25高数考研张宇 -- 公式总结(更新中)

1. 两个重要极限

(1) lim⁡x→0sin⁡xx=1\lim _{x \rightarrow 0} \frac{\sin x}{x}=1limx→0​xsinx​=1, 推广形式 lim⁡f(x)→0sin⁡f(x)f(x)=1\lim _{f(x) \rightarrow 0} \frac{\sin f(x)}{f(x)}=1limf(x)→0​f(x)sinf(x)​=1.

(2) lim⁡x→∞(1+1x)x=e\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^x=\mathrm{e}limx→∞​(1+x1​)x=e, 推广形式 lim⁡x→0(1+x)1x=e,lim⁡f(x)→∞[1+1f(x)]f(x)=e\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\mathrm{e}, \lim _{f(x) \rightarrow \infty}\left[1+\frac{1}{f(x)}\right]^{f(x)}=\mathrm{e}limx→0​(1+x)x1​=e,limf(x)→∞​[1+f(x)1​]f(x)=e

2. 常用的等价无穷小量及极限公式

(1) 当 x→0x \rightarrow 0x→0 时,常用的等价无穷小

(1) x∼sin⁡x∼tan⁡x∼arcsin⁡x∼arctan⁡x∼ln⁡(1+x)∼ex−1x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim \ln (1+x) \sim \mathrm{e}^x-1x∼sinx∼tanx∼arcsinx∼arctanx∼ln(1+x)∼ex−1.(2) 1−cos⁡x∼12x2,1−cos⁡bx∼b2x2(b≠0)1-\cos x \sim \frac{1}{2} x^2, 1-\cos ^b x \sim \frac{b}{2} x^2(b \neq 0)1−cosx∼21​x2,1−cosbx∼2b​x2(b=0).(3) ax−1∼xln⁡a(a>0a^x-1 \sim x \ln a(a>0ax−1∼xlna(a>0, 且 a≠1)a \neq 1)a=1).(4) (1+x)α−1∼αx(α≠0)(1+x)^\alpha-1 \sim \alpha x (\alpha \neq 0)(1+x)α−1∼αx(α=0).

(2) 当 n→∞n \rightarrow \inftyn→∞ 或 x→∞x \rightarrow \inftyx→∞ 时,常用的极限公式

(1) lim⁡n→∞nn=1,lim⁡n→∞an=1(a>0)\lim _{n \rightarrow \infty} \sqrt[n]{n}=1, \lim _{n \rightarrow \infty} \sqrt[n]{a}=1(a>0)limn→∞​nn​=1,limn→∞​na​=1(a>0).(2) lim⁡x→∞anxn+an−1xn−1+⋯+a1x+a0bmxm+bm−1xm−1+⋯+b1x+b0={anbm,n=m,0,nm,\lim _{x \rightarrow \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}=\left\{\begin{array}{ll}\frac{a_n}{b_m}, & n=m, \\ 0, & nm,\end{array}\right.limx→∞​bm​xm+bm−1​xm−1+⋯+b1​x+b0​an​xn+an−1​xn−1+⋯+a1​x+a0​​=⎩⎨⎧​bm​an​​,0,∞,​n=m,nm,​ 其中 an,bma_n, b_man​,bm​ 均不

为 0 .

(3) lim⁡n→∞xn={0,∣x∣<1,∞,∣x∣>1,1,x=1, 不存在, x=−1;lim⁡n→∞enx={0,x<0,+∞,x>0,1,x=0.\lim _{n \rightarrow \infty} x^n=\left\{\begin{array}{ll}0, & |x|<1, \\ \infty, & |x|>1, \\ 1, & x=1, \\ \text { 不存在, } & x=-1 ;\end{array} \lim _{n \rightarrow \infty} \mathrm{e}^{n x}= \begin{cases}0, & x<0, \\ +\infty, & x>0, \\ 1, & x=0 .\end{cases}\right.limn→∞​xn=⎩⎨⎧​0,∞,1, 不存在, ​∣x∣<1,∣x∣>1,x=1,x=−1;​limn→∞​enx=⎩⎨⎧​0,+∞,1,​x<0,x>0,x=0.​(4) 若 lim⁡g(x)=0,lim⁡f(x)=∞\lim g(x)=0, \lim f(x)=\inftylimg(x)=0,limf(x)=∞, 且 lim⁡g(x)f(x)=A\lim g(x) f(x)=Alimg(x)f(x)=A, 则有

lim⁡[1+g(x)]f(x)=eA.

\lim [1+g(x)]^{f(x)}=\mathrm{e}^A .

lim[1+g(x)]f(x)=eA.

3. x→0x \rightarrow 0x→0 时常见的麦克劳林公式

sin⁡x=x−13!x3+o(x3),cos⁡x=1−12!x2+14!x4+o(x4),tan⁡x=x+13x3+o(x3),arcsin⁡x=x+13!x3+o(x3),arctan⁡x=x−13x3+o(x3),ln⁡(1+x)=x−12x2+13x3+o(x3),ex=1+x+12!x2+13!x3+o(x3),(1+x)a=1+ax+a(a−1)2!x2+o(x2).

\begin{aligned}

& \sin x=x-\frac{1}{3 !} x^3+o\left(x^3\right), \quad \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), \quad \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), \quad \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+a x+\frac{a(a-1)}{2 !} x^2+o\left(x^2\right) .

\end{aligned}

​sinx=x−3!1​x3+o(x3),cosx=1−2!1​x2+4!1​x4+o(x4),tanx=x+31​x3+o(x3),arcsinx=x+3!1​x3+o(x3),arctanx=x−31​x3+o(x3),ln(1+x)=x−21​x2+31​x3+o(x3),ex=1+x+2!1​x2+3!1​x3+o(x3),(1+x)a=1+ax+2!a(a−1)​x2+o(x2).​

当 x→0x \rightarrow 0x→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∼x36,tan⁡x−x∼x33,x−ln⁡(1+x)∼x22x-\sin x \sim \frac{x^3}{6}, \quad \tan x-x \sim \frac{x^3}{3}, \quad x-\ln (1+x) \sim \frac{x^2}{2}x−sinx∼6x3​,tanx−x∼3x3​,x−ln(1+x)∼2x2​, arcsin⁡x−x∼x36,x−arctan⁡x∼x33\arcsin x-x \sim \frac{x^3}{6}, \quad x-\arctan x \sim \frac{x^3}{3}arcsinx−x∼6x3​,x−arctanx∼3x3​.

4. 基本导数公式

(xμ)′=μxμ−1(μ为常数),(ax)′=axln⁡a(a>0,a≠1),(log⁡ax)′=1xln⁡a(a>0,a≠1),(ln⁡x)′=1x,(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x,(arcsin⁡x)′=11−x2,(arccos⁡x)′=−11−x2,(tan⁡x)′=sec⁡2x,(cot⁡x)′=−csc⁡2x,(arctan⁡x)′=11+x2,(arccot⁡x)′=−11+x2,(sec⁡x)′=sec⁡xtan⁡x,(csc⁡x)′=−csc⁡xcot⁡x,[ln⁡(x+x2+1)]′=1x2+1,,[ln⁡(x+x2−1)]′=1x2−1

\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 \neq 1), \\ \\

\left(\log _a x\right)^{\prime}=\frac{1}{x \ln a}(a>0, a \neq 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}, & (\operatorname{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}

(xμ)′=μxμ−1(μ为常数),(loga​x)′=xlna1​(a>0,a=1),(sinx)′=cosx,(arcsinx)′=1−x2​1​,(tanx)′=sec2x,(arctanx)′=1+x21​,(secx)′=secxtanx,[ln(x+x2+1​)]′=x2+1​1​,,​(ax)′=axlna(a>0,a=1),(lnx)′=x1​,(cosx)′=−sinx,(arccosx)′=−1−x2​1​,(cotx)′=−csc2x,(arccotx)′=−1+x21​,(cscx)′=−cscxcotx,[ln(x+x2−1​)]′=x2−1​1​​

三角函数六边形记忆法:

注: 变限积分求导公式.

设 F(x)=∫φ2(x)φ1(x)f(t)dtF(x)=\int_{\varphi_2(x)}^{\varphi_1(x)} f(t) \mathrm{d} tF(x)=∫φ2​(x)φ1​(x)​f(t)dt, 其中 f(x)f(x)f(x) 在 [a,b][a, b][a,b] 上连续, 可导函数 φ1(x)\varphi_1(x)φ1​(x) 和 φ2(x)\varphi_2(x)φ2​(x) 的值域在 [a,b][a, b][a,b] 上, 则在函数 φ1(x)\varphi_1(x)φ1​(x) 和 φ2(x)\varphi_2(x)φ2​(x) 的公共定义域上有:

F′(x)=ddx[∫φ1(x)φ2(x)f(t)dt]=f[φ2(x)]φ2′(x)−f[φ1(x)]φ1′(x).

F^{\prime}(x)=\frac{\mathrm{d}}{\mathrm{d} x}\left[\int_{\varphi_1(x)}^{\varphi_2(x)} f(t) \mathrm{d} t\right]=f\left[\varphi_2(x)\right] \varphi_2^{\prime}(x)-f\left[\varphi_1(x)\right] \varphi_1^{\prime}(x) .

F′(x)=dxd​[∫φ1​(x)φ2​(x)​f(t)dt]=f[φ2​(x)]φ2′​(x)−f[φ1​(x)]φ1′​(x).

5. 几个重要函数的麦克劳林展开式

(1) ex=1+x+12!x2+⋯+1n!xn+o(xn)\mathrm{e}^x=1+x+\frac{1}{2 !} x^2+\cdots+\frac{1}{n !} x^n+o\left(x^n\right)ex=1+x+2!1​x2+⋯+n!1​xn+o(xn).

(2) sin⁡x=x−13!x3+⋯+(−1)n1(2n+1)!x2n+1+o(x2n+1)\sin x=x-\frac{1}{3 !} x^3+\cdots+(-1)^n \frac{1}{(2 n+1) !} x^{2 n+1}+o\left(x^{2 n+1}\right)sinx=x−3!1​x3+⋯+(−1)n(2n+1)!1​x2n+1+o(x2n+1).

(3) cos⁡x=1−12!x2+14!x4−⋯+(−1)n1(2n)!x2n+o(x2n)\cos x=1-\frac{1}{2 !} x^2+\frac{1}{4 !} x^4-\cdots+(-1)^n \frac{1}{(2 n) !} x^{2 n}+o\left(x^{2 n}\right)cosx=1−2!1​x2+4!1​x4−⋯+(−1)n(2n)!1​x2n+o(x2n).

(4) 11−x=1+x+x2+⋯+xn+o(xn),∣x∣<1\frac{1}{1-x}=1+x+x^2+\cdots+x^n+o\left(x^n\right),|x|<11−x1​=1+x+x2+⋯+xn+o(xn),∣x∣<1.

(5) 11+x=1−x+x2−⋯+(−1)nxn+o(xn),∣x∣<1\frac{1}{1+x}=1-x+x^2-\cdots+(-1)^n x^n+o\left(x^n\right),|x|<11+x1​=1−x+x2−⋯+(−1)nxn+o(xn),∣x∣<1.

(6) ln⁡(1+x)=x−x22+x33−⋯+(−1)n−1xnn+o(xn),−1

(7) (1+x)a=1+ax+a(a−1)2!x2+⋯+a(a−1)⋯(a−n+1)n!xn+(1+x)^a=1+a x+\frac{a(a-1)}{2 !} x^2+\cdots+\frac{a(a-1) \cdots(a-n+1)}{n !} x^n+(1+x)a=1+ax+2!a(a−1)​x2+⋯+n!a(a−1)⋯(a−n+1)​xn+ o(xn)o\left(x^n\right)o(xn).

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)y=f(x)y=f(x) 上任意一点 (x,f(x))(x, f(x))(x,f(x)) 处的曲率为

K=∣y′′∣[1+(y′)2]32.

K=\frac{\left|y^{\prime \prime}\right|}{\left[1+\left(y^{\prime}\right)^2\right]^{\frac{3}{2}}} \text {. }

K=[1+(y′)2]23​∣y′′∣​. (2) (参数方程) {x=x(t),y=y(t)\left\{\begin{array}{l}x=x(t), \\ y=y(t)\end{array}\right.{x=x(t),y=y(t)​ 上任意一点的曲率为

K=∣x′(t)y′′(t)−y′(t)x′′(t)∣{[x′(t)]2+[y′(t)]2}32.

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}}} .

K={[x′(t)]2+[y′(t)]2}23​∣x′(t)y′′(t)−y′(t)x′′(t)∣​.

参数方程求导:

参数方程 {x=φ(t)y=ψ(t)\left\{\begin{array}{l}x=\varphi(t) \\ y=\psi(t)\end{array}\right.{x=φ(t)y=ψ(t)​

dydx=dy/dtdx/dt=ψ′(t)φ′(t),令其为F(t),

\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)},令其为F(t),\\

dxdy​=dx/dtdy/dt​=φ′(t)ψ′(t)​,令其为F(t),

d2ydx2=d(dydx)dx=d(dydx)/dtdx/dt=ψ′′(t)φ′(t)−ψ′(t)φ′′(t)[φ′(t)]3=d(F(t))/dtdx/dt=F′(t)φ′(t)

\frac{d^{2} y}{d x^{2}}=\frac{d\left(\frac{d y}{d x}\right)}{d x}=\frac{d\left(\frac{d y}{d x}\right) / d t}{d x / d t}=\frac{\psi^{\prime \prime}(t) \varphi^{\prime}(t)-\psi^{\prime}(t) \varphi^{\prime \prime}(t)}{\left[\varphi^{\prime}(t)\right]^{3}} = \frac{d(F(t))/dt}{dx/dt} = \frac{F^{\prime}(t)}{\varphi^{\prime}(t)}

dx2d2y​=dxd(dxdy​)​=dx/dtd(dxdy​)/dt​=[φ′(t)]3ψ′′(t)φ′(t)−ψ′(t)φ′′(t)​=dx/dtd(F(t))/dt​=φ′(t)F′(t)​

可以记最后那个简单的式子

(2) 曲率半径

R=1K(K≠0)

R=\frac{1}{K}(K \neq 0)

R=K1​(K=0)