数学要背的内容

常用泰勒公式:

\(\sin x = x - {\Large\frac{x^3}{3!}} + o(x^3)\)

\(\cos x = 1 - {\Large \frac{x^2}{2!} + \frac{x^4}{4!}} + o(x^4)\)

\(\tan x = x + {\Large\frac{x^3}{3!}} + o(x^3)\)

\(\arcsin x = x + {\Large\frac{x^3}{3!}} + o(x^3)\)

\(\arctan x = x - {\Large\frac{x^3}{3!}} + o(x^3)\)

\(\ln (1 + x) = x - {\Large \frac{x^2}{2!} + \frac{x^3}{3!}} + o(x^3)\)

\(e^x = 1 + x + {\Large \frac{x^2}{2!} + \frac{x^3}{3!}} + o(x^3)\)

\((1 + x)^a = 1 + ax + {\Large \frac{a(a - 1)}{2!}}x^2 + o(x^2)\)

常用等价替换:

当 x->0 时，有：

\(\sin x \sim x，\tan x \sim x，\arcsin x \sim x，\arctan x \sim x\)

\(1-\cos x \sim {\Large \frac{x^2}{2}} ， a^x-1 \sim x\ln a ，e^x-1 \sim x\)

\(\ln(1+x) \sim x ，(1+x)^a-1 \sim ax\)

当 \(x\rightarrow\infty\) 时，有：

重要极限:

\(\underset{x\rightarrow \infty}{\lim}(1 + \frac{1}{x})^x = e\)

\(\underset{x\rightarrow \infty}{\lim}{\Large \frac{\sin x}{x}} = 1\)

三角函数公式：

基本求导公式

\((x^a)' = ax^{a - 1}, a为常数\)

\((a^x)' = a^x\ln a,(a>0, a\neq 1)\)

\((e^x)' = e^x\)

\((\log _ax)' = {\Large \frac{1}{x\ln a}},(a>0, a\neq 1）\)

\((\ln|x|)' = \Large \frac{1}{x}\)

\((\sin x)' = \cos x\)

\((\cos x)' = -\sin x\)

\((\arcsin x)'=\Large \frac{1}{\sqrt{1 - x^2}}\)

\((\arccos x)' = -\Large \frac{1}{\sqrt{1 - x^2}}\)

\((\tan x)' = \sec ^2x\)

\((\cot x)' = -\csc ^2x\)

\((\arctan x)' = \Large \frac{1}{1 + x^2}\)

\((arccot x)' = -\Large \frac{1}{1 + x^2}\)

\((\sec x)'= \sec x \tan x\)

\((\csc x)' = -\csc x \cot x\)

\((\ln(x + \sqrt{x^2 + 1}))' = \Large \frac{1}{\sqrt{x^2 + 1}}\)

\((\ln(x + \sqrt{x^2 - 1}))' = \Large \frac{1}{\sqrt{x^2 - 1}}\)