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一、数列

1. 等差数列

$$a_n=a_1+(n-1)d$$ $$S_n=\frac{n}{2}\left[2a_1+(n-1)d\right]=\frac{n}{2}(a_1+a_n)=n{a}_1+\frac{n(n-1)}{2}d$$

2.等比数列

$$a_n=a_1{q}^{n-1}$$ $$S_n=\frac{a_1(1-q^n)}{1-q}$$

3. 前n项和

$$1+2+\cdots +n=\frac{n(n+1)}{2}$$ $$1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$$ $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\cdots +\frac{1}{n\times (n+1)}=\frac{n}{n+1}$$

二、三角函数

1. 基本关系

$$1+{\tan }^2\alpha ={\sec }^2\alpha$$ $$1+{\cot }^2\alpha ={\csc }^2\alpha$$ $$a\sin x+b\sin x=\sqrt{a^2+b^2}\sin (x+\phi )$$

2. 诱导公式

$\frac{\pi }{2}-\alpha$$\frac{\pi }{2}+\alpha$$\pi -\alpha$$\pi +\alpha$$\frac{3\pi }{2}-\alpha$$\frac{3\pi }{2}+\alpha$$2\pi -\alpha$$\sin \theta$$\cos \alpha$$\cos \alpha$$\sin \alpha$$-\sin \alpha$$-\cos \alpha$$-\cos \alpha$$-\sin \alpha$$\cos \theta$$\sin \alpha$$-\sin \alpha$$-\cos \alpha$$-\cos \alpha$$-\sin \alpha$$\sin \alpha$$\cos \alpha$$\tan \theta$$\cot \alpha$$-\cot \alpha$$-\tan \alpha$$\tan \alpha$$\cot \alpha$$-\cot \alpha$$-\tan \alpha$$\cot \theta$$\tan \alpha$$-\tan \alpha$$-\cot \alpha$$\cot \alpha$$\tan \alpha$$-\tan \alpha$$-\cot \alpha$

3. 倍角公式

$$\sin 3\alpha =-4{\sin }^3\alpha +3\sin \alpha$$ $$\cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha$$ $$\tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }$$ $$\cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha }$$

4. 半角公式

$$\tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha }=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}$$ $$\cot \frac{\alpha }{2}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}$$

5. 和差公式

$$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$ $$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$$ $$\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }$$ $$\cot (\alpha \pm \beta )=\frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \mp \cot \alpha }$$

6. 积化和差

$$\sin \alpha \cos \beta =\frac{1}{2}[\sin (\alpha +\beta )+\sin (\alpha -\beta )]$$ $$\cos \alpha \sin \beta =\frac{1}{2}[\sin (\alpha +\beta )-\sin (\alpha -\beta )]$$ $$\cos \alpha \cos \beta =\frac{1}{2}[\cos (\alpha +\beta )+\cos (\alpha -\beta )]$$ $$\sin \alpha \sin \beta =\frac{1}{2}[\cos (\alpha -\beta )-\cos (\alpha +\beta )]$$

7. 和差化积

$$\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$$ $$\sin \alpha -\sin \beta =2\sin \frac{\alpha -\beta }{2}\cos \frac{\alpha +\beta }{2}$$ $$\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$$ $$\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$

8. 万能公式

$$当\mu =\tan \frac{x}{2}(-\pi <x<\pi )，则\sin x=\frac{2\mu }{1+{\mu }^2}$$

三、一元二次方程

1. 韦达定理

$$x_1+x_2=-\frac{b}{a}$$ $$x_1{x}_2=\frac{c}{a}$$

2. 抛物线顶点

$$设y=a{x}^2+bx+c，则顶点：p(-\frac{b}{2a},c-\frac{b^2}{4a})$$

3. 点到直线距离

$$\frac{|A{x}_0+B{y}_0+C|}{\sqrt{A^2+B^2}}$$

四、因式分解

$$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$ $$a^3-b^3=(a-b)(a^2+ab+b^2)$$ $$a^3+b^3=(a+b)(a^2-ab+b^2)$$ $$n为正整数：a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots +a{b}^{n-2}+b^{n-1})$$ $$n为正偶数：a^n-b^n=(a+b)(a^{n-1}-a^{n-2}b+\cdots +a{b}^{n-2}-b^{n-1})$$ $$n为正奇数：a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+\cdots -a{b}^{n-2}+b^{n-1})$$ $$二项式定理：(a+b{)}^n=\underset{k=0}{\overset{n}{\lim }}{C}_n^k{a}^{n-k}{b}^k$$

五、阶乘

$$(2n)!!=2\cdot 4\cdot 6\dots (2n)=2^n\cdot n!$$ $$(2n-1)!!=1\cdot 3\cdot 5\dots (2n-1)$$

六、不等式

1. 基础

$$|a\pm b|\le |a|+|b|$$ $$||a|+|b||\le |a-b|$$ $$|{\int }_a^{b}f(x)dx|\le {\int }_a^b|f(x)|dx$$

2. 基础

$$\sqrt{ab}\le \frac{a+b}{2}\le \sqrt{\frac{a^2+b^2}{2}}(a,b>0)$$

3. 重要

$$设a>b>0，则\{\begin{array}{cc}{a}^n>b^n,& n>0\\ a^n<b^n,& n<0\end{array}$$

4. 低频

$$若0<a<x<b，0<c<y<d，则\frac{c}{b}<\frac{y}{x}<\frac{d}{a}$$

5. 重要

$$\sin x<x<\tan x(0<x<\frac{\pi }{2})$$

6. 重要

$$\sin x<x(x>0)$$

7. 低频

$$\arcsin x\le x\le \arcsin x(0\le x\le 1)$$

8. 重要

$$e^x\ge x+1$$

9. 重要

$$x-1\ge \ln x(x>0)$$

10. 低频

$$\frac{1}{1+x}<\ln (1+\frac{1}{x})<\frac{1}{x}(x>0)$$

七、泰勒公式

$$\sin x=x-\frac{x^3}{3!}+o(x^3)$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+o(x^4)$$ $$\arcsin x=x+\frac{x^3}{3!}+o(x^3)$$ $$\tan x=x+\frac{x^3}{3}+o(x^3)$$ $$\arctan x=x-\frac{x^3}{3}+o(x^3)$$ $$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+o(x^3)$$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+o(x^3)$$ $$(1+x{)}^{\alpha }=1+\alpha x+\frac{\alpha (\alpha -1)}{2!}{x}^2+o(x^2)$$ $$\frac{1}{1-x}=1+x+x^2+x^3+o(x^3)$$ $$\sqrt{1+x^2}=1+\frac{x^2}{2}-\frac{x^4}{8}+o(x^4)$$ $$y=f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n$$

八、等价无穷小

1. 基础

$$\sin x\sim x、\tan x\sim x、\arcsin x\sim x、\arcsin x\sim x、\ln (1+x)\sim x、e^x-1\sim x$$

2. 重要

$$a^x-1\sim x\ln a、1-\cos x\sim \frac{1}{2}{x}^2、(1+x{)}^{\alpha }-1\sim \alpha x、1-{\cos }^{\alpha }x\sim \frac{a}{2}{x}^2$$

3. 两个重要极限

$$\underset{x⟶\infty }{\lim }\frac{\sin x}{x}=1$$ $$\underset{x⟶\infty }{\lim }(1+\frac{1}{x}{)}^x=e$$

九、导数公式

1. 基本求导公式

$$({\log }_{\alpha }x{)}^{\prime }=\frac{1}{x\ln \alpha }(\alpha >0,\alpha \ne 0)$$ $$(\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}$$ $$(\tan x{)}^{\prime }={\sec }^{2}x$$ $$(\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$$ $$(\cot x{)}^{\prime }=-{\csc }^{2}x$$ $$(\arctan x{)}^{\prime }=\frac{1}{1+x^2}$$ $$(arccotx{)}^{\prime }=-\frac{1}{1+x^2}$$ $$(\sec x{)}^{\prime }=\sec x\tan x$$ $$(\csc x{)}^{\prime }=-\csc x\cot x$$ $${\left[\ln \left(x+\sqrt{x^2+1}\right)\right]}^{\prime }=\frac{1}{\sqrt{x^2+1}}$$ $${\left[\ln \left(x+\sqrt{x^2-1}\right)\right]}^{\prime }=\frac{1}{\sqrt{x^2-1}}$$

2. 参数方程

$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ $$\frac{d^{2}y}{d{x}^2}=\frac{\frac{d(\frac{dy}{dt})}{dt}}{\frac{dx}{dt}}=\frac{y^{\prime \prime }(t)x^{\prime }(t)-x^{\prime \prime }(t)y^{\prime }(t)}{{\left[x^{\prime }\left(t\right)\right]}^3}$$

3. 反函数

$$y_x^{\prime }=\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=\frac{1}{x_y^{\prime }}$$ $$y_{xx}^{\prime \prime }=\frac{d^{2}y}{d{x}^2}=\frac{d(\frac{dy}{dx})}{dx}=\frac{d(\frac{1}{x_y^{\prime }})}{dx}=\frac{d(\frac{1}{x_y^{\prime }})}{dy}\cdot \frac{1}{x_y^{\prime }}=-\frac{x_{yy}^{\prime \prime }}{(x_y^{\prime }{)}^3}$$

4. 变限积分求导公式

$$F(x)={\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(t)dt$$ $$F^{\prime }(x)=f[{\phi }_2(x)]{\phi }_2^{\prime }(x)-f[{\phi }_1(x)]{\phi }_1^{\prime }(x)$$

5. 莱布尼茨公式

$$(uv{)}^{(n)}=\sum _{k=0}^n{C}_n^k{u}^{(n-k)}{v}^{(k)}$$

6. 曲率

$$曲率：k=\frac{|y^{\prime \prime }|}{(1+y^{\prime 2}{)}^{\frac{3}{2}}}$$ $$极坐标：k=\frac{r^2+2r^{\prime 2}-r{r}^{\prime \prime }}{(r^2+r^{\prime 2}{)}^{\frac{3}{2}}}$$ $$曲率中心：(x-\frac{y^{\prime }(y^{\prime 2}+1)}{y^{\prime \prime }},x+\frac{y^{\prime 2}+1}{y^{\prime \prime }})$$

十、积分公式

1. 不定积分

$$\int \frac{1}{x}dx=\ln |x|+C$$ $$\int a^{x}dx=\frac{a^x}{\ln a}+C(a>0且a\ne 0)$$ $$\int \tan xdx=-\ln |\cos x|+C$$ $$\int \cot xdx=\ln |\sin x|+C$$ $$\int \frac{1}{\cos x}dx=\ln |\sec x+\tan x|+C$$ $$\int \frac{dx}{\sin x}=\ln |\csc x-\cot x|+C$$ $$\int {\sec }^{2}xdx=\tan x+C$$ $$\int {\csc }^{2}xdx=-\cot x+C$$ $$\int \sec x\tan xdx=\sec x+C$$ $$\int \csc x\cot xdx=-\csc x+C$$ $$\int \frac{1}{1+x^2}dx=\arctan x+C$$ $$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C(a>0)$$ $$\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C$$ $$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C(a>0)$$ $$\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln (x+\sqrt{x^2+a^2})+C$$ $$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln |x+\sqrt{x^2-a^2}|+C(|x|>|a|)$$ $$\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$$ $$\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln |\frac{x+a}{x-a}|+C$$ $$\int \sqrt{a^2-x^2}dx=\frac{a^2}{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^2-x^2}+C(a>|x|\ge 0)$$ $$\int {\sin }^{2}xdx=\frac{x}{2}-\frac{\sin 2x}{4}+C$$ $$\int {\cos }^{2}xdx=\frac{x}{2}+\frac{\sin 2x}{4}+C$$ $$\int {\tan }^{2}xdx=\tan x-x+C$$ $$\int {\cot }^{2}xdx=-\cot x-x+C$$

2. 定积分

$$偶函数：{\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$$ $$奇函数：{\int }_{-a}^a=0$$ $${\int }_0^{\pi }xf(\sin x)dx=\frac{\pi }{2}{\int }_0^{\pi }f(\sin x)dx=\pi {\int }_0^{\frac{\pi }{2}}f(\sin x)dx$$ $${\int }_0^{\frac{\pi }{2}}f(\sin x)dx={\int }_0^{\frac{\pi }{2}}f(\cos x)dx$$ $${\int }_0^{\frac{\pi }{2}}f(\sin x,\cos x)dx={\int }_0^{\frac{\pi }{2}}f(\cos x,\sin x)dx$$ $${\int }_a^{b}f(x)dx={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}f(\frac{a+b}{2}+\frac{b-a}{2}\sin t)\cdot \frac{b-a}{2}\cos tdt$$ $${\int }_1^{xy}f(t)dt=x{\int }_1^{y}f(t)dt+y{\int }_1^{x}f(t)dt$$

3. 区间再现

$${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$$ $${\int }_a^{b}f(x)dx=\frac{1}{2}{\int }_a^b[f(x)+f(a+b-x)]dx$$ $${\int }_a^{b}f(x)dx={\int }_a^{\frac{a+b}{2}}[f(x)+f(a+b-x)]dx$$

4. 华里士公式

$${\int }_0^{\frac{\pi }{2}}{\sin }^{n}xdx={\int }_0^{\frac{\pi }{2}}{\cos }^{n}xdx=\{\begin{array}{cc}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{2}{3}\cdot 1& n为大于1的奇数\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{1}{2}\cdot \frac{\pi }{2}& n为正偶数\end{array}$$ $${\int }_0^{\pi }{\sin }^{n}xdx=\{\begin{array}{cc}2\cdot \frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{2}{3}\cdot 1& n为大于1的奇数\\ 2\cdot \frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{1}{2}\cdot \frac{\pi }{2}& n为正偶数\end{array}$$ $${\int }_0^{\pi }{\cos }^{n}xdx=\{\begin{array}{cc}0& n为正奇数\\ 2\cdot \frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{1}{2}\cdot \frac{\pi }{2}& n为正偶数\end{array}$$ $$\{\begin{array}{c}{\int }_0^{2\pi }{\sin }^{n}xdx\\ {\int }_0^{2\pi }{\cos }^{n}xdx\end{array}=\{\begin{array}{cc}0& n为正奇数\\ 4\cdot \frac{n-1}{n}\cdot \frac{n-3}{n-2}\dots \frac{1}{2}\cdot \frac{\pi }{2}& n为正偶数\end{array}$$

5. 区间简化公式

$${\int }_a^{b}f(x)dx={\int }_0^1(b-a)f\left[a+(b-a)t\right]dt$$ $${\int }_{-a}^{a}f(x)dx={\int }_0^a\left[f(x)+f(-x)\right]dx(a>0)$$

6. 应用

① 面积：

$$S={\int }_a^b|y_1(x)-y_2(x)|dx$$ $$S=\frac{1}{2}{\int }_{\alpha }^{\beta }|r_1^2(\theta )-r_2^2\theta |d\theta$$ $$S={\int }_a^{b}y(t)dx(t)={\int }_{\alpha }^{\beta }|y(t)\cdot x^{\prime }(t)|dt$$ $$椭圆面积公式：S=\pi ab$$

② 体积

$$绕x轴：V={\int }_a^b\pi y^2(x)dx$$ $$绕y轴：V=2\pi {\int }_a^{b}x|y(x)|dx$$

③ 均值

$$\bar{y}=\frac{1}{b-a}{\int }_a^{b}y(x)dx，注：平均值是积分中值定理中的\xi$$

④弧长

$$s={\int }_a^b\sqrt{1+[y^{\prime }(x){]}^2}dx$$ $$s={\int }_{\alpha }^{\beta }\sqrt{[x^{\prime }(t){]}^2+[y^{\prime }(t){]}^2}dt$$ $$s={\int }_{\alpha }^{\beta }\sqrt{[r(\theta ){]}^2+[r^{\prime }(\theta ){]}^2}d\theta$$

⑤表面积

$$y(x)绕x轴：S=2\pi {\int }_a^b|y(x)|\sqrt{1+[y^{\prime }(x){]}^2}dx$$ $$y(t)绕x轴：S=2\pi {\int }_a^b|y(t)|\sqrt{[x^{\prime }(t){]}^2+[y^{\prime }(t){]}^2}dx$$

⑥形心

$$\bar{x}=\frac{\underset{D}{\iint }xd\sigma }{\underset{D}{\iint }d\sigma }=\frac{{\int }_a^{b}dx{\int }_0^{f(x)}xdy}{{\int }_a^{b}dx{\int }_0^{f(x)}dy}=\frac{{\int }_a^{b}xf(x)dx}{{\int }_a^{b}f(x)dx}$$ $$\bar{y}=\frac{\underset{D}{\iint }yd\sigma }{\underset{D}{\iint }d\sigma }=\frac{{\int }_a^{b}dx{\int }_0^{f(x)}ydy}{{\int }_a^{b}dx{\int }_0^{f(x)}dy}=\frac{\frac{1}{2}{\int }_a^b{f}^2(x)dx}{{\int }_a^{b}f(x)dx}$$

⑦平行截面面积

$$V={\int }_b^{a}S(x)dx$$

十一、中值定理

1. 二重积分中值定理

$$\underset{D}{\iint }f(x,y)d\sigma =f(\mu ,\xi )\cdot \sigma ,(\mu ,\xi )\in D$$

十二、函数图像

1. 心形线

$$r=a(1-cos\theta )(a>0)$$ $$r=a(1+cos\theta )(a>0)$$

2. 玫瑰线

$$r=asin3\theta (a>0)$$

3. 阿基米德螺线

$$r=a\theta (a>0,\theta >=0)$$

4. 伯努利双扭线

$$r²=2a²cos2\theta$$ $$r²=a²cos2\theta$$ $$r²=a²sin2\theta$$

5. 摆线

$$\{\begin{array}{c}x=r(t-sint)\\ y=r(1-cost)\end{array}$$

6. 星形线

$$x⅔+y⅔=r⅔$$ $$\{\begin{array}{c}x=rcos³t\\ y=rsin³t\end{array}$$

十三、微分方程

1. 齐次线性微分方程的通解

若$p^2-4q>0$，则${\lambda }_1\ne {\lambda }_2$是特征方程的两个不等实根，则通解为： $$y=C_1{e}^{{\lambda }_{1}x}+C_2{e}^{{\lambda }_{2}x}$$若$p^2-4q=0$，则${\lambda }_1={\lambda }_2$是特征方程的两个相等实根，则通解为： $$y=\left(C_1+C_{2}x\right)e^{\lambda x}$$若$p^2-4q<0$，设$\alpha \pm \beta i$是特征方程的一堆共轭复根，则通解为： $$y=e^{\alpha x}\left(C_1\cos \beta x+C_2\sin \beta x\right)$$

2. 非齐次线性微分方程的特解

当自由项$f(x)=P_n(x)e^{\alpha x}$时，特解要设为： $$y^{\ast }=e^{\alpha x}{Q}_n(x)x^k$$其中： $$\{\begin{array}{c}{e}^{\alpha x}照抄\\ Q_n(x)为x的n次一般多项式\\ k=\{\begin{array}{cc}0,& \alpha \ne {\lambda }_1,\alpha \ne {\lambda }_2\\ 1,& \alpha ={\lambda }_1或\alpha ={\lambda }_2\\ 2,& \alpha ={\lambda }_1={\lambda }_2\end{array}\end{array}$$当自由项$f(x)=e^{\alpha x}\left[P_m(x)\cos \beta x+P_n(x)\sin \beta x\right]$时，特解要设为： $$y^{\ast }=e^{\alpha x}\left[Q_l^{(1)}(x)\cos \beta x+Q_l^{(2)}(x)\sin \beta x\right]x^k$$ 其中： $$\{\begin{array}{c}{e}^{\alpha x}照抄\\ l=\max (m,n),Q_l^{(1)}(x),Q_l^{(2)}(x)分别为x的两个不同的l次一般多项式\\ k=\{\begin{array}{cc}0,& \alpha \pm \beta i不是特征根\\ 1,& \alpha \pm \beta i是特征根\end{array}\end{array}$$