考研必备数学公式大全（数学二）（高等数学篇）

由于公式太多，汇总成一篇容易打不开，所以分三篇。整篇链接：https://blog.csdn.net/zhaohongfei_358/article/details/106039576章节链接基础回顾篇https://blog.csdn.net/zhaohongfei_358/article/details/119929920高等数学篇https://blog.csdn.net/zhaohongf

iioSnail

34253人浏览 · 2021-08-26 13:59:35

iioSnail · 2021-08-26 13:59:35 发布

由于公式太多，汇总成一篇容易打不开，所以分三篇。整篇链接：https://blog.csdn.net/zhaohongfei_358/article/details/106039576

章节	链接
基础回顾篇	https://blog.csdn.net/zhaohongfei_358/article/details/119929920
高等数学篇	https://blog.csdn.net/zhaohongfei_358/article/details/119929988
线性代数篇	https://blog.csdn.net/zhaohongfei_358/article/details/119930063

文章目录

极限相关公式

数列极限递推式

$\begin{aligned} & a_{n+1} = f(a_n) \\\\ 结论一： & f'(x) > 0 ， \begin{cases} a_2 > a_1 \implies \{ a_n \} \nearrow单调递增 \\ a_2 < a_1 \implies \{ a_n \} \searrow单调递减 \end{cases} \\\\ 结论二（压缩映像原理）：& \exist k \in (0,1)，使得 |f'(x)| \le k \implies {a_n} 收敛 \end{aligned}$

重要极限公式

$\begin{aligned} & \lim_{x \to 0^+} x^\alpha \ln x = 0 ~~~~~ 其中 \alpha >0 \\\\ & \lim_{x \to 0^+} x^\alpha (\ln x)^k = 0 ~~~~~ 其中 \alpha >0 ，k>0 \\\\ & \lim_{x \to +\infty} x^\alpha e^{-\delta x} = 0 ~~~~~ 其中 \alpha >0 ，\delta >0 \\\\ & \lim_{x\to 0} \frac{\sin x}{x} = 1 ~~ \implies \lim_{\phi (x) \to 0} \frac{\sin \phi (x)}{\phi (x)} =1 ~~~~~其中\phi (x) \neq 0 \\ \\ & \lim_{x \to 0} (1+x)^{\frac{1}{x}} = e ~~ \implies \lim_{\phi (x) \to 0} (1+\phi (x))^{\frac{1}{\phi (x)}} = e ~~~~~ 其中\phi (x) \neq 0 \\\\ & \lim_{n \to \infty} \sqrt[n]{n} = 1 \\\\ & \lim_{n \to \infty} \sqrt[n]{a} = 1 ~~~（常数a>0）\\\\ \end{aligned}$

常用等价无穷小

$\begin{aligned} & x \to 0 时，\\\\ & \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim (e^x - 1) \sim \ln(1+x) \sim x ~~,~~ 1- \cos x \sim \frac{1}{2} x^2 ~~, \\ \\ & (1+x)^a - 1 \sim ax ~~,~~ a^x - 1 \sim x\ln a ~~~(a>0,a\neq1) \end{aligned}$

1^∞ 型

$\lim u^v = e^{\lim (u-1)v} ~~~~~~(其中 \lim u=1,\lim v=\infty，即 1^\infty 型)$

导数相关公式

导数定义

$\begin{aligned} & f'(x_0) = \lim\limits_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \\ \\ & f'(x_0) = \lim\limits_{x \to x_0} \frac{f(x) - f(x_0)}{x-x_0} \end{aligned}$

微分定义

$\begin{aligned} & \Delta y = f(x_0 + \Delta x) - f(x_0) \\ \\ & \Delta y = A\Delta x + o(\Delta x) \\ \\ & A\Delta x = f'(x_0) \Delta x \end{aligned}$

连续，可导及可微关系

一元函数

多元函数

导数四则运算

$\begin{aligned} & [u(x) \pm v(x)]' = u'(x) \pm v'(x) \\ \\ & [u(x)v(x)]' = u'(x)v(x) + u(x)v'(x) \\ \\ & [u(x)v(x)w(x)]' = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x) \\ \\ & \begin{bmatrix}\frac{u(x)}{v(x)} \end{bmatrix}' = \frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2} \\ \\ \end{aligned}$

复合函数求导

${ f[g(x)] \}' = f'[g(x)]g'(x)$

反函数求导

$\begin{aligned} & y = f(x), x = \varphi(y) \implies \varphi ' (y) = \frac{1}{f'(x)} \\ \\ & y'_x = \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{x'_y} \\ \\ & y^{''}_{xx} = \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac{d(\frac{1}{x'_y})}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{dy}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{1}{x'_y} = \frac{-x^{''}_{yy}}{(x'_y)^3} \end{aligned}$

参数方程求导

$\begin{aligned} & \begin{cases} x = \varphi (t) \\ y = \psi (t) \end{cases} \\\\ & \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi ' (t)}{\varphi ' (t)} \\ \\ & \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac {d(\frac{dy}{dx})/dt}{dx/dt} = \frac{\psi '' (t) \varphi '(t) - \psi '(t) \varphi '' (t) }{[\varphi ' (t)]^3} \end{aligned}$

变限积分求导公式

$\begin{aligned} & 设 F(x) = \int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt, 则 \\ \\ & F'(x) = \frac{d}{dx}\begin{bmatrix}\int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt \end{bmatrix} = f[\varphi _2(x)]\varphi '_2(x) - f[\varphi_1(x)]\varphi '_1(x) \end{aligned}$

基本初等函数的导数公式（❤❤❤）

$\begin{aligned} & (x^a)' = a x^{a-1} ~~~~~(a为常数) \\ \\ & (a^x)' = a^x \ln a \\ \\ & (e^x)' = e^x \\ \\ & (log_a x)' = \frac{1}{x \ln a} ~~~~~(a>0, a \ne 1) \\ \\ & (\ln x)' = \frac{1}{x} \\ \\ & (\sin x)' = \cos x \\ \\ & (\cos x)' = -\sin x \\ \\ & (\arcsin x)' = \frac{1}{\sqrt{1-x^2}} \\ \\ & (\arccos x)' = - \frac{1}{\sqrt{1-x^2}} \\ \\ & (\tan x)' = \sec ^2 x \\ \\ & (\cot x)' = - \csc^2 x \\ \\ & (\arctan x)' = \frac{1}{1+x^2} \\ \\ & (arccot ~ x)' = - \frac{1}{1+x^2} \\ \\ & (\sec x)' = \sec x \cdot \tan x \\ \\ & (\csc x)' = - \csc x \cdot \cot x \\ \\ & [\ln (x+\sqrt{x^2+1})]' = \frac{1}{\sqrt{x^2+1}} \\ \\ & [\ln (x+\sqrt{x^2-1})]' = \frac{1}{\sqrt{x^2-1}} \\ \\ \end{aligned}$

高阶导数的运算

$\pm v ]^{(n)} = u^{(n)} \pm v^{(n)}$
$\begin{aligned} (uv)^{(n)} & = u^{(n)}v + C_n^1 u^{(n-1)}v' + C_n^2 u^{(n-2)}v'' + ... + C_n^k u^{(n-k)}v^{(k)} + ... + C_n^{n-1} u'v^{(n-1)} + uv^{(n)} \\ & = \displaystyle\sum_{k=0}^n C_n^k u^{(n-k)}v^{(k)} \end{aligned}$

常用初等函数的n阶导数公式

$\begin{aligned} & (a^x)^{(n)} = a^x (\ln a)^n \\ \\ & (e^x)^{(n)} = e^x \\ \\ & (\sin kx)^{(n)} = k^n \sin (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\cos kx)^{(n)} = k^n \cos (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\ln x) ^ {(n)} = (-1)^{n-1} \frac{(n-1)!}{x^n} ~~~~~(x>0) \\ \\ & [\ln(1+x)]^{(n)} = (-1)^{n-1} \frac{(n-1)!}{(x+1)^n} ~~~~~(x>-1) \\ \\ & [(x+x_0)^m]^{(n)} = m(m-1)(m-2)\cdotp\cdotp\cdotp\cdot (m-n+1)(x+x_0)^{m-n} \\ \\ & (\frac{1}{x+a})^{(n)} = \frac{(-1)^n \cdot n!}{(x+a)^{n+1}} \\ \\ \end{aligned}$

极值判别条件

$\begin{cases} 1.~~f'(x)左右异号 \implies \begin{cases} 左正右负 \implies 极大值 \\ 左负右正 \implies 极小值 \end{cases} \\ \\ 2.~~f'(x)=0, f''(x)\ne 0 \implies \begin{cases} f''(x) < 0 \implies 极大值 \\ f''(x)>0 \implies 极小值 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ，f^{(n)}(x) \ne 0， n为偶数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 极大值 \\ f^{(n)}(x) > 0 \implies 极小值 \end{cases} \end{cases}$

凹凸性判定

$\begin{aligned} 1.&\begin{cases} f(\frac{x_1+x_2}{2}) < \frac{f(x_1)+f(x_2)}{2} \implies 凹 \\\\ f(\frac{x_1+x_2}{2}) > \frac{f(x_1)+f(x_2)}{2} \implies 凸 \end{cases} \\\\ 2.&\begin{cases} f''(x) > 0 \implies 凹 \\\\ f''(x) < 0 \implies 凸 \end{cases} \end{aligned}$

拐点判别条件

$\begin{cases} 1.~~f''(x)左右异号 \implies \begin{cases} 左负右正 \implies 凸 \to 凹 \\ 左正右负 \implies 凹 \to 凸 \end{cases} \\ \\ 2.~~f''(x)=0, f'''(x)\ne 0 \implies \begin{cases} f'''(x) < 0 \implies 凹 \to 凸 \\ f'''(x)>0 \implies 凸 \to 凹 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ，f^{(n)}(x) \ne 0， n为奇数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 凹 \to 凸 \\ f^{(n)}(x) > 0 \implies 凸 \to 凹 \end{cases} \end{cases}$

斜渐近线

$\lim_{x \to +\infty} \frac{f(x)}{x} = a ~~~~\lim_{x \to + \infty}(f(x)-ax) = b \implies 斜渐近线为： y=ax+b$

曲率

$\begin{aligned} 密切圆半径 ~~~~~ & r = \frac{(1+y'^2)^{\frac{3}{2}}}{|y''|} \\ \\ 曲率 ~~~~~ &K = \frac{1}{r} = \frac{|y''|}{(1+y'^2)^{\frac{3}{2}}} \\ \\ 曲率圆 ~~~~~& (X-(x-\frac{y'(1+y'^2)}{y''^2}))^2 +(Y-(y+\frac{1+y'^2}{y''^2})) = (\frac{(1+y'^2)^{\frac{3}{2}}}{|y''|})^2 \end{aligned}$

积分相关公式

定积分的精确定义

$\begin{aligned} & \int_a^b f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(a+\frac{b-a}{n}i)\frac{b-a}{n} \\ \\ \\ 常用：& \int_0^1 f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ & \int_0^k f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^{kn} f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ \\ 二重定积分精确定义：& \iint\limits_D f(x,y) d\sigma = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(a+\frac{b-a}{n}i, c+\frac{d-c}{n}j) \cdot \frac{b-a}{n} \cdot \frac{d-c}{n} \\ \\ \\ 常用：&\int_0^1 \int_0^1 f(x,y) dxdy = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(\frac{i}{n}, \frac{j}{n})\cdot \frac{1}{n^2} \end{aligned}$

分布积分公式

$\begin{aligned} & \int u dv = uv - \int vdu \\ & \int uv^{(n+1)}dx = uv^{(n)} - u'v^{(n-1)} + u''v^{(n-2)} - ... + (-1)^nu^{(n)}v + (-1)^{n+1}\int u^{(n+1)}vdx \end{aligned}$

分部积分表格法

$\int f(x) g(x) dx\\\\\\$

$\def\arraystretch{2} \begin{array}{c:c:c:c:c} f(x) & f'(x) & f''(x) & \cdots & f^{(n)}(x) \\ \hline g(x) & g_1(x) & g_2(x) & \cdots & g_n(x) \\ \end{array}$

$\int f(x) g(x) dx = f(x)g_1(x) - f'(x)g_2(x) + f''(x)g_3(x) - \cdots \pm \int f^{(n)}(x) g_n(x) dx \\ \\ 其中 g_1(x) 代表g(x)的积分，依次类推。公式为斜对角线加减交替。$

区间再现公式

$\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$

华里士公式

$\int_0^\frac{\pi}{2} \sin ^n x dx = \int_0^\frac{\pi}{2} \cos ^n x dx = \begin{cases} \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\cdot\cdot\cdot\cdot \frac{1}{2} \cdot \frac{\pi}{2} ~~~~~ n 为正偶数 \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\cdot\cdot\cdot\cdot \frac{2}{3} ~~~~~ n 为大于1的奇数 \\ \end{cases}$

敛散性判别公式

$\begin{aligned} & \int_1^{+\infty} \frac{1}{x^p} dx \implies & \begin{cases} p>1 \implies 收敛 \\ p \le 1 \implies 发散 \end{cases} \\\\ & \int_0^1 \frac{1}{x^p} dx \implies & \begin{cases} p < 1 \implies 收敛 \\ p \ge 1 \implies 发散 \end{cases} \\\\ \end{aligned}$

基本积分公式

$\begin{aligned} & 以下公式中，\alpha 与 a 均为常数，除声明者外，a>0 \\ \\ & \int x^\alpha dx = \frac{1}{\alpha +1} x^{\alpha + 1} + C ~~~~~(\alpha \ne -1) \\ \\ & \int \frac{1}{x} dx = \ln |x| + C \\ \\ & \int a^x dx = \frac{a^x}{\ln a} + C ~~~~~(a >0 , a\ne 1) \\ \\ & \int e^x dx = e^x + C \\ \\ & \int \sin x dx = -\cos x + C \\ \\ & \int \cos x dx = \sin x +C \\ \\ & \int \tan x dx = - \ln |\cos x| + C \\ \\ & \int \cot xdx = \ln |\sin x| + C \\ \\ & \int \sec x dx = \ln |\sec x + \tan x| + C \\ \\ & \int \csc x dx = \ln |\csc x - \cot x| + C \\ \\ & \int \sec ^2 x dx = \tan x + C \\ \\ & \int \csc^2x dx = - \cot x + C \\ \\ & \int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan \frac{x}{a} + C \\ \\ & \int \frac{1}{a^2-x^2} dx = \frac{1}{2a} \ln |\frac{a+x}{a-x}| + C \\ \\ & \int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin \frac{x}{a} + C \\ \\ & \int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln |x+\sqrt{x^2 \pm a^2}| + C \end{aligned}$

重要积分公式

$\begin{aligned} & \int_{-\infty}^{+\infty} e^{-x^2} dx = 2\int_{0}^{+\infty} e^{-x^2} dx = \sqrt{\pi} \\ \\ & \int_{0}^{+\infty} x^n e^{-x} dx = n! \\ \\ & \int_{-a}^{a} f(x) dx = \int_0^a [f(x)+f(-x)]dx \\ \\ & \int_0^\pi xf(\sin x) dx = \frac{\pi}{2} \int_0^\pi f(\sin x)dx = \pi \int_0^{\frac{\pi}{2}} f(\sin x) dx \\ \\ & \int_a^b f(x) dx = (b-a) \int_0^1 f[a+(b-a)x] dx \end{aligned}$

有理函数的拆分

$\begin{aligned} & \frac{P_n(x)}{(a_1 x+b_1)(a_2 x+b_2)(a_3 x+b_3)} = \frac{A_1}{a_1 x+b_1}+\frac{A_2}{a_2 x+b_2}+\frac{A_3}{a_2 x+b_2} \\ \\ & \frac{P_n(x)}{Q_m(x)(ax+b)^3} = \frac{A(x)}{Q_m(x)} + \frac{A_1}{(ax+b)^3} + \frac{A_2}{(ax+b)^2} + \frac{A_3}{(ax+b)} \\ \\ & \frac{P_n(x)}{Q_m(x)(ax^2+bx+c)^3} = \frac{A(x)}{Q_m(x)} + \frac{A_1x+B_1}{(ax^2+bx+c)^3} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + \frac{A_3x+B_3}{(ax^2+bx+c)} \\ \\ \end{aligned}$

积分求平均值

$\frac{\int_a^b f(x) dx}{b-a}$

定积分应用

定积分求平面图形面积

$\begin{aligned} & y=y_1(x) 与 y=y_2(x)，及x=a，x=b（a<b）所围成的平面图形面积：\\ \\ & S= \int_a^b |y_1(x)-y_2(x)|dx \\ \\ \\ & 曲线 r=r_1(\theta) 与 r=r_2(\theta) 与两射线 \theta = \alpha 与 \theta = \beta （0<\beta - \alpha \le 2\pi）所围成的曲边扇形的面积：\\ \\ & S = \frac{1}{2}\int_\alpha^\beta |{r_1}^2(\theta) - {r_2}^2(\theta)|d\theta \end{aligned}$

定积分求旋转体的体积

$\begin{aligned} & 曲线 y=y(x)与x=a，x=b(a<b)及x轴围成的曲边梯形绕x轴旋转一周所得到的旋转体的体积 \\ \\ & V = \pi\int_a^b y^2(x) dx \\ \\ \\ & 曲线y=y_1(x) \ge 0 与 y = y_2(x) \ge 0 及 x=a，x=b（a<b）所围成的平面图形绕x轴旋转一周所的到的旋转体的体积 \\ \\ & V = \pi \int_a^b |{y_1}^2(x) - {y_2}^2(x)| dx \\ \\ \\ & 曲线 y=y(x) 与 x=a,x=b（0\le a < b）及x轴围成的曲边梯形绕y轴旋转一周所得到的的旋转体的体积 \\ \\ & V_y = 2\pi \int_a^b x|y(x)|dx \\ \\ \\ & 曲线y=y_1(x)与y=y_2(x)及x=a，x=b（0\le a \le b）所围成的圆形绕y轴旋转一周所成的旋转体的体积 \\\\ & V= 2\pi \int_a^b x |{y_1}(x) - {y_2}(x)| dx \end{aligned}$

平面曲线的弧长

$\begin{aligned} & L = \int_a^b \sqrt{1+[y'(x)]^2}dx \\ \\ & L = \int_\alpha^\beta \sqrt{[x'(t)]^2 + [y'(t)]^2}dt \\ \\ & L = \int_\alpha^\beta \sqrt{[r(\theta)]^2+[r'(\theta)]^2}d\theta \end{aligned}$

旋转曲面的面积

$\begin{aligned} & 曲线y=y(x)在区间[a,b]上的曲线弧段绕x轴旋转一周所得到的旋转曲面的“面积” \\\\ & S = 2\pi \int_a^b |y(x)| \sqrt{1+[y'(x)]^2}dx \\ \\ & S = 2\pi \int_a^b |y(t)| \sqrt{[x'(t)]^2 + [y'(t)]^2} dt \\ \\ & S = 2\pi \int_\alpha^\beta r\sin \theta \sqrt{r^2 + (r')^2} d\theta \end{aligned}$

平面截面面积为已知的立体体积

$\begin{aligned} & 在区间[a,b]上，垂直于x轴的平面截立体\Omega所得到截面面积为x的连续函数A(x)，则\Omega的体积为 \\ \\ & V = \int_a^b A(x)dx \end{aligned}$

变力沿直线做功

$\begin{aligned} & 设力函数为F(x) （a\le x \le b），则物体沿x轴从点a移动到点b时，变力F(x)所做的功为 \\\\ & W = \int_a^bF(x)dx \end{aligned}$

抽水做功

在这里插入图片描述

$\begin{aligned} & W = \rho g \int_a^b xA(x)dx ~~~~~\\\\ &（\rho为水的密度，g为重力加速度，A(x)为水平截面面积） \end{aligned}$

水压力

在这里插入图片描述

$\begin{aligned} & P = \rho g \int_a^b x[f(x)-h(x)]dx ~~~~~\\\\ &（\rho为水的密度，g为重力加速度，f(x)-h(x)是矩形条的宽度，dx是矩形条的高度） \end{aligned}$

质心

直线段的质心（一维）

$\begin{aligned} & \bar{x} = \frac{\int_a^b x \rho(x)dx}{\int_a^b \rho(x)dx} \\\\ & \rho(x) 为不均匀物体的密度与x的函数关系 \end{aligned}$

不均匀薄片质心（二维）

$\begin{aligned} & \bar{x} = \frac{M_y}{M} = \frac{\iint\limits_D x \rho(x,y)dxdy}{\iint\limits_D \rho(x,y) dxdy} \\\\ & \bar{y} = \frac{M_x}{M} = \frac{\iint\limits_D y \rho(x,y)dxdy}{\iint\limits_D \rho(x,y) dxdy} \\\\ & \rho(x,y) 为不均匀物体的密度与x,y的函数关系 \end{aligned}$

形心

$\begin{aligned} & \bar{x} = \frac{M_y}{M} = \frac{\iint\limits_D x dxdy}{\iint\limits_D dxdy} \\\\ & \bar{y} = \frac{M_x}{M} = \frac{\iint\limits_D y dxdy}{\iint\limits_D dxdy} \\\\ & 质量均匀的情况下，质心与形心重合，即舍去密度\rho(x,y) \end{aligned}$

质量

$\iint\limits_D \rho (x,y)dxdy$

转动惯量

$\begin{aligned} & I_x = \iint\limits_D y^2 \rho(x,y)dxdy \\\\ & I_y = \iint\limits_D x^2 \rho(x,y)dxdy \\\\ \end{aligned}$

物理公式

$\begin{aligned} 浮力公式：~~~ & F_{浮} = \rho_{_液}gV_{_排} \\\\ 压强：~~~& P = \frac{F}{S} ~~~~~~~(F为压力，S为受力面积)\\\\ 压强与气体体积成反比：~~~& PV=k ~~~~~(k=nRT，n是分子个数，R为常数，T为温度) \\\\ 水深h处的压强：~~~& P=\rho gh \\\\ 在水中的压力：~~~& F_{_压} = P_{_压}S_{_{受压面积}} = \rho g h S_{_{受压面积}} \\ \\ 力：~~~& F = G_{_{重力}} = mg_{_{重力加速度}} \\ \\ 做功：~~~& W_功 = F_{_力} S_{_{距离}} = mgh_{_{高度}} = \rho g h V =\rho g s h^2 \\ \\ 密度公式：~~~& \rho = \frac{m}{V} \\\\ 引力：~~~ & F = G \frac{m_1 m_2}{r^2} ~~~~~~~(质量为m_1,m_2相距为r的两质点间的引力大小) \end{aligned}$

泰勒公式

$\begin{aligned} f(x) & = \frac{f(a)}{0!} + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + ... + \frac{f^{(n)}(a)}{n!} (x-a)^n + R_n(x) \\ \\ & = \lim_{n \to \infty} \displaystyle\sum_{i=0}^n \frac{f^{(n)}(a)}{n!} (x-a)^n \end{aligned}$

拉格朗日余项的泰勒公式

$\frac{f(a)}{0!} + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + ... + \frac{f^{(n)}(a)}{n!} (x-a)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} ~~~~~其中\xi 介于x,a之间.$

佩亚诺余项的泰勒公式

$\frac{f(a)}{0!} + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + ... + \frac{f^{(n)}(a)}{n!} (x-a)^n + o((x-a)^n)$

常用的泰勒展开式

$\begin{aligned} & \\ & \sin x = x - \frac{x^3}{3!} + o(x^3) \\ \\ & \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + o(x^4) \\ \\ & \arcsin x = x + \frac{x^3}{3!} + o(x^3) \\ \\ \\ & \tan x = x + \frac{x^3}{3} + o (x^3) \\ \\ & \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} + o(x^5) \\ \\ \\ & (1+x)^\alpha = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2!} x^2 + o(x^2) \\ \\ & \frac{1}{1-x} = 1 + x + x^2 + x^3 + o(x^3) \\ \\ & \frac{1}{1+x} = 1 - x + x^2 - x^3 + o(x^3) \\ \\ & \ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + o(x^4) \\ \\ & \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + o(x^6) \\ \\ & e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + o(x^3) \end{aligned}$

口诀：指对连，三角断，三角对数隔一换，三角指数有感叹，反三角它同又乱
指对连：指数函数、对数函数，都是12345连续的
三角断：三角函数的展开式是135,246这样不连续的
三角对数隔一换：三角函数和对数函数的符号隔一个换一次
三角指数有感叹：三角函数和指数函数中分母有阶层（感叹号）
反三角它同又乱：反三角函数的和三角函数第一项相同，第二项为相反数

中值定理

罗尔定理

$\begin{cases} （1）[a,b]上连续 \\ （2）(a,b) 内可导 \\ （3）f(a) = f(b) \end{cases} \implies则\exists \xi \in (a,b)，使得 f'(\xi)=0$

罗尔定理推论

$f^{(n)}(x)=0 至多k个根 \implies 则 f^{(n-1)}(x)=0至多k+1个根$

罗尔定理证明题辅助函数构造

$\begin{aligned} & f''(x) + g(x)f'(x) = 0 \implies F(x) = f'(x) e^{\int g(x)dx} \\ \\ & f(x) + g(x)\int_0^x f(t)dt = 0 \implies F(x) = \int_0^x f(t) dt \cdot e^{\int g(x)dx} \\ \\ & f'(x) + g(x)[f(x)-1] =0 \implies F(x) = [f(x) -1] \cdot e^{\int g(x)dx} \\ \\ & \cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot \end{aligned}$

微分方程构造罗尔定理辅助函数

$\begin{aligned} & 欲证：F[\xi, f(\xi), f'(\xi)] = 0 \\ \\ & 1) 求微分方程 F(x, y, y') =0的通解H(x,y) =C \\ \\ & 2) 设辅助函数：g(x) = H(x, f(x)) \end{aligned}$

拉格朗日中值定理

$\begin{aligned} & 设f(x)满足 \begin{cases} （1）[a,b]上连续 \\ （2）(a,b) 内可导 \\ \end{cases} \\ \\ & \implies则\exists \xi \in (a,b)，使得 f(b)-f(a) = f'(\xi)(b-a)，或者写成 \\ \\ & f'(\xi) = \frac{f(b)-f(a)}{b-a} \end{aligned}$

柯西中值定理

$\begin{aligned} & 设f(x)，g(x)满足 \begin{cases} （1）[a,b]上连续 \\ （2）(a,b) 内可导 \\ （3）g'(x) \neq 0 \end{cases} \\ \\ & \implies则\exists \xi \in (a,b)，使得 \frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)} \end{aligned}$

积分中值定理

$\begin{aligned} & f(x)在[a,b]上连续 \implies 存在 \eta \in [a,b], 使得 \\ & \int_a^b f(x) dx = f(\eta)(b-a) \\ \\ \\ & f(x)，g(x)在[a,b]上连续，且g(x)不变号 \implies \\ & \int_a^b f(x)g(x)dx = f(\eta)\int_a^b g(x)dx \\\\\\ & 二重积分中值定理，D上连续，A为D的面积 \implies \\ & \iint\limits_D f(x,y) dxdy = f(\xi, \eta)A \end{aligned}$

多元微积分相关公式

多元微分定义

$\begin{aligned} 定义：& \Delta z = A \Delta x + B \Delta y + o (\rho) ~~~~~~ (\rho = \sqrt{(\Delta x)^2 + (\Delta y) ^2}) \\ \\ 全增量：& \Delta z =f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0) \\ \\ 线性增量：& A\Delta x + B \Delta y ~~~~~~ 其中 ~~ A=f'_x(x_0, y_0)，B=f'_y(x_0, y_0) \\ \\ 可微判定：& \lim_{\underset{\Delta y \to 0}{\Delta x \to 0}} \frac{\Delta z - (A\Delta x + B \Delta y)}{\sqrt{(\Delta x)^2 + (\Delta y )^2}} \implies \begin{cases} = 0 \implies 可微 \\ \ne 0 \implies 不可微 \end{cases} \\ \\ \end{aligned}$

多元隐函数求导

$\begin{aligned} & \frac{\partial z}{\partial x} = - \frac{F'_x}{F'_z} \\ \\ & \frac{\partial z}{\partial y} = - \frac{F'_y}{F'_z} \\ \end{aligned}$

极坐标下二重积分计算法

$\underset{D}{\iint} f(x,y)d\sigma = \int_\alpha^\beta d\theta \int_{r_1(\theta)}^{r_2(\theta)}f(r\cos \theta, r\sin\theta)rdr$

隐函数存在定理

$\begin{aligned} & F函数F(x,y,z)在点P(x_0,y_0,z_0)的某一邻域内具有连续偏导数 \\ & F(x_0, y_0, z_0) =0， F'_z(x_0, y_0, z_0) \ne 0 \\ \implies & 方程F(x,y,z)=0 在点(x_0, y_0, z_0)的某一邻域内能唯一确定\\ &一个连续且具有连续偏导数的函数 z=f(x,y) \end{aligned}$

多元函数极值判定

$\begin{aligned} 1.&~~~ 求出同时满足 f'_x(x,y) =0，f'_y(x,y) =0的“一组”(x_0,y_0) \\ \\ 2. &~~~ 记 \begin{cases} f''_{xx}(x_0,y_0) =A \\ f''_{xy}(x_0,y_0) =B \\ f''_{yy}(x_0,y_0)=C \end{cases} ~~则 \Delta=B^2-AC \begin{cases} <0 \implies 极值 \begin{cases} A<0 \implies 极大值 \\ A>0 \implies 极小值 \end{cases} \\ >0 \implies 非极值 \\ =0 \implies 方法失效，另谋他法 \end{cases} \end{aligned}$

拉格朗日数乘法求最值

$\begin{aligned} & 求 u = f(x,y,z) 在条件\begin{cases} \phi(x,y,z) =0 \\ \psi(x,y,z) = 0 \end{cases} 的最值 \\\\\\ 1.&~~~ 构造辅助函数~~ F(x,y,z,\lambda, \mu) = f(x,y,z) + \lambda \phi(x,y,z) + \mu \psi(x,y,z) \\\\ 2.&~~~ 得 \begin{cases} F_x' = f_x' + \lambda\phi _x' +\mu \psi_x' = 0 \\ F_y' = f_y' + \lambda\phi _y' +\mu \psi_y' = 0 \\ F_z' = f_z' + \lambda\phi _z' +\mu \psi_z' = 0 \\ F_{\lambda}' = \phi(x,y,z) = 0 \\ F_{\mu}' = \psi(x,y,z) = 0 \\ \end{cases} \\\\ 3.& ~~~解方程得备选点 P_i, i=1,2,3, \cdots,n \\\\ 4.& ~~~求f(P_i) ，取最大值为 u_{max}，最小值为u_{min} \end{aligned}$

多重积分的应用

空间曲面的面积

$\begin{aligned} & \\ & A = \iint\limits_D \sqrt{1+(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2} ~ dxdy \end{aligned}$

微分方程

一阶线性微分方程

$\begin{aligned} & y' + p(x)y = q(x) ~~~~~其中p(x),q(x)为连续函数 \implies \\\\ &y = e^{-\int p(x)dx} (\int e^{\int p(x)dx} \cdot q(x) dx + C) \end{aligned}$

二阶常系数齐次线性微分方程的通解

$\begin{aligned} & y'' + py' + qy =0 ~~~~~~~~p,q为常数 \\ \\ \xRightarrow{特征方程为} & ~~ \lambda^2 + p\lambda + q = 0 \\ \\ \implies & \begin{cases} p^2 - 4q>0 , \xRightarrow{通解为} y=C_1 e^{\lambda _1x} + C_2 e^{\lambda _2 x} \\ \\ p^2 - 4q=0 , \xRightarrow{通解为} y = (C_1+C_2x)e^{\lambda x}\\ \\ p^2 - 4q<0 , \lambda_{1,2}=\alpha \pm i\beta ~~~ \xRightarrow{通解为} y = e^{\alpha x} (C_1 \cos \beta x + C_2\sin \beta x) \end{cases} \end{aligned}$

三阶常系数齐次线性微分方程的通解

$\begin{aligned} & y''' + py'' + qy' +ry =0 ~~~~~~~~p,q,r为常数 \\ \\ \xRightarrow{特征方程为} & ~~ \lambda^3 + p\lambda^2 + q\lambda + r = 0 \\ \\ \implies & \begin{cases} \lambda_1,\lambda_2,\lambda_3\in R, 且\lambda_1 \ne \lambda_2 \ne \lambda_3, \xRightarrow{通解为} y=C_1 e^{\lambda _1x} + C_2 e^{\lambda _2 x} + C_3 e^{\lambda _3 x} \\ \\ \lambda_1,\lambda_2,\lambda_3\in R, 且\lambda_1 = \lambda_2 \ne \lambda_3 \xRightarrow{通解为} y = (C_1+C_2x)e^{\lambda_1 x} + C_3 e^{\lambda_3 x}\\ \\ \lambda_1,\lambda_2,\lambda_3\in R, 且\lambda_1 = \lambda_2 = \lambda_3 \xRightarrow{通解为} y = (C_1+C_2x + C_3x^2)e^{\lambda_1 x}\\ \\ \lambda_1 \in R, \lambda_{2,3}=\alpha \pm i\beta ~~~ \xRightarrow{通解为} y = C_1e^{\lambda_1x} + e^{\alpha x} (C_2 \cos \beta x + C_3\sin \beta x) \end{cases} \end{aligned}$

二阶常系数非齐次线性微分方程的特解

$\begin{aligned} & y'' + py'+qy = f(x) \\ \\ （1）& 自由项f(x)=P_n(x)e^{ax} ~时，特解为 ~ y^*= e^{ax}Q_n(x) x^k \\ \\ 其中 & \begin{cases} e^{ax} ~~照抄 \\ Q_n(x) 为x的n次一般多项式 \\ k = \begin{cases} 0，~~~ a\ne \lambda_1, a \ne \lambda_2 \\ 1，~~~ a = \lambda_1 或a=\lambda_2 \\ 2，~~~ a = \lambda_1 = \lambda_2 \end{cases} \end{cases} \\ \\ \\ (2) & 自由项~ f(x) = e^{ax}[P_m(x) \cos \beta x + P_n(x) \sin \beta x] ~时， \\ \\ & y^* = e^{ax} [ Q_l^{(1)}(x) \cos \beta x + Q_l^{(2)}(x)\sin \beta x]x^k \\ \\ 其中，& \begin{cases} e^{ax} ~~照抄 \\ l=max\{ m, n \} ，Q_l^{(1)}(x), Q_l^{(2)}(x)分别为x的两个不同的l次一般多项式 \\ \\ k = \begin{cases} 0，~~~ a \pm \beta i ~不是特征根 \\ 1，~~~ a \pm \beta i ~是特征根 \\ \end{cases} \end{cases} \end{aligned}$

“算子法”求二阶常系数非齐次线性微分方程的特解

$\begin{aligned} & y^{(n)} + a_1 y^{(n-1)} + a_2 y^{(n-2)} + \cdots + a_n y = f(x) \\\\\\ 记：& \frac{d}{dx} = D, ~~~D表示求导，\frac{1}{D} 表示积分 \\ \\ & F(D) = D^n + a_1D^{n-1} + a_2D^{n-2} + \cdots + a_{n-1}D + a_n \\\\ 则，特解~~~& y^* = \frac{1}{F(D)} f(x) \\ \\ \\ & (1) ~~~ y'' + py' + qy = ae^{kx} \\\\ & 则 ~~y^*= \frac{1}{D^2 + pD + q} ae^{kx} \\\\ & = \frac{1}{k^2+pk+q} ae^{kx} ~~~~~（将D换成k） \\ \\ & = x \frac{1}{2k+p} ae^{kx} ~~~~~~(上式分母为零，提x，求导) ~~ 以此类推 \\ \\ \\ & (2)~~~ y'' + py' + qy = \sin a x (或\cos a x) \\\\ & 则 ~~y*= \frac{1}{D^2 + pD + q} \sin a x \\\\ & = \frac{1}{-a^2+pD+q} \sin a x ~~~~~~(将D^2换成-a^2，若为0与(1)一样，提x求导) \\ \\ & = \frac{pD-(q-a^2)}{p^2D^2 - (q-a^2)^2} \sin \alpha x ~~~~~~(若分母包含D，则通过(a+b)(a-b)得出D^2) \\ \\ \\ & (3) ~~~ y '' + a_1 y ' + a_2 y = P_n(x) \\\\ & y^* = \frac{1}{D^2 + a_1D + a_2} P_n(x) \\\\ & = \frac{1}{D^u} \frac{1}{1 - (kD)^m} P_n(x) ~~（大概化成这种形式）\\\\ & = \frac{1}{D^u} (1 + kD + (kD)^2 + \cdots)P_n(x) ~~~~~~（\frac{1}{1 - (kD)^m}泰勒展开）\\\\ & = Q_n(x) \\\\\\ & (4)~~~~ y'' + a_1y' + a_2y = e^{kx} f(x) \\\\ & = e^{kx} \frac{1}{F(D+k)} f(x) ~~~~~~~（然后看右边，根据f(x)的形式，转(1)(2)(3)方法求解） \end{aligned}$