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  • 函数m,v在定义域内可,有: 1、(m±v)`=m`±v` 2、(mv)`=m`v+mv` 3、(m/v)`=(m`v-mv`)/v^2 注意:公式3要求v≠0。

    函数m,v在定义域内可导,有:

    1、(m±v)`=m`±v`
    2、(mv)`=m`v+mv`
    3、(m/v)`=(m`v-mv`)/v^2
    注意:公式3要求v≠0。

     

    展开全文
  • 四则运算法则6.基本导数与微分表7.复合函数,反函数,隐函数以及参数方程所确定的函数的微分法8.常用高阶导数公式9.微分中值定理,泰勒公式10.洛必达法则11.泰勒公式12.函数单调性的判断13.渐近线的求法14.函数凹凸...

    机器学习的数学基础

    高等数学

    1.导数定义:

    导数和微分的概念

    2.左右导数导数的几何意义和物理意义

    3.函数的可导性与连续性之间的关系

    4.平面曲线的切线和法线

    切线方程 :
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    法线方程:
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    5.四则运算法则

    6.基本导数与微分表

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    7.复合函数,反函数,隐函数以及参数方程所确定的函数的微分法

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    8.常用高阶导数公式

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    9.微分中值定理,泰勒公式

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    10.洛必达法则

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    11.泰勒公式

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    12.函数单调性的判断

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    13.渐近线的求法

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    14.函数凹凸性的判断

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    15.弧微分

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    16.曲率

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    17.曲率半径

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    展开全文
  • 面(体)积公式 一元二次方程基础 极坐标方程与直角坐标转换 切线与法线方程 因式分解公式 阶乘与双阶乘 函数的奇偶性 排列组合 等差数列 等比数列 ...导数四则运算 复合函数求导 反函数求导 参数方程

    文章目录

    基础回顾

    面(体)积公式

    S=4πR2V=43πR3V=13sh     (sh)S=πabS=12lr=12r2θ     lrθ(π) \begin{aligned} & \\ & 球表面积公式:S= 4\pi R^2 \\ \\ & 球体积公式:V = \frac{4}{3}\pi R^3 \\ \\ & 圆锥体积公式:V=\frac{1}{3} sh ~~~~~(s为圆锥底面积,h为圆锥的高) \\\\ & 椭圆面积公式: S=\pi ab \\ \\ & 扇形面积公式: S= \frac{1}{2}l r = \frac{1}{2}r^2\theta ~~~~~(其中l为弧长,r为半径,\theta为夹角(用\pi表示)) \end{aligned}

    一元二次方程基础

    ax2+bx+c=0     (a0)    x1,2=b±b24ac2ax1+x2=ba       x1x2=caΔ=b24ac    {Δ>0Δ=0Δ<0线 y=ax2+bx+c(b2a,cb24a) \begin{aligned} & \\ & 一元二次方程:ax^2 + bx + c =0 ~~~~~(a \ne 0) \\ \\ & 根的公式 ~~~~ x_{1,2} = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\ \\ & 韦达定理: x_1 + x_2 = -\frac{b}{a} ~~~~~~~ x_1 x_2 = \frac{c}{a} \\ \\ & 判别式: \Delta=b^2 - 4ac \implies \begin{cases} \Delta >0,两个不等实根 \\ \Delta =0,两个相等实根 \\ \Delta <0,两个共轭的复根(无实根) \\ \end{cases} \\\\ & 抛物线~ y=ax^2 + bx + c 的顶点:(-\frac{b}{2a}, c-\frac{b^2}{4a}) \end{aligned}

    极坐标方程与直角坐标转换

    {x=ρcosθy=ρsinθ    x2+y2=ρ2ρ2=x2+y2    tanθ=yx \begin{aligned} &直角坐标化极坐标 \begin{cases} x = \rho \cos \theta \\ y = \rho \sin \theta \end{cases} \implies x^2+y^2=\rho ^2 \\\\ &极坐标化直角坐标 :\rho ^2 = x^2+y^2 \implies \tan \theta = \frac{y}{x} \\\\ \end{aligned}

    切线与法线方程

    线yy0xx0=f(x0)    yy0=f(x0)(xx0)线yy0xx0=1f(x0)     ,yy0=1f(x0)(xx0) \begin{aligned} & 切线方程: \frac{y - y_0}{x - x_0} = f'(x_0) ~~~~,即 y-y_0 = f'(x_0)(x-x_0) \\ \\ & 法线方程: \frac{y - y_0}{x - x_0} = -\frac{1}{f'(x_0)}~~~~~,即y-y_0 = -\frac{1}{f'(x_0)}(x-x_0) \end{aligned}

    因式分解公式

    (a+b)2=a2+2ab+b2(ab)2=a22ab+b2(a+b+c)2=a2+b2+c2+2ab+2ac+2bc(a+b)3=a3+3a2b+3ab2+b3(ab)3=a33a2b+3ab2b3(a+b)(ab)=a2b2a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)anbn=(ab)(an1+an2b++abn2+bn1)(a+b)n=k=0nCnkakbnk=an+nan1b+n(n1)2!an1b2++n(n1)(nk+1)k!ankbk++nabn1+bn \begin{aligned} & \\ & (a+b)^2 = a^2 + 2ab+b^2 \\ \\ & (a-b)^2 = a^2 - 2ab + b^2 \\ \\ & (a+b+c)^2 =a^2+b^2+c^2 + 2ab+2ac+2bc \\\\ & (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ \\ & (a-b)^3 = a^3 - 3a^2b+3ab^2-b^3 \\ \\ & (a+b)(a-b) = a^2 - b^2 \\ \\ & a^3 + b^3 = (a+b) (a^2 -ab + b^2) \\ \\ & a^3-b^3 = (a-b)(a^2+ab+b^2) \\ \\ & a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1}) \\ \\ & (a+b)^n = \sum_{k=0}^n C_n^ka^kb^{n-k} = a^n + na^{n-1}b + \frac{n(n-1)}{2!}a^{n-1}b^2 + \cdots + \frac{n(n-1)\cdots(n-k+1)}{k!}a^{n-k}b^k + \cdots + nab^{n-1} + b^n \end{aligned}

    阶乘与双阶乘

    n!=1×2×3×...×n     (0!=1)(2n)!!=2×4×6×...×(2n)=2nn!(2n1)!!=1×3×5...×(2n1) \begin{aligned} & n! = 1\times2\times3\times ... \times n ~~~~~(规定0!=1) \\ \\ & (2n)!! = 2\times4\times6\times ... \times (2n) = 2^n \cdot n! \\ \\ & (2n-1)!! = 1\times3\times5...\times(2n-1) \end{aligned}

    函数的奇偶性

    [a,a]f(x)=12[f(x)f(x)]+12[f(x)+f(x)] \begin{aligned} & 定义在[-a,a]上的任一函数,可以表示为一个奇函数与一个偶函数之和: \\ \\ & f(x) = \frac{1}{2}[f(x)-f(-x)] + \frac{1}{2}[f(x) + f(-x)] \end{aligned}

    排列组合

    Anm=n(n1)(n2)(nm+1)=n!(nm)!Cnm=Anmm!=n(n1)(nm+1)m!=n!m!(nm)! \begin{aligned} A_n^m & = n(n-1)(n-2)\cdots(n-m +1) \\\\ & = \frac{n!}{(n-m)!} \\\\ \\ C_n^m & = \frac{A_n^m}{m!} = \frac{n(n-1)\cdots(n-m + 1)}{m!} \\\\ & = \frac{n!}{m!(n-m)!} \end{aligned}

    等差数列

    an=a1+(n1)dSn=na1+n(n1)2d        nNSn=n(a1+an)2 \begin{aligned} & a_n = a_1 + (n-1)d \\ \\ & S_n = na_1 + \frac{n(n-1)}{2}d ~~~~~~~~ n \in N^* \\ \\ & S_n = \frac{n(a_1+a_n)}{2} \\ \\ \end{aligned}

    等比数列

    an=a1qn1Sn=a1(1qn)1q      (q1) \begin{aligned} & a_n = a_1 \cdot q^{n-1} \\ \\ & S_n = \frac{a_1(1-q^n)}{1-q} ~~~~~~(q\neq 1) \end{aligned}

    常用数列前n项和

    k=1nk=1+2+3++n=n(n+1)2k=1n(2k1)=1+3+5++(2n1)=n2k=1nk2=12+22+32++n2=n(n+1)(2n+1)6k=1nk3=13+23+33++n3=[n(n+1)2]2=(k=1nk)2k=1nk(k+1)=1×2+2×3+3×4++n(n+1)=n(n+1)(n+2)3k=1n1k(k+1)=11×2+12×3+13×4++1n(n+1)=nn+1 \begin{aligned} & \\ & \sum_{k=1}^n k = 1 + 2+3+\cdots + n=\frac{n(n+1)}{2} \\ \\ & \sum_{k=1}^n (2k-1) = 1+ 3 + 5 + \cdots + (2n-1) = n^2 \\ \\ & \sum_{k=1}^n k^2 = 1^2+2^2+3^2+\cdots +n^2 = \frac{n(n+1)(2n+1)}{6} \\ \\ & \sum_{k=1}^n k^3 = 1^3 + 2^3 +3^3 +\cdots + n^3 = [\frac{n(n+1)}{2}]^2 = (\sum_{k=1}^n k)^2 \\ \\ & \sum_{k=1}^n k(k+1) = 1 \times 2 + 2 \times 3 + 3 \times 4 + \cdots + n(n+1) = \frac{n(n+1)(n+2)}{3} \\\\ & \sum_{k=1}^n \frac{1}{k(k+1)} = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1} \end{aligned}

    不等式

    2aba2+b2a±ba+bababa1±a2±±ana1+a2++anabf(x)dxabf(x)dx     (a<b)aba+b2a2+b22     (a,b>0)abc3a+b+c3a2+b2+c23     (a,b,c>0)a1a2anna1+a2+...+anna12+a22+...+an2n    a1,a2,...an>0a1=a2=...=anxyxpp+xqq     (x,y,p,q>0,1p+1q=1)(ac+bd)2(a2+b2)(c2+d2)(a1b1+a2b2+a3b3)2(a12+a22+a32)(b12+b22+b32)[abf(x)g(x)dx]2abf2(x)dxabg2(x)dxsinx<x<tanx     (0<x<π2)arctanxxarcsinx     (0x1)x+1exlnxx111+x<ln(1+1x)<1x     (x>0) \begin{aligned} & \\ & 2 |ab| \le a ^ 2 + b^2 \\ \\ & |a \pm b| \le |a| + |b| \\ \\ & | |a| - |b| | \le |a-b| \\ \\ & |a_1 \pm a_2 \pm \cdot\cdot\cdot\cdot \pm a_n| \le |a_1| + |a_2| + \cdot\cdot\cdot + |a_n| \\ \\ & |\int_a^b f(x) dx| \le \int_a^b |f(x)| dx ~~~~~(a<b) \\ \\ & \sqrt{ab} \le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} ~~~~~(a,b>0) \\ \\ & \sqrt[3]{abc} \le \frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} ~~~~~(a,b,c>0) \\ \\ & \sqrt[n]{a_1a_2\cdot\cdot\cdot a_n} \le \frac{a_1+a_2+...+a_n}{n} \le \sqrt{\frac{{a_1}^2+{a_2}^2 + ... + {a_n}^2}{n}} ~~~~(a_1,a_2,...a_n > 0,等号当且仅当 a_1 = a_2 = ... = a_n时成立) \\ \\ & xy \le \frac{x^p}{p} + \frac{x^q}{q} ~~~~~(x,y,p,q>0, \frac{1}{p}+\frac{1}{q}=1) \\ \\ & (ac+bd)^2 \le (a^2+b^2)(c^2+d^2) \\ \\ & (a_1 b_1 + a_2 b_2 + a_3 b_3)^2 \le ({a_1}^2 + {a_2}^2 + {a_3}^2)({b_1}^2 + {b_2}^2 + {b_3}^2) \\ \\ & [\int_a^b f(x)\cdot g(x) dx]^2 \le \int_a^bf^2(x)dx \cdot\int_a^bg^2(x)dx \\ \\ & \sin x < x < \tan x ~~~~~(0<x<\frac{\pi}{2}) \\ \\ & \arctan x \le x \le \arcsin x ~~~~~(0\le x \le 1) \\ \\ & x+1 \le e^x \\ \\ & \ln x \le x-1 \\ \\ & \frac{1}{1+x} < \ln (1+\frac{1}{x}) < \frac{1}{x} ~~~~~(x>0) \end{aligned}

    三角函数公式

    诱导公式

    sin(α)=sinαcos(α)=cosαsin(π2α)=cosαcos(π2α)=sinαsin(π2+α)=cosαcos(π2+α)=sinαsin(πα)=sinαcos(πα)=cosαsin(π+α)=sinαcos(π+α)=cosα  kπ2  kαsin(cos)(kπ2±α) \begin{aligned} & \sin (-\alpha) = -\sin \alpha \\ \\ & \cos (-\alpha) = \cos \alpha \\ \\ & \sin (\frac{\pi}{2} - \alpha) = \cos \alpha \\ \\ & \cos (\frac{\pi}{2} - \alpha) = \sin \alpha \\ \\ & \sin (\frac{\pi}{2} + \alpha) = \cos \alpha \\ \\ & \cos (\frac{\pi}{2} + \alpha) = - \sin \alpha \\ \\ & \sin (\pi - \alpha) = \sin \alpha \\ \\ & \cos (\pi - \alpha) = - \cos \alpha \\ \\ & \sin (\pi + \alpha) = - \sin \alpha \\ \\ & \cos (\pi + \alpha) = - \cos \alpha \\ \\ \\ & 奇变偶不变,符号看象限 \\ & 奇指 ~~ k\cdot\frac{\pi}{2} ~~ 中 k \\ & 看象限是指:将 \alpha 看成锐角,然后看 \sin_{_{(变之前的,或cos)}}(k\cdot\frac{\pi}{2} \pm \alpha) 的符号 \end{aligned}

    平方关系

    1+tan2α=sec2α1+cot2α=csc2αsin2α+cos2α=1 \begin{aligned} & \\ & 1 + \tan ^2 \alpha = \sec^2 \alpha \\ \\ & 1 + \cot^2 \alpha = \csc ^2 \alpha \\ \\ & \sin^2 \alpha + \cos ^2 \alpha = 1 \end{aligned}

    两角和与差的三角函数

    sin(α+β)=sinαcosβ+cosαsinβcos(α+β)=cosαcosβsinαsinβsin(αβ)=sinαcosβcosαsinβcos(αβ)=cosαcosβ+sinαcosβtan(α+β)=tanα+tanβ1tanαtanβtan(αβ)=tanαtanβ1+tanαtanβ \begin{aligned} & \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \\ & \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \\ & \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \\ \\ & \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \cos \beta \\ \\ & \tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{ 1- \tan \alpha \tan \beta} \\ \\ & \tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1+ \tan \alpha \tan \beta} \end{aligned}

    积化和差公式

    cosαcosβ=12[cos(α+β)+cos(αβ)]cosαsinβ=12[sin(α+β)sin(αβ)]sinαcosβ=12[sin(α+β)+sin(αβ)]sinαsinβ=12[cos(α+β)cos(αβ)] \begin{aligned} & \cos \alpha \cos \beta = \frac{1}{2} [\cos (\alpha + \beta) + cos (\alpha - \beta)] \\ \\ & \cos \alpha \sin \beta = \frac{1}{2} [\sin(\alpha + \beta) - \sin (\alpha - \beta)] \\ \\ & \sin \alpha \cos \beta = \frac {1}{2} [\sin (\alpha + \beta) + \sin (\alpha - \beta)] \\ \\ & \sin \alpha \sin \beta = - \frac{1}{2} [\cos (\alpha + \beta) - \cos (\alpha - \beta)] \end{aligned}
    记忆口诀:
    积化和差得和差,余弦在后要想加。异名函数取正弦,正弦相乘取负号。

    和差化积公式

    sinα+sinβ=2sinα+β2cosαβ2sinαsinβ=2cosα+β2sinαβ2cosα+cosβ=2cosα+β2cosαβ2cosαcosβ=2sinα+β2sinαβ2 \begin{aligned} & \sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2} \\ \\ & \sin \alpha - \sin \beta = 2 \cos \frac{\alpha + \beta}{2}\sin \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} \end{aligned}
    记忆口诀:
    正加正,正在前,正减正,余在前;余加余,余并肩;余减余,负正弦

    倍角公式

    sin2α=2sinαcosαcos2α=cos2αsin2α=12sin2α=2cos2α1sin3α=4sin3α+3sinαcos3α=4cos3α3cosαsin2α=1cos2α2cos2α=1+cos2α2tan2α=2tanα1tan2αcot2α=cot2α12cotα \begin{aligned} & \sin 2\alpha = 2\sin \alpha \cos \alpha \\ \\ & \cos 2\alpha = \cos ^2 \alpha - \sin ^2 \alpha = 1- 2\sin^2 \alpha = 2 \cos^2\alpha -1 \\ \\ & \sin 3\alpha = -4 \sin^3 \alpha + 3\sin \alpha \\ \\ & \cos 3 \alpha = 4\cos^3\alpha -3 \cos \alpha \\ \\ & \sin^2 \alpha = \frac{1-\cos 2\alpha}{2} \\ \\ & \cos^2 \alpha = \frac{1+\cos 2\alpha}{2} \\ \\ & \tan 2\alpha = \frac{2\tan \alpha}{1-\tan ^2\alpha} \\ \\ & \cot 2\alpha = \frac{\cot ^2 \alpha -1}{2\cot \alpha} \end{aligned}

    半角公式

    sin2α2=1cosα2cos2α2=1+cosα2sinα2=±1cosα2cosα2=±1+cosα2tanα2=1cosαsinα=sinα1+cosα=±1cosα1+cosαcotα2=sinα1cosα=1+cosαsinα=±1+cosα1cosα \begin{aligned} & \sin^2 \frac{\alpha}{2} = \frac{1- \cos \alpha}{2} \\ \\ & \cos^2 \frac{\alpha}{2} = \frac{1+\cos \alpha}{2} \\ \\ & \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1- \cos \alpha}{2}} \\ \\ & \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+ \cos \alpha}{2}} \\ \\ & \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}} \end{aligned}

    万能公式

    sinα=2tanα21+tan2α2cosα=1tan2α21+tan2α2 \begin{aligned} & \sin \alpha = \frac{2 \tan \frac{\alpha}{2}}{1+\tan^2 \frac{\alpha}{2}} \\ \\ & \cos \alpha = \frac{1- \tan^2 \frac{\alpha}{2}}{1+ \tan^2 \frac{\alpha}{2}} \end{aligned}

    其他公式

    1+sinα=(sinα2+cosα2)21sinα=(sinα2cosα2)2 \begin{aligned} & 1 + \sin \alpha = (\sin \frac{\alpha}{2} + \cos \frac{\alpha}{2}) ^2 \\ \\ & 1 - \sin \alpha = (\sin \frac{\alpha}{2} - \cos \frac{\alpha}{2}) ^2 \end{aligned}

    反三角函数恒等式

    arcsinx+arccosx=π2arctanx+arccot x=π2sin(arccosx)=1x2cos(arcsinx)=1x2sin(arcsinx)=xarcsin(sinx)=xcos(arccosx)=xarccos(cosx)=xarccos(x)=πarccosx \begin{aligned} & \arcsin x + \arccos x = \frac{\pi}{2} \\ \\ & \arctan x + arccot ~x= \frac{\pi}{2} \\ \\ & \sin(\arccos x) = \sqrt{1-x^2} \\ \\ & \cos(\arcsin x) = \sqrt{1- x^2} \\ \\ & \sin(\arcsin x) = x \\ \\ & \arcsin (\sin x) = x \\ \\ & \cos (\arccos x) = x \\ \\ & \arccos (\cos x) =x \\ \\ & \arccos (-x) = \pi - \arccos x \end{aligned}

    极限相关公式

    数列极限递推式

    an+1=f(an)f(x)>0{a2>a1    {an}a2<a1    {an}k(0,1)使f(x)k    an \begin{aligned} & a_{n+1} = f(a_n) \\\\ 结论一: & f'(x) > 0 , \begin{cases} a_2 > a_1 \implies \{ a_n \} \nearrow单调递增 \\ a_2 < a_1 \implies \{ a_n \} \searrow单调递减 \end{cases} \\\\ 结论二(压缩映像原理):& \exist k \in (0,1),使得 |f'(x)| \le k \implies {a_n} 收敛 \end{aligned}

    重要极限公式

    limx0+xαlnx=0     α>0limx0+xα(lnx)k=0     α>0k>0limx+xαeδx=0     α>0δ>0limx0sinxx=1      limϕ(x)0sinϕ(x)ϕ(x)=1     ϕ(x)0limx0(1+x)1x=e      limϕ(x)0(1+ϕ(x))1ϕ(x)=e     ϕ(x)0limnnn=1limnan=1   a>0 \begin{aligned} & \lim_{x \to 0^+} x^\alpha \ln x = 0 ~~~~~ 其中 \alpha >0 \\\\ & \lim_{x \to 0^+} x^\alpha (\ln x)^k = 0 ~~~~~ 其中 \alpha >0 ,k>0 \\\\ & \lim_{x \to +\infty} x^\alpha e^{-\delta x} = 0 ~~~~~ 其中 \alpha >0 ,\delta >0 \\\\ & \lim_{x\to 0} \frac{\sin x}{x} = 1 ~~ \implies \lim_{\phi (x) \to 0} \frac{\sin \phi (x)}{\phi (x)} =1 ~~~~~其中\phi (x) \neq 0 \\ \\ & \lim_{x \to 0} (1+x)^{\frac{1}{x}} = e ~~ \implies \lim_{\phi (x) \to 0} (1+\phi (x))^{\frac{1}{\phi (x)}} = e ~~~~~ 其中\phi (x) \neq 0 \\\\ & \lim_{n \to \infty} \sqrt[n]{n} = 1 \\\\ & \lim_{n \to \infty} \sqrt[n]{a} = 1 ~~~(常数a>0)\\\\ \end{aligned}

    常用等价无穷小

    x0sinxtanxarcsinxarctanx(ex1)ln(1+x)x  ,  1cosx12x2  ,(1+x)a1ax  ,  ax1xlna   (a>0,a1) \begin{aligned} & x \to 0 时,\\\\ & \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim (e^x - 1) \sim \ln(1+x) \sim x ~~,~~ 1- \cos x \sim \frac{1}{2} x^2 ~~, \\ \\ & (1+x)^a - 1 \sim ax ~~,~~ a^x - 1 \sim x\ln a ~~~(a>0,a\neq1) \end{aligned}

    1^∞ 型

    limuv=elim(u1)v      (limu=1,limv=1) \lim u^v = e^{\lim (u-1)v} ~~~~~~(其中 \lim u=1,\lim v=\infty,即 1^\infty 型)

    导数相关公式

    导数定义

    f(x0)=limΔx0f(x0+Δx)f(x0)Δxf(x0)=limxx0f(x)f(x0)xx0 \begin{aligned} & f'(x_0) = \lim\limits_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \\ \\ & f'(x_0) = \lim\limits_{x \to x_0} \frac{f(x) - f(x_0)}{x-x_0} \end{aligned}

    微分定义

    Δy=f(x0+Δx)f(x0)Δy=AΔx+o(Δx)AΔx=f(x0)Δx \begin{aligned} & \Delta y = f(x_0 + \Delta x) - f(x_0) \\ \\ & \Delta y = A\Delta x + o(\Delta x) \\ \\ & A\Delta x = f'(x_0) \Delta x \end{aligned}

    连续,可导及可微关系

    一元函数

    可微
    连续
    可导

    多元函数

    一阶偏导数连续
    可微
    连续
    可导

    导数四则运算

    [u(x)±v(x)]=u(x)±v(x)[u(x)v(x)]=u(x)v(x)+u(x)v(x)[u(x)v(x)w(x)]=u(x)v(x)w(x)+u(x)v(x)w(x)+u(x)v(x)w(x)[u(x)v(x)]=u(x)v(x)u(x)v(x)[v(x)]2 \begin{aligned} & [u(x) \pm v(x)]' = u'(x) \pm v'(x) \\ \\ & [u(x)v(x)]' = u'(x)v(x) + u(x)v'(x) \\ \\ & [u(x)v(x)w(x)]' = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x) \\ \\ & \begin{bmatrix}\frac{u(x)}{v(x)} \end{bmatrix}' = \frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2} \\ \\ \end{aligned}

    复合函数求导

    {f[g(x)]}=f[g(x)]g(x) \{ f[g(x)] \}' = f'[g(x)]g'(x)

    反函数求导

    y=f(x),x=φ(y)    φ(y)=1f(x)yx=dydx=1dxdy=1xyyxx=d2ydx2=d(dydx)dx=d(1xy)dx=d(1xy)dydydx=d(1xy)dy1xy=xyy(xy)3 \begin{aligned} & y = f(x), x = \varphi(y) \implies \varphi ' (y) = \frac{1}{f'(x)} \\ \\ & y'_x = \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{x'_y} \\ \\ & y^{''}_{xx} = \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac{d(\frac{1}{x'_y})}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{dy}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{1}{x'_y} = \frac{-x^{''}_{yy}}{(x'_y)^3} \end{aligned}

    参数方程求导

    {x=φ(t)y=ψ(t)dydx=dy/dtdx/dt=ψ(t)φ(t)d2ydx2=d(dydx)dx=d(dydx)/dtdx/dt=ψ(t)φ(t)ψ(t)φ(t)[φ(t)]3 \begin{aligned} & \begin{cases} x = \varphi (t) \\ y = \psi (t) \end{cases} \\\\ & \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi ' (t)}{\varphi ' (t)} \\ \\ & \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac {d(\frac{dy}{dx})/dt}{dx/dt} = \frac{\psi '' (t) \varphi '(t) - \psi '(t) \varphi '' (t) }{[\varphi ' (t)]^3} \end{aligned}

    变限积分求导公式

    F(x)=φ1(x)φ2(x)f(t)dt,F(x)=ddx[φ1(x)φ2(x)f(t)dt]=f[φ2(x)]φ2(x)f[φ1(x)]φ1(x) \begin{aligned} & 设 F(x) = \int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt, 则 \\ \\ & F'(x) = \frac{d}{dx}\begin{bmatrix}\int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt \end{bmatrix} = f[\varphi _2(x)]\varphi '_2(x) - f[\varphi_1(x)]\varphi '_1(x) \end{aligned}

    基本初等函数的导数公式(❤❤❤)

    (xa)=axa1     (a)(ax)=axlna(ex)=ex(logax)=1xlna     (a>0,a1)(lnx)=1x(sinx)=cosx(cosx)=sinx(arcsinx)=11x2(arccosx)=11x2(tanx)=sec2x(cotx)=csc2x(arctanx)=11+x2(arccot x)=11+x2(secx)=secxtanx(cscx)=cscxcotx[ln(x+x2+1)]=1x2+1[ln(x+x21)]=1x21 \begin{aligned} & (x^a)' = a x^{a-1} ~~~~~(a为常数) \\ \\ & (a^x)' = a^x \ln a \\ \\ & (e^x)' = e^x \\ \\ & (log_a x)' = \frac{1}{x \ln a} ~~~~~(a>0, a \ne 1) \\ \\ & (\ln x)' = \frac{1}{x} \\ \\ & (\sin x)' = \cos x \\ \\ & (\cos x)' = -\sin x \\ \\ & (\arcsin x)' = \frac{1}{\sqrt{1-x^2}} \\ \\ & (\arccos x)' = - \frac{1}{\sqrt{1-x^2}} \\ \\ & (\tan x)' = \sec ^2 x \\ \\ & (\cot x)' = - \csc^2 x \\ \\ & (\arctan x)' = \frac{1}{1+x^2} \\ \\ & (arccot ~ x)' = - \frac{1}{1+x^2} \\ \\ & (\sec x)' = \sec x \cdot \tan x \\ \\ & (\csc x)' = - \csc x \cdot \cot x \\ \\ & [\ln (x+\sqrt{x^2+1})]' = \frac{1}{\sqrt{x^2+1}} \\ \\ & [\ln (x+\sqrt{x^2-1})]' = \frac{1}{\sqrt{x^2-1}} \\ \\ \end{aligned}

    高阶导数的运算

    [u±v](n)=u(n)±v(n) [u \pm v ]^{(n)} = u^{(n)} \pm v^{(n)}
    (uv)(n)=u(n)v+Cn1u(n1)v+Cn2u(n2)v+...+Cnku(nk)v(k)+...+Cnn1uv(n1)+uv(n)=k=0nCnku(nk)v(k) \begin{aligned} (uv)^{(n)} & = u^{(n)}v + C_n^1 u^{(n-1)}v' + C_n^2 u^{(n-2)}v'' + ... + C_n^k u^{(n-k)}v^{(k)} + ... + C_n^{n-1} u'v^{(n-1)} + uv^{(n)} \\ & = \displaystyle\sum_{k=0}^n C_n^k u^{(n-k)}v^{(k)} \end{aligned}

    常用初等函数的n阶导数公式

    (ax)(n)=ax(lna)n(ex)(n)=ex(sinkx)(n)=knsin(kx+nπ2)(coskx)(n)=kncos(kx+nπ2)(lnx)(n)=(1)n1(n1)!xn     (x>0)[ln(1+x)](n)=(1)n1(n1)!(x+1)n     (x>1)[(x+x0)m](n)=m(m1)(m2)(mn+1)(x+x0)mn(1x+a)(n)=(1)nn!(x+a)n+1 \begin{aligned} & (a^x)^{(n)} = a^x (\ln a)^n \\ \\ & (e^x)^{(n)} = e^x \\ \\ & (\sin kx)^{(n)} = k^n \sin (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\cos kx)^{(n)} = k^n \cos (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\ln x) ^ {(n)} = (-1)^{n-1} \frac{(n-1)!}{x^n} ~~~~~(x>0) \\ \\ & [\ln(1+x)]^{(n)} = (-1)^{n-1} \frac{(n-1)!}{(x+1)^n} ~~~~~(x>-1) \\ \\ & [(x+x_0)^m]^{(n)} = m(m-1)(m-2)\cdotp\cdotp\cdotp\cdot (m-n+1)(x+x_0)^{m-n} \\ \\ & (\frac{1}{x+a})^{(n)} = \frac{(-1)^n \cdot n!}{(x+a)^{n+1}} \\ \\ \end{aligned}

    极值判别条件

    {1.  f(x)    {        2.  f(x)=0,f(x)0    {f(x)<0    f(x)>0    3.  f(x)f(n1)(x)=0f(n)(x)0n    {f(n)(x)<0    f(n)(x)>0     \begin{cases} 1.~~f'(x)左右异号 \implies \begin{cases} 左正右负 \implies 极大值 \\ 左负右正 \implies 极小值 \end{cases} \\ \\ 2.~~f'(x)=0, f''(x)\ne 0 \implies \begin{cases} f''(x) < 0 \implies 极大值 \\ f''(x)>0 \implies 极小值 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ,f^{(n)}(x) \ne 0, n为偶数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 极大值 \\ f^{(n)}(x) > 0 \implies 极小值 \end{cases} \end{cases}

    凹凸性判定

    1.{f(x1+x22)<f(x1)+f(x2)2    f(x1+x22)>f(x1)+f(x2)2    2.{f(x)>0    f(x)<0     \begin{aligned} 1.&\begin{cases} f(\frac{x_1+x_2}{2}) < \frac{f(x_1)+f(x_2)}{2} \implies 凹 \\\\ f(\frac{x_1+x_2}{2}) > \frac{f(x_1)+f(x_2)}{2} \implies 凸 \end{cases} \\\\ 2.&\begin{cases} f''(x) > 0 \implies 凹 \\\\ f''(x) < 0 \implies 凸 \end{cases} \end{aligned}

    拐点判别条件

    {1.  f(x)    {        2.  f(x)=0,f(x)0    {f(x)<0    f(x)>0    3.  f(x)f(n1)(x)=0f(n)(x)0n    {f(n)(x)<0    f(n)(x)>0     \begin{cases} 1.~~f''(x)左右异号 \implies \begin{cases} 左负右正 \implies 凸 \to 凹 \\ 左正右负 \implies 凹 \to 凸 \end{cases} \\ \\ 2.~~f''(x)=0, f'''(x)\ne 0 \implies \begin{cases} f'''(x) < 0 \implies 凹 \to 凸 \\ f'''(x)>0 \implies 凸 \to 凹 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ,f^{(n)}(x) \ne 0, n为奇数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 凹 \to 凸 \\ f^{(n)}(x) > 0 \implies 凸 \to 凹 \end{cases} \end{cases}

    斜渐近线

    limx+f(x)x=a    limx+(f(x)ax)=b    线y=ax+b \lim_{x \to +\infty} \frac{f(x)}{x} = a ~~~~\lim_{x \to + \infty}(f(x)-ax) = b \implies 斜渐近线为: y=ax+b

    曲率

         r=(1+y2)32y     K=1r=y(1+y2)32     (X(xy(1+y2)y2))2+(Y(y+1+y2y2))=((1+y2)32y)2 \begin{aligned} 密切圆半径 ~~~~~ & r = \frac{(1+y'^2)^{\frac{3}{2}}}{|y''|} \\ \\ 曲率 ~~~~~ &K = \frac{1}{r} = \frac{|y''|}{(1+y'^2)^{\frac{3}{2}}} \\ \\ 曲率圆 ~~~~~& (X-(x-\frac{y'(1+y'^2)}{y''^2}))^2 +(Y-(y+\frac{1+y'^2}{y''^2})) = (\frac{(1+y'^2)^{\frac{3}{2}}}{|y''|})^2 \end{aligned}

    积分相关公式

    定积分的精确定义

    abf(x)dx=limni=1nf(a+bani)ban01f(x)dx=limni=1nf(in)1n0kf(x)dx=limni=1knf(in)1nDf(x,y)dσ=limni=1nj=1nf(a+bani,c+dcnj)bandcn0101f(x,y)dxdy=limni=1nj=1nf(in,jn)1n2 \begin{aligned} & \int_a^b f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(a+\frac{b-a}{n}i)\frac{b-a}{n} \\ \\ \\ 常用:& \int_0^1 f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ & \int_0^k f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^{kn} f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ \\ 二重定积分精确定义:& \iint\limits_D f(x,y) d\sigma = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(a+\frac{b-a}{n}i, c+\frac{d-c}{n}j) \cdot \frac{b-a}{n} \cdot \frac{d-c}{n} \\ \\ \\ 常用:&\int_0^1 \int_0^1 f(x,y) dxdy = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(\frac{i}{n}, \frac{j}{n})\cdot \frac{1}{n^2} \end{aligned}