x\(\to\)0时的等价无穷小:
\(\large\sin x\sim\arcsin x\sim\tan x\sim\arctan x\sim\ln (1+x)\sim e^x-1\sim\ln(x+\sqrt{1+x^2})\sim x\)
\(\large\displaystyle x-\sin x\sim\arcsin x-x\sim\frac{1}{6}x^3\)
\(\large\displaystyle\tan x-x\sim x-\arctan x\sim\frac{1}{3}x^3\)
\(\large\displaystyle 1-\cos x\sim x-\ln(1+x)\sim\frac{1}{2}x^2\)
\(\large(1+x)^\alpha-1\sim\alpha x\)
\(\large\displaystyle\tan x-\sin x\sim\frac{1}{2}x^3\)
\(\large\displaystyle\log _a(1+x)\sim\frac{x}{\ln a}\)
常用三角函数恒等式:
\(\large\displaystyle\sin\alpha+\sin\beta=2\sin(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})\)
\(\large\displaystyle\sin\alpha-\sin\beta=2\sin(\frac{\alpha-\beta}{2})\cos(\frac{\alpha+\beta}{2})\)
\(\large\displaystyle\cos\alpha+\cos\beta=2\cos(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})\)
\(\large\displaystyle\cos\alpha-\cos\beta=-2\sin(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})\)
\(\large\displaystyle\sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\)
\(\large\displaystyle\sin\beta\cos\alpha=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)]\)
\(\large\displaystyle\cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)]\)
\(\large\displaystyle\sin\alpha\sin\beta=-\frac{1}{2}[\cos(\alpha+\beta)-\cos(\alpha-\beta)]\)
\(\large\displaystyle\sin\alpha =\frac{2\tan\frac{\alpha}{2}}{1+\tan^{2}\frac{\alpha}{2}}\)
\(\large\displaystyle\cos\alpha =\frac{1-\tan^{2}\frac{\alpha}{2}}{1+\tan^{2}\frac{\alpha}{2}}\)
\(\large\displaystyle\tan\alpha =\frac{2\tan\frac{\alpha}{2}}{1-\tan^{2}\frac{\alpha}{2}}\)
\(\large\displaystyle a\sin\alpha+b\cos\alpha=\sqrt{a^2+b^2}\sin\left(\alpha+\varphi\right),\tan\varphi=\frac{b}{a}\)
重点导数公式:
\(\large\displaystyle\tan^{\prime}x=\sec^2x\qquad\cot^{\prime}x=-\csc^2x\qquad (a^x)^{\prime}=a^x\ln a\,(a>0,a\neq 1)\)
\(\large\displaystyle\sec^{\prime} x=\sec x\vdot\tan x\qquad\csc^{\prime} x=-\csc x\vdot\cot x\qquad\log^{\prime}_{a}x=\frac{1}{x\ln a}\left(a>0,a\neq1\right)\)
\(\large\displaystyle\arcsin^{\prime}x=\frac{1}{\sqrt{1-x^{2}}}\qquad\arccos^{\prime}x=-\frac{1}{\sqrt{1-x^{2}}}\)
\(\large\displaystyle\arctan^{\prime}x=\frac{1}{1+x^{2}}\qquad\mathrm{arccot}^{\prime}x=-\frac{1}{1+x^{2}}\)
泰勒公式及常用泰勒展开:
\(\large\displaystyle f(x)=\sum_{i=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k+\frac{f^{(k+1)}(\xi)}{(k+1)!}(x-x_0)^{k+1}\)
\(\large\displaystyle e^x=\sum_{n=0}^\infty\frac{x^n}{n!}=1+x+\frac1{2!}x^2+\cdots\, ,x\in(-\infty,+\infty)\)
\(\large\displaystyle\sin x=\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac1{3!}x^3+\frac1{5!}x^5+\cdots,x\in(-\infty,+\infty)\)
\(\large\displaystyle\cos x=\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac1{2!}x^2+\frac1{4!}x^4+\cdots,x\in(-\infty,+\infty)\)
\(\large\displaystyle\ln(1+x)=\sum_{n=0}^\infty (-1)^n\frac{x^{n+1}}{n+1}=x-\frac12x^2+\frac13x^3+\cdots,x\in(-1,1]\)
\(\large\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\cdots,x\in(-1,1)\)
\(\large\displaystyle (1+x)^\alpha=1+\sum_{n=1}^\infty\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}x^n=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\cdots,x\in(-1,1)\)
\(\large\displaystyle\arctan x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}=x-\frac13x^3+\frac15x^5+\cdots+x\in[-1,1]\)
\(\large\displaystyle\arcsin x=\sum_{n=0}^\infty\frac{(2n)!x^{2n+1}}{4^n(n!)^2(2n+1)}=x+\frac16x^3+\frac3{40}x^5+\frac5{112}x^7+\frac{35}{1152}x^9+\cdots+,x\in(-1,1)\)
重点不定积分公式:
\(\large\displaystyle\int\sec^{2}x\mathrm{d}x=\tan x+C\qquad\int\csc^{2}x\mathrm{d}x=-\cot x+C\)
\(\large\displaystyle\int\tan x\mathrm{d}x=-\ln|\cos x|+C\qquad\int\cot x\mathrm{d}x=\ln\left|\sin x\right|+C\)
\(\large\displaystyle\int\sec x\mathrm{d}x=\ln\left|\sec x+\tan x\right|+C\qquad\int\csc x\mathrm{d}x=\ln\left|\csc x-\cot x\right|+C\)
\(\large\displaystyle\int\frac{\mathrm{d}x}{\sqrt{a^{2}-x^{2}}}=\arcsin\frac{x}{a}+C\qquad\int\frac{\mathrm{d}x}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C\)
\(\large\displaystyle\int\frac{\mathrm{d}x}{a^{2}-x^{2}}=\frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right|+C\qquad\int\frac{\mathrm{d}x}{\sqrt{x^{2}\pm a^{2}}}=\ln\left|x+\sqrt{x^{2}\pm a^{2}}\right|+C\)
\(\large\displaystyle\int\sqrt{a^{2}-x^{2}}\mathrm{d}x=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}arc\sin\frac{x}{a}+C\)
\(\large\displaystyle\int\sqrt{x^{2}\pm a^{2}}dx=\frac{x}{2}\sqrt{x^{2}\pm a^{2}}\pm\frac{a^{2}}{2}\ln\left|x+\sqrt{x^{2}\pm a^{2}}\right|+C\)
\(\large\displaystyle\int\mathrm{sh}x\mathrm{d}x=\mathrm{ch}x+C\qquad\int\mathrm{ch}x\mathrm{d}x=\mathrm{sh}x+C\)
重点定积分公式:
设\(\large f\left( x\right)\) 为连续函数
(1)\(\large \displaystyle{\int }_{-a}^{a}f\left( x\right) \mathrm{d}x = {\int }_{0}^{a}\left\lbrack {f\left( x\right) + f\left( {-x}\right) }\right\rbrack \mathrm{d}x\)
(2)\(\large \displaystyle{\int }_{-a}^{a}f\left( x\right) \mathrm{d}x = \left\{ \begin{array}{ll} 2{\displaystyle\int }_{0}^{a}f\left( x\right) \mathrm{d}x, & f\left( x\right) \text{ 是偶函数 } \\ 0, & f\left( x\right) \text{ 是奇函数 } \end{array}\right.\)
(3)\(\large \displaystyle{\int }_{0}^{\frac{\pi }{2}}f\left( {\sin x}\right) \mathrm{d}x = {\int }_{0}^{\frac{\pi }{2}}f\left( {\cos x}\right) \mathrm{d}x\)
(4)\(\large \displaystyle{\int }_{0}^{\pi }{xf}\left( {\sin x}\right) \mathrm{d}x = \frac{\pi }{2}{\int }_{0}^{\pi }f\left( {\sin x}\right) \mathrm{d}x\)
(5)\(\large \displaystyle f\left( {x + L}\right) = f\left( x\right) ,\left( {L > 0}\right)\) ,则 \(\large \displaystyle{\int }_{0}^{L}f\left( x\right) \mathrm{d}x = {\int }_{-\frac{L}{2}}^{\frac{L}{2}}f\left( x\right) \mathrm{d}x = {\int }_{a}^{a + L}f\left( x\right) \mathrm{d}x\)
(6)\(\large \displaystyle{\int }_{0}^{\frac{\pi }{2}}{\left( \sin x\right) }^{n}\mathrm{\;d}x = {\int }_{0}^{\frac{\pi }{2}}{\left( \cos x\right) }^{n}\mathrm{\;d}x = \left\{ \begin{array}{ll} \displaystyle\frac{\left( {n - 1}\right) !!}{n!!} \vdot \frac{\pi }{2}, & \text{ 当 }n\text{ 为偶数时 } \\ \displaystyle\frac{\left( {n - 1}\right) !!}{n!!}, & \text{ 当 }n\text{ 为奇数时 } \end{array}\right.\)
\(\large\text{此公式在定积分计算中十分有用,应记住. 当}n\text{为偶数时, }n!\,!\text{表示所有偶数 (不大于 }n\text{)连乘积.}\)
\(\large n\text{为奇数时,}n!\,!\text{表示所有奇数(不大于}n\text{)的连乘积.}\)
八个枢轴变量:
\(\large\text{对于单正态总体}X\sim N(0,1)\text{有:}\)
\(\large \displaystyle \frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)\quad\left(\overline{X}\sim N\left(\mu,\frac{\sigma^{2}}{n}\right)\right)\)
\(\large \displaystyle \frac{\sum_{i=1}^n(X_i-\mu)^2}{\sigma^2}\sim\chi^2(n)\)
\(\large \displaystyle \frac{(n-1)S^2}{\sigma^2}\sim\chi^2(n-1)\left[\frac{\sum_{i=1}^n(X_i-\overline{X})^2}{\sigma^2}\sim\chi^2(n-1)\right]\)
\(\large \displaystyle \frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t(n-1)\)
\(\large\text{对于双正态总体}X{\sim}N(\mu_1,\sigma_1^2),Y{\sim}N(\mu_2,\sigma_2^2)\text{有:}\)
\(\large \displaystyle \frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\sim N(0,1)\)
\(\large \displaystyle \frac{\overline{X}-\overline{Y}-(\mu_1-\mu_2)}{S_\omega\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\sim t(n_1+n_2-2)\)
\(\large \displaystyle \text{其中}S_{\omega}=\sqrt{\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}}\)
\(\large \displaystyle \frac{n_2\sigma_2^2\sum_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sigma_1^2\sum_{i=1}^{n_2}(Y_i-\mu_2)^2}\sim F(n_1,n_2)\)
\(\large \displaystyle \frac{\sigma_2^2S_1^2}{\sigma_1^2S_2^2}\sim F(n_1-1,n_2-1)\)