首页 思维导图 详情

高数I【上】思维导图

2024-07-16 19:11:22 36 举报
AI智能生成
该思维导图包含高等数学上册绝大部分知识点以及例题总结，还在不断完善中，后续会更新高等数学下册的相关内容，你还在等什么^-^
学习笔记
高等数学
大学生
作者其他创作
大纲/内容
不定积分
不定积分
原函数与不定积分
原函数：设<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>是定义在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上的函数，若存在函数<span class="equation-text" contenteditable="false" data-index="2" data-equation="F(x)"><span></span><span></span></span>使得对任意<span class="equation-text" contenteditable="false" data-index="3" data-equation="x\in I"><span></span><span></span></span>都有<span class="equation-text" contenteditable="false" data-index="4" data-equation="F'(x)=f(x)"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="5" data-equation="\mathrm{d}F(x)=f(x)\mathrm{d}x"><span></span><span></span></span>，则称<span class="equation-text" contenteditable="false" data-index="6" data-equation="F(x)"><span></span><span></span></span>是<span class="equation-text" contenteditable="false" data-index="7" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="8" data-equation="I"><span></span><span></span></span>上的一个原函数
原函数存在定理（连续性与原函数存在）：若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上连续，则若<span class="equation-text" data-index="2" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" data-index="3" data-equation="I"><span></span><span></span></span>上存在原函数
不定积分：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上的全体原函数称为<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>在的不定积分，写作<span class="equation-text" contenteditable="false" data-index="3" data-equation="\int f(x)\, \mathrm{d}x=F(x)+C"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="4" data-equation="C"><span></span><span></span></span>为积分常数
不定积分的性质
四则运算法则，注意<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int kf(x)\mathrm{d}x= k\int f(x)\mathrm{d}x"><span></span><span></span></span>
基本积分公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\alpha\,\mathrm{d}x=\alpha x+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\frac{1}{x}\,\mathrm{d}x=\ln |x|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int x^{\mu}\mathrm{d}x= \frac{x^{\mu + 1}}{\mu + 1}+C（\mu \neq-1）"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int a^x\mathrm{d}x=\frac{a^x}{\ln a}+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan&nbsp;{\frac{x}{a}}+C"><span></span><span></span></span>，特例<span class="equation-text" data-index="1" data-equation="\int\frac{1}{1+x^2}\,\mathrm{d}x=\arctan x+C， (a=1)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\frac{1}{x^2-a^2}\,\mathrm{d}x=\frac{1}{2a}\ln |{\frac{x-a}{x+a}}|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin&nbsp;{\frac{x}{a}}+C"><span></span><span></span></span>，特例<span class="equation-text" data-index="1" data-equation="\int \frac{1}{\sqrt{1-x^2}}\,\mathrm{d}x=\arcsin x+C，(a=1)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int&nbsp;\frac{1}{\sqrt{x^2+a^2}}\,\mathrm{d}x=\ln |x+\sqrt{x^2+a^2}|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int&nbsp;\frac{1}{\sqrt{x^2-a^2}}\,\mathrm{d}x=\ln&nbsp;|x+\sqrt{x^2-a^2}|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \sqrt{ a^2-x^2}\mathrm{d}x=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin {\frac{x}{a}}+C"><span></span><span></span></span>，特例<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int\sqrt{&nbsp;1-x^2}\mathrm{d}x=\frac{x}{2}\sqrt{1-x^2}+\frac{1}{2}\arcsin&nbsp;x+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\sqrt{&nbsp;1+x^2}\mathrm{d}x=\frac{x}{2}\sqrt{1+x^2}+\frac{1}{2}\ln(x+\sqrt{1+x^2})+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \tan x\mathrm{d}x=-\ln|\cos x|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\cot x\mathrm{d}x=\ln|\sin&nbsp;x|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\sec x\mathrm{d}x=\ln|\sec&nbsp;x+\tan x|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\csc&nbsp;x\mathrm{d}x=-\ln|\csc&nbsp;x-\cot&nbsp;x|+C"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \ln x \ \mathrm{d}x=x\ln x-x+C"><span></span><span></span></span>
可积性与连续性
函数连续则必可积，但可积函数不一定是连续的
换元积分法
第一类换元法（凑微分法）
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(u)"><span></span><span></span></span>的原函数为<span class="equation-text" contenteditable="false" data-index="1" data-equation="F(u)"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="u=φ(x)"><span></span><span></span></span>可导，则有<span class="equation-text" contenteditable="false" data-index="3" data-equation="\int f[φ(x)]φ'(x)\,\mathrm{d}x=\int f[φ(x)]\mathrm{d}φ(x)=F[φ(x)]+C"><span></span><span></span></span>
第二类换元法
设函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=φ(t)"><span></span><span></span></span>可导，且<span class="equation-text" contenteditable="false" data-index="1" data-equation="φ'(t)\neq0"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="f[φ(t)]φ'(t)"><span></span><span></span></span>有原函数<span class="equation-text" contenteditable="false" data-index="3" data-equation="\phi(t)"><span></span><span></span></span>，则有<span class="equation-text" contenteditable="false" data-index="4" data-equation="\int f(x)\mathrm{d}x=[\int f[φ(t)]φ'(t)\mathrm{d} t ]=\phi[φ^{-1}(x)]+C"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="5" data-equation="t=φ^{-1}(x)"><span></span><span></span></span>
根式代换：例如令<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt{x}=t"><span></span><span></span></span>
三角代换：例如令<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=a\sin t \ (-\frac{\pi}{2}&lt;t&lt;\frac{\pi}{2})"><span></span><span></span></span>
倒代换：例如<span class="equation-text" contenteditable="false" data-index="0" data-equation="x= \frac{1}{t}"><span></span><span></span></span>
分部积分法
设函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="u(x),v(x)"><span></span><span></span></span>有连续导数，则有<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int u(x)v'(x)\mathrm{d}x=u(x)v(x)-\int v(x)u'(x)\mathrm{d}x"><span></span><span></span></span>，称为分部积分公式，写作<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int u \,\mathrm{d}v=uv-\int v\,\mathrm{d}u"><span></span><span></span></span>
递推法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \cos^nx\mathrm{d}x=\frac{1}{n}\sin x\cos^{n-1}x+\frac{n-1}{n}\int \cos^{n-2}x\mathrm{d}x"><span></span><span></span></span>；<span class="equation-text" contenteditable="false" data-index="1" data-equation="\ \ \ \ \ \ \ \ \ \ \ \ \ \int \sin^nx\mathrm{d}x=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int \sin^{n-2}x\mathrm{d}x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="n\in N^+"><span></span><span></span></span>
有理函数及三角函数有理式的积分
有理函数的积分
有理函数：即两个多项式的商所表示的函数，形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{P(x)}{Q(x)}=\frac{a_nx^n+a_{n-1}x^{n-1}+·······+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+······+b_1x+b_0}"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="m,n"><span></span><span></span></span>为正整数，当<span class="equation-text" contenteditable="false" data-index="2" data-equation="n&lt;m"><span></span><span></span></span>时为真分式；当<span class="equation-text" contenteditable="false" data-index="3" data-equation="n\geq m"><span></span><span></span></span>时为假分式
部分分式：以下有理真分式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A}{x-a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A}{(x-a)^n}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{Mx+N}{x^2+px+q}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{Mx+N}{(x^2+px+q)^n}"><span></span><span></span></span>
任意一个有理真分式都可以分解为若干个部分分式之和
待定系数法
赋值法
注意：若出现形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\frac{1}{x^2+px+q}"><span></span><span></span></span>的式子，应利用公式<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int\frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan&nbsp;{\frac{x}{a}}+C"><span></span><span></span></span>
三角函数有理式积分
三角函数有理式记作<span class="equation-text" contenteditable="false" data-index="0" data-equation="R(\sin x ,\cos x)"><span></span><span></span></span>，此类积分可通过万能代换<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=\tan \frac{x}{2}"><span></span><span></span></span>转换为<span class="equation-text" contenteditable="false" data-index="2" data-equation="u"><span></span><span></span></span>的积分，当然，当求<span class="equation-text" contenteditable="false" data-index="3" data-equation="\sin ^2x，\cos^2x，\sin x\cos x"><span></span><span></span></span>此类平方和二倍角时可令<span class="equation-text" contenteditable="false" data-index="4" data-equation="u=\tan x"><span></span><span></span></span>
对形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int&nbsp;\sin^{\alpha}x&nbsp;\cos&nbsp;^{\beta}x"><span></span><span></span></span>，解法如图：
定积分
定积分
定义
设函数<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="[a,b]" contenteditable="false"><span></span><span></span></span>上有界，将区间<span class="equation-text" data-index="2" data-equation="[a,b]" contenteditable="false"><span></span><span></span></span>分割为<span class="equation-text" data-index="3" data-equation="n" contenteditable="false"><span></span><span></span></span>个小区间<span class="equation-text" data-index="4" data-equation="[x_{i-1},x_i]" contenteditable="false"><span></span><span></span></span>，小区间长度为<span class="equation-text" data-index="5" data-equation="Δ_i=x_{i}-x_{i-1}" contenteditable="false"><span></span><span></span></span>，则有<span class="equation-text" data-index="6" data-equation="\int_{a}^{b} f(x)\, \mathrm{d}x=\lim_{\lambda \to \ 0}\sum_{i=1}^{n}f(ξ_i) Δ_i" contenteditable="false"><span></span><span></span></span>，称<span class="equation-text" data-index="7" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="8" data-equation="[a,b]"><span></span><span></span></span>上可积，其中<span class="equation-text" data-index="9" data-equation="a,b" contenteditable="false"><span></span><span></span></span>为积分下限和积分上限
默认<span class="equation-text" contenteditable="false" data-index="0" data-equation="a&lt;b"><span></span><span></span></span>，若<span class="equation-text" contenteditable="false" data-index="1" data-equation="a&gt;b"><span></span><span></span></span>，则有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int_a^bf(x)\mathrm{d}x=-\int_b^af(x)\mathrm{d}x"><span></span><span></span></span>，特殊情况：当<span class="equation-text" contenteditable="false" data-index="3" data-equation="a=b"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="4" data-equation="\int _a^af(x)\mathrm{d}x=0"><span></span><span></span></span>
性质
线性运算性质
保号性
估值定理：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上可积，且<span class="equation-text" contenteditable="false" data-index="2" data-equation="m&lt;f(x)&lt;M"><span></span><span></span></span>，有<span class="equation-text" contenteditable="false" data-index="3" data-equation="m(b-a)\leq\int_a^bf(x)\mathrm{d}x\leq M(b-a)"><span></span><span></span></span>
区间上的可加性：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{a}^{b}&nbsp;f(x)\,&nbsp;\mathrm{d}x=\int_{a}^{c}&nbsp;f(x)\,&nbsp;\mathrm{d}x+\int_{c}^{b}&nbsp;f(x)\,&nbsp;\mathrm{d}x"><span></span><span></span></span>
积分中值定理：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，存在一点<span class="equation-text" contenteditable="false" data-index="2" data-equation="ξ\in[a,b]"><span></span><span></span></span>，使得<span class="equation-text" contenteditable="false" data-index="3" data-equation="\int_a^bf(x)\mathrm{d}x=f(ξ)(b-a)"><span></span><span></span></span>
意义
定积分是一个数值，仅与被积函数和积分区间相关
几何意义：
定积分变上限的函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Φ(x)=\int_a^xf(t)\mathrm{d}t"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="x\in[a,b]"><span></span><span></span></span>
定积分变上限函数的导数<br>
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，则有<span class="equation-text" contenteditable="false" data-index="2" data-equation="Φ'(x)=\frac{\mathrm{d}}{\mathrm{d}x}\int_a^xf(t)\mathrm{d}t=f(x)"><span></span><span></span></span>
特殊情况：若<span class="equation-text" contenteditable="false" data-index="0" data-equation="a(x),b(x)"><span></span><span></span></span>可导，则有<span class="equation-text" contenteditable="false" data-index="1" data-equation="F'(x)=\frac{\mathrm{d}}{\mathrm{d}x}\int _{a(x)}^{b(x)}f(t)\mathrm{d}t= f[b(x)]b'(x)-f[a(x)]a'(x)"><span></span><span></span></span>
原函数存在定理
定积分变上限的函数是被积函数的一个原函数，即<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x)= \int f(x) \mathrm{d}x=\int_a^xf(t)\mathrm{d}t+C"><span></span><span></span></span>
定积分变限函数的导数
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>连续，则<span class="equation-text" data-index="1" data-equation="(\int_{v(x)}^{u(x)}f(t)\mathrm{d}t)'=f[u(x)]u'(x)-f[v(x)]v'(x)" contenteditable="false"><span></span><span></span></span>
微积分基本公式（牛顿-莱布尼茨公式）
若<span class="equation-text" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，则有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int_a^bf(x)\mathrm{d}x=F(b)-F(a)=F(x)|_a^b=[F(x)]_a^b"><span></span><span></span></span>
定积分的换元法与分部积分法
换元法
定积分的换元公式：若<span class="equation-text" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int_a^bf(x)\mathrm{d}x=\int_{\alpha}^{\beta}f[φ(t)]φ'(t)\mathrm{d}t"><span></span><span></span></span>
分部积分法
与不定积分的分部积分类似：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b&nbsp;u&nbsp;\,\mathrm{d}v=uv|_a^b-\int_a^b&nbsp;v\,\mathrm{d}u"><span></span><span></span></span>
奇偶函数的积分性质
被积函数为奇函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{-a}^af(x)\mathrm{d}x=0"><span></span><span></span></span>
被积函数为偶函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{-a}^af(x)\mathrm{d}x=2\int_0^af(x)\mathrm{d}x"><span></span><span></span></span>
反常积分（广义积分）
无穷积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="p"><span></span><span></span></span>积分：<span class="equation-text" data-index="1" data-equation="\int_a^{+\infty}\frac{1}{x^p}\mathrm{d}x\begin{cases}p&gt;1,收敛 \\p \leq1,发散\\\end{cases}" contenteditable="false"><span></span><span></span></span>
瑕积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="q"><span></span><span></span></span>积分：<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int_a^b\frac{1}{(x-a)^q}\mathrm{d}x\begin{cases}q<1,收敛\\q\geq1,发散\\\end{cases}"><span></span><span></span></span>​​​​​
反常积分的审敛法
先判断有几处瑕点，一次处理一个瑕点
对被积函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>进行等价无穷小和乘以非零常数，<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>的敛散性不变
伽马函数：含参变量反常积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="Γ(\alpha)=\int_0^{+\infty}x^{\alpha-1}e^{-x}\mathrm{d}x \, \,\, (a&gt;0)"><span></span><span></span></span>，该积分收敛，特别地当<span class="equation-text" contenteditable="false" data-index="1" data-equation="\alpha=n-1"><span></span><span></span></span>时，有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int_0^{+\infty}x^ne^{-x}\mathrm{d}x=n!"><span></span><span></span></span>
定积分的应用
定积分的元素法（微元法）
设整体量为<span class="equation-text" contenteditable="false" data-index="0" data-equation="U"><span></span><span></span></span>，部分量为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\mathrm{d}U"><span></span><span></span></span>，则有<span class="equation-text" contenteditable="false" data-index="2" data-equation="U=\sum_{i=1}^n\mathrm{d}U=\int_a^b|f(x)|\mathrm{d}x"><span></span><span></span></span>
平面图形的面积和立体体积
平面图形的面积
直角坐标系下
面积<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\int_a^b[y_{上}(x)-y_{下}(x)]\mathrm{d}x"><span></span><span></span></span>
参数方程下
参数方程<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=φ(t)&nbsp;\\y=\psi(t)\end{cases}，A=\int_{a}^{b}y\mathrm{d}x=\int_{\alpha}^{\beta}\psi(t)φ'(t)\mathrm{d}t"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="a=φ(\alpha)"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="b=φ(\beta)"><span></span><span></span></span>
极坐标系下
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\int_{\alpha}^{\beta}\frac{1}{2}φ^2(\theta)\mathrm{d}\theta"><span></span><span></span></span>，其中极坐标方程为<span class="equation-text" contenteditable="false" data-index="1" data-equation="r=φ(\theta)"><span></span><span></span></span>
笛卡尔心形线：<span class="equation-text" contenteditable="false" data-index="0" data-equation="ρ=a(1+\cos θ&nbsp;)"><span></span><span></span></span>
三叶玫瑰线：<span class="equation-text" contenteditable="false" data-index="0" data-equation="ρ=a\sin3θ&nbsp;"><span></span><span></span></span>
伯努利双纽线：<span class="equation-text" contenteditable="false" data-index="0" data-equation="r^2=a^2\cos2θ&nbsp;"><span></span><span></span></span>
立体的体积
已知截面面积的立体体积
立体体积<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b A(x)\mathrm{d}x"><span></span><span></span></span>，截面面积<span class="equation-text" contenteditable="false" data-index="1" data-equation="A(x)"><span></span><span></span></span>为连续函数
旋转体的体积
曲边图形绕<span class="equation-text" contenteditable="false" data-index="0" data-equation="x"><span></span><span></span></span>轴旋转：<span class="equation-text" contenteditable="false" data-index="1" data-equation="V=\int_a^b\pi f^2(x)\mathrm{d}x"><span></span><span></span></span>
曲边图形绕<span class="equation-text" contenteditable="false" data-index="0" data-equation="y"><span></span><span></span></span>轴旋转：相应地有<span class="equation-text" contenteditable="false" data-index="1" data-equation="V=\int _c^d\pi [φ(y)]^2\mathrm{d}y= \int_a^b2\pi ·xf(x)\mathrm{d}x"><span></span><span></span></span>
平面曲线的弧长与曲率
平面曲线弧长计算原理
利用曲线的内接折线长的极限来定义平面曲线的弧长<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mathrm{d}s=\sqrt{(\mathrm{d}x)^2+(\mathrm{d}y)^2} \,，s=\int_a^b\sqrt{(\mathrm{d}x)^2+(\mathrm{d}y)^2}=\int ^{b}_{a}\sqrt{1+(y')^2}\mathrm{d}x"><span></span><span></span></span>
平面曲线弧长计算
直角坐标方程情况
设曲线<span class="equation-text" data-index="0" data-equation="\overset{\frown} {AB}" contenteditable="false"><span></span><span></span></span>的方程为<span class="equation-text" data-index="1" data-equation="y=f(x)" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="2" data-equation="a\leq x\leq b" contenteditable="false"><span></span><span></span></span>，若<span class="equation-text" data-index="3" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="4" data-equation="[a,b]" contenteditable="false"><span></span><span></span></span>上有连续的一阶导数，则<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overset{\frown} {AB}"><span></span><span></span></span>的弧长为<span class="equation-text" contenteditable="false" data-index="6" data-equation="s=\int _{a}^{b}\sqrt{1+(y')^2}\mathrm{d}x"><span></span><span></span></span>
参数方程情况
平面曲线参数方程为<span class="equation-text" data-index="0" data-equation="\begin{cases}x=φ(t)&nbsp;\\y=\psi(t)\end{cases}，t\in[\alpha,\beta]"><span></span><span></span></span>，则曲线弧长为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int_{\alpha}^{\beta}\sqrt{[φ'(t)]^2+[\psi'(t)]^2&nbsp;}\mathrm{d}t=\int_{\alpha}^{\beta}\sqrt{(x')^2+(y')^2}\mathrm{d}t"><span></span><span></span></span>
极坐标方程情况
曲线极坐标方程为<span class="equation-text" contenteditable="false" data-index="0" data-equation="r=φ(\theta)，\theta\in[\alpha,\beta]"><span></span><span></span></span>，转换为参数方程<span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{cases}x=r\cos \theta\\y=r \sin \theta\end{cases}"><span></span><span></span></span>，曲线弧长即为<span class="equation-text" contenteditable="false" data-index="2" data-equation="\int_{\alpha}^{\beta}\sqrt{(x')^2+(y')^2}\mathrm{d}\theta=\int_{\alpha}^{\beta}\sqrt{r^2+(r')^2}\mathrm{d}\theta"><span></span><span></span></span>
平面曲线弧长的曲率
曲率概念（适用于直角坐标系和参数方程）
曲率是用来度量曲线弯曲程度的量，则有曲率<span class="equation-text" contenteditable="false" data-index="0" data-equation="k=|\frac{\mathrm{d}\alpha}{\mathrm{d}s}|=\frac{|y''|}{[1+(y')^2]^{\frac{3}{2}}}"><span></span><span></span></span>
曲率半径
曲率半径<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho=\frac{1}{k}"><span></span><span></span></span>
曲率圆
在曲线上找一点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M"><span></span><span></span></span>，过点<span class="equation-text" contenteditable="false" data-index="1" data-equation="M"><span></span><span></span></span>作曲线的法线，在曲线凹侧法线上取点<span class="equation-text" contenteditable="false" data-index="2" data-equation="D"><span></span><span></span></span>，使得<span class="equation-text" contenteditable="false" data-index="3" data-equation="|DM|=\rho=\frac{1}{k}"><span></span><span></span></span>，以<span class="equation-text" contenteditable="false" data-index="4" data-equation="D"><span></span><span></span></span>为圆心，<span class="equation-text" contenteditable="false" data-index="5" data-equation="\rho"><span></span><span></span></span>为半径作圆，则该圆为曲线在点<span class="equation-text" contenteditable="false" data-index="6" data-equation="M"><span></span><span></span></span>的曲率圆
曲率中心
即曲率圆的圆心点<span class="equation-text" contenteditable="false" data-index="0" data-equation="D"><span></span><span></span></span>
旋转曲面的面积
将由平面曲线旋转得到的旋转曲面的侧面积化为微圆柱体求其侧面积，面积元素为<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="\mathrm{d}A=2\pi y\mathrm{d}s"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="\mathrm{d}s"><span></span><span></span></span>为弧微分，故旋转曲面的面积为<span class="equation-text" contenteditable="false" data-index="2" data-equation="A=2\pi \int_a^by\sqrt{1+(y')^2}\mathrm{d}x"><span></span><span></span></span>
空间解析几何与向量代数
空间直角坐标系
向量模
子主题
两点间的距离公式
空间两点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M_1(x_1,y_1,z_1)"><span></span><span></span></span>和<span class="equation-text" contenteditable="false" data-index="1" data-equation="M_2(x_2,y_2,z_2)"><span></span><span></span></span>之间的距离为<span class="equation-text" contenteditable="false" data-index="2" data-equation="d=|M_1M_2|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}"><span></span><span></span></span>
曲面与空间曲线的一般方程
曲面的一般方程
在空间直角坐标系中，若曲面<span class="equation-text" contenteditable="false" data-index="0" data-equation="S"><span></span><span></span></span>上任意一点的坐标都满足三元方程<span class="equation-text" contenteditable="false" data-index="1" data-equation="F(x,y,z)=0"><span></span><span></span></span>，且相应的坐标满足该方程的点也一定在曲面<span class="equation-text" contenteditable="false" data-index="2" data-equation="S"><span></span><span></span></span>上，则该方程称为曲面<span class="equation-text" contenteditable="false" data-index="3" data-equation="S"><span></span><span></span></span>的一般方程
空间曲线的一般方程
空间曲线可以看作两个曲面的交线，设两曲面<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_1,S_2"><span></span><span></span></span>相交于一条空间曲线<span class="equation-text" contenteditable="false" data-index="1" data-equation="C"><span></span><span></span></span>，则其上的坐标必定满足<span class="equation-text" contenteditable="false" data-index="2" data-equation="\begin{cases}F(x,y,z)=0\\G(x,y,z)=0\end{cases}"><span></span><span></span></span>，且相应的满足该方程组的点必定在这条曲线上，则称方程组为空间曲线<span class="equation-text" contenteditable="false" data-index="3" data-equation="C"><span></span><span></span></span>的一般方程
平面
法线向量
点法式
一个点和一个法线向量确定一个平面
点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M_0(x_0,y_0,z_0)"><span></span><span></span></span>，​法线向量​<span class="equation-text" contenteditable="false" data-index="1" data-equation="n=(A,B,C)"><span></span><span></span></span>，设面内任意一点​<span class="equation-text" contenteditable="false" data-index="2" data-equation="M(x,y,z)"><span></span><span></span></span>，​<span class="equation-text" contenteditable="false" data-index="3" data-equation="n·M_0M=0"><span></span><span></span></span>，两向量垂直​​，点法式方程​<span class="equation-text" contenteditable="false" data-index="4" data-equation="A(x-x_0)+B(y-y_0)+C(z-z_0)=0"><span></span><span></span></span>​
两平面的夹角
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\leq\theta\leq\frac{\pi}{2}"><span></span><span></span></span>​​
通过两平面的法向量的夹角求解
点到平面的距离公式
平面外一点<span class="equation-text" contenteditable="false" data-index="0" data-equation="P_0(x_0,y_0,z_0)"><span></span><span></span></span>，​平面<span class="equation-text" contenteditable="false" data-index="1" data-equation="Ax+By+Cz+D=0"><span></span><span></span></span>，&nbsp; &nbsp; &nbsp; &nbsp; 距离​<span class="equation-text" contenteditable="false" data-index="2" data-equation="d=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A_2+B^2+C^2}}"><span></span><span></span></span>​​​​
空间直线及其方程
一般式
两平面相交<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\end{cases}"><span></span><span></span></span>​​​
对称式
方向向量
一个 向量与该直线平行
直线过点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M_0(x_0,y_0,z_0)"><span></span><span></span></span>，方向向量​<span class="equation-text" contenteditable="false" data-index="1" data-equation="s=(m,n,p)"><span></span><span></span></span>​，&nbsp;<span class="equation-text" contenteditable="false" data-index="2" data-equation="\frac{x-x_0}{m}=\frac{y-y_0}{n}=\frac{z-z_0}{p}"><span></span><span></span></span>​​​
参数式
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x-x_0}{m}=\frac{y-y_0}{n}=\frac{z-z_0}{p}=t"><span></span><span></span></span>，则​​<span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{cases}x=x_0+mt\\y=y_0+nt\\z=z_0+pt\end{cases}"><span></span><span></span></span>
两直线的夹角
两方向向量的夹角，<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\leq\theta\leq\frac{\pi}{2}"><span></span><span></span></span>​
直线与平面的夹角
即直线与投影的夹角，<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\leq\theta\leq\frac{\pi}{2}"><span></span><span></span></span>​，通过直线向量的方向向量和平面的法向量的夹角
球面
设球面球心为点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M_0(x_0,y_0,z_0)"><span></span><span></span></span>，半径为<span class="equation-text" contenteditable="false" data-index="1" data-equation="R"><span></span><span></span></span>，点<span class="equation-text" contenteditable="false" data-index="2" data-equation="M(x,y,z)"><span></span><span></span></span>在球面上，则该球面的方程为<span class="equation-text" contenteditable="false" data-index="3" data-equation="(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2"><span></span><span></span></span>
一般地，形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="A(x^2+y^2+z^2)+Dx+Ey+Fz+G=0"><span></span><span></span></span>的三元二次方程表示的图形可能为一个球面、一个点、或者一个虚球面
柱面
平行于定直线<span class="equation-text" contenteditable="false" data-index="0" data-equation="L"><span></span><span></span></span>，并沿定曲线<span class="equation-text" contenteditable="false" data-index="1" data-equation="C"><span></span><span></span></span>移动的直线<span class="equation-text" contenteditable="false" data-index="2" data-equation="l"><span></span><span></span></span>形成的曲面称为柱面，<span class="equation-text" contenteditable="false" data-index="3" data-equation="C"><span></span><span></span></span>称为柱面的准线，<span class="equation-text" contenteditable="false" data-index="4" data-equation="L"><span></span><span></span></span>称为柱面的母线
对于准线在坐标面上，且母线平行于坐标轴的柱面，设该柱面<span class="equation-text" contenteditable="false" data-index="0" data-equation="S"><span></span><span></span></span>方程为<span class="equation-text" contenteditable="false" data-index="1" data-equation="F(x,y)=0"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="2" data-equation="F(y,z)=0"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="3" data-equation="F(z,x)=0"><span></span><span></span></span>
二次柱面
抛物柱面
圆柱面
椭圆柱面
双曲柱面
旋转曲面
母线，旋转轴
圆锥面
双曲面
二次曲面
子主题
空间曲线与曲面的参数方程
空间曲线的参数方程
两种曲线方程的互化
曲面的参数方程
点的柱面坐标和球面坐标
投影柱面和投影曲线
向量的概念和运算
概念
定义
模
平行
负向量
零向量
方向角与方向余弦
运算
加法
三角形法则，平行四边形法则
数乘
投影
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Prj_u \ r=|r|\cos \theta"><span></span><span></span></span>​
数量积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a·b=|a||b|\cos \theta \\ \ \ \ \ \ \ \ =|a|Prj_a\ b=|b|Prj_b \ a"><span></span><span></span></span>​
向量积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|a×b|=|a||b|\sin \theta"><span></span><span></span></span>
向量及向量运算的坐标表示
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a=(a_1,a_2,a_3)"><span></span><span></span></span>，​<span class="equation-text" contenteditable="false" data-index="1" data-equation="b=(b_1,b_2,b_3)"><span></span><span></span></span>，&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;​​内积<span class="equation-text" contenteditable="false" data-index="2" data-equation="a·b=(a_1b_1,a_2b_2,a_3b_3)"><span></span><span></span></span>​​​
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a×b=\begin{vmatrix}i&amp;j&amp;k\\a_1&amp;a_2&amp;a_3\\b_1&amp;b_2&amp;b_3\end{vmatrix}"><span></span><span></span></span>​
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\cos \theta=\frac{a·b}{|a||b|}"><span></span><span></span></span>
平面和直线的方程
平面方程
点到平面的距离
直线方程
线面间的夹角
点到直线的距离和直线与直线的距离
特殊公式
立方和/立方差&nbsp; ：&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^3+1=(x+1)(x^2-x+1) \ /\ x^3-1=(x-1)(x^2+x+1)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a^n-b^n=(a-b)(a^0b^{n-1}+a^1b^{n-2}+a^2b^{n-3}+……+a^{n-2}b^1+a^{n-1}b^0)"><span></span><span></span></span>
和差化积：<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin \ x-sin\ a=2sin\ ({x-a\over2})cos\ ({x+a\over 2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan \ {\pi x\over2 }=tan\ ({\pi \over 2}-{\pi&nbsp;\over2 }t)，t=1-x，x\longrightarrow1，t\longrightarrow0"><span></span><span></span></span>
常用不等关系：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x}{1+x}&lt;\ln(1+x)&lt;x，x&gt;0"><span></span><span></span></span>
三角函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\tan^2 \ x=\sec^2\ x-1/\cot^2 \ x=\csc^2\ x-1"><span></span><span></span></span>
糖水不等式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{a+m}{b+m}&gt;\frac{a}{b}"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="b&gt;a&gt;0"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="m&gt;0"><span></span><span></span></span>
扇形面积公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="S_扇=\frac{1}{2}lR=\frac{1}{2}\theta R^2"><span></span><span></span></span>
扇形弧长公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="l=\alpha R"><span></span><span></span></span>
函数的极限
收敛数列
<b>ε</b>-N定义
对任给的ε＞0，要使|<span class="equation-text" data-index="0" data-equation="a_n" contenteditable="false"><span></span><span></span></span>-<span class="equation-text" contenteditable="false" data-index="1" data-equation="a"><span></span><span></span></span>|=(……带n)&lt;<b>ε</b>
只要n&gt;(……带<b>ε</b>)
设N≥(……带<b>ε</b>）
得n&gt;N，证明极限是某个数
性质
唯一性：若{<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n"><span></span><span></span></span>}收敛，则极限唯一
有界性：若{<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_n"><span></span><span></span></span>}收敛，则必有界
夹逼准则：对原数列进行放缩，使得不等式左右两边式子的极限相等，则原数列的极限与两边相等
单调有界准则：单调递增数列有上界必有极限，单调递减数列有下界必有极限，有极限⇒数列收敛
证明单调性
作差法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="a_{n+1}-"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="a_n" contenteditable="false"><span></span><span></span></span>
相除法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\left(\frac{a_{n+1}}{a_n} \right)"><span></span><span></span></span>
证明有界
特殊数列极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n \to \infty}x^n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n \to \infty}(1+{1\over n})^n=e"><span></span><span></span></span>
函数极限
ε-X定义（适用于<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\to \infty"><span></span><span></span></span>或+∞或-∞）<br>
ε-δ定义（适用于<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\rightarrow x_0"><span></span><span></span></span>）
对任意的ε&gt;0，若有<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)-A|=……&lt;ε"><span></span><span></span></span>
则有<span class="equation-text" contenteditable="false" data-index="0" data-equation="|x-x_0|&lt;(……带\epsilon)"><span></span><span></span></span>
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="\delta=(……带\epsilon)"><span></span><span></span></span>，使得<span class="equation-text" contenteditable="false" data-index="1" data-equation="0&lt;|x-x_0|&lt;\delta"><span></span><span></span></span>
则有<span class="equation-text" data-index="0" data-equation="|f(x)-A|&lt;\epsilon" contenteditable="false"><span></span><span></span></span>，所以<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{x \to \ x_0}f(x)=A"><span></span><span></span></span>
性质
唯一性：若<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to \ x_0}f(x)"><span></span><span></span></span>存在，则极限唯一
局部有界性：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}f(x)" contenteditable="false"><span></span><span></span></span>存在，则在<span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span></span></span>的去心邻域Ů<span class="equation-text" contenteditable="false" data-index="2" data-equation="(x_0)"><span></span><span></span></span>，f(x)有界
夹逼准则：设<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}g(x)=\lim_{x \to \ x_0}h(x)=A" contenteditable="false"><span></span><span></span></span>，若存在<span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span></span></span>的去心邻域Ů<span class="equation-text" data-index="2" data-equation="(x_0,\delta_0)" contenteditable="false"><span></span><span></span></span>，使得对任意<span class="equation-text" data-index="3" data-equation="x\in" contenteditable="false"><span></span><span></span></span>&nbsp;Ů<span class="equation-text" data-index="4" data-equation="(x_0,\delta_0)" contenteditable="false"><span></span><span></span></span>，总有<span class="equation-text" data-index="5" data-equation="g(x)\leq f(x)\leq h(x)" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="6" data-equation="\lim_{x \to \ x_0}f(x)=A"><span></span><span></span></span>
重要函数极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to \ 0}{\sin\ x \over x}=1"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\lim_{x \to \infty}(1+{1\over x})^x=e" contenteditable="false"><span></span><span></span></span>或化为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{x&nbsp;\to&nbsp;\ 0}(1+x)^{1\over x}=e"><span></span><span></span></span>
求函数极限方法
相除法
因式分解相消
构造特殊极限法
换元法
有理化
等价无穷小及其性质
左右极限相等法
三角函数法
夹逼定理
构造法
对数法
海涅定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to a}f(x)=L\iff\forall x_n \to a \,,\lim_{n \to \infty}f(x_n)=L"><span></span><span></span></span>，其中数列<span class="equation-text" contenteditable="false" data-index="1" data-equation="{\{x_n\}}\subset f(x)"><span></span><span></span></span>的定义域，<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_n\not=a"><span></span><span></span></span>
无穷小量与无穷大量
无穷小量
性质
有限个无穷小的代数和仍为无穷小
有限个无穷小的乘积仍为无穷小
无穷小与有界函数的乘积仍为无穷小
无穷小量的阶
高阶无穷小：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}{\alpha(x)\over\beta(x)}=0" contenteditable="false"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="x\longrightarrow x_0" contenteditable="false"><span></span><span></span></span>时<span class="equation-text" data-index="2" data-equation="\alpha(x)" contenteditable="false"><span></span><span></span></span>是<span class="equation-text" contenteditable="false" data-index="3" data-equation="\beta(x)"><span></span><span></span></span>的高阶无穷小
同阶无穷小：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}{\alpha(x)\over\beta(x)}=c\not=0"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="x\longrightarrow x_0"><span></span><span></span></span>时<span class="equation-text" data-index="2" data-equation="\alpha(x)"><span></span><span></span></span>是<span class="equation-text" data-index="3" data-equation="\beta(x)"><span></span><span></span></span>的同阶无穷小
等价无穷小：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}{\alpha(x)\over\beta(x)}=1"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="x\longrightarrow x_0"><span></span><span></span></span>时<span class="equation-text" data-index="2" data-equation="\alpha(x)"><span></span><span></span></span>是<span class="equation-text" data-index="3" data-equation="\beta(x)"><span></span><span></span></span>的等价无穷小
k阶无穷小：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}{\alpha(x)\over\beta^k(x)}=c\not=0，k&gt;0"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="x\longrightarrow x_0"><span></span><span></span></span>时<span class="equation-text" data-index="2" data-equation="\alpha(x)"><span></span><span></span></span>是<span class="equation-text" data-index="3" data-equation="\beta(x)"><span></span><span></span></span>的k阶无穷小
常用等价无穷小
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\longrightarrow0"><span></span><span></span></span>时
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sin\ x\sim x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\tan\ x\sim x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1-\cos\ x\sim {1\over2 }x^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\arcsin\ x\sim x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\arctan\ x\sim x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln (1+x)\sim x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a^x-1\sim x\ln a"><span></span><span></span></span>，特例&nbsp;<span class="equation-text" data-index="1" data-equation="e^x-1\sim x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+x)^a-1\sim ax"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt[m]{x+1}-1\sim {x\over m}"><span></span><span></span></span>​
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\tan\ x-\sin\ x\sim \frac{1}{2}x^3"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x-\sin \ x\sim\frac{1}{6}x^3"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\tan \ x-x\sim\frac{1}{3}x^3"><span></span><span></span></span>
无穷大量
M-δ定义&nbsp; &nbsp;(适用于<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\to x_0"><span></span><span></span></span>)
M-X定义&nbsp; (适用于<span class="equation-text" contenteditable="false" data-index="0" data-equation="x \to \infty"><span></span><span></span></span>)
函数的连续性
连续性定义
定义一：若<span class="equation-text" data-index="0" data-equation="\lim_{x \to \ x_0}f (x)= \lim _{x \to x_0^+}f(x)=\lim _{x \to x_0^-}f(x)=f(x_0)"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="2" data-equation="x_0"><span></span><span></span></span>连续
定义二：若<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{Δx \to 0}Δy=\lim_{Δx \to 0}[f(x_0+Δx)-f(x_0)]=0"><span></span><span></span></span>，则称<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_0"><span></span><span></span></span>处连续
间断点
第一类间断点
可去间断点：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\lim_{x \to \ x_0} f (x)=A\neq f(x_0)"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="3" data-equation="f(x_0)"><span></span><span></span></span>不存在
跳跃间断点：<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span>存在左右极限但不相等，</span></span><span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x_0^-)\neq f(x_0^+)"><span></span><span></span></span>
第二类间断点
无穷间断点：<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span></span></span>有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\lim_{x \to \ x_0} f(x)= \infty"><span></span><span></span></span>
震荡间断点
复合函数的连续性
设函数<span class="equation-text" data-index="0" data-equation="u=φ(x)" contenteditable="false"><span></span><span></span></span>当<span class="equation-text" data-index="1" data-equation="x\to x_0" contenteditable="false"><span></span><span></span></span>时极限为<span class="equation-text" data-index="2" data-equation="a" contenteditable="false"><span></span><span></span></span>，且在<span class="equation-text" data-index="3" data-equation="x_0" contenteditable="false"><span></span><span></span></span>连续，函数<span class="equation-text" data-index="4" data-equation="y=f(u)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="5" data-equation="u=a" contenteditable="false"><span></span><span></span></span>连续，则有<span class="equation-text" contenteditable="false" data-index="6" data-equation="\lim_{x \to \ x_0}f[φ(x)]=f[\lim_{x \to \ x_0}φ(x)]=f[φ(x_0)]"><span></span><span></span></span>
即外函数为连续函数时，函数符号<span class="equation-text" contenteditable="false" data-index="0" data-equation="f"><span></span><span></span></span>与极限符号<span class="equation-text" data-index="1" data-equation="\lim_{x \to \ x_0}" contenteditable="false"><span></span><span></span></span>可以调换顺序
初等函数的连续性：一切初等函数在其定义区间内都连续
闭区间上的连续函数定理
最值定理
有界定理
介值定理
零点定理（介值定理的特殊情况）
开区间上的连续函数定理（闭区间推广）
有界性
零点定理
导数与微分
导数
定义
函数在某一点上的导数
当自变量<span class="equation-text" contenteditable="false" data-index="0" data-equation="x"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>处取得增量<span class="equation-text" contenteditable="false" data-index="2" data-equation="Δx"><span></span><span></span></span>时，若极限<span class="equation-text" contenteditable="false" data-index="3" data-equation="\lim_{Δx \to \ 0}\tfrac{f(x_0+Δx)-f(x_0)}{Δx}"><span></span><span></span></span>存在，则称函数<span class="equation-text" contenteditable="false" data-index="4" data-equation="y=f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="5" data-equation="x_0"><span></span><span></span></span>处可导，此处的导数记作<span class="equation-text" contenteditable="false" data-index="6" data-equation="f'(x_0)"><span></span><span></span></span>
导函数
设函数<span class="equation-text" data-index="0" data-equation="y=f(x)" contenteditable="false"><span></span><span></span></span>在开区间<span class="equation-text" data-index="1" data-equation="I" contenteditable="false"><span></span><span></span></span>内可导，有<span class="equation-text" data-index="2" data-equation="f'(x)=\lim_{Δx \to \ 0}\tfrac{f(x_0+Δx)-f(x_0)}{Δx}" contenteditable="false"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="3" data-equation="\lim_{x \to \ x_0}\frac{f(x)-f(x_0)}{x-x_0}"><span></span><span></span></span>
利用导函数的定义求函数在某一点处的导数
单侧导数
左导数和右导数称为单侧导数，有<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x)"><span></span><span></span></span>存在<span class="equation-text" contenteditable="false" data-index="1" data-equation="\Longleftrightarrow f_-'(x_0)=f_+'(x_0)"><span></span><span></span></span>
可导性与连续性
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x"><span></span><span></span></span>可导，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="3" data-equation="x"><span></span><span></span></span>连续
求导法则
基本初等函数求导
<span class="equation-text" data-index="0" data-equation="(x^\mu)'=\mu · x^{\mu-1}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a^x)'=a^x\ln a \ (a&gt;0 \ ,a\neq1)"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(\log_ax)'=\frac{1} {x\ln a} \ (a&gt;0 \ ,a\neq1)" contenteditable="false"><span></span><span></span></span>
三角函数导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\sin\ x)'=\cos \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\cos \ x)'=-\sin\ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\tan \ x)'=\sec^2 \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\cot \ x)'=-\csc^2 \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\csc \ x)'=-\csc\ x \cot \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\sec\ x)'=\sec\ x \tan\ x"><span></span><span></span></span>
反函数导数
反函数求导法则
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=f(y)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上严格单调、可导，则其反函数<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=f^{-1}(x)"><span></span><span></span></span>在相应的区间内也可导，即<span class="equation-text" contenteditable="false" data-index="3" data-equation="[f^{-1}(x)]'=\frac{1}{f'(y)}"><span></span><span></span></span>
反三角函数导数（需在特定区间内才成立）&nbsp;
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\arcsin\ x)'=\frac{1}{\sqrt{1-x^2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\arccos \ x)'=-\frac{1}{\sqrt{1-x^2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\arctan\ x)'=\frac{1}{1+x^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\ arccot \ x)'=-\frac{1}{1+x^2}"><span></span><span></span></span>
复合函数导数
链式法则
即<span class="equation-text" contenteditable="false" data-index="0" data-equation="[f(g(x))]'=f'(u)·g'(x)"><span></span><span></span></span>，其中<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=g(x)"><span></span><span></span></span>
幂指函数导数
幂指函数形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="u(x)^{v(x)}"><span></span><span></span></span>
双曲函数导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\sinh \ x)'=\cosh \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\cosh\ x)'=\sinh \ x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\tanh\ x)'=\frac{1}{(\cosh \ x)^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\coth \ x)'=-\frac{1}{(\sinh\ x)^2}"><span></span><span></span></span>
反双曲函数导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\ arcsinh\ x)'=\frac{1}{\sqrt{1+x^2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\ arccosh\ x)'=\frac{1}{\sqrt{x^2-1}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\ arctanh\ x)'=\frac{1}{1-x^2}=(\ arccoth \ x)'"><span></span><span></span></span>
高阶导数
二阶和二阶以上的导数统称高阶导数
n阶导数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>的<span class="equation-text" contenteditable="false" data-index="1" data-equation="n-1"><span></span><span></span></span>阶导函数<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{(n-1)}(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="3" data-equation="x"><span></span><span></span></span>的导数，称为<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="5" data-equation="x"><span></span><span></span></span>的<span class="equation-text" contenteditable="false" data-index="6" data-equation="n"><span></span><span></span></span>阶导数，记为<span class="equation-text" contenteditable="false" data-index="7" data-equation="f^{(n)}(x)"><span></span><span></span></span>
常用<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>阶导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=x^\mu"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y^{(n)}=(x^\mu)^{(n)}=\mu(\mu-1)······(\mu-n+1)x^{\mu-n}"><span></span><span></span></span>，特例<span class="equation-text" contenteditable="false" data-index="2" data-equation="(\frac{1}{x})^{(n)}=\frac{(-1)^nn!}{x^{n+1}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=a^x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y^{(n)}=(a^x)^{(n)}=a^x(\ln a)^n"><span></span><span></span></span>，特例<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=e^x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="3" data-equation="(e^x)^{(n)}=e^x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=\ln x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y^{(n)}=(\ln x)^{(n)}=(-1)^{n-1}\frac{(n-1)!}{x^n}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=\sin x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y^{(n)}=(\sin x)^{(n)}=\sin \ (x+n·\frac{\pi}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=\cos x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y^{(n)}=(\cos x)^{(n)}=\cos \ (x+n·\frac{\pi}{2})"><span></span><span></span></span>
替换自变量
已知<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(u)n"><span></span><span></span></span>阶可导，<span class="equation-text" contenteditable="false" data-index="1" data-equation="y=f(ax+b)"><span></span><span></span></span>的<span class="equation-text" contenteditable="false" data-index="2" data-equation="n"><span></span><span></span></span>阶导数<span class="equation-text" contenteditable="false" data-index="3" data-equation="y^{(n)}=a^n·f^{(n)}(ax+b)"><span></span><span></span></span>
莱布尼茨公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(u·v)^{(n)}=\sum_{k=0}^n C_n^ku^{(n-k)}v^{(k)}=C_n^0u^{(n)}v+C_n^1u^{(n-1)}v'+C_n^2u^{(n-2)}v''+······+C_n^ku^{(n-k)}v^{(k)}+····"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="··+uv^{(n)}"><span></span><span></span></span>
隐函数的导数
隐函数定义：相对于显函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>，形如<span class="equation-text" contenteditable="false" data-index="1" data-equation="F(x,y)=0"><span></span><span></span></span>的函数称为隐函数
隐函数求导法则：同时对方程两边x，y求导
对数求导法：先在等式两侧取对数，再对两侧同时求导
参数方程确定的导数
参数方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=φ(t) \\y=\psi(t)\end{cases},t\in[\alpha,\beta]"><span></span><span></span></span>
参数方程的求导法则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}=\frac{\psi'(t)}{φ'(t)}=f'(x)"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{d^2y}{dx^2}=\frac{[f'(x)]'}{φ'(t)}"><span></span><span></span></span>
函数的微分
定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="1" data-equation="x"><span></span><span></span></span>的微分(或函数的微分)为<span class="equation-text" contenteditable="false" data-index="2" data-equation="dy=f'(x)dx"><span></span><span></span></span>
微分的几何意义
导数与微分的关系
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>可微的充要条件是函数<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="3" data-equation="x_0"><span></span><span></span></span>可导，即<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="5" data-equation="x_0"><span></span><span></span></span>可导与可微等价
基本初等函数的微分公式
微分的运算法则
四则运算法则
复合函数的微分法则
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="u=g(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x"><span></span><span></span></span>可微，<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=f(u)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="3" data-equation="u"><span></span><span></span></span>可微，则复合函数<span class="equation-text" contenteditable="false" data-index="4" data-equation="y=f[g(x)]"><span></span><span></span></span>的微分为<span class="equation-text" contenteditable="false" data-index="5" data-equation="dy=f'(u)du=f'(u)g'(x)dx"><span></span><span></span></span>
一阶微分形式不变性：不管<span class="equation-text" contenteditable="false" data-index="0" data-equation="u"><span></span><span></span></span>是自变量还是中间变量（如<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=g(x)"><span></span><span></span></span>），对于<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=f(u)"><span></span><span></span></span>，函数的微分表达式都有形式不变性，即<span class="equation-text" contenteditable="false" data-index="3" data-equation="\mathrm{d}y=f'(u)\mathrm{d}u"><span></span><span></span></span>
微分的简单应用
函数的近似计算：若函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>处可微，则有<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)\approx f(x_0)+f'(x_0)(x-x_0)"><span></span><span></span></span>
导数的应用
微分中值定理
费马引理
设函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>的某邻域<span class="equation-text" contenteditable="false" data-index="2" data-equation="U(x_0)"><span></span><span></span></span>内有定义，并且在<span class="equation-text" contenteditable="false" data-index="3" data-equation="x_0"><span></span><span></span></span>处可导，则对任意的<span class="equation-text" contenteditable="false" data-index="4" data-equation="x\in U(x_0)"><span></span><span></span></span>，有<span class="equation-text" contenteditable="false" data-index="5" data-equation="f(x)\leq f(x_0)"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="6" data-equation="f(x)\geq f(x_0)"><span></span><span></span></span>，那么<span class="equation-text" contenteditable="false" data-index="7" data-equation="f'(x_0)=0"><span></span><span></span></span>
罗尔定理
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在闭区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，在开区间<span class="equation-text" contenteditable="false" data-index="2" data-equation="(a,b)"><span></span><span></span></span>上可导，且<span class="equation-text" contenteditable="false" data-index="3" data-equation="f(a)=f(b)"><span></span><span></span></span>，则在<br><span class="equation-text" contenteditable="false" data-index="4" data-equation="(a,b)"><span></span><span></span></span>内有一点<span class="equation-text" contenteditable="false" data-index="5" data-equation="ξ(a&lt;ξ&lt;b)"><span></span><span></span></span>，使得<span class="equation-text" contenteditable="false" data-index="6" data-equation="f'(ξ)=0"><span></span><span></span></span>
拉格朗日中值定理
函数<span class="equation-text" data-index="0" data-equation="f(x)"><span></span><span></span></span>在闭区间<span class="equation-text" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，在开区间<span class="equation-text" data-index="2" data-equation="(a,b)"><span></span><span></span></span>上可导，则在<span class="equation-text" data-index="3" data-equation="(a,b)"><span></span><span></span></span>内有一点<span class="equation-text" data-index="4" data-equation="ξ"><span></span><span></span></span>，使得<span class="equation-text" data-index="5" data-equation="\frac{f(b)-f(a)}{b-a}=f'(ξ)"><span></span><span></span></span>
利用拉格朗日中值定理证明不等式
利用不等式构造函数
求导，判别导数正负
利用导数正负确定函数单调性，产生大于小于
有限增量定理
设为<span class="equation-text" data-index="0" data-equation="[x,x+Δx]" contenteditable="false"><span></span><span>为</span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>的子区间，自然符合拉格朗日中值定理，所以有<span class="equation-text" contenteditable="false" data-index="2" data-equation="\frac{f(x+Δx)-f(x)}{Δx}=f'(x+\thetaΔx),\ (0&lt;\theta&lt;1)"><span></span><span></span></span>，若记<span class="equation-text" contenteditable="false" data-index="3" data-equation="Δy"><span></span><span></span></span>为<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x+Δx)-f(x)"><span></span><span></span></span>，则原式可改写为<span class="equation-text" contenteditable="false" data-index="5" data-equation="Δy=f'(x+\thetaΔx)Δx,(0&lt;\theta&lt;1)"><span></span><span></span></span>
柯西中值定理
函数<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在闭区间<span class="equation-text" data-index="1" data-equation="[a,b]" contenteditable="false"><span></span><span></span></span>上连续，在开区间<span class="equation-text" data-index="2" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>上可导，对任意<span class="equation-text" data-index="3" data-equation="x\in(a,b)" contenteditable="false"><span></span><span></span></span>都有<span class="equation-text" data-index="4" data-equation="F'(x)\not=0" contenteditable="false"><span></span><span></span></span>，则在<span class="equation-text" data-index="5" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内有一点<span class="equation-text" data-index="6" data-equation="ξ" contenteditable="false"><span></span><span></span></span>，使得<span class="equation-text" contenteditable="false" data-index="7" data-equation="\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f'(ξ)}{F'(ξ)}"><span></span><span></span></span>成立
利用柯西中值定理证明某一等式
设出辅助函数
确定辅助函数的导数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\not=0"><span></span><span></span></span>
按照柯西中值定理构造式子
将式子转化为需要证明的等式
函数单调性与曲线凹凸性
函数的单调性
导数法判定函数单调性
若在<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b)"><span></span><span></span></span>内<span class="equation-text" contenteditable="false" data-index="1" data-equation="f'(x)&lt;0"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="3" data-equation="[a,b]"><span></span><span></span></span>上严格单调递减
若在<span class="equation-text" data-index="0" data-equation="(a,b)"><span></span><span></span></span>内<span class="equation-text" data-index="1" data-equation="f'(x)&gt;0"><span></span><span></span></span>，则<span class="equation-text" data-index="2" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="3" data-equation="[a,b]"><span></span><span></span></span>上严格单调递增
曲线的凹凸性
定义
设<span class="equation-text" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" data-index="1" data-equation="I"><span></span><span></span></span>上连续，如果对<span class="equation-text" data-index="2" data-equation="I"><span></span><span></span></span>上任意两点<span class="equation-text" data-index="3" data-equation="x_1、x_2"><span></span><span></span></span>，恒有<span class="equation-text" data-index="4" data-equation="f(\frac{x_1+x_2}{2})&lt;\frac{f(x_1)+f(x_2)}{2}"><span></span><span></span></span>，则称<span class="equation-text" data-index="5" data-equation="f(x)"><span></span><span></span></span>的图形在<span class="equation-text" data-index="6" data-equation="I"><span></span><span></span></span>上是凹的
设<span class="equation-text" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" data-index="1" data-equation="I"><span></span><span></span></span>上连续，如果对<span class="equation-text" data-index="2" data-equation="I"><span></span><span></span></span>上任意两点<span class="equation-text" data-index="3" data-equation="x_1、x_2"><span></span><span></span></span>，恒有<span class="equation-text" data-index="4" data-equation="f(\frac{x_1+x_2}{2})&gt;\frac{f(x_1)+f(x_2)}{2}"><span></span><span></span></span>，则称<span class="equation-text" data-index="5" data-equation="f(x)"><span></span><span></span></span>的图形在<span class="equation-text" data-index="6" data-equation="I"><span></span><span></span></span>上是凸的
判定定理
若在<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b)"><span></span><span></span></span>内<span class="equation-text" contenteditable="false" data-index="1" data-equation="f''(x)&gt;0"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="3" data-equation="[a,b]"><span></span><span></span></span>上的图形是凹的
若在<span class="equation-text" data-index="0" data-equation="(a,b)"><span></span><span></span></span>内<span class="equation-text" data-index="1" data-equation="f''(x)&lt;0"><span></span><span></span></span>，则<span class="equation-text" data-index="2" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="3" data-equation="[a,b]"><span></span><span></span></span>上的图形是凸的
拐点：函数凹凸性发生变化的点，形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x,y)"><span></span><span></span></span>
函数的极值与最值
函数的极值
概念：使得<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x)=0"><span></span><span></span></span>的点称为驻点，驻点和导数不存在的点称为函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>的临界点
一阶导数判别法
<span style="font-size:inherit;">设</span><span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">是连续函数</span><span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">的一个临界点，且</span><span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">在某一邻域</span><span class="equation-text" contenteditable="false" data-index="3" data-equation="Ů(x_0，δ)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">上可导，则当</span><span class="equation-text" contenteditable="false" data-index="4" data-equation="x\in(x_0-\delta，x_0)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">，</span><span class="equation-text" contenteditable="false" data-index="5" data-equation="f'(x)&gt;0"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">；</span><span class="equation-text" contenteditable="false" data-index="6" data-equation="x\in(x_0，x_0+\delta)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">，</span><span class="equation-text" contenteditable="false" data-index="7" data-equation="f'(x)&lt;0"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">时，</span><span class="equation-text" contenteditable="false" data-index="8" data-equation="f(x)"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">在</span><span class="equation-text" contenteditable="false" data-index="9" data-equation="x_0"><span></span><span></span></span><span style="font-size:inherit; color:rgb(115, 92, 69); font-family:微软雅黑; text-align:left; font-style:normal; font-weight:normal; background-color:rgb(252, 249, 234); display:inline !important;">处取极大值&nbsp; &nbsp;&nbsp;</span><br>
条件同上，则当<span class="equation-text" data-index="0" data-equation="x\in(x_0-\delta，x_0)"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="f'(x)&lt;0"><span></span><span></span></span>；<span class="equation-text" data-index="2" data-equation="x\in(x_0，x_0+\delta)"><span></span><span></span></span>，<span class="equation-text" data-index="3" data-equation="f'(x)&gt;0"><span></span><span></span></span>时，<span class="equation-text" data-index="4" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" data-index="5" data-equation="x_0"><span></span><span></span></span>处取极小值
二阶导数判别法
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>为函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>的驻点，且<span class="equation-text" contenteditable="false" data-index="2" data-equation="f''(x)\neq0"><span></span><span></span></span>，则当<span class="equation-text" contenteditable="false" data-index="3" data-equation="f''(x_0)&lt;0"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x_0)"><span></span><span></span></span>取极大值
当<span class="equation-text" data-index="0" data-equation="f''(x_0)&gt;0"><span></span><span></span></span>时，<span class="equation-text" data-index="1" data-equation="f(x_0)"><span></span><span></span></span>取极小值
函数的最值
定义：连续函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>内仅有一个临界点<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_0"><span></span><span></span></span>，若<span class="equation-text" contenteditable="false" data-index="3" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="4" data-equation="x_0"><span></span><span></span></span>处取极小（大）值，则<span class="equation-text" contenteditable="false" data-index="5" data-equation="f(x_0)"><span></span><span></span></span>为最小（大）值
利用函数最值证明不等式
利用所要证的不等式构造函数，将证明不等式转化为证明函数最值问题
求导，利用单调性求出最值
回代，证明不等式成立
函数图像的描绘
函数的渐近线
铅直渐近线：若有<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to x^+_0}f(x)=\infty"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{x \to x_0^-}f(x)=\infty"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="x = x_0"><span></span><span></span></span>为曲线<span class="equation-text" contenteditable="false" data-index="3" data-equation="f(x)"><span></span><span></span></span>的一条铅直渐近线
水平渐近线：若有<span class="equation-text" data-index="0" data-equation="\lim_{x \to +\infty}f(x)=b" contenteditable="false"><span></span><span></span></span>或<span class="equation-text" data-index="1" data-equation="\lim_{x\to -\infty}f(x)= b" contenteditable="false"><span></span><span></span></span>（<span class="equation-text" contenteditable="false" data-index="2" data-equation="b"><span></span><span></span></span>为常数），则<span class="equation-text" contenteditable="false" data-index="3" data-equation="x= x_0"><span></span><span></span></span>为曲线<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x)"><span></span><span></span></span>的一条水平渐近线
斜渐近线
利用待定系数法先设出斜渐近线的直线方程，再代入求解
洛必达法则
定义：设<span class="equation-text" data-index="0" data-equation="\lim_{n \to \ a}f(x)=0/\infty，\lim_{n \to \ a}g(x)=0/\infty" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="f'(x),g'(x)" contenteditable="false"><span></span><span></span></span>都存在且<span class="equation-text" data-index="2" data-equation="g'(x)\neq0" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="3" data-equation="\lim_{n \to \ a}\frac{f'(x)}{g'(x)}=A/\infty" contenteditable="false"><span></span><span></span></span>（<span class="equation-text" contenteditable="false" data-index="4" data-equation="A"><span></span><span></span></span>为常数）
未定式：把两个无穷小或无穷大之比的极限称作未定式（形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n \to \ a}\frac{f(x)}{g(x)}"><span></span><span></span></span>），常见类型为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{0}{0}"><span></span><span></span></span>型或<span class="equation-text" contenteditable="false" data-index="2" data-equation="\frac{\infty }{\infty}"><span></span><span></span></span>型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0·\infty"><span></span><span></span></span>型
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to\ 0^+}x^p\ln x(p&gt;0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\infty-\infty"><span></span><span></span></span>型
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to\ 0}(\frac{1}{x}-\frac{1}{e^x-1})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0^0"><span></span><span></span></span>型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1^\infty"><span></span><span></span></span>型
对数极限法，转化为<span class="equation-text" contenteditable="false" data-index="0" data-equation="0·\infty"><span></span><span></span></span>型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\infty^0"><span></span><span></span></span>型
注意：当式子过于复杂时，不可暴力使用洛必达，应先利用等价无穷小化简
泰勒公式
泰勒中值定理：设<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在含有<span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span></span></span>的某个开区间<span class="equation-text" data-index="2" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内有<span class="equation-text" data-index="3" data-equation="n-1" contenteditable="false"><span></span><span></span></span>阶导数，则当<span class="equation-text" data-index="4" data-equation="x\in(a,b)" contenteditable="false"><span></span><span></span></span>时，<span class="equation-text" data-index="5" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>的<span class="equation-text" data-index="6" data-equation="n" contenteditable="false"><span></span><span></span></span>阶泰勒公式为<span class="equation-text" contenteditable="false" data-index="7" data-equation="f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+······+\frac{f^{(n)}(x_0)}{n!}(x-"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="8" data-equation="x_0)^n+o[(x-x_0)^n]"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="9" data-equation="f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+······+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="10" data-equation="\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x)^{n+1}，\xi\in(x,x_0)"><span></span><span></span></span>
称<span class="equation-text" contenteditable="false" data-index="0" data-equation="o[(x-x_0)^n]"><span></span><span></span></span>为佩亚诺余项
称<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}"><span></span><span></span></span>为拉格朗日余项
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+······+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n"><span></span><span></span></span>去掉余项后的式子称为函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="2" data-equation="x =x _0"><span></span><span></span></span>处的n阶泰勒多项式
麦克劳林公式：条件同上，取<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0=0"><span></span><span></span></span>，有<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+······+\frac{f^{(n)}(0)}{n!}x^n+o(x^n)"><span></span><span></span></span>
佩亚诺余项化为<span class="equation-text" contenteditable="false" data-index="0" data-equation="o(x^n)"><span></span><span></span></span>
拉格朗日余项化为<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{f^{(n+1)}(\theta x)}{(n+1)!}x^{n+1}"><span></span><span></span></span>
常见麦克劳林公式，即<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x =0"><span></span><span></span></span>处的泰勒公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sin \ x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-···+(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}+o(x^{2n-1})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\cos \ x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-···+(-1)^{n}\frac{x^{2n}}{(2n)!}+o(x^{2n})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+···+\frac{x^n}{n!}+o(x^n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln\ (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-···+(-1)^{n-1}\frac{x^n}{n}+o(x^n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+···+\frac{\alpha(\alpha-1)···(\alpha-n+1)}{n!}x^n+o(x^n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{1-x}=1+x+x^2+···+x^n+o(x^n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\tan \ x=x+\frac{1}{3}x^3+o(x^3)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\arcsin \ x=x+\frac{1}{6}x^3+o(x^3)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\arctan \ x=x-\frac{1}{3}x^3+o(x^3)"><span></span><span></span></span>
利用泰勒公式求极限
乘除位置用等价无穷小
加减位置上下同阶
消掉低阶量
忽略高阶量
找全同阶量
确定阶数找前后不能抵消的最低次幂
α&nbsp;β<b>&nbsp;</b>φ μ Γ Δ<b>&nbsp;</b>ξ θ ρ～Φ<b>&nbsp;ε&nbsp;&nbsp;</b>δ<b>&nbsp;</b>≥&nbsp;∈&nbsp;≤&nbsp;≠ ョ≈&nbsp;∀ ⇒&nbsp;⊆ &nbsp;→&nbsp; ∞&nbsp;&nbsp;Ů&nbsp;×<br>
评论
0 条评论
下一页
为你推荐
查看更多
高情商
高情商
上庄
上庄
数电开票流程图
数电开票流程图
I/O概述
I/O概述
高情商
高情商
自助上单
自助上单
自助上单
自助上单
自助上单
自助上单
I/O
I/O
.i.
.i.