高数考研 -- 公式总结(更新中)

1. 两个重要极限

(1) ${\lim }_{x\to 0}\frac{\sin x}{x}=1$, 推广形式 ${\lim }_{f(x)\to 0}\frac{\sin f(x)}{f(x)}=1$.
(2) ${\lim }_{x\to \infty }{\left(1+\frac{1}{x}\right)}^x=\mathrm{e}$, 推广形式 ${\lim }_{x\to 0}(1+x{)}^{\frac{1}{x}}=\mathrm{e},{\lim }_{f(x)\to \infty }{\left[1+\frac{1}{f(x)}\right]}^{f(x)}=\mathrm{e}$

2. 常用的等价无穷小量及极限公式

(1) 当 $x\to 0$ 时,常用的等价无穷小

• (1) $x\sim \sin x\sim \tan x\sim \arcsin x\sim \arctan x\sim \ln (1+x)\sim {\mathrm{e}}^x-1$.
• (2) $1-\cos x\sim \frac{1}{2}{x}^2,1-{\cos }^{b}x\sim \frac{b}{2}{x}^2(b\ne 0)$.
• (3) $a^x-1\sim x\ln a(a>0$, 且 $a\ne 1)$.
• (4) $(1+x{)}^{\alpha }-1\sim \alpha x(\alpha \ne 0)$.

(2) 当 $n\to \infty$ 或 $x\to \infty$ 时,常用的极限公式

• (1) ${\lim }_{n\to \infty }\sqrt[n]{n}=1,{\lim }_{n\to \infty }\sqrt[n]{a}=1(a>0)$.
• (2) ${\lim }_{x\to \infty }\frac{a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0}{b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0}=\{\begin{array}{ll}\frac{a_n}{b_m},& n=m,\\ 0,& n<m,\\ \infty ,& n>m,\end{array}$ 其中 $a_n,b_m$ 均不

为 0 .

• (3) ${\lim }_{n\to \infty }{x}^n=\{\begin{array}{ll}0,& |x|<1,\\ \infty ,& |x|>1,\\ 1,& x=1,\\ \text{ 不存在, }& x=-1;\end{array}{\lim }_{n\to \infty }{\mathrm{e}}^{nx}=\{\begin{array}{ll}0,& x<0,\\ +\infty ,& x>0,\\ 1,& x=0.\end{array}$
• (4) 若 $\lim g(x)=0,\lim f(x)=\infty$, 且 $\lim g(x)f(x)=A$, 则有
  $$\lim [1+g(x){]}^{f(x)}={\mathrm{e}}^A.$$

3. $x\to 0$ 时常见的麦克劳林公式

$$\begin{array}{rl}& \sin x=x-\frac{1}{3!}{x}^3+o\left(x^3\right),\cos x=1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4+o\left(x^4\right),\\ & \\ & \tan x=x+\frac{1}{3}{x}^3+o\left(x^3\right),\arcsin x=x+\frac{1}{3!}{x}^3+o\left(x^3\right),\\ & \\ & \arctan x=x-\frac{1}{3}{x}^3+o\left(x^3\right),\ln (1+x)=x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3+o\left(x^3\right),\\ & \\ & {\mathrm{e}}^x=1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+o\left(x^3\right),(1+x{)}^a=1+ax+\frac{a(a-1)}{2!}{x}^2+o\left(x^2\right).\end{array}$$

当 $x\to 0$ 时,由以上公式可以得到以下几组“差函数”的等价无穷小代换式:

$x-\sin x\sim \frac{x^3}{6},\tan x-x\sim \frac{x^3}{3},x-\ln (1+x)\sim \frac{x^2}{2}$, $\arcsin x-x\sim \frac{x^3}{6},x-\arctan x\sim \frac{x^3}{3}$.

4. 基本导数公式

$$\begin{array}{ll}{\left(x^{\mu }\right)}^{\prime }=\mu x^{\mu -1}(\mu 为常数),& {\left(a^x\right)}^{\prime }=a^x\ln a(a>0,a\ne 1),\\ & \\ {\left({\log }_{a}x\right)}^{\prime }=\frac{1}{x\ln a}(a>0,a\ne 1),& (\ln x{)}^{\prime }=\frac{1}{x},\\ & \\ (\sin x{)}^{\prime }=\cos x,& (\cos x{)}^{\prime }=-\sin x,\\ & \\ (\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}},& (\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}},\\ & \\ (\tan x{)}^{\prime }={\sec }^{2}x,& (\cot x{)}^{\prime }=-{\csc }^{2}x,\\ & \\ (\arctan x{)}^{\prime }=\frac{1}{1+x^2},& (\mathrm{arccot}x{)}^{\prime }=-\frac{1}{1+x^2},\\ & \\ (\sec x{)}^{\prime }=\sec x\tan x,& (\csc x{)}^{\prime }=-\csc x\cot x,\\ & \\ {\left[\ln \left(x+\sqrt{x^2+1}\right)\right]}^{\prime }=\frac{1}{\sqrt{x^2+1}},,& {\left[\ln \left(x+\sqrt{x^2-1}\right)\right]}^{\prime }=\frac{1}{\sqrt{x^2-1}}\end{array}$$
三角函数六边形记忆法:

注: 变限积分求导公式.
设 $F(x)={\int }_{{\phi }_2(x)}^{{\phi }_1(x)}f(t)\mathrm{d}t$, 其中 $f(x)$ 在 $[a,b]$ 上连续, 可导函数 ${\phi }_1(x)$ 和 ${\phi }_2(x)$ 的值域在 $[a,b]$ 上, 则在函数 ${\phi }_1(x)$ 和 ${\phi }_2(x)$ 的公共定义域上有:
$$F^{\prime }(x)=\frac{\mathrm{d}}{\mathrm{d}x}\left[{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(t)\mathrm{d}t\right]=f\left[{\phi }_2(x)\right]{\phi }_2^{\prime }(x)-f\left[{\phi }_1(x)\right]{\phi }_1^{\prime }(x).$$

5. 几个重要函数的麦克劳林展开式

(1) ${\mathrm{e}}^x=1+x+\frac{1}{2!}{x}^2+\cdots +\frac{1}{n!}{x}^n+o\left(x^n\right)$.

(2) $\sin x=x-\frac{1}{3!}{x}^3+\cdots +(-1{)}^n\frac{1}{(2n+1)!}{x}^{2n+1}+o\left(x^{2n+1}\right)$.

(3) $\cos x=1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4-\cdots +(-1{)}^n\frac{1}{(2n)!}{x}^{2n}+o\left(x^{2n}\right)$.

(4) $\frac{1}{1-x}=1+x+x^2+\cdots +x^n+o\left(x^n\right),|x|<1$.

(5) $\frac{1}{1+x}=1-x+x^2-\cdots +(-1{)}^n{x}^n+o\left(x^n\right),|x|<1$.

(6) $\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +(-1{)}^{n-1}\frac{x^n}{n}+o\left(x^n\right),-1<x⩽1$.

(7) $(1+x{)}^a=1+ax+\frac{a(a-1)}{2!}{x}^2+\cdots +\frac{a(a-1)\cdots (a-n+1)}{n!}{x}^n+$ $o\left(x^n\right)$.

6. 曲率和曲率半径计算公式

(1) 曲率

• (1) (非参数方程) 曲线 $y=f(x)$ 上任意一点 $(x,f(x))$ 处的曲率为
  $$K=\frac{\left|y^{\prime \prime }\right|}{{\left[1+{\left(y^{\prime }\right)}^2\right]}^{\frac{3}{2}}}\text{. }$$
• (2) (参数方程) $\{\begin{array}{l}x=x(t),\\ y=y(t)\end{array}$ 上任意一点的曲率为
  $$K=\frac{\left|x^{\prime }(t)y^{\prime \prime }(t)-y^{\prime }(t)x^{\prime \prime }(t)\right|}{{\left\{{\left[x^{\prime }(t)\right]}^2+{\left[y^{\prime }(t)\right]}^2\right\}}^{\frac{3}{2}}}.$$
  参数方程求导:
  参数方程 $\{\begin{array}{l}x=\phi (t)\\ y=\psi (t)\end{array}$

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{{\psi }^{\prime }(t)}{{\phi }^{\prime }(t)},令其为F(t),$$
$$\frac{d^{2}y}{d{x}^2}=\frac{d\left(\frac{dy}{dx}\right)}{dx}=\frac{d\left(\frac{dy}{dx}\right)/dt}{dx/dt}=\frac{{\psi }^{\prime \prime }(t){\phi }^{\prime }(t)-{\psi }^{\prime }(t){\phi }^{\prime \prime }(t)}{{\left[{\phi }^{\prime }(t)\right]}^3}=\frac{d(F(t))/dt}{dx/dt}=\frac{F^{\prime }(t)}{{\phi }^{\prime }(t)}$$
可以记最后那个简单的式子

(2) 曲率半径
$$R=\frac{1}{K}(K\ne 0)$$