简)
\end{cases}
$$
## 2. 函数的表达形式
1. **显函数**
$$
y = x,y=x^2
$$
2. **隐函数**
$$
f(x,y)=0
$$
3. **参数函数(方程)**
$$
f(x)=
\begin{cases}
x = \sin t\\
y = \cos t
\end{cases}
$$
4. **分段函数(求导)**
$$
y=|x|=\begin{cases}
x, &x = x>0\\
0, &x = 0&分段点\\
-x, &x<0
\end{cases}\\
x\in R
$$
$$
f(x)=\begin{cases}
\sin x, &00且a\neq 1.x\in R)
$$
### 指数函数的关系试
1. $$
e^m \cdot e^n = e^{m+e}
$$
2. $$
\frac{e^{m}}{e^{n}}=e^{m-n}
$$
3. $$
(e^m) = e^{m}\cdot n
$$
4. $$
(ab)^m = a^m \cdot a^m
$$
5. $$
(\frac{a}{b})^m = \frac{a^m}{b^m}
$$
### 对数函数的关系试
1. $$
\ln a\cdot b = \ln a + \ln b
$$
2. $$
\ln \frac{a}{b} = \ln a - \ln b
$$
3. $$
lnx^k = k\cdot\ln x(只有对数函数可以做到)
$$
4. $$
e^{\ln x} =x\\
\ln e^x = x
$$
5. $$
\ln e = 1
$$
$$
\log _{a}x(a>0且a\neq1,真数x恒大于0,a是底数)\\
\log _{e}x=\ln x\\
\begin{cases}
\ln 1 =0\\
\ln e =1
\end{cases}
$$
$$
\log _{a}x(a>1)
$$
$$
\log _{e}x=\ln x\\
\begin{cases}
\ln +\infty & \to +\infty\\
\ln 0 & \to -\infty
\end{cases}
$$
$$
\lg x = \log_{10}x
$$
### 三角函数
#### sin正弦
#### cos余弦
#### tan正切
#### cot余切
$$
y=\cot x = \frac{1}{\tan x}\\
\cot 30^{\circ} = \cot \frac{\pi}{6} = \frac{1}{\tan\frac{\pi}{6}}= \frac{3}{\sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}
$$
#### sec正割
$$
y=\sec x\\
secx = \frac{1}{\cos x}\\
$$
#### csc余割
$$
y=\csc x\\
cscx = \frac{1}{\sin x}\\
$$
### 倍角公式
$$
\sin2x=2\sin x \cdot\cos x\\
\cos 2x = \cos^2x - \sin^2x\\
降幂公式\\
=1-2\sin^2x \to \sin^2 = \frac{1-\cos2x}{2}\\
=2cos^2x-1 \to cos^2x = \frac{1+\cos2x}{2}
$$
### 反三角函数
$$
\left.\begin{matrix}
y=\arcsin x\\
y=\arccos x
\end{matrix}\right\}
定义域-1\leqslant x \leqslant 1
$$
$$
y=\arctan x\\
\left\{\begin{matrix}
\arctan+\infty \to \frac{\pi}{2}\\
\arctan-\infty \to -\frac{\pi}{2}
\end{matrix}\right.
$$
$$
$$