# [笔记] 高数笔记

## 函数极限

$\lim\limits_{x\rightarrow x_0} f(x)=A$

$\sin(x)<x<\tan(x)\\ \Rightarrow \frac{\sin(x)}{x}<\frac{\sin(x)}{\sin(x)}=1,\frac{\sin(x)}{x}>\frac{\sin(x)}{\tan(x)} =\cos(x)\\ \Rightarrow \lim\limits_{x\to 0} \frac{\sin(x)}{x}=1$

## 导数与斜率

$(u \pm v)' = u' \pm v'\\ (uv)'= u'v + v'u\\ (\frac uv)'=\frac{(u'v-v'u)}{v^2}\\ (c \cdot f(x))'=c \cdot f'(x) \\ (f(g(x)))'=f'(g(x)) \cdot g'(x)$

$f''(x_0)=\lim \limits_{\Delta x\to 0} \frac{f'(x+x_0)-f'(x_0)}{\Delta x}=\lim \limits_{\Delta x\to 0} \frac{f(x_0+2\Delta x)-2f(x_0+\Delta x)+f(x_0)}{\Delta x ^2}$，记做 $\frac{\mathrm{d}^2 y }{\mathrm{d}x^2}$

$e=\lim\limits_{n \to \infty}(1+\frac{1}{n})^n=2.718281828459\cdots$

$(e^x)'=e^x,(\ln(x))'=\frac{1}{x}$

$f(x)$$g(x)$$a$ 点处为 $0$ ，即 $0/0$ 类型

$\lim\limits_{x\to a}\frac{f(x)}{g(x)}=\lim\limits_{x\to a} \frac{\frac{f(x)-f(a)}{x-a}}{\frac{g(x)-g(a)}{x-a}}=\lim\limits_{x\to a}\frac{f'(x)}{g'(x)}$

$x_{next}=x-\frac{f(x)}{f'(x)}$

$x<eps$ 终止。

## 积分

### 定积分

$S=\int_{a}^b f(x)\mathrm{d}x$

\begin{aligned} \int_0^cx^2\mathrm{d}x &=\lim\limits_{n\to \infty} \sum\limits_{k=1}^n (k\frac{c}{n})^2 \cdot \frac{c}{n}\\ &=\lim\limits_{n\to \infty} (\frac{c}{n})^3 \sum\limits_{k=1} k^2\\ &=\frac{c^3\cdot n\cdot (n+1) \cdot (2n+1)}{n^3\cdot 6}\\ &=\frac{c^3(2n^3-3n^3+n)}{6n^3}\\ &=\frac{c^3}{3}\\ \int_0^c x^n\mathrm{d}x &=\frac{c^{n+1}}{n+1} \end{aligned}

\begin{aligned}\\ \int_a^b f'(x)\mathrm{d}x &= \lim\limits_{n \to \infty}\sum\limits_{0\leq k<n}f'(x_k)\mathrm{d}x\\ &=\lim\limits_{n \to \infty}\sum\limits_{0\leq k<n}\frac{f(x_{k+1})-f(x_k)}{x_{k+1}-x_k}(x_{k+1}-x_k)\\ &=\lim\limits_{n \to \infty}\sum\limits_{0\leq k<n} f(x_{k+1})-f(x_k)\\ &=\lim\limits_{n \to \infty}f(x_{n-1})-f(x_0)\\ &=f(b)-f(a)\\ &=f(x)|_a^b \end{aligned}\\

### 自适应辛普森积分法

inline double f(double x) {}
inline double simpson(double l,double r)
{return (f(l)+4*f((l+r)/2)+f(r))*(r-l)/6;}
inline double integral(double l,double r,double ans) {
register double md=(l+r)/2;
register double LL=simpson(l,md),RR=simpson(md,r);
if(fabs(LL+RR-ans)<=15*eps) return LL+RR-(LL+RR-ans)/15;
return integral(l,md,LL)+integral(md,r,RR);
}

integral(l,r,simpson(l,r));


### 不定积分

$\int f(x)\mathrm{d}x=g(x)+c$

## 函数最优化

$f(x)$ 存在导数，则有更好的方法：

### 无约束函数极值

$f(x)=(x_1-x_2-2)^2+(x_2-1)^2$ 的最小值。

$\nabla f(x)=(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2})=(2(x_1-x_2-2),-2(x_1-x_2-2)+2(x_2-1))=(0,0)$

### 拉格朗日乘数法

$(x,y,z)$ 使得 $(x-4)^2+y^2+z^2$ 最小，并且 $x+y+z=3, 2x+y+z=4$

## 多项式逼近

### 泰勒展开

$f(x) = a_0 + a_1(x - x_0) + a_2(x - x_0)^2 + a_3(x - x_0)^3 + ...$

\begin{aligned}\\e^x&=\sum\limits_{n=0}^\infty \frac{x^n}{n!} &x\in(-\infty,+\infty)\\\ln(x+1)&=\sum\limits_{n=0}^\infty \frac{(-1)^n}{n+1} x^{n+1} &x\in(-1,1]\\\sin(x)&=\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1} &x\in (-\infty,+\infty)\\\cos(x)&=\sum\limits_{n=0}^\infty\frac{(-1)^n}{2n!}x^{2n} &x\in (-\infty,+\infty)\\\frac{1}{1-x}&=\sum\limits_{n=0}^\infty x^n &|x|<1\\\frac{1}{1+x}&=\sum\limits_{n=0}^\infty (-1)^nx^n &|x|<1\\\end{aligned}\\

### 欧拉公式

$e^{i\theta}=\cos(\theta)+i\sin(\theta)$ （可以由上面的展开式看出）

2020.01.24

posted @ 2020-01-24 17:25  LuitaryiJack  阅读(1613)  评论(0编辑  收藏  举报