高等数学基础公式

数学基础公式

导数和积分

ps: 积分需要加上C

积分	原函数	求导
$$ \frac{1}{\alpha + 1} x^{\alpha+1} \ (\alpha \neq -1) $$	$$ x^\alpha $$	$$ \alpha(x)^{\alpha - 1} $$
$$\frac{a^x}{\ln a} $$	$$ a^x $$	$$a^xln {a} $$
\	$$ log_a{x} $$	$$ \frac{1}{x \ln a} $$
$$ x\ln x - x $$	$$ \ln{|x}| $$	$$ \frac{1}{x} $$
$$ -\cos x $$	$$\sin x $$	$$ \cos x $$
\	$$\arcsin x $$	$$\frac{1}{\sqrt{1-x^2}} $$
\	$$\arccos x $$	$$-\frac{1}{\sqrt{1-x^2}} $$
$$ -\ln |\cos x| $$	$$\tan x $$	$$ sec^2 x $$
$$ \ln |\sin x| $$	$$\cot x $$	$$ -csc^2 x $$
\	$$ \frac{1}{a}\arctan \frac{x}{a} $$	$$ \frac{1}{a^2 + x^2}$$
\	$$\operatorname{arccot} x$$	$$ -\frac{1}{1 + x^2}$$
$$ \ln|\sec x + \tan x| $$	$$ \sec x $$	$$ \sec x \tan x $$
$$ \ln|\csc x - \cot x| $$	$$ \csc x $$	$$ -\csc x \cot x $$
\	$$\ln{x + \sqrt{x^2 + a^2}}$$	$$ \frac{1}{\sqrt{x^2 + a^2}}$$
\	$$\ln{x + \sqrt{x^2 - a^2}}$$	$$ \frac{1}{\sqrt{x^2 - a^2}}$$
$$ \frac{1}{2a}\ln{|\frac{x-a}{x+a}|} $$	$$ \frac{1}{x^2 - a^2} $$	\
$$ \frac{a^2}{2}\arcsin{\frac{x}{a}} + \frac{x}{2}\sqrt{a^2 - x^2} $$	$$ \sqrt{a^2 - x^2 } $$	\
$$ \frac{x}{2} - \frac{\sin 2x}{4} $$	$$ \sin x^2 $$	\
$$ \frac{x}{2} + \frac{\sin 2x}{4} $$	$$ \cos x^2 $$	\
$$ \tan x - x $$	$$ \tan x^2 $$	\
$$ -\cot x - x $$	$$ \cot x^2 $$	\

泰勒展开式

\[y = f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \]

\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!} + o(x^n) , \ -\infty < x < +\infty \]

\[\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + \dots + (-1)^n x^n + o(x^n) ,\ -1 < x < 1 \]

\[\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots + x^n + o(x^n) ,\ -1 < x < 1 \]

\[\ln(1 + x) = \sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} + \dots + (-1)^n \frac{x^n}{n} + o(x^n) ,\ -1 < x < 1 \]

\[\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n + 1}}{(2n + 1)!} = x - \frac{x^3}{3} + \frac{x^5}{5} + \dots + (-1)^n \frac{x^{2n + 1}}{(2n + 1)!} + o(x^n) , \ -\infty < x < +\infty \]

\[\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2} + \frac{x^4}{4} + \dots + (-1)^n \frac{x^{2n}}{(2n)!} + o(x^n) , \ -\infty < x < +\infty \]

\[(1 + x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \dots + \frac{\alpha (\alpha-1)\dots(\alpha-n+1)}{n!}x^n + o(x^n) \]

\[\tan x = x + \frac{1}{3} x^3 + \dots... \]

\[\arcsin x = x + \frac{1}{6} x^3 + \dots... \]

\[\arctan x = x - \frac{1}{3} x^3 + \dots... \]

三角代数换元

被积函数包含特殊根式

\[\sqrt{a^2 - x^2} → x = a\sin t, \quad |t| < \frac{\pi}{2} \]

\[\sqrt{a^2 + x^2} → x = a\tan t, \quad |t| < \frac{\pi}{2} \]

\[\sqrt{x^2 - a^2} → x = a\sec t, \quad 若\: {x>0 ,\ 0 < t < \frac{\pi}{2}}, 若\: {x<0 ,\ \frac{\pi}{2}< t < \pi} \]

\[\int e^{ax}\sin b dx = \ \frac{\ \begin{vmatrix} \ (e^{ax})' & (\sin bx)' \\ e^{ax} & \sin bx \ \end{vmatrix}\ }{a^2 + b^2} + C \]

\[\int e^{ax}\cos b dx = \ \frac{\ \begin{vmatrix} \ (e^{ax})' & (\cos bx)' \\ e^{ax} & \cos bx \ \end{vmatrix}\ }{a^2 + b^2} + C \]

有理函数积分

将 \(\frac{P_n(x)}{Q_m(x)}\) 拆分成若干最简有理分式之和

\[\frac{A}{ax + b}, \frac{A_k}{(ax+b)^k}, \frac{Ax+B}{(px^2+qx+r)}, \frac{A_kx+B_k}{(px^2+qx+r)^k} \]

微积分基础公式

常用极限与等价无穷小

\[\lim_{x \to 0} \frac{\sin x}{x} = 1 \]

\[\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \]

\[\lim_{x \to 0} \frac{e^x - 1}{x} = 1 \]

\[\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \]

\[\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \]

当 \(x \to 0\) 时:

函数	等价无穷小
\(\sin x\)	\(x\)
\(\tan x\)	\(x\)
\(\arcsin x\)	\(x\)
\(\arctan x\)	\(x\)
\(1 - \cos x\)	\(\frac{x^2}{2}\)
\(e^x - 1\)	\(x\)
\(\ln(1+x)\)	\(x\)
\((1+x)^\alpha - 1\)	\(\alpha x\)

基本求导法则

\[(u \pm v)' = u' \pm v' \]

\[(uv)' = u'v + uv' \]

\[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}, \quad v \neq 0 \]

\[(f(g(x)))' = f'(g(x))g'(x) \]

\[y = y(x), \quad \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad \text{参数方程求导} \]

\[y = f^{-1}(x), \quad (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} \]

基本积分法则

\[\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx \]

\[\int k f(x) dx = k \int f(x) dx \]

\[\int f(g(x))g'(x) dx = \int f(u)du, \quad u = g(x) \]

\[\int u\,dv = uv - \int v\,du \]

\[\int_a^b f(x)dx = F(b) - F(a), \quad F'(x)=f(x) \]

\[\frac{d}{dx}\int_a^x f(t)dt = f(x) \]

\[\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(t)dt = f(\beta(x))\beta'(x)-f(\alpha(x))\alpha'(x) \]

多元微积分

若 \(z=f(x,y)\):

\[dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy \]

\[\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n}\right)^T \]

方向导数:

\[D_{\mathbf{u}}f(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{u}, \quad \|\mathbf{u}\| = 1 \]

二阶泰勒展开:

\[f(\mathbf{x}+\mathbf{h}) \approx f(\mathbf{x}) + \nabla f(\mathbf{x})^T\mathbf{h} + \frac{1}{2}\mathbf{h}^T H_f(\mathbf{x})\mathbf{h} \]

其中 \(H_f\) 为 Hessian 矩阵:

\[ H_f = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} \]

线性代数基础公式

向量与内积

向量内积:

\[\mathbf{a}\cdot\mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta \]

向量范数:

\[\|\mathbf{x}\|_2 = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \]

常用范数:

\[\|\mathbf{x}\|_1 = \sum_{i=1}^{n}|x_i|, \quad \|\mathbf{x}\|_\infty = \max_i |x_i| \]

Cauchy-Schwarz 不等式:

\[|\mathbf{a}\cdot\mathbf{b}| \leq \|\mathbf{a}\|\|\mathbf{b}\| \]

三角不等式:

\[\|\mathbf{a}+\mathbf{b}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\| \]

矩阵基本运算

矩阵乘法:

\[(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \]

转置:

\[(A+B)^T = A^T + B^T \]

\[(AB)^T = B^T A^T \]

逆矩阵:

\[AA^{-1} = A^{-1}A = I \]

\[(AB)^{-1} = B^{-1}A^{-1} \]

\[(A^T)^{-1} = (A^{-1})^T \]

行列式

\[\det(AB) = \det(A)\det(B) \]

\[\det(A^T) = \det(A) \]

\[\det(A^{-1}) = \frac{1}{\det(A)}, \quad \det(A) \neq 0 \]

二阶行列式:

\[\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \]

三阶行列式:

\[\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = a_1b_2c_3 + a_2b_3c_1 + a_3b_1c_2 - a_3b_2c_1 - a_2b_1c_3 - a_1b_3c_2 \]

秩、线性方程组与逆

矩阵可逆条件:

\[A \text{ 可逆} \Longleftrightarrow \det(A) \neq 0 \Longleftrightarrow \operatorname{rank}(A)=n \]

线性方程组:

\[A\mathbf{x}=\mathbf{b} \]

若 \(A\) 可逆:

\[\mathbf{x}=A^{-1}\mathbf{b} \]

解的判定:

\[\operatorname{rank}(A)=\operatorname{rank}(A|\mathbf{b}) \Rightarrow \text{有解} \]

\[\operatorname{rank}(A)<\operatorname{rank}(A|\mathbf{b}) \Rightarrow \text{无解} \]

若有解且 \(\operatorname{rank}(A)=n\)，则解唯一；若有解且 \(\operatorname{rank}(A)<n\)，则有无穷多解。

特征值与特征向量

特征方程:

\[A\mathbf{v}=\lambda\mathbf{v}, \quad \mathbf{v}\neq \mathbf{0} \]

\[\det(\lambda E - A)=0 \]

迹与特征值:

\[\operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}=\sum_{i=1}^{n}\lambda_i \]

行列式与特征值:

\[\det(A)=\prod_{i=1}^{n}\lambda_i \]

若 \(A\) 可对角化:

\[A=PDP^{-1} \]

其中 \(D\) 为特征值构成的对角矩阵，\(P\) 的列向量为对应特征向量。

正交投影与最小二乘

向量 \(\mathbf{b}\) 在非零向量 \(\mathbf{a}\) 上的投影:

\[\operatorname{proj}_{\mathbf{a}}\mathbf{b} = \frac{\mathbf{a}^T\mathbf{b}}{\mathbf{a}^T\mathbf{a}}\mathbf{a} \]

若矩阵 \(A\) 的列向量线性无关，投影矩阵为:

\[P = A(A^TA)^{-1}A^T \]

最小二乘问题:

\[\min_{\mathbf{x}}\|A\mathbf{x}-\mathbf{b}\|_2^2 \]

正规方程:

\[A^TA\hat{\mathbf{x}}=A^T\mathbf{b} \]

若 \(A^TA\) 可逆:

\[\hat{\mathbf{x}}=(A^TA)^{-1}A^T\mathbf{b} \]