考研高等数学（数一）
2026-04-28 00:51:37 0 举报AI智能生成
数一的基础要点
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考研数学
考研数学数一
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大纲/内容
前置知识<br>
三角函数
和差公式
和差化积
积化和差
基本公式<br>
<br><span class="equation-text" data-index="0" data-equation="\sin(x+y)=\sin x\cos y+\sin y\cos x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sin(x-y)=\sin x\cos y - \sin y\cos x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos (x+y) = \cos x\cos y - \sin x\sin y" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos (x-y) = \cos x\cos y + \sin x\sin y" contenteditable="false"><span></span><span></span></span>
万能公式
<br><span class="equation-text" data-index="0" data-equation="\sin a=\frac{2\tan \frac{a}{2}}{1+tan^2 \frac{a}{2}}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos a=\frac{1-tan^2 \frac{a}{2}}{1+tan^2 \frac{a}{2}}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\tan a=\frac{2\tan\frac{a}{2}}{1-tan^2 \frac{a}{2}}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="1+tan^2\frac{a}{2}=\frac{\cos^2(a/2)+\sin^2(a/2)}{\cos^2(a/2)}=\frac{1}{cos^2(a/2)}=\sec^2 \frac{a}{2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="1-tan^2\frac{a}{2}=\frac{\cos^2(a/2)-\sin^2(a/2)}{\cos^2(a/2)}=\frac{\cos a}{cos^2(a/2)}" contenteditable="false"><span></span><span></span></span>
半角公式
<br><span class="equation-text" data-index="0" data-equation="\sin \frac{a}{2}=\pm \sqrt \frac{1-\cos a}{2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos \frac{a}{2}=\pm \sqrt \frac{1+\cos a}{2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\tan \frac{a}{2}=\pm \sqrt \frac{1-\cos a}{1+\cos a}=\frac{\sin a}{1+\cos a}=\frac{1-\cos a}{\sin a}" contenteditable="false"><span></span><span></span></span>
二倍角公式
<br><span class="equation-text" data-index="0" data-equation="\sin 2a=2\sin a\cos a" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos 2a=\cos^2a-\sin^2a=2\cos^2a-1=1-2\sin^2a" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\tan 2a=\frac{2\tan a}{1-\tan^2a}" contenteditable="false"><span></span><span></span></span>
正割(sec)与正切(tan)<br>
<br><span class="equation-text" data-index="0" data-equation="\sec ^2x=\tan^2x+1 \\ \frac{\sin^2x+\cos^2x}{\cos^2x}=\frac{\sin^2x}{\cos^2x}+1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\tan'x=\sec^2x=1+\tan^2x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sec 'x=\tan x\sec x" contenteditable="false"><span></span><span></span></span>
三角恒等式<br>
<br><span class="equation-text" data-index="0" data-equation="\cos^2x+\sin^2x=1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sin^2x+\cos^2x=(\sin ^2x+\cos ^2x)^2=\sin^4x+\cos^4 x-2\sin ^2x\cos^2x" contenteditable="false"><span></span><span></span></span>
积分使用<br><span class="equation-text" data-index="0" data-equation="\cos^2x=\frac{1}{2}(\cos 2a+1)" contenteditable="false"><span></span><span></span></span>
欧拉恒等公式<br><span class="equation-text" data-index="0" data-equation="e^{ix}=\cos x+i\sin x" contenteditable="false"><span></span><span></span></span><br>
辅助角公式<br>
<br><span class="equation-text" data-index="0" data-equation="a\cos x+b\sin x=\sin (x+\varphi); \\\tan \varphi=\frac{a}{b}, \sin \varphi=\frac{a}{\sqrt{a^2+b^2}},\cos \varphi=\frac{b}{\sqrt{a^2+b^2}};" contenteditable="false"><span></span><span></span></span>
正余弦转换
<br><span class="equation-text" data-index="0" data-equation="\sin (x+\pi/2)=\cos x" contenteditable="false"><span></span><span></span></span>
反三角函数图像<br>
<br>
正割/余割/余切函数<br>
<span class="equation-text" data-index="0" data-equation="\csc x=\sin x^{-1}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\sec x= \cos x^{-1}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\cot x =\tan x^{-1}" contenteditable="false"><span></span><span></span></span><br>
弧与弧度<br>
弧长=弧度 x 半径<br><span class="equation-text" data-index="0" data-equation="l=ar" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="2\pi (rad)=360^\circ \rightarrow 1 \bold{rad}=360/(2\pi)" contenteditable="false"><span></span><span></span></span>
不等式<br>
调几算方<br><span class="equation-text" data-index="0" data-equation="\frac{2}{1/a+1/b}=\frac{2ab}{a+b}\leq \sqrt{ab}\leq \frac{a+b}{2} \leq \sqrt{(a^2+b^2)/2}" contenteditable="false"><span></span><span></span></span><br>
算数 >= 几何 <br><span class="equation-text" data-index="0" data-equation="\frac{1}{2}(x+\frac{1}{x})\geq \sqrt{x*\frac{1}{x}}=1" contenteditable="false"><span></span><span></span></span>
完全平方·<br><span class="equation-text" data-index="0" data-equation="ab\leq (a^2+b^2)/2" contenteditable="false"><span></span><span></span></span>
高数常用不等式<br><span class="equation-text" data-index="0" data-equation="x/(1+x)<\ln (x+1)<x [x>0]" contenteditable="false"><span></span><span></span></span>
数列求和<br>
等差数列 <br><span class="equation-text" data-index="0" data-equation="S=a_0+...+a_{n-1}+a_n,a_n=a_1+nd \implies S=(n+1)a_0+(0+...+n)d=(n+1)a_0+\frac{n(n+1)}{2}d" contenteditable="false"><span></span><span></span></span><br>
等比数列<br><span class="equation-text" data-index="0" data-equation="S=a_0+...+a_n,a_n=a_0q^n(q\neq 0) \implies S=a_0+...+a_0q^n=a_0(1+...+q^n)=a_0\frac{1-q^{n+1}}{1-q}" contenteditable="false"><span></span><span></span></span>
SP：当数列从 1下标开始时<br><span class="equation-text" data-index="0" data-equation="等差:na_1+\frac{n(n-1)}{2}d,等比:a_1\frac{1-q^{n}}{1-q}" contenteditable="false"><span></span><span></span></span><br>
多项式<br>
二项式定理
<br><span class="equation-text" data-index="0" data-equation="x^n-1=(1+x+...+x^{n-1})(x-1)" contenteditable="false"><span></span><span></span></span>
除与被除
几何图形<br>
扇形<br>
面积：<span class="equation-text" data-index="0" data-equation="S=\frac{1}{2} r l" contenteditable="false"><span></span><span></span></span> = 1/2 [半径 <span class="equation-text" data-index="1" data-equation="\times" contenteditable="false"><span></span><span></span></span> 弧长]<br>
弧长 <span class="equation-text" data-index="0" data-equation="L" contenteditable="false"><span></span><span></span></span><br>= n（圆心角）× π（圆周率）× r（半径）/180<br>=α (圆心角弧度数)× r（半径）<br>
球体<br>
表面积 <span class="equation-text" data-index="0" data-equation="4\pi r^2" contenteditable="false"><span></span><span></span></span><br>
体积 <span class="equation-text" data-index="0" data-equation="\frac{3}{4}\pi r^3" contenteditable="false"><span></span><span></span></span><br>
锥体
表面积 = 侧面+底面<br>
体积 = <span class="equation-text" data-index="0" data-equation="\frac{1}{3} \times" contenteditable="false"><span></span><span></span></span>底面积 <span class="equation-text" data-index="1" data-equation="\times" contenteditable="false"><span></span><span></span></span> 高 <br>
圆
面积 S=πr²
周长 C=2πr
椭圆<br>
面积<br><span class="equation-text" data-index="0" data-equation="S=\pi ab" contenteditable="false"><span></span><span></span></span>
周长<br><span class="equation-text" data-index="0" data-equation="L=2\pi b+4(a-b)" contenteditable="false"><span></span><span></span></span><br>
标准方程<br><span class="equation-text" data-index="0" data-equation="\frac{x^2}{a^2}+\frac{y^2}{b^2}=1" contenteditable="false"><span></span><span></span></span>
焦点<br><span class="equation-text" data-index="0" data-equation="F_1=(-c,0) , F_2=(c_0,0) [x轴时，a>b>0]" contenteditable="false"><span></span><span></span></span><br>
a,b,c关系<br><span class="equation-text" data-index="0" data-equation="a^2-b^2=c^2" contenteditable="false"><span></span><span></span></span><br>
各种“心”
重心
质心
平面域 <br><span class="equation-text" data-index="0" data-equation="\bar x = \iint_D x\rho(x,y) d\sigma/m" contenteditable="false"><span></span><span></span></span><br>
空间体 <br><span class="equation-text" data-index="0" data-equation="\bar x=\iiint_\Omega x\rho(x,y,z)dv/m" contenteditable="false"><span></span><span></span></span><br>
曲线段<br> <span class="equation-text" data-index="0" data-equation="\bar x=\int_C xf(x,y,z)ds/m" contenteditable="false"><span></span><span></span></span><br>
曲面片<br> <span class="equation-text" data-index="0" data-equation="\bar x=\iint_\Sigma\rho(x,y,z) dS/m" contenteditable="false"><span></span><span></span></span><br>
形心<br>
虚数
<span class="equation-text" data-index="0" data-equation="i^2=-1" contenteditable="false"><span></span><span></span></span>
欧拉恒等公式<br><span class="equation-text" data-index="0" data-equation="e^{ix}=\cos x+i\sin x" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="e^{i\pi}+1=0 [x=\pi 时]" contenteditable="false"><span></span><span></span></span>
函数<br>
一元函数
连续性<br>
定义<br>
自变量增量趋于0，函数值增量趋于0<br><span class="equation-text" data-index="0" data-equation="当\Delta x \rightarrow 0时，对应的\Delta y \rightarrow 0" contenteditable="false"><span></span><span></span></span>
记作<br><span class="equation-text" data-index="0" data-equation="\lim_{\Delta x \rightarrow 0} \Delta y = \lim_{\Delta x \rightarrow 0}[f(x_0+\Delta x)-f(x_0)]=0" contenteditable="false"><span></span><span></span></span>
一般证明<br>
左右极限存在且等于该点的值<br>
连续函数
在该区间上的每一个点都连续的函数
单侧导数存在 <span class="equation-text" data-index="0" data-equation="\implies" contenteditable="false"><span></span><span></span></span> 单侧连续, 所以左右导数同时存在则函数在 <span class="equation-text" data-index="1" data-equation="x_0" contenteditable="false"><span></span><span></span></span> 处必然连续
定理
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow x_0^-}f(x)=f(x_0)=\lim_{x\rightarrow x_0^+}f(x)" contenteditable="false"><span></span><span></span></span><br>
复合函数连续性
性质<br>
连续与有界性<br><span class="equation-text" data-index="0" data-equation="f(x)在[a,b]连续，则其在(a,b)上有界" contenteditable="false"><span></span><span></span></span>
连续与左右极限<br><span class="equation-text" data-index="0" data-equation="f(x)在(a,b)连续，则\lim_{x\rightarrow a+} f(x) 与\lim_{x\rightarrow b-} f(x)均存在" contenteditable="false"><span></span><span></span></span><br>
连续与绝对值<br><span class="equation-text" data-index="0" data-equation="f(x)连续 \implies |f(x)|连续" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases}1, x\geq1\\-1,x<1 \end{cases}" contenteditable="false"><span></span><span></span></span>
连续与可导性<br><span class="equation-text" data-index="0" data-equation="f'(x)连续\implies f(x)可导" contenteditable="false"><span></span><span></span></span>
连续与可积性<br><span class="equation-text" data-index="0" data-equation="f'(x)可积 \implies f(x)连续" contenteditable="false"><span></span><span></span></span><br>
间断点
定义
<br><span class="equation-text" data-index="0" data-equation="x=x_0没有定义" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x=x_0处有定义，且\lim_{x\rightarrow x_0}f(x) 不存在" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x=x_0处有定义，且\lim_{x\rightarrow x_0}f(x) \neq f(x_0)" contenteditable="false"><span></span><span></span></span>
类型<br>
第一类间断点
左右极限存在，且 <span class="equation-text" data-index="0" data-equation="x_0" contenteditable="false"><span></span><span></span></span> 为 <span class="equation-text" data-index="1" data-equation="f(x)" contenteditable="false"><span></span><span></span></span> 间断点
类别
可去间断点<br>
跳跃间断点<br>
第二类间断点
无穷间断点<br>
振荡间断点<br>
其他<br>
易错问题<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases}0,x\leq 0\\1 ,x>0\end{cases}当x=0时是否为间断点" contenteditable="false"><span></span><span></span></span>
闭区间性质<br>
有界性/收敛<br>
区间内有界<br><span class="equation-text" data-index="0" data-equation="在[a,b]区间内，\exist M,m 使得 m\leq f(x)\leq M【区间内必有上下界】" contenteditable="false"><span></span><span></span></span><br>
收敛【数列中讨论】<br><span class="equation-text" data-index="0" data-equation="单调+有界 \implies 收敛" contenteditable="false"><span></span><span></span></span><br>
特殊函数
<br><span class="equation-text" data-index="0" data-equation="|\arcsin x|\leq \frac{\pi}{2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="|\arctan x|< \frac{\pi}{2}" contenteditable="false"><span></span><span></span></span>
最大最小值定理<br>
零点定理<br>
介值定理<br>
一致连续性<br>
函数性态<br>
单调性
条件<br><span class="equation-text" data-index="0" data-equation="y=f(x)在[a,b]连续,(a,b)可导" contenteditable="false"><span></span><span></span></span>
结论<br><span class="equation-text" data-index="0" data-equation="如果在(a,b)内f'(x)\geq 0,且等号仅在有限多个点处成立，则函数在[a,b]单调增加；f'(x)\leq 0则减少" contenteditable="false"><span></span><span></span></span>
经典错误
<span class="equation-text" data-index="0" data-equation="函数连续，f(x_0)'>0，则\exist\delta>0使得 f(x)在(0,\delta)单增" contenteditable="false"><span></span><span></span></span>
凹凸性
几何判断<br>
平均值比较法<br><span class="equation-text" data-index="0" data-equation="凹：f(\frac{x_1+x_2}{2})<\frac{f(x_1)+f(x_2)}{2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="凸：f(\frac{x_1+x_2}{2})>\frac{f(x_1)+f(x_2)}{2}" contenteditable="false"><span></span><span></span></span>
导数判断<br>
条件<br><span class="equation-text" data-index="0" data-equation="f(x)在[a,b]上连续，在(a,b)内有一阶和二阶导数" contenteditable="false"><span></span><span></span></span><br>
结论<br><span class="equation-text" data-index="0" data-equation="(a,b)内f''(x)>0,则f(x)在[a,b]为凹；f''(x)<0则为凸" contenteditable="false"><span></span><span></span></span><br>
奇偶性<br>
偶函数<br><span class="equation-text" data-index="0" data-equation="f(-x)=f(x)" contenteditable="false"><span></span><span></span></span><br>
幂函数<br><span class="equation-text" data-index="0" data-equation="x^2,|x|" contenteditable="false"><span></span><span></span></span>
三角<br><span class="equation-text" data-index="0" data-equation="\cos x" contenteditable="false"><span></span><span></span></span>
抽象函数<br><span class="equation-text" data-index="0" data-equation="f(x)+f(-x)" contenteditable="false"><span></span><span></span></span>
奇函数<br><span class="equation-text" data-index="0" data-equation="f(-x)=-f(x)" contenteditable="false"><span></span><span></span></span><br>
三角<br><span class="equation-text" data-index="0" data-equation="\sin x,\tan x,\arcsin x,\arctan x" contenteditable="false"><span></span><span></span></span>
对数<br><span class="equation-text" data-index="0" data-equation="\ln\frac{1-x}{1+x}, \ln(x+\sqrt{1+x^2})" contenteditable="false"><span></span><span></span></span>
指数<br><span class="equation-text" data-index="0" data-equation="(e^2-1)/(e^2+1)" contenteditable="false"><span></span><span></span></span><br>
抽象函数<br><span class="equation-text" data-index="0" data-equation="f(x)-f(-x)" contenteditable="false"><span></span><span></span></span><br>
几何性质
奇函数：原点对称<br>
偶函数：y轴对称
导数与原函数奇偶<br>
【可导】原函数的导数，奇偶性与原函数相反<br><span class="equation-text" data-index="0" data-equation="f(x)可导,f(x)为奇（偶）函数\implies f'(x)为偶（奇）函数" contenteditable="false"><span></span><span></span></span>
连续【可积】函数的原函数不一定与其奇偶相反<br><span class="equation-text" data-index="0" data-equation="f(x)为连续偶函数，则其原函数中有唯一一个奇函数F(x)+C(C=0) \\ f(x)为连续奇函数，则原函数均为 偶函数" contenteditable="false"><span></span><span></span></span><br>
周期性
定义<br>
<br><span class="equation-text" data-index="0" data-equation="f(x+T)=f(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_0^T f(t)dt=0" contenteditable="false"><span></span><span></span></span>
其他性质
【可导】周期函数导数也为周期函数, 且周期不变<br><span class="equation-text" data-index="0" data-equation="f(x+T)=f(x)\implies f'(x)=f'(x+T)" contenteditable="false"><span></span><span></span></span><br>
【可积】周期函数积分为0时，其原函数也为周期函数<br><span class="equation-text" data-index="0" data-equation="\int_x^{x+T} f(x)dx =0" contenteditable="false"><span></span><span></span></span><br>
证明<br><span class="equation-text" data-index="0" data-equation="F(x)=\int_a^xf(x)dx;\\F(x+T)=\int_a^x f(x)dx+\int_x^{x+T}f(x)dx;\\F(x)=F(x+T)时\implies \int_x^{x+T}f(x)dx;=0" contenteditable="false"><span></span><span></span></span><br>
拐点<br>
定义
定义<br><span class="equation-text" data-index="0" data-equation="f(x)在区间I上连续，x_0为I上的点，如果曲线经过(x_0,f(x_0))，曲线凹凸发生改变，则称该点位拐点" contenteditable="false"><span></span><span></span></span><br>
计算<br>
步骤<br><span class="equation-text" data-index="0" data-equation="1)求f''(x) \\2)令f''(x)=0,求出区间内实根【以及不存在】的点\\3)对于每一个实根或则二阶导数不存在的点，检查二阶导两侧是否异号" contenteditable="false"><span></span><span></span></span>
判定<br>
必要条件
<br><span class="equation-text" data-index="0" data-equation="f''(x_0)=0" contenteditable="false"><span></span><span></span></span>
第一充分条件
【两侧判定法】<br><span class="equation-text" data-index="0" data-equation="f''(x_0)=0; 左右异号为拐点；同号则不是" contenteditable="false"><span></span><span></span></span>
第二充分条件
【三阶导数判断法】<br><span class="equation-text" data-index="0" data-equation="f''(x_0)=0,f'''(x_0)\neq0，是拐点;f'''(x_0)=0 不能判断" contenteditable="false"><span></span><span></span></span><br>
第三充分条件
<br><span class="equation-text" data-index="0" data-equation="f''(x_0)=...=f^{(n-1)}(x_0)=0; f^{(n)}\neq 0" contenteditable="false"><span></span><span></span></span>
n为偶数<br>不是拐点<br>
n为奇数<br>拐点<br>
驻点<br>
定义<br><span class="equation-text" data-index="0" data-equation="f'(x)为0的点" contenteditable="false"><span></span><span></span></span>
极值与最值<br>
极值
定义<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)在x_0某邻域有定义，对于去心邻域内任一x，有f(x)<f(x_0)或f(x)>f(x_0)\\则称f(x_0)是函数的一个极值" contenteditable="false"><span></span><span></span></span>
计算
求出全部驻点与不可导点，观察这些点左右邻近的情况从而判断极值情况<br>
判定
必要条件
<br><span class="equation-text" data-index="0" data-equation="如果f(x)在x_0可导且在x_0取得极值,则f'(x_0)=0" contenteditable="false"><span></span><span></span></span>
第一充分条件<br>
一阶导存在且左右异号【两侧判断法】<br><span class="equation-text" data-index="0" data-equation="f(x)在x_0连续，且在x_0某【去心邻域】可导\\若左邻域f'(x)>0,右邻域f'(x)<0则f(x)在x_0取得最大值；反之取得最小值；\\若在去心邻域内符号不变，则x_0处无极值" contenteditable="false"><span></span><span></span></span>
第二充分条件<br>
二阶导数存在且不等于0【二阶正负判断法】<br><span class="equation-text" data-index="0" data-equation="f(x)在x_0处具有二阶导数，且f'(x_0)=0,f''(x_0)\neq 0\\1)f''(x_0)<0,极大值 \\2)f''(x_0)>0,极小值" contenteditable="false"><span></span><span></span></span>
第三充分条件<br>
<br><span class="equation-text" data-index="0" data-equation="f'(x_0)=f''(x_0)=...=f^{(n-1)}(x_0)=0; f^{(n)}\neq 0" contenteditable="false"><span></span><span></span></span>
n为偶数<br>f(x)在x0有极值，其中n阶导>0极小值，n阶导<0极大值<br>
n为奇数<br>无极值<br>
最值
求出 f(x) 在 (a,b) 内的驻点和不可导点<br>
求出 驻点 和 不可导点 以及 端点处的函数值<br>
取这些函数值的极大值或者极小值<br>
一元连续函数区间内部的唯一极值点，为该区间的最值点<br>
渐近线
水平渐近线<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow \infty}f(x)=y_0" contenteditable="false"><span></span><span></span></span>
垂直渐近线<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow x_0}f(x)=\infty, x_0一般为间断点" contenteditable="false"><span></span><span></span></span>
斜渐近线<br>
构造法【常用泰勒公式】<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow \infty}[f(x)-(ax+b)]=0,注：a,b为常数" contenteditable="false"><span></span><span></span></span>
极限法<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow +\infty} y/x = a" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow +\infty} (y-ax)=b" contenteditable="false"><span></span><span></span></span>
函数图形描绘<br>
<br><span class="equation-text" data-index="0" data-equation="1）求性质：奇偶性、周期性；求导数：一阶二阶" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="2）求出驻点与拐点，并划分区间" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="3）根据一二阶导数值确定升降、凹凸性" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="4）确定渐近线与其他变化趋势" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="5）计算f'(x),f''(x)不存在点的函数值，被划分的区间补充一些点，使得曲线尽可能光滑" contenteditable="false"><span></span><span></span></span>
点/区间书写格式<br>
极值点<br>
驻点
零点<br>
间断点<br>
<br><span class="equation-text" data-index="0" data-equation="x = x_0" contenteditable="false"><span></span><span></span></span>
极值<br>
<br><span class="equation-text" data-index="0" data-equation="f(x_0)=a" contenteditable="false"><span></span><span></span></span>
单调区间<br>
<br><span class="equation-text" data-index="0" data-equation="(-\infty,a),(b,+\infty)" contenteditable="false"><span></span><span></span></span>
拐点
<br><span class="equation-text" data-index="0" data-equation="(x_0,f(x_0))" contenteditable="false"><span></span><span></span></span>
常见函数类型<br>
符号函数<br>
取整函数
复合函数
反函数
计算步骤<br>
y=f(x)在定义域上是否为单调函数<br>
如果是单调函数<br>逆运算求出x=g(y)<br>
常见反函数<br>
反三角函数
<br><span class="equation-text" data-index="0" data-equation="x=\sin y;y=\arcsin x;[-\pi/2\leq y\leq \pi/2]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x=\cos y;y=\arccos x;[0\leq y\leq \pi]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x=\tan y;y=\arctan x;[-\infty<x<+\infty;-\pi/2<y<\pi/2]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x=\cot y;y=arccot x;[-\infty<x<+\infty;0<y<\pi]" contenteditable="false"><span></span><span></span></span>
几何性质
关于x=y对称<br>
初等函数<br>
定义域<br>
函数不等式证明
单调性
最值
拉格朗日中值定理
泰勒公式<br>
凹凸性<br>
基本不等式
多元函数<br>
平面点集<br>
邻域<br>
点与点集关系<br>
内点<br>
外点<br>
边界点<br>
聚点<br>
<br><span class="equation-text" data-index="0" data-equation="对于任意\delta>0,点P的去心邻域内总有E中的点，则称P是E的聚点" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="点集E的聚点P，可以属于E，也可以不属于E" contenteditable="false"><span></span><span></span></span>
n元函数定义
<br><span class="equation-text" data-index="0" data-equation="D是\bold R^n的一个非空子集,称映射f:D\rightarrow \bold R^n为定义在D上的n元函数" contenteditable="false"><span></span><span></span></span>
连续性<br>
定义<br><span class="equation-text" data-index="0" data-equation="\lim_{(x,y)\rightarrow (x_0,y_0)}f(x,y)=f(x_0,y_0)" contenteditable="false"><span></span><span></span></span><br>
间断点
闭区间性质<br>
有界性与最值定理<br>
介值定理<br>
一致连续性定理<br>
极值/极值点
定理1
<br><span class="equation-text" data-index="0" data-equation="函数在(x_0,y_0)偏导存在，且为极值点则f_x(x_0,y_0)=f_y(x_0,y_0)=0" contenteditable="false"><span></span><span></span></span>
驻点<br>
<br><span class="equation-text" data-index="0" data-equation="f_x(x_0,y_0)=f_y(x_0,y_0)=0" contenteditable="false"><span></span><span></span></span>
判断方法
充分条件<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="函数z在点(x_,y_0)某邻域内连续且有【二阶连续偏导】\\又f'_x=f'_y=0,令f''_{xx}=A,f''_{xy}=f_{yx}=B, f''_{yy}=C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="AC-B^2>0【A<0极大值，A>0极小值】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="AC-B^2<0【没有极值】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="AC-B^2=0【可能有极值，也可能没有】" contenteditable="false"><span></span><span></span></span>
证明
同济版高数下p125<br>
几何法
保号性判断法
性质
<span class="equation-text" data-index="0" data-equation="若函数f(x,y)在(x_0,y_0)处取得极小值，则f(x_0,y_0)在x=x_0取得极小值，在y=y_0处取得极小值" contenteditable="false"><span></span><span></span></span>
条件极值求解
拉格朗日乘法<br>
定义<br>
原函数<br><span class="equation-text" data-index="0" data-equation="f(x,y,z)" contenteditable="false"><span></span><span></span></span><br>
约束条件<br><span class="equation-text" data-index="0" data-equation="\varphi(x,y,z)=0" contenteditable="false"><span></span><span></span></span>
求解步骤
构造函数<br><span class="equation-text" data-index="0" data-equation="F(x,y,z,\lambda)=f(x,y,z)-\lambda \varphi(x,y,z)" contenteditable="false"><span></span><span></span></span><br>
列出所有偏导方程<br>
解方程<br>
极坐标求解
特征
<span class="equation-text" data-index="0" data-equation="D=\{(x,y)|(x-a)^2+(y-b)^2\leq c \}" contenteditable="false"><span></span><span></span></span>
不等式
最值点
<span class="equation-text" data-index="0" data-equation="多元连续函数有界闭区间内部的唯一极值点，不一定为该区间的最值点" contenteditable="false"><span></span><span></span></span>
极限/重极限
一元函数极限
定义<br>
函数<br><span class="equation-text" data-index="0" data-equation="函数去心领域有定义，存在常数A，对于任意\xi,总存在正数\delta,使当|x-x_0|<\delta 时，对应函数值f(x)满足|f(x)-A|<\xi" contenteditable="false"><span></span><span></span></span><br>
书写<br><span class="equation-text" data-index="0" data-equation="注：f(x_0)=\infty 意味着极限不存在，但极限为无穷" contenteditable="false"><span></span><span></span></span>
极限存在证明<br>
数列极限<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow +\infty}x_n=a \Leftrightarrow \lim_{k\rightarrow +\infty}x_{2k-1}=\lim_{k\rightarrow +\infty}x_{2k}=a" contenteditable="false"><span></span><span></span></span>
函数极限
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow x_0}f(x)=A \iff\lim_{x\rightarrow x_0^-}f(x)=\lim_{x\rightarrow x_0^-}f(x)=A" contenteditable="false"><span></span><span></span></span>
常见分左右极限讨论
分段函数
指数
arctan<br>
海涅定理
数列极限为 a【接近但不相等, 极限是因为 x是趋于 a而不等于a的情况】<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow a}f(x)=b \iff取f(x)定义域内的任意数列\{a_n\},\lim_{n\rightarrow \infty} a_n=a,且a_n\neq a有\lim_{n\rightarrow \infty}f(a_n)=b" contenteditable="false"><span></span><span></span></span>
极限性质<br>
局部有界性<br>
保号性
极限值与无穷小关系<br>
极限与绝对值
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow a}f(x)=a \implies \lim_{x\rightarrow a}|f(x)|=a" contenteditable="false"><span></span><span></span></span>
无穷小/无穷大
性质
无穷小有限和积<br><span class="equation-text" data-index="0" data-equation="无穷小有限+(\times)=无穷小" contenteditable="false"><span></span><span></span></span><br>
无穷小 x 有界函数 = 无穷小<br><span class="equation-text" data-index="0" data-equation="\lim_{x->0}x\sin x=0" contenteditable="false"><span></span><span></span></span><br>
阶大小关系<br>
高阶<br>
同阶<br>
等价
k阶<br>
阶的比较
直接比较法【极限比较法】<br>
泰勒展开<br>
定义法【洛必达法,用 <span class="equation-text" data-index="0" data-equation="O(x^k)" contenteditable="false"><span></span><span></span></span> 进行比较】
求导定阶【微分阶】<br>
<br><span class="equation-text" data-index="0" data-equation="若 \lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=1,则 \int_{0}^{\varphi(x)} f(t)dt \sim \int_0^{\varphi(x)} g(t)dt【\varphi(x) \rightarrow 0】" contenteditable="false"><span></span><span></span></span>
积分等价<br>
结论法<br>
不定式<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{0}{0} 或\frac{\infty}{\infty}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="0 \times \infty=\frac{1}{\infty}\times \infty" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="1^{\infty}" contenteditable="false"><span></span><span></span></span>
注：这里的 1 必须是常数而不是极限值<br><span class="equation-text" data-index="0" data-equation="(1+\alpha)^\beta,\lim \alpha=0,\lim \beta=\infty" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="=e^{\lim \alpha \beta}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\infty^0 = e^{0\ln \infty}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="0^0=e^{0\ln0}【\ln 0\rightarrow -\infty】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\infty-\infty = \infty \cdot0" contenteditable="false"><span></span><span></span></span>
六大求极限方法<br>
极限有理运算<br>
限制条件<br>
1）加间极限必须存在<br>2）商的分母值不能为0<br>3）幂次必须为正整数<br>
常见错误<br>
lim(A+B)=C,不能推出lim A+limB=C[A,B可能分别极限均不存在]<br>
夹逼原理<br>
<br><span class="equation-text" data-index="0" data-equation="\forall n,\frac{1}{n+1}<\ln(1+\frac{1}{n})<\frac{1}{n}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty}(\frac{1}{n^2+1}+\frac{2}{n^2+1}+...+\frac{n}{n^2+1})" contenteditable="false"><span></span><span></span></span>
等价无穷小<br>
限制条件<br>
<span class="equation-text" data-index="0" data-equation="1)设\alpha\sim \tilde{\alpha}, \beta\sim \tilde{\beta},且\lim \frac{\tilde{\beta}}{\tilde{\alpha}}存在则\lim \frac{{\beta}}{{\alpha}}=\lim \frac{\tilde{\beta}}{\tilde{\alpha}} \\ 2) \beta=\alpha+o(\alpha)[充要条件]" contenteditable="false"><span></span><span></span></span>
常用公式
函数<br>
幂等价
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0](1+x)^{\alpha}-1 \sim \alpha x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0，\alpha 为奇数](x-1)^{\alpha}+1 \sim -\alpha x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[f(x)\rightarrow 0](1+f(x))^{\varphi(x)}-1\sim \varphi(x) f(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] (1+x)^{\frac{1}{x}}\sim e" contenteditable="false"><span></span><span></span></span>
指/对数<br>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] e^x-1\sim x\ln e=x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0]a^x-1\sim x\ln a" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0]\ln(1+x)\sim x\frac{1}{\ln e}=x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 1]\ln x=\ln [1+(x-1)]\sim x-1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0} \frac{\log a^{(1+x)}}{x}=\frac{1}{\ln a}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[t\rightarrow 0] t-\ln(1+t)\sim\frac{1}{2}t^2+o(t^2)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] \ln(1-t)+t\sim\frac{1}{2}t^2+o(t^2)" contenteditable="false"><span></span><span></span></span>
三角等价<br>
<br><span class="equation-text" data-index="0" data-equation="\tan x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] x - \arctan x \sim \frac{1}{3}x^3 +o(x^3)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] \tan x - x\sim \frac{1}{3}x^3 +o(x^3)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+o(x^7)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] 1-\cos x\sim \frac{1}{2}x^2+o(x^2)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="1-cos^\alpha x\sim \frac{\alpha}{2} x^2" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] 1-\sqrt[n]{\cos x}= 1-\sqrt[n]{(\cos x-1)+1}\sim -[\frac{1}{n}(\cos x-1)]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sin x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] sinx \sim x +o(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] x - \sin x\sim \frac{1}{6}x^3 +o(x^3)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x\rightarrow 0] \arcsin x - x\sim \frac{1}{6}x^3 +o(x^3)" contenteditable="false"><span></span><span></span></span>
数列
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty}\sqrt[n]{n} = 1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty} \sqrt[n]{a}=1(a>0)" contenteditable="false"><span></span><span></span></span>
积分等价<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=1 ,则\int_0^xf(t)dt\sim \int_0^x g(t)dt" contenteditable="false"><span></span><span></span></span>
阶的比较
直接比较法【极限比较法】<br>
泰勒展开<br>
求导定阶【微分阶】<br>
积分等价<br>
结论法<br>
真题
x 趋近于 0<br><span class="equation-text" data-index="0" data-equation="\sqrt{\cos 2x-1}=e^{\frac{1}{2}\ln{\cos 2x}}-1=\frac{1}{2}\ln{\cos 2x}=\frac{1}{2}\ln{(\cos 2x-1+1)}=\cos 2x-1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty} e-(1+\frac{1}{n})^n\sim \frac{e}{2n}" contenteditable="false"><span></span><span></span></span>
中值定理<br>
罗尔定理<br>
前提<br>
费马引理<br><span class="equation-text" data-index="0" data-equation="f(x)再x_0某领域有定义，且再x_0处可导,如果对于任意x\in U(x_0)有f(x)\leq f(x_0)[反之同理],则f'(x_0)=0" contenteditable="false"><span></span><span></span></span>
限制条件<br>
<br><span class="equation-text" data-index="0" data-equation="1)闭区间[a,b]连续 \\ 2) 开区间(a,b)可导 \\ 3)区间端点处函数值相等,f(a)=f(b)" contenteditable="false"><span></span><span></span></span>
结果<br>
<br><span class="equation-text" data-index="0" data-equation="存在一点\xi \in(a,b),f'(\xi)=0" contenteditable="false"><span></span><span></span></span>
拉格朗日中值定理[微中]<br>
限制条件<br>
<br><span class="equation-text" data-index="0" data-equation="1)闭区间[a,b]连续 \\ 2) 开区间(a,b)可导" contenteditable="false"><span></span><span></span></span>
公式<br>
<br><span class="equation-text" data-index="0" data-equation="F(\varphi(x_1))-F(\varphi(x_2))=\int_{\varphi(x_2)}^{\varphi(x_1)} f(x)dx=f(\xi)(\varphi(x_1)-\varphi(x_2))=F'(\xi)(\varphi(x_1)-\varphi(x_2))" contenteditable="false"><span></span><span></span></span>
柯西中值定理<br>
限制条件<br>
<br><span class="equation-text" data-index="0" data-equation="1)闭区间[a,b]连续 \\ 2) 开区间(a,b)可导 \\ 3)对任一x\in(a,b),F'(x)\neq0" contenteditable="false"><span></span><span></span></span>
公式
<br><span class="equation-text" data-index="0" data-equation="\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f'(\xi)}{F'(\xi)}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{f(b)-f(a)}{f'(\xi)}=b-a=\frac{F(b)-F(a)}{F'(\xi)}" contenteditable="false"><span></span><span></span></span>
常见处理式<br>
<br><span class="equation-text" data-index="0" data-equation="\lim (e^{\sin x}-e^x)/(\sin x-x) [1]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim x^2(a^{\frac{1}{x}}-a^{\frac{1}{x+1}})[\ln a]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim[\sin \sqrt{x+1}-\sin \sqrt{x}][0]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim (1+1/t)^{t}-e=\lim e^{t\ln(1+1/t)}-e" contenteditable="false"><span></span><span></span></span>
洛必达法则
限制条件<br>
着重讨论 0/0 未定式情况，而 ∞/∞ 则可以通过分子分母同时倒数来转化为 0/0 型<br><span class="equation-text" data-index="0" data-equation="1) 当x\rightarrow a时，f(x)以及F(x)都趋于0\\ 2)在点a的去心邻域内，f'(x)及F'(x)均存在且F'(x)\neq 0 \\3)\lim_{x\rightarrow a}\frac{f'(x)}{F'(x)}存在(或为无穷大)" contenteditable="false"><span></span><span></span></span>
推导来源<br>
柯西中值定理<br>
泰勒公式
使用思考<br>
是否对于任意形式都满足泰勒公式<br>
泰勒公式与泰勒级数、泰勒展开式之间的关系<br>
常用公式[麦克劳林展式]<br>
<br><span class="equation-text" data-index="0" data-equation="\sin x=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-...+(-1)^{m-1}\frac{x^{2m-1}}{(2m-1)!}+R_{2m}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos x=1-\frac{1}{2!}x^2+...+(-1)^m\frac{x^{2m}}{(2m)!}+R_{2m+1}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\ln (1+x) = x-\frac{1}{2}x^2+\frac{1}{3}x^3...+(-1)^{n-1}\frac{1}{n}x^n+R_n(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" 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+R_n(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="e^x=1+x+\frac{x^2}{2}+...+\frac{x^n}{n!}+R_n(x)" contenteditable="false"><span></span><span></span></span>
基本形式<br>
<br><span class="equation-text" data-index="0" data-equation="在x_0处有n阶导数，则存在x_0的某领域对于领域内的任一x,有\\f(x)=f(x_0)+(x-x_0)f'(x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+...+\frac{f^{(n)}}{n!}(x-x_0)^n" contenteditable="false"><span></span><span></span></span>
系数 an 与 f(x)的n阶导关系<br><span class="equation-text" data-index="0" data-equation="由于为使用多项式近似表达原函数P_n(x)=a_0+a_1(x-x_0)+..+a_n(x-x_0)^n ,\\ 且无论是函数值或者直到n阶导函数都相等，n!*a_n=f(x)^{(n)} 即a_n=\frac{1}{n!}f(x)^{(n)}" contenteditable="false"><span></span><span></span></span>
【变形式子】<br><span class="equation-text" data-index="0" data-equation="f(x+x_0)=f(x_0)+xf'(x_0)+\frac{f''(x_0)}{2!}x^2+...+\frac{f^{(n)}(x_0)}{n!}x^n" contenteditable="false"><span></span><span></span></span>
课外内容
<br><span class="equation-text" data-index="0" data-equation="e^{ix}=\cos x+i\sin x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="e^{i\pi}+1=0" contenteditable="false"><span></span><span></span></span>
导数定义<br>
极限定义式<br><span class="equation-text" data-index="0" data-equation="\lim _{x\rightarrow x_0}f'(x_0)=\lim_ {x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}=\lim_ {\Delta x\rightarrow 0} \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}" contenteditable="false"><span></span><span></span></span>
有理化<br>
求极限思路<br>
乘积形式/指对数
等价<br>
导数简单/积分上限函数<br>
洛必达
高阶式子/加减不能等价代换/复杂三角函数<br>
泰勒【部分展开】<br>
导数形式
求导定义
带有根号形式的分式<br>
幂等价式<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0}(1+x)^{\alpha}-1 \sim \alpha x" contenteditable="false"><span></span><span></span></span>
三角函数化简<br>
有理化<br>
幂指函数<br>
幂指恒等式 <span class="equation-text" data-index="0" data-equation="x^y = e^{y\ln x}" contenteditable="false"><span></span><span></span></span><br>
幂等价式<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0}(1+x)^{\alpha}-1 \sim \alpha x" contenteditable="false"><span></span><span></span></span>
不定式<br><span class="equation-text" data-index="0" data-equation="(1+\alpha)^\beta,\lim \alpha=0,\lim \beta=\infty；e^{\lim \alpha \beta}" contenteditable="false"><span></span><span></span></span>
其他题型<br>
确定极限式参数<br>
难题总结
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 1^{-}} e^{\frac{1}{x^2-1}}/(x-1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0}\frac{sin(x^2sin\frac{1}{x})}{x}" contenteditable="false"><span></span><span></span></span>
夹逼定理求解<br><span class="equation-text" data-index="0" data-equation="sinx\leq x\implies0\leq |\lim_{x\rightarrow 0}\frac{sin(x^2sin\frac{1}{x})}{x}|\leq |\lim_{x\rightarrow 0}\frac{x^2sin\frac{1}{x}}{x}|" contenteditable="false"><span></span><span></span></span><br>
多元函数极限
定义<br>
<br><span class="equation-text" data-index="0" data-equation="n元函数f(P)的定义域为D，P_0为D的聚点，如果存在常数A，对于任意\xi,总存在正数\delta,使当P_0\in D\cap U^\circ(P_0,\delta) 时，对应函数值f(P)满足|f(P)-A|<\xi" contenteditable="false"><span></span><span></span></span>
存在证明<br>
常见求极限方法<br>
夹逼定理<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{(x.y)\rightarrow (0,0)}\frac{x^2y}{x^2+y^2}\leq\lim \frac{x^2+y^2}{x^2+y^2}|y|=\lim y=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{(x.y)\rightarrow (0,0)} \frac{x^2+y^2}{|x|+|y|}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{(x.y)\rightarrow (0,0)} \frac{xy^2\sin(xy)}{x^2+y^4}" contenteditable="false"><span></span><span></span></span>
反证法<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{(x.y)\rightarrow (0,0)} xy/(x^2+y^2)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{(x.y)\rightarrow (0,0)} xy^2/(x^2+y^4)" contenteditable="false"><span></span><span></span></span>
计算方法<br>
极限判断<br>
<span class="equation-text" data-index="0" data-equation="\frac{高阶}{低阶} ，考虑 0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="0 \leq |f(x,y)| \leq \lim g(x)=0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\frac{低阶}{高阶} ，考虑 \infty" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\frac{同阶}{同阶} ，考虑 不存在" contenteditable="false"><span></span><span></span></span>
数列极限<br>
单调有界准则
定积分定义<br>
<br><span class="equation-text" data-index="0" data-equation="\int_{a}^{b} f(x)\, \mathrm{d}x= \begin{matrix} \lim_{\lambda \to \ 0} \sum_{i=1}^{n} f(\xi_i)\Delta x_i \end{matrix}" contenteditable="false"><span></span><span></span></span>
n项和数列极限
变化部分<br><span class="equation-text" data-index="0" data-equation="例\sum_{i=1}^nn/(n^2+i)=\frac{1}{n}\lim \frac{1}{(1+i/n^2)}\\i为变化部分与主体部分n^2不是统一量,从而采用夹逼" contenteditable="false"><span></span><span></span></span>
根号n次幂次项和
<br><span class="equation-text" data-index="0" data-equation="\sqrt[n]{a_1^n+...+a_m^n}=max\{a_i\}" contenteditable="false"><span></span><span></span></span>
n项连乘<br>
夹逼原理<br>
取对数化为n项和<br>
例题<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow\infty} \sqrt[n]{n!}/n=1/e" contenteditable="false"><span></span><span></span></span><br>
递推关系定义数列<br>
定义
数列<br><span class="equation-text" data-index="0" data-equation contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="存在常数a,对于任意正数\xi,总存在N,当n>N,使得|x_n-a|<\xi都成立" contenteditable="false"><span></span><span></span></span>
a为数列极限=数列收敛于a<br>
充要条件
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow +\infty}x_n=a \Leftrightarrow \lim_{k\rightarrow +\infty}x_{2k-1}=\lim_{k\rightarrow +\infty}x_{2k}=a" contenteditable="false"><span></span><span></span></span>
收敛性质<br>
数列收敛，则极限唯一【唯一性】
数列收敛，则一定有界【有界性】
【保号性】<br><span class="equation-text" data-index="0" data-equation="\lim x_n=a,且a>0(或a<0)则存在正整数N，当n>N时，x_n>0(或x_n<0)" contenteditable="false"><span></span><span></span></span><br>
【子序列关系】<br>如果数列收敛，则其任一子数列也收敛，且极限也为 a<br>
无穷级数<br>
定义<br>
级数
<br><span class="equation-text" data-index="0" data-equation="\sum_{i=1}^\infty u_i=u_1+..+u_i" contenteditable="false"><span></span><span></span></span>
收敛/发散
级数与部分数列和 Sn 同收敛发散<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty} s_n=s=\sum_{i=1}^\infty u_i=\lim_{n->\infty}\sum_{i=1}^n u_i" contenteditable="false"><span></span><span></span></span><br>
常见级数<br>
等比(几何)级数<br><span class="equation-text" data-index="0" data-equation="\sum_{i=1}^\infty aq_i=a+...+aq^i+..." contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="|q|<1 收敛 \lim s_n=a/(1-q)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="|q|\geq 1 发散[极限为无穷，极限不存在]" contenteditable="false"><span></span><span></span></span>
p 级数<br>
调和级数<br><span class="equation-text" data-index="0" data-equation="\sum_{i=1}^\infty \frac{1}{n_i} = 1+\frac{1}{2}+...+\frac{1}{n}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sum_{i=1}^\infty \frac{1}{n^p}=\begin{cases}发散,p\leq 1\\收敛,p>1\end{cases}" contenteditable="false"><span></span><span></span></span>
基本性质<br>
性质一：级数每项乘以不为0的常数，收敛性不改变<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty} \sum ku_i=k\lim_{n->\infty} \sum u_i=k\lim_{n->\infty} S_n" contenteditable="false"><span></span><span></span></span><br>
性质二：收敛级数加减仍收敛；两个收敛级数可以逐项相加或相减<br><span class="equation-text" data-index="0" data-equation="\sum u_i+\sum v_i=\sum u_i\pm v_i" contenteditable="false"><span></span><span></span></span><br>
性质三：级数中修改有限项，不会改变级数收敛性<br>【即真正决定敛散性的不是有限项】<br>
性质四：如果原级数收敛，则加括号仍收敛<br>【逆否：如果加括号后级数发散，则原级数发散】<br>
性质五 (<b>必要</b>条件)<br>若级数收敛，则其一般项 un 趋于零<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow\infty} u_n=0" contenteditable="false"><span></span><span></span></span><br>
注：必要而非充分，例如 调和级数<br><span class="equation-text" data-index="0" data-equation="1+1/2+...+1/n+..." contenteditable="false"><span></span><span></span></span><br>
收敛数列必有界
敛散性判断<br>
柯西审敛原理 (<b>充要条件</b>)<br><span class="equation-text" data-index="0" data-equation="\forall \xi,\exist N,n>N时,\forall 正整数p，|u_{n+1}+...+u_{n+p}|<\xi" contenteditable="false"><span></span><span></span></span><br>
常数项级数审敛法<br>
正向级数<br>
<b>(充要条件</b>) 部分和数列 Sn 有界<br>
比较审敛法<br>
推论<br><span class="equation-text" data-index="0" data-equation="u_i\leq v_i, \exist N,n\geq N时有u_n" contenteditable="false"><span></span><span></span></span><br>
比较审敛法【极限形式】<br>
v 收敛 -> u 收敛【k倍】<br><span class="equation-text" data-index="0" data-equation="\lim u_n/v_n=l (0\leq l < +\infty)" contenteditable="false"><span></span><span></span></span>
v 发散 -> u 发散【发散->发散】<br><span class="equation-text" data-index="0" data-equation="\lim u_n/v_n=l(l>0)" contenteditable="false"><span></span><span></span></span><br>
常见比较级数
p 级数<br>
等比级数<br>
比值审敛法【达朗贝尔判别法】<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty} u_{n+1}/u_{n} = \rho \\ \rho<1收敛,\rho >1 发散，\rho=1都有可能" contenteditable="false"><span></span><span></span></span>
根值审敛法【柯西判别法】
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty} \sqrt{ u_n }=\rho\\\rho<1收敛,\rho >1 发散，\rho=1都有可能" contenteditable="false"><span></span><span></span></span>
极限审敛法<br>
级数发散<br><span class="equation-text" data-index="0" data-equation="\lim nu_n=l>0(或+\infty) 级数发散\\\lim u_n/\frac{1}{n}，与调和级数比较，发散->发散" contenteditable="false"><span></span><span></span></span>
级数收敛<br><span class="equation-text" data-index="0" data-equation="p>1,而\lim n^pu_n=l(0\leq l<+\infty)\\与收敛p级数比较，收敛->收敛" contenteditable="false"><span></span><span></span></span><br>
积分审敛法<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)在[1,+\infty)上单减，非负的连续函数，且a_n=f(n),则\sum_1^{\infty} a_n与\int_1^{+\infty}f(x)dx同敛散" contenteditable="false"><span></span><span></span></span>
交错级数
定义<br>
各项正负交错<br><span class="equation-text" data-index="0" data-equation="\lim_{n=1}^{\infty}(-1)^{n-1}u_n" contenteditable="false"><span></span><span></span></span><br>
莱布尼兹定理<br>
交错级数收敛 <b>充分条件</b><br><span class="equation-text" data-index="0" data-equation="1) u_n \geq u_{n+1}\\2)\lim_{n\rightarrow \infty}u_n=0\\则 s\leq u_1,余项|r_n|\leq u_{n+1}" contenteditable="false"><span></span><span></span></span>
绝对收敛/条件收敛<br>
定义<br>
绝对收敛（充分条件）<br>对于级数 un 各项绝对值所构成的正向级数 |un| 收敛则称为绝对收敛<br>
条件收敛<br>如果原级数收敛而其绝对值所构成的正向级数发散，则称为条件收敛【部分交错级数】<br>
相关定理
如果级数<b>绝对收敛</b>，则级数必然收敛<br>
绝对收敛性质<br>
绝对收敛级数具有可交换性<br>
绝对收敛级数的乘法<br>
加减收敛性
绝对收敛+条件收敛=条件收敛<br>
绝对收敛+绝对收敛=绝对收敛<br>
条件收敛+条件收敛=两种都有可能<br>
幂级数<br>
前置概念
无穷级数
表达式<br><span class="equation-text" data-index="0" data-equation="u_1(x)+...+u_n(x)+..." contenteditable="false"><span></span><span></span></span>
收敛点/发散点<br>
收敛域/发散域 [极限收敛点的全体]<br>
收敛区间 (-R,R)<br>
定义<br>
表达式<br><span class="equation-text" data-index="0" data-equation="\sum_{n=0}^\infty a_nx^n=a_0+..+a_nx^n+..." contenteditable="false"><span></span><span></span></span>
幂级数系数 a0,..an<br>
阿贝尔定理<br>
推论
幂级数收敛半径<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty}|a_{n+1}/a_n|=\rho,其中a_n,a_{n+1}为相邻两项的系数，\\则幂级数收敛半径为,R= \begin{cases} 1/\rho,\rho\neq0 \\ +\infty,\rho=0 \\ 0,\rho=+\infty \end{cases}" contenteditable="false"><span></span><span></span></span>
求解方法
公式<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty}|a_{n+1}/a_n|=\rho, R=1/\rho" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty}\sqrt[n]{|a_n|}=\rho, R=1/\rho" contenteditable="false"><span></span><span></span></span>
对于只有偶次或几次项幂级数<br>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty}|a_{n+1}/a_n|=\rho, R=\sqrt{1/\rho}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{n->\infty}\sqrt[n]{|a_n|}=\rho, R=\sqrt{1/\rho}" contenteditable="false"><span></span><span></span></span>
阿贝尔定理<br>
<br><span class="equation-text" data-index="0" data-equation="\sum_{n=0}^{\infty} a_nx^n当 x=x_0(x\neq 0)收敛，当|x|<|x_0|时，收敛" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sum_{n=0}^{\infty} a_nx^n当 x=x_0(x\neq 0)发散，当|x|>|x_0|时，发散" contenteditable="false"><span></span><span></span></span>
推论得出收敛半径对称
幂级数运算<br>
性质1 幂级数的和函数s(x)在其收敛域上连续<br>
性质2 幂级数和函数s(x)在其收敛域上可积，并有逐项积分公式<br><span class="equation-text" data-index="0" data-equation="\int_0^x s(t)dt=\int_0^x[\sum a_nt^n]=\sum\int_0^x a_nt^ndt=\sum \frac{a_n}{n+1}x^{n+1}" contenteditable="false"><span></span><span></span></span><br>
性质3 幂级数和函s(x)在其收敛区间可导，且有逐项求导公式<br><span class="equation-text" data-index="0" data-equation="s'(x)=(\sum a_nx^n)'=\sum(a_nx^n)'=\sum na_nx^{n-1}" contenteditable="false"><span></span><span></span></span><br>
推论：幂级数和函数s(x)在其收敛区间具有<b>任意阶</b>导数<br>
四则运算法则<br>两个收敛的幂级数四则运算后收敛半径为R=min(R1,R2)<br>
函数展开为幂级数
泰勒级数/麦克劳林级数<br>
形如下面的幂级数叫做 函数f(x) 在 x0 处的幂级数<br><span class="equation-text" data-index="0" data-equation="\sum_{n=0}^{\infty} \frac{1}{n!} f^{(n)}(x_0)(x-x_0)^n,x\in U(x_0)" contenteditable="false"><span></span><span></span></span>
x0 = 0处为麦克劳林级数<br><span class="equation-text" data-index="0" data-equation="\sum_{n=0}^{\infty} \frac{1}{n!} f^{(n)}(0)x^n,x\in U(x_0)" contenteditable="false"><span></span><span></span></span><br>
泰勒展开式/麦克劳林展开式<br>
展开式【等式】叫做 函数f(x) 在 x0 处的泰勒展开式 <br><span class="equation-text" data-index="0" data-equation="f(x)=\sum_{n=0}^{\infty} \frac{1}{n!} f^{(n)}(x_0)(x-x_0)^n,x\in U(x_0)" contenteditable="false"><span></span><span></span></span>
函数f(x) 能在收敛于 (-r,r) 内展开为x的幂级数，称为麦克劳林级数<br><span class="equation-text" data-index="0" data-equation="f(x)=\sum_{n=0}^{\infty} \frac{1}{n!} f^{(n)}(x_0)(x-x_0)^n,x\in U(x_0)" contenteditable="false"><span></span><span></span></span>
泰勒展开<b>充要条件</b><br>在某邻域 U(x0) 内，f(x)的泰勒公式中的余项Rn(x),当 n 趋近于 ∞ 时，极限为0<br>
展开方法<br>
直接展开法【步骤】
求出f(x)的各阶导数，f'(x)...f''(x)..., 如果某阶导数不存在则停止，也就不能展开为幂级数<br>
判断余项是否为0<br>
间接展开法
利用已知幂级数进行【四则、逐项求导、逐项积分】及变量代换方法<br>
泰勒级数/公式/展开式<br>
微分方程的幂级数解法
常用展开式<br>
正负交替
1/(1+x) (-1<x<1)<br><span class="equation-text" data-index="0" data-equation="1/(1+x)=1-x+x^2-...+(-1)^nx^n+...=\sum_{n=0}^\infty (-1)^nx^n" contenteditable="false"><span></span><span></span></span>
运用【几何分布期望计算】<br><span class="equation-text" data-index="0" data-equation="1+...+nx^{n-1}+...=(x+...+x^n+...)'=(\frac{x}{1-x})'=\frac{1}{(1-x)^2}" contenteditable="false"><span></span><span></span></span><br>
sin x (负无穷，正无穷)<br><span class="equation-text" data-index="0" data-equation="\sin x=\sum_{n=0}^\infty (-1)^{n}x^{2n+1}/(2n+1)!" contenteditable="false"><span></span><span></span></span><br>
cos x （负无穷，正无穷）<br><span class="equation-text" data-index="0" data-equation="\cos x=\sum_{n=0}^\infty (-1)^nx^{2n}/(2n)!" contenteditable="false"><span></span><span></span></span>
ln x (-1<x<=1)<br><span class="equation-text" data-index="0" data-equation="\ln(1+x)=\sum_{n=0}^\infty (-1)^{n-1}x^n/n" contenteditable="false"><span></span><span></span></span><br>
arctan x<br><span class="equation-text" data-index="0" data-equation="\arctan x=\int_0^x dt/(1+t^2)=\int_0^x (\sum_{n=0}^\infty(-1)^nt^{2n})dt=\sum_{n=0}^{\infty} (-1)^nx^{2n+1}/(2n+1)" contenteditable="false"><span></span><span></span></span>
系数恒正
1/(1-x) (-1<x<1)<br><span class="equation-text" data-index="0" data-equation="1/(1-x)=1+x+x^2+...+x^n+...=\sum_{n=0}^\infty x^n" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="e^x" contenteditable="false"><span></span><span></span></span>(负无穷，正无穷)<br><span class="equation-text" data-index="1" data-equation="e^x=\sum_{n=0}^\infty x^n/n!" contenteditable="false"><span></span><span></span></span>
运用【泊松分布期望计算】<br><span class="equation-text" data-index="0" data-equation="\sum_{k=1}^\infty \frac{\lambda^{k-1}}{(k-1)!}=e^{\lambda}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="(1+x)^a" contenteditable="false"><span></span><span></span></span> (-1<x<1)<br><span class="equation-text" data-index="1" data-equation="(1+x)^a=\sum_{n=0}^\infty a(a-1)...(a-n+1)/n! *x^n" contenteditable="false"><span></span><span></span></span>
傅里叶级数<br>
前置概念<br>
周期函数<br><span class="equation-text" data-index="0" data-equation="y=A\sin(\omega t+\varphi);\\A振幅，\omega 角频率，\varphi 初相" contenteditable="false"><span></span><span></span></span><br>
收敛定理[狄利克雷]<br>
<br><span class="equation-text" data-index="0" data-equation="设f(x)在[-\pi,\pi]上的分段单调函数，除有限个第一类间断点外都是连续的，则f(x)的傅里叶级数在[-\pi,\pi]处处收敛" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(1)f(x),x为连续点\\(2)[f(x^-)+f(x^+)]/2,x为间断点\\(3)[f(-\pi^+)+f(\pi^-)]/2,当x=\pm\pi" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="周期为 2\pi 函数展开" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="周期为 2l 函数展开" contenteditable="false"><span></span><span></span></span>
常考题型<br>
收敛定理<br>
函数展开为傅里叶级数<br>
不定积分<br>
一元函数不定积分<br>
原函数<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="F'(x)=f(x)或dF(x)=f(x)dx" contenteditable="false"><span></span><span></span></span>
存在原函数的导数性质<br>
f(x)不一定连续<br>
f(x)不一定是初等函数
F(x)不一定是初等函数
由原函数定义，F'(x)=f(x)，因而F(x)连续<br>
不定积分定义
<br><span class="equation-text" data-index="0" data-equation="\int f(x)dx,\int 积分号，f(x)被积函数,f(x)dx被积表达式,x积分变量" contenteditable="false"><span></span><span></span></span>
积分表<br>
三角函数<br>
<br><span class="equation-text" data-index="0" data-equation="\int \cos xdx=\sin x+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="-\int \sin xdx=\cos x+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \sec ^2xdx=\int \frac{dx}{\cos^2x}=\tan x+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="-\int \csc ^2xdx=-\int \frac{dx}{\sin^2x}=\cot x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \sec^3 xdx=\frac{1}{2}(\sec x \tan x+\ln|\sec x+\tan x|)+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \sec x\tan xdx=\sec x+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="-\int \csc x\cot xdx=\csc x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \tan xdx=-\ln|\cos x| +C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \cot xdx=\ln|\sin x|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \sec xdx=\ln |\sec x+\tan x|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \csc xdx=\ln|\csc x-\cot x|+C" contenteditable="false"><span></span><span></span></span>
幂函数
<br><span class="equation-text" data-index="0" data-equation="\int x^{-1}dx=\int \frac{dx}{x}=\ln|x|+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int x^ndx=\frac{x^{n+1}}{n+1}+C(n\neq-1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int kdx=kx+C" contenteditable="false"><span></span><span></span></span>
指数函数
<br><span class="equation-text" data-index="0" data-equation="\int a^xdx=\frac{a^x}{\ln a}+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int e^xdx=e^x+C" contenteditable="false"><span></span><span></span></span>
平方和/差
<br><span class="equation-text" data-index="0" data-equation="\int \frac{dx}{1+x^2}=\arctan x+C" contenteditable="false"><span></span><span></span></span>
注意 x 所代表的函数性<br><span class="equation-text" data-index="0" data-equation="\int \frac{dx}{a^2+p^2(x)}=\arctan \frac{p(x)}{a}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \frac{dx}{\sqrt{1^2-x^2}}= \arcsin x+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="-\int \frac{dx}{\sqrt{a^2-x^2}}=\arccos \frac{x}{a}+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{dp(x)}{\sqrt{a^2-p^2(x)}}=\int \frac{d\frac{p(x)}{a}}{\sqrt{1-\frac{p(x)^2}{a^2}}} =\arcsin \frac{p(x)}{a}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \frac{dx}{\sqrt{a^2-x^2}}= \int \frac{d\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} = \arcsin \frac{x}{a}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln|x+\sqrt{x^2-a^2}|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln|x+\sqrt{x^2+a^2}|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int \sqrt{a^2-x^2}dx = \frac{a^2}{2} \arcsin \frac{x}{a} + \frac{x}{2} \sqrt{a^2-x^2} +C" contenteditable="false"><span></span><span></span></span>
分式 -> 对数<br><span class="equation-text" data-index="0" data-equation="\int \frac{dx}{x}=\ln|x|+C" contenteditable="false"><span></span><span></span></span><br>
大在下，小在上【个人结论，很好用】<br><span class="equation-text" data-index="0" data-equation="2\int \frac{1}{t^2-1}dt=2\int \frac{1}{(t+1)(t-1)}dt=\int (\frac{t+1}{t-1})(\frac{t+1-(t-1)}{(t+1)^2})=\ln\frac{|1-x|}{|1+x|}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{1}{(t+a)(t-b)}dt=\frac{1}{a+b}\int (\frac{t+a}{t-b})(\frac{(t+a)-(a-b)}{(t+a)^2})=\frac{1}{a+b}\ln|\frac{t-b}{t+a}|=\frac{1}{a+b}[\ln(t+a)-\ln(t-b)]" contenteditable="false"><span></span><span></span></span>
sin x+cos x 分式配凑<br><span class="equation-text" data-index="0" data-equation="\int_0^{\pi/2} \frac{\sin x}{\sin x+ \cos x}dx=\frac{1}{2}\int_0^{\pi/2} [\frac{\sin x- \cos x}{\sin x + \cos x}+1]dx=\frac{1}{2}[-\ln(\sin x+\cos x)+x]|_0^{\pi/2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{dx}{x\ln^2x}=-\frac{1}{\ln x}+C" contenteditable="false"><span></span><span></span></span>
性质
<br><span class="equation-text" data-index="0" data-equation="\int [af(x)+ bg(x)]dx=a\int f(x)dx+b\int g(x) dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[\int [af(x)+ bg(x)]dx]'=af(x)+bg(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int kf(t)dx=k\int f(t)dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int Cdx=C(b-a)" contenteditable="false"><span></span><span></span></span>
积分中值定理<br>
<br><span class="equation-text" data-index="0" data-equation="\int_0^x f(x)dx=xf(\xi)" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="(0\leq \xi\leq x)" contenteditable="false"><span></span><span></span></span>
抵消原理<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dx}[\int f(x)dx]=f(x) ;d[\int f(x)dx]=f(x)dx" contenteditable="false"><span></span><span></span></span>
补充
<br><span class="equation-text" data-index="0" data-equation="\int_a^af(x)dx=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^b f(x)dx=-\int_b^a f(x)dx" contenteditable="false"><span></span><span></span></span>
求解方法<br>
主要积分法
分部积分
<br><span class="equation-text" data-index="0" data-equation="\int u(x)v'(x)dx=[u(x)v(x)]-\int u'(x)v(x)dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int udv=uv-\int vdu" contenteditable="false"><span></span><span></span></span>
常见情形<br>
<br><span class="equation-text" data-index="0" data-equation="两类不同函数相乘不定积分" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int p_n(x)e^{ax}dx=\frac{1}{a}\int p_n(x)de^{ax}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="p_n(x)e^{ax}、p_n(x)\sin ax、p_n(x)\cos ax 同一类：将多项式以外先凑进微分号" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int p_n(x)\ln xdx=\int \ln xdp_{n1}(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="p_n(x)\ln x，p_n(x)\arctan x,p_n(x) arcsin x, 对数反三角先将多项式往后凑" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int e^{ax}\sin \beta xdx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int e^{ax}\sin \beta xdx,\int e^{ax}\cos \beta xdx, 两次积分，将指数凑入微分号[移星大法]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int p(e^x)\arctan e^xdx, 有先指数凑入微分号" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="把e^x当作t, 得到p_n(t)\arctan tdt形式" contenteditable="false"><span></span><span></span></span>
高中口诀：<b>反对幂指三</b><br>
第一类换元法(配凑法)<br>
条件<br>
<br><span class="equation-text" data-index="0" data-equation="\varphi(t)在[\alpha,\beta]上有连续导数，且值域为R=[a,b]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\varphi(\alpha)=a,\varphi(\beta)=b" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int f(x)dx=\int f[\varphi(t)]\varphi'(t)dt=F(\varphi(t))+C=[\int f(u)du]|_{u=\varphi(t)}" contenteditable="false"><span></span><span></span></span>
常见配凑形式
<br><span class="equation-text" data-index="0" data-equation="dx=d(x\pm c)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int Cd(x+a)=C(x+a)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int dx=\int (p_n(x)+1-p_n(x))dx" contenteditable="false"><span></span><span></span></span>
根式<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{1}{\sqrt{x}}dx=2d\sqrt{x}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{e^x}{2\sqrt{e^x+C}}dx=d\sqrt{e^x+C}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{P'(x)}{2\sqrt{P(x)+C}}dx=d\sqrt{P(x)+C}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{1}{x}dx=\frac{1}{n*x^n}dx^n" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{1}{\sqrt{x(a-x)}}dx=\int \frac{1}{\sqrt{\frac{a^2}{4}-(x-\frac{a}{2})^2}}dx=\arcsin t" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{1}{2\sqrt{1+x^n}}dx^n=d(\sqrt{1+x^n})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int x\sqrt{1+x}dx=\int(x+1-1)\sqrt{1+x}d(1+x)=\int [t\sqrt t-\sqrt t]dt" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \sqrt{1+x}dx=\int td(t^2-1)=\int 2t^2dt" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \sqrt{e^x\pm 1}dx=\int td\ln(t^2\pm 1)【t=\sqrt{e^x\pm 1}】=\int \frac{t^2}{t^2\pm1}dt;\\\frac{1}{t^2+1}dt\Longrightarrow \arctan t;\frac{1}{t^2-1}dt \Longrightarrow \ln|\frac{t-1}{t+1}|" contenteditable="false"><span></span><span></span></span>
分式
<br><span class="equation-text" data-index="0" data-equation="\int \frac{1}{x+x^n}dx=\int \frac{1}{x(1+x^{n-1})}dx=\frac{1}{n-1}\int \frac{1}{x^{n-1}(x^{n-1}+1)}dx^{n-1}=\frac{1}{n-1}\int \frac{1}{u(u+1)}du" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{1}{(t+a)(t-b)}dt=\frac{1}{a+b}\int (\frac{t+a}{t-b})(\frac{(t+a)-(t-b)}{(t+a)^2})=\frac{1}{a+b}\ln|\frac{t-b}{t+a}|=\frac{1}{a+b}[\ln(t-b)-\ln(t+a)]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{t+1}{t^2+t+1}dt=\frac{1}{2}\int \frac{2t+1+1}{t^2+t+1}dt=\frac{1}{2} \ln|t^2+t+1|+\int \frac{1}{(t+\frac{1}{2})^2+(\frac{\sqrt{3}}{2})^2}d(t+\frac{1}{2})=" contenteditable="false"><span></span><span></span></span>
三角<br>
<br><span class="equation-text" data-index="0" data-equation="\int \frac{dx}{1+\sin x}" contenteditable="false"><span></span><span></span></span>
第二类换元法(反函数法)
三种常见变量代换<br>
<br><span class="equation-text" data-index="0" data-equation="\sqrt{a^2-x^2},令x=a\sin t或a\cos t" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sqrt{a^2+x^2},令x=a\tan t" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sqrt{x^2 - a^2},令x=a\sec t" contenteditable="false"><span></span><span></span></span>
常见可积函数<br>
有理函数积分<br>
有理分式<br>
<br><span class="equation-text" data-index="0" data-equation="两个多项式的商\frac{P(x)}{Q(x)}" contenteditable="false"><span></span><span></span></span>
真分式<br>
<br><span class="equation-text" data-index="0" data-equation="P(x)<Q(x)(x次数)" contenteditable="false"><span></span><span></span></span>
假分式
<br><span class="equation-text" data-index="0" data-equation="P(x)>Q(x)(x次数)" contenteditable="false"><span></span><span></span></span>
化简<br>
<span class="equation-text" data-index="0" data-equation="\frac{P(x)}{Q(x)}=\frac{P_1(x)}{Q_1(x)}+\frac{P_2(x)}{Q_2(x)}" contenteditable="false"><span></span><span></span></span>
最简形式<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{P_1(x)}{(x-a)^k}；\frac{P_2(x)}{(x^2+px+q)^l}(p^2-4q<0)" contenteditable="false"><span></span><span></span></span>
化简方法<br>
三角函数万能公式法(不推荐使用)<br>
对数化简
化乘除为加减
常见多项式配凑<br>
<br><span class="equation-text" data-index="0" data-equation="(1+x^n)=(1+x)(1-x+x^2-..+x^{n-1})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(1+x)^n=C_n^0+..+C_n^nx^n=(1+x)(C_n^0+..+C_{n-1}^{n-1}x^{n-1})" contenteditable="false"><span></span><span></span></span>
三角有理式<br>
<br><span class="equation-text" data-index="0" data-equation="R(-\sin x,\cos x)=-R(\sin x,\cos x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="u = \cos x,凑 d\cos x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="R(\sin x,-\cos x)=-R(\sin x,\cos x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{1}{\cos x \varphi(sinx)}dx =-\frac{1}{\cos^2 x\varphi(sinx)}d\sin x =\frac{1}{\sin^2 x \varphi(\sin x)}d\sin x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="u = \sin x,凑 d\sin x" contenteditable="false"><span></span><span></span></span>
简单无理函数<br>
<br><span class="equation-text" data-index="0" data-equation="\int R(x,\sqrt[n]{\frac{ax+b}{cx+d}})dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="分式(子)有理化" contenteditable="false"><span></span><span></span></span>
根式代换/无理式换元<br>
<br><span class="equation-text" data-index="0" data-equation="\sqrt[n]{ax+b}或\sqrt[n]{\frac{ax+b}{cx+d}}" contenteditable="false"><span></span><span></span></span>
一般技巧<br>
拆项 裂项<br>
同乘 同除<br>
升幂 降幂<br>
三角恒等变换
积分重现 抵消<br>
不定积分存在定理<br>(可积性判断)<br>
连续必有原函数<br><span class="equation-text" data-index="0" data-equation="如果函数f(x)在区间I连续，则在区间I存在可导函数F(x)，使得任一x\in I都有F'(x)=f(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="函数在有第一类或无穷间断点区间内，必不存在原函数" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="函数有振荡间断点时不一定存在原函数" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases} 2x\sin \frac{1}{x}-\cos x ,x\neq 0,\\ 0, x=0 \end{cases} 存在原函数" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases} \frac{1}{x}\sin \frac{1}{x} ,x\neq 0,\\ 0, x=0 \end{cases} 不存在原函数" contenteditable="false"><span></span><span></span></span>
微分方程<br>
常微分方程
定义
表示未知函数、未知函数导数与自变量之间关系的方程
概念<br>
方程的阶
通解
特解
初值条件/初值问题
微分方程积分曲线
积分因子法
类型
可分离变量微分方程
一阶<br><span class="equation-text" data-index="0" data-equation="y'=f(x,y)" contenteditable="false"><span></span><span></span></span>
一般形式<br><span class="equation-text" data-index="0" data-equation="\int g(y)dy=\int f(x)dx；G(y)=F(x)+C； \phi'(x)=\frac{F'(x)}{G'(y)}=\frac{f(x)}{g(y)}" contenteditable="false"><span></span><span></span></span>
换元<br>
<br><span class="equation-text" data-index="0" data-equation="y'=\cos(x+y)" contenteditable="false"><span></span><span></span></span>
齐次方程<br>
前提
<br><span class="equation-text" data-index="0" data-equation="dy/dx=\varphi(y/x)" contenteditable="false"><span></span><span></span></span>
求解
令 y=ux<br>
<br><span class="equation-text" data-index="0" data-equation="dy/dx=u+x(du/dx)=\varphi(u)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x/dx=(\varphi(u)-u)/du" contenteditable="false"><span></span><span></span></span>
可化为齐次方程形式<br>
<br><span class="equation-text" data-index="0" data-equation="dy/dx=(ax+by+c)/(a_1x+b_1y+c_1)\\ c=c1=0时为齐次" contenteditable="false"><span></span><span></span></span>
非齐次化为齐次<br>
一阶线性微分方程<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="y'+P(x)y=Q(x)" contenteditable="false"><span></span><span></span></span>
类型<br>
齐次
可分类变量还原<br><span class="equation-text" data-index="0" data-equation="dy/dx=-P(x)y\iff dy/y=-P(x)dx" contenteditable="false"><span></span><span></span></span><br>
通解公式<br><span class="equation-text" data-index="0" data-equation="y=Ce^{-\int P(x)dx}(C=\pm e^{C_1})" contenteditable="false"><span></span><span></span></span>
非齐次
常数变易法<br><span class="equation-text" data-index="0" data-equation="将C换做u(x),y=ue^{-\int P(x)dx}\\变形得: u=\int Q(x)e^{\int P(x)dx}dx+C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y=(\int Q(x)e^{\int P(x)dx}dx【非齐次特解】+C【齐次通解】)e^{-\int P(x)dx}" contenteditable="false"><span></span><span></span></span>
反解
<br><span class="equation-text" data-index="0" data-equation="y'=\frac{1}{xy+y^3}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(x-\sin y)dy+\tan ydx=0" contenteditable="false"><span></span><span></span></span>
证明
伯努利方程
定义<br>
形如<br><span class="equation-text" data-index="0" data-equation="y'+p(x)y=Q(x)y^n" contenteditable="false"><span></span><span></span></span>
解法<br>
<br><span class="equation-text" data-index="0" data-equation="令u=y^{1-n},将原方程还原为一阶线性微分方程" contenteditable="false"><span></span><span></span></span>
全微分方程
定义<br>
<br><span class="equation-text" data-index="0" data-equation="du(x,y)=P(x,y)dx+Q(x,y)dy" contenteditable="false"><span></span><span></span></span>
解法<br>
偏积分
凑微分<br>
线积分<br>
充要条件
<br><span class="equation-text" data-index="0" data-equation="P(x,y)dx+Q(x,y)dy=0, \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}" contenteditable="false"><span></span><span></span></span>
可降阶高阶微分方程
直接可积型<br><span class="equation-text" data-index="0" data-equation="y^{(n)}=f(x)" contenteditable="false"><span></span><span></span></span><br>
半隐<br><span class="equation-text" data-index="0" data-equation="y''=f(x,y');f(x,p)" contenteditable="false"><span></span><span></span></span><br>
解法<br><span class="equation-text" data-index="0" data-equation="设y'=p,自然y''=p'\\即转换为 p'=f(x,p)，化为一阶线微方" contenteditable="false"><span></span><span></span></span>
全隐<br><span class="equation-text" data-index="0" data-equation="y''=f(y,y');f(y,p)" contenteditable="false"><span></span><span></span></span><br>
解法<br><span class="equation-text" data-index="0" data-equation="y'=p,y''=p\frac{dp}{dy}\rightarrow p\frac{dp}{dy}=f(y,p) \\ 分离变量积分" contenteditable="false"><span></span><span></span></span>
高阶线性微分方程<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="y''=f(x,y,y')型" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=f(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y''+P(x)y'+Q(x)y=f(x)" contenteditable="false"><span></span><span></span></span>
类型
齐次
常系数齐次线性微分方程<br><span class="equation-text" data-index="0" data-equation="y''+py'+qy=0【p、q为常数】" contenteditable="false"><span></span><span></span></span><br>
解法<br>
<br><span class="equation-text" data-index="0" data-equation="指数函数容易满足该形式,设y=e^{rx}\\带入后化简得到(r^2+pr+q)e^{rx}=0" contenteditable="false"><span></span><span></span></span>
特征方程<br><span class="equation-text" data-index="0" data-equation="r^2+pr+q=0" contenteditable="false"><span></span><span></span></span><br>
三种根情况<br>
<br><span class="equation-text" data-index="0" data-equation="有解且r_1=r_2【p^2-4q=0】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y=(C_1+C_2x)e^{r_1x}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="有解且r_1\neq r_2【p^2-4q>0】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y=C_1e^{r_1x}+C_2e^{r_2x}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="【p^2-4q<0】\\r_1=\alpha+\beta i,r_2=\alpha - \beta i" contenteditable="false"><span></span><span></span></span>
无解，两个共轭复根<br><span class="equation-text" data-index="0" data-equation="y=e^{\alpha x}(C_1cos\beta x+C_2\sin\beta x)" contenteditable="false"><span></span><span></span></span>
非齐次
常系数齐次线性微分方程<br><span class="equation-text" data-index="0" data-equation="y''+py'+qy=f(x)【p、q为常数】" contenteditable="false"><span></span><span></span></span><br>
形式<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)=e^{\lambda x}P_m(x)型" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f(x)=e^{\lambda x}[{P_l(x)\cos \omega x+Q_n(x)\sin \omega x}] 型" contenteditable="false"><span></span><span></span></span>
特解
<br><span class="equation-text" data-index="0" data-equation="y^*=x^kQ_m(x)e^{\lambda x}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y^*=x^ke^{\alpha x}[R_m^{(1)}(x)\cos \omega x+R_m^{(2)}(x)\sin \omega x] 型" contenteditable="false"><span></span><span></span></span>
解结构
齐次方程解=非齐次方程两个特解的差<br><span class="equation-text" data-index="0" data-equation="y(x)=y_2^*(x)-y_1^*(x)" contenteditable="false"><span></span><span></span></span>
非齐次方程通解形式=齐次方程两个线性无关特解+非齐次特解<br><span class="equation-text" data-index="0" data-equation="y=C_1y_1(x)+C_2y_2(x)+y^*(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="y_1^*,y_2^*为y''+py'+qy=f_1(x),y''+py'+qy=f_2(x)的特解\\y_1^*+y_2^*是方程y''+py'+qy=f_1(x)+f_2(x)的特解" contenteditable="false"><span></span><span></span></span>
常数变易法
欧拉方程<br>
定义<br>
形如<br><span class="equation-text" data-index="0" data-equation="x^ny^{(n)}+p_1x^{n-1}y^{(n-1)}+...+p_ny=f(x)" contenteditable="false"><span></span><span></span></span>
求解
令x=<span class="equation-text" data-index="0" data-equation="e^t" contenteditable="false"><span></span><span></span></span>或t=ln x,可将上述欧拉方程化为 <b>线性常系数微分方程</b><br><span class="equation-text" data-index="1" data-equation="x^ky^{(k)}=D(D-1)\cdot \cdot(D-k+1)y" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="D为 \frac{d}{dt},对t的微分算子" contenteditable="false"><span></span><span></span></span>
真题<br>
2021，(二)，13<br>
性质
解的叠加原理<br>
解的结构
导数/偏导<br>
一元函数导数<br>
引例
直线运动速度<br>
切线问题
定义
<div>定义式【极限形式: 必须是 <b><font color="#B71C1C">动点 - 定点/自变量 </font></b>】<br></div><span class="equation-text" data-index="0" data-equation="\lim_{\Delta x \rightarrow0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x \rightarrow0}\frac{f(\Delta x+x_0)-f(x_0)}{\Delta x}=\lim_{ x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}" contenteditable="false"><span></span><span></span></span>
导数与微分<br><span class="equation-text" data-index="0" data-equation="\lim_{ x \rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}-f'(x_0)=0 \implies \lim_{ x \rightarrow x_0}\frac{f(x)-f(x_0)-f'(x_0)(x-x_0)}{x-x_0}=0" contenteditable="false"><span></span><span></span></span><br>
极限 lim表示 与 无穷小表示法<br><span class="equation-text" data-index="0" data-equation="\lim_{\Delta x \rightarrow0}\frac{f(\Delta x+x_0)-f(x_0)}{\Delta x}=f'(x_0)\iff\frac{f(\Delta x+x_0)-f(x_0)}{\Delta x}=f'(x_0)+o(\Delta x)" contenteditable="false"><span></span><span></span></span><br>
书写规范<br><span class="equation-text" data-index="0" data-equation="记作:y'|_{x=x_0}, \frac{dy}{dx}|_{x=x_0},或\frac{df(x)}{dx}|_{x=x_0}" contenteditable="false"><span></span><span></span></span>
单侧导数<br><span class="equation-text" data-index="0" data-equation="\lim_{h\rightarrow 0^-}\frac{f(x_0+h)-f(x_0)}{h}=f_{-}'(x_0)【左导数】" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="单侧导数存在 \implies 单侧连续, 所以左右导数同时存在则函数在x_0处必然连续" contenteditable="false"><span></span><span></span></span>
<b>某点导数</b> 存在<b> 充要条件</b><br><span class="equation-text" data-index="0" data-equation="f_{-}'(x_0)=f_{+}'(x_0)【左右导数存在且相等】" contenteditable="false"><span></span><span></span></span><br>
求导法则
四则运算<br>
<br><span class="equation-text" data-index="0" data-equation="[v(x)\pm u(x)]'=u'(x)\pm v'(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[u(x)v(x)]'=u'(x)v(x)+u(x)v'(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(uvw)'=u'vw+uv'w+uvw'" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[\frac{u(x)}{v(x)}]'=\frac{u'(x)v(x)-u(x)v'(x)}{v^2(x)}" contenteditable="false"><span></span><span></span></span>
反函数导数<br>
公式<br><span class="equation-text" data-index="0" data-equation="[f^{-1}(x)]'=\frac{1}{f'(y)} 即\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}" contenteditable="false"><span></span><span></span></span>
证明 1<br>
<br><span class="equation-text" data-index="0" data-equation="y=f(x),x=f^{-1}(y)=g(y);" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dx}x=\frac{d}{dx}g(y)\implies 1=\frac{d}{dy}\frac{dy}{dx}g(y)【链式求导】\implies \frac{d}{dy}g'(y)=\frac{dx}{dy}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[f^{-1}(y)]_y'=g'_y(y)=\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}=\frac{1}{f'_x(x)}\implies [f^{-1}(x)]'_x=\frac{1}{f'(y)_y}" contenteditable="false"><span></span><span></span></span>
证明 2<br>
y 是关于 x 的 f 映射<br><span class="equation-text" data-index="0" data-equation="\frac{dy}{dx}=f'(x)_x=y'_x" contenteditable="false"><span></span><span></span></span>
x 是关于 y 的 f^(-1) 逆映射<br><span class="equation-text" data-index="0" data-equation="\frac{dx}{dy}=[f^{-1}(y)]_y=g(y)'_x" contenteditable="false"><span></span><span></span></span>
微商形式相等<br><span class="equation-text" data-index="0" data-equation="\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\iff f(x)'_x=\frac{1}{f^{-1}(y)_y}" contenteditable="false"><span></span><span></span></span><br>
复合函数求导<br>
微商法则<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d^2y}{dx^2}=\frac{d}{dx}\cdot \frac{dy}{dx}=\frac{d}{dy}\cdot \frac{dt}{dx}\cdot \frac{dy}{dx}" contenteditable="false"><span></span><span></span></span>
微分形式不变性
链式求导法则<br>
隐函数求导<br>
隐函数求导公式<br>
两端直接求导【暴力】<br>
对数求导法<br>
<br><span class="equation-text" data-index="0" data-equation="(\ln y)'=y'/y \implies y(\ln y)' = y'" contenteditable="false"><span></span><span></span></span>
参数方程求导<br>
一阶导数<br><span class="equation-text" data-index="0" data-equation="dy/dx=y'(t)/x'(t);" contenteditable="false"><span></span><span></span></span>
二阶导数<br><span class="equation-text" data-index="0" data-equation="d^2y/dx^2=[y''(t)x'(t)-x''(t)y'(t)]/x'^3(t)" contenteditable="false"><span></span><span></span></span><br>
性质<br>
奇偶性<br>
可到原函数的导数，奇偶性与原函数相反<br><span class="equation-text" data-index="0" data-equation="f(x)可导,f(x)为奇函数\implies f'(x)为偶函数" contenteditable="false"><span></span><span></span></span>
连续函数的原函数不一定与其奇偶相反<br><span class="equation-text" data-index="0" data-equation="f(x)为连续偶函数，则其原函数中有唯一一个奇函数F(x)+C(C=0)" contenteditable="false"><span></span><span></span></span><br>
周期性
求导公式<br>
三角与反三角
<span class="equation-text" data-index="0" style data-equation="\sin'x=\cos x" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\cos 'x=-\sin x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\tan 'x=\frac{\cos^2x+\sin^2x}{\cos^2x}=1+\tan^2x=sec^2x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cot 'x=-\frac{\sin^2x+\cos^2x}{sin^2x}=-(1+\cot^2x)=-\csc^2x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sec'x=\tan x\sec x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\csc'x=-\csc x\cot x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\arctan 'x=\frac{1}{\tan'y}=\frac{1}{\sec ^2y}=\frac{1}{1+tan^2y}=\frac{1}{1+x^2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="arccot 'x=-\frac{1}{\cot'y}=-\frac{1}{\csc ^2y}=-\frac{1}{1+\cot^2y}=-\frac{1}{1+x^2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\arcsin 'x=\frac{1}{\sin'y}=\frac{1}{\cos y}=\frac{1}{\sqrt{1-sin^2y}}=\frac{1}{\sqrt{1-x^2}}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\arccos 'x=\frac{1}{\cos'y}=-\frac{1}{\sin y}=-\frac{1}{\sqrt{1-cos^2y}}=-\frac{1}{\sqrt{1-x^2}}" contenteditable="false"><span></span><span></span></span>
对数与指数
<br><span class="equation-text" data-index="0" data-equation="(a^x)'=a^x\ln a" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(e^x)'=e^x\ln e=e^x" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(\log_ax)'=\frac{1}{x\ln a}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\ln'x=\frac{1}{x\ln e}=\frac{1}{x}" contenteditable="false"><span></span><span></span></span>
利用反函数记忆<br><span class="equation-text" data-index="0" data-equation="log_ax = y, x = a^y; \implies\frac{1} {a^y\ln a} =\frac{1}{x\ln a}" contenteditable="false"><span></span><span></span></span>
其他常用<br>
<br><span class="equation-text" data-index="0" data-equation="(x^a)' =ax^{a-1}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="C'=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[n\ln(1+\frac{1}{n})]'=[\ln(1+\frac{1}{n})-\frac{1}{1+n}]=\frac{1}{n}-\frac{1}{2!}\frac{1}{n^2}+o(\frac{1}{n^2})-\frac{1}{1+n}\sim \frac{1}{n^2}" contenteditable="false"><span></span><span></span></span>
高阶导数<br>
定义<br>
二阶导数<br>
<br><span class="equation-text" data-index="0" data-equation="a=\frac{dv}{dt}=(\frac{ds}{dt})\frac{d}{dt}=(s')'" contenteditable="false"><span></span><span></span></span>
n阶导数<br>
<br><span class="equation-text" data-index="0" data-equation="y''',y^{(4)},..,y^{(n)}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d^3y}{dx^3},...,\frac{d^ny}{dx^n}" contenteditable="false"><span></span><span></span></span>
计算
n阶求导四则<br>
<br><span class="equation-text" data-index="0" data-equation="(u\pm v)^{(n)}=u^{(n)}\pm v^{(n)}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(uv)^{(n)}=\sum_{k=0}^nC_n^ku^{(n-k)}v^{(k)}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(u+v)^n=\sum_{k=0}^nC_n^ku^{n-k}v^k" contenteditable="false"><span></span><span></span></span>
常见函数的n阶导数
<br><span class="equation-text" data-index="0" data-equation="\sin ^{(n)}x=\sin(x+n \cdot \frac{\pi}{2})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sin'x=\cos x=\sin(x+\frac{\pi}{2}) 由此类推" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\cos^{(n)}x=\cos(x+n\cdot \frac{\pi}{2})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\ln^{(n)}(1+x)=(-1)^{n-1}\frac{(n-1)!}{(1+x)^n}" contenteditable="false"><span></span><span></span></span>
与泰勒公式联系<br><span class="equation-text" data-index="0" data-equation="带入泰勒展式中\frac{f^{(n)}}{n!}(x-x_0)^n=(-1)^{n-1}\frac{1}{n}(x)^n" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="[x^{\mu}]^{(n)}=\frac{\mu!}{(\mu-n)!}x^{\mu-n}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(e^{-x})^{(n)}=(\frac{1}{e^x})^{(n)}=(-1)^n(e^{-1})" contenteditable="false"><span></span><span></span></span>
归纳法<br>
函数奇偶性<br>
泰勒展式
相关变化率
导数与可微<br>
<br>
<br><span class="equation-text" data-index="0" data-equation="f_{-}'(x_0)=f_{+}'(x_0)【左右导数存在且相等】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow x_0^-}f(x)=f(x_0)=\lim_{x\rightarrow x_0^+}f(x)" contenteditable="false"><span></span><span></span></span><br>
变上限求导<br>
<br><span class="equation-text" data-index="0" data-equation="\int_{\varphi(x)}^{\psi(x)}f(x)dx=\varphi'(x)f(\varphi(x))-\psi'(x)f(\psi(x))" contenteditable="false"><span></span><span></span></span>
微分不变性<br><span class="equation-text" data-index="0" data-equation="\int_{\varphi(x)}^{\psi(x)}f(x,t)dx=\int_{\varphi(x)}^{\psi(x)}\frac{\partial f(x,t)}{\partial x}dt+\varphi'(x)f(x,\varphi(x))-\psi'(x)f(x,\psi(x))" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(\int_a^bf(x,t)dt)'=\int_a^b\frac{\partial f(x,t)}{\partial x}dt" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dx}\int_0^x tf(t)dt=xf(x)" contenteditable="false"><span></span><span></span></span>
x<font color="#B71C1C"> 在外部就拆开，在函数内部就换元</font><br><span class="equation-text" data-index="0" data-equation="\int _a^{\varphi(x)} g(x)+f(\psi(x)t)dt" contenteditable="false"><span></span><span></span></span>
中值定理
性质<br>
极限保号性】<br>推导来源导数定义<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)在x_0可导，导函数大于0，则\exist \sigma >0,使得 f(x)>f(x_0),x\in(x_0,x_0+\sigma)" contenteditable="false"><span></span><span></span></span>
导函数与连续
可导与连续性
函数在某点可导，但导函数不一定连续【导数 ≠ 导函数】<br>
<span class="equation-text" data-index="0" data-equation="f(x) 在 x = a 处 n 阶可导，即 n-1 阶导函数连续（可利用求导公式求出 n-1 阶导函数）" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="f(x) 在 x = a 处有n阶连续导函数，则f(x)在x=a出可以进行n阶求导" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases}|x|^\alpha \sin\frac{1}{x},x\neq0 \\ 0, x=0 \end{cases}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="1) 在x=0处连续 \iff \alpha >0\\ 2) 在x=0处可导\iff \alpha>1; \\3) 在x=0处 f'(x)连续\iff \alpha > 2" contenteditable="false"><span></span><span></span></span>
连续与有界性
导数与原函数奇偶<br>
可到原函数的导数，奇偶性与原函数相反<br><span class="equation-text" data-index="0" data-equation="f(x)可导,f(x)为奇函数\implies f'(x)为偶函数" contenteditable="false"><span></span><span></span></span>
连续函数的原函数不一定与其奇偶相反<br><span class="equation-text" data-index="0" data-equation="f(x)为连续偶函数，则其原函数中有唯一一个时奇函数F(x)+C(C=0)" contenteditable="false"><span></span><span></span></span><br>
导函数的两个特性<br>
<span class="equation-text" data-index="0" data-equation="若 f(x) 在区间 I 上可导，则器导函数在 I 上不存在第一类间断点" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="【介值性】若 f(x) 在 [a,b] 上可导，且 f'_{-}(a)\neq f'_+(a)，\\\mu介于f'_+(a)与f'_{-}(b)之间的任何值，则至少存在 \xi \in (a,b) 使得 f'(\xi) = \mu" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="[a,b] 可导，且f'(x)\neq 0,则在[a,b]上 恒成立 f'(x) > 0或 f'(x)<0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="[a,b] 可导，且f'_+(a)f'_{-}(a)<0,则存在 \xi \in (a,b) 使得 f'(\xi) = 0" contenteditable="false"><span></span><span></span></span>
绝对值
<span class="equation-text" data-index="0" data-equation="f(x)在x_0连续，且|f(x)|在x_0处可导，则f(x)在x_0处可导" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="若\varphi(x)在x=a处连续，则\varphi(x)|x-a|在x=a处可导的充要条件为 \varphi(a)=0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="前提:若f(x)连续，则\\当f(x_0)\neq 0时，f(x)在x_0处可导 \iff |f(x)|在x_0处可导\\ 当f(x_0)=0时，|f(x)|在x_0处可导 \iff f'(x_0)=0" contenteditable="false"><span></span><span></span></span>
导数极限定理<br>
定义<br><span class="equation-text" data-index="0" data-equation="f(x)在x_0=a处连续，\lim_{x\rightarrow x_0} f'(x)=A \\ \implies f(x)在x_0处可导，且f'(x_0)=A" contenteditable="false"><span></span><span></span></span><br>
导函数连续<br><span class="equation-text" data-index="0" data-equation="f_{+}'(x_0) =f'(x_0+)时右连续，左连续同理" contenteditable="false"><span></span><span></span></span><br>
导函数右极限<br><span class="equation-text" data-index="0" data-equation="令f'(x)=g(x),则f'(x_0+)=\lim_{x\rightarrow x_0^+}g(x)" contenteditable="false"><span></span><span></span></span><br>
右导数<br><span class="equation-text" data-index="0" data-equation="f_+'(x_0)=\lim_{x\rightarrow x_0^+}\frac{f(x)-f(x_0)}{x-x_0}" contenteditable="false"><span></span><span></span></span><br>
右导数 ≠ 导函数的右极限<br><span class="equation-text" data-index="0" data-equation="f_{+}'(x_0) \neq f'(x_0+)，例:x^2\sin \frac{1}{x}在x\rightarrow0处" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="f_{+}'(x_0)存在但 f'(x_0+)不存在" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f_{+}'(x_0) 不存在 f'(x_0+)存在" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="若 f(x)在[x_0,x_0+\sigma) 上连续，在(x_0, x_0+\sigma)内可导\\且\lim_{x\rightarrow x_0^+}f'(x)存在，则f'_+(x_0)=\lim_{x\rightarrow x_0^+}f'(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="f(x)在x_0连续，U(x_0)可导，导数左右极限相等且等于a，则其f'(x_0)=a" contenteditable="false"><span></span><span></span></span>
导函数与原函数有界性
<span class="equation-text" data-index="0" data-equation="f'(x)在(a,b)上有界，则f(x)在(a,b)上有界" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="f(x)在(a,b)上无界，f'(x)在(a,b)上一定无界" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="在无穷区间上，以f(x)或f'(x)无界为条件分别推不出他们关于有界与无界的结论" contenteditable="false"><span></span><span></span></span>
导函数与函数凹凸性<br>
<br><span class="equation-text" data-index="0" data-equation="f'(x)在(a,b)上单调递减，f(x)在(a,b)上为凸函数" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f'(x)在(a,b)上单调递增，则f(x)在(a,b)上为凹函数" contenteditable="false"><span></span><span></span></span>
题型<br>
利用导数定义求极限
利用导数定义求导数<br>
利用导数定义判定可导性
多元函数偏导<br>
偏导与可微
<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="f_x(x_0,y_0)=\lim_{\Delta x\rightarrow 0}\frac{f(x_0,y_0+\Delta x)-f(x_0,y_0)}{\Delta x}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="记作：\frac{\partial z}{\partial x}|_{x=x_0,y=y_0},\frac{\partial f}{\partial x}|_{x=x_0,y=y_0},z_x|_{x=x_0,y=y_0},f_x(x_0,y_0)" contenteditable="false"><span></span><span></span></span>
高阶偏导数
混合偏导数
书写<br><span class="equation-text" data-index="0" data-equation="\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="如果二阶混合偏导存在且在区域D连续，则该区域内相等\\即(\frac{\partial ^2z}{\partial y\partial x})=(\frac{\partial ^2z}{\partial x\partial y})" contenteditable="false"><span></span><span></span></span>
高阶偏导数<br>
<br><span class="equation-text" data-index="0" data-equation="偏导连续则混合偏导与求导次序无关" contenteditable="false"><span></span><span></span></span>
拉普拉斯方程<br>
<br><span class="equation-text" data-index="0" data-equation="z=\ln \sqrt{x^2+y^2}满足\frac{\partial ^2z}{\partial x^2}+\frac{\partial ^2z}{\partial y^2}=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="u=\frac{1}{r}满足\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}+\frac{\partial ^2u}{\partial z^2}=0,其中r=\sqrt{x^2+y^2+z^2}" contenteditable="false"><span></span><span></span></span>
隐函数求导<br>
存在定理1<br>
<br><span class="equation-text" data-index="0" data-equation="1)F(x,y)在点P(x_0,y_0)邻域有连续偏导\\2)F(x_0,y_0)=0,F_y(x_0,y_0)【分母限制】\neq 0\\则能确定有连续导数的函数y=f(x),满足条件y=f(x_0)\\并有\frac{dy}{dx}=-\frac{F_x}{F_y}" contenteditable="false"><span></span><span></span></span>
存在定理2<br>
<br><span class="equation-text" data-index="0" data-equation="1)F(x,y,z)在点P(x_0,y_0,z_0)邻域有连续偏导\\2)F(x_0,y_0,z_0)=0,F_z(x_0,y_0,z_0)\neq 0\\则能确定有连续导数的函数z=f(x,y),满足条件z_0=f(x_0,y_0)\\并有\frac{dz}{dx}=-\frac{F_x}{F_z}" contenteditable="false"><span></span><span></span></span>
存在定理3/方程组情形
<br><span class="equation-text" data-index="0" data-equation="1)F(x,y,u,v),G(x,y,u,v)在P(x_0,y_0,u_0,v_0)某邻域有连续偏导\\2)F(P)=0,G(P)=0,其偏导组成的函数行列式(雅可比式）不等于0" contenteditable="false"><span></span><span></span></span>
具体点求偏导值<br>
微分/全微分<br>
一元函数微分<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="\Delta y=f(x_0+\Delta x)-f(x_0)=A\Delta x+o(x)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A是不依赖于\Delta x的常数" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="记为dy=A\Delta x，\Delta y=dy+o(dy)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="主部:dy, \Delta y\approx dy" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{\Delta y}{\Delta x}\approx \frac{dy}{dx}" contenteditable="false"><span></span><span></span></span>
充分必要条件<br>
可微即可导，可导即可微【"微商"】<br><span class="equation-text" data-index="0" data-equation="A=\lim_{\Delta x\rightarrow0}\frac{\Delta y}{\Delta x}=f'(x)" contenteditable="false"><span></span><span></span></span>
微分公式与运算法则<br>
<br><span class="equation-text" data-index="0" data-equation="求导：F'(x)=f(x) \\求微： d(F(x))=f(x)dx" contenteditable="false"><span></span><span></span></span>
微分形式不变性
变换自变量，微分形式不会改变<br>
弧长与曲率<br>
弧微分<br>
<span class="equation-text" data-index="0" data-equation="简记：ds=\sqrt{(dx)^2+(dy)^2}=\sqrt{1+(\frac{dy}{dx})^2}dx=\sqrt{1+y'^2}dx" contenteditable="false"><span></span><span></span></span>
曲率<br>
公式
<br><span class="equation-text" data-index="0" data-equation="K=|\frac{\Delta \alpha}{\Delta s}|" contenteditable="false"><span></span><span></span></span>
一般方程<br>
<br><span class="equation-text" data-index="0" data-equation="(1)\tan \alpha=y',y''=\sec^2 \alpha\frac{d\alpha}{dx}\implies\frac{d\alpha}{dx}=\frac{y''}{1+y'^2}\\(2) ds=\sqrt{1+y'^2}dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="K=|\frac{\Delta \alpha}{\Delta s}|=\frac{|y''|}{(1+y'^2)^{\frac{3}{2}}}" contenteditable="false"><span></span><span></span></span>
参数方程<br>
<br><span class="equation-text" data-index="0" data-equation="\begin{cases} x=u(t)\\y=v(t) \end{cases} v'/u'=g(u);(v'/u')'=g'_t(u)=\frac{v''u'-v'u''}{u'^2}" contenteditable="false"><span></span><span></span></span>
利用参数方程求导法<br><span class="equation-text" data-index="0" data-equation="K=|\Delta\alpha /\Delta s|=|y_x''|/{(1+y_x'^2)^{\frac{3}{2}}}=|\frac{d^2y}{dx^2}|/{[1+(\frac{dy}{dx})^2])^{\frac{3}{2}}}=\frac{|u'v''-u''v'|}{[u'^2+v'^2]^\frac{3}{2}}" contenteditable="false"><span></span><span></span></span>
曲率圆
曲率半径<br><span class="equation-text" data-index="0" data-equation="r=\frac{1}{K}" contenteditable="false"><span></span><span></span></span><br>
圆的曲率<br><span class="equation-text" data-index="0" data-equation="K=\frac{1}{r}" contenteditable="false"><span></span><span></span></span><br>
方程根的存在性及个数
根
代数理解【解方程】<br>
几何理解【函数交点】
存在性<br>
零点定理<br>
罗尔定理<br>
个数
单调性<br>
罗尔定理推论
含参方程<br>
实根个数<br>
参数范围<br>
多元函数全微分
相关概念<br>
偏增量<br>
全增量<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)=A\Delta x+B\Delta y+o(\rho)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A,B是不依赖\Delta x和\Delta y，只与x,y有关，\rho=\sqrt{(\Delta x)^2+(\Delta y)^2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="全微分dz=A\Delta x+B\Delta y=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y" contenteditable="false"><span></span><span></span></span>
充分/必要条件<br>
微分与偏导关系<br>
除此之外，连续 -> 极限存在<br>
充要条件
<span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow x_0,y\rightarrow y_0}\frac{f(x+x_0,y+y_0)-f(x_0,y_0)-Adx-Bdy}{\sqrt{x^2+y^2}}=o(\rho)=0" contenteditable="false"><span></span><span></span></span>
x→0 y→0和(x,y)→(0,0)的区别
必要条件<br>
如果可微分则偏导必定"<b>存在</b>"<br>
实例<br>
可微，则<span class="equation-text" data-index="0" data-equation="\lim \frac{f(x,y)}{x^2+y^2}=0【错误，取f(x,y)=1】" contenteditable="false"><span></span><span></span></span><br>
充分条件<br>
偏导"<b>连续"</b>则该点可微<br>
<span class="equation-text" data-index="0" data-equation="\lim_{(x,y) \rightarrow (x_0,y_0)} f_x'(x,y)=f_x'(x_0,y_0)且\lim_{(x,y) \rightarrow (x_0,y_0)} f_y'(x,y)=f_y'(x_0,y_0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="注：\lim_{x \rightarrow x_0} f_x'(x,y_0)=f_x'(x_0,y_0)" contenteditable="false"><span></span><span></span></span>
实例<br>
<span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0,y\rightarrow 0}\frac{f(x,y)}{x^2+y^2} \exist ，则可微" contenteditable="false"><span></span><span></span></span>
求导法则<br>
链式求导法<br>
一元与多元复合
<br><span class="equation-text" data-index="0" data-equation="[u=\varphi(t),v=\psi(t),z=f(u,v)]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{\partial z}{\partial t}=\frac{\partial z}{\partial u}\frac{du}{dt}+\frac{\partial z}{\partial v}\frac{dv}{dt}" contenteditable="false"><span></span><span></span></span>
多元与多元复合<br>
<br><span class="equation-text" data-index="0" data-equation="u=\varphi(x,y),v=\psi(x,y),z=f(u,v)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{du}{dx}+\frac{\partial z}{\partial v}\frac{dv}{dx},\frac{dz}{dy}=\frac{\partial z}{\partial u}\frac{du}{dy}+\frac{\partial z}{\partial v}\frac{dv}{dy}" contenteditable="false"><span></span><span></span></span>
其他情形<br>
<br><span class="equation-text" data-index="0" data-equation="u=\varphi(x,y),z=f(u,x,y)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="注:\frac{\partial z}{\partial x}\neq \frac{\partial f}{\partial x};\frac{\partial z}{\partial x}是函数所有关于x的偏导数，而\frac{\partial f}{\partial x}则只是求出u,y作为不变时对x的偏导数" contenteditable="false"><span></span><span></span></span>
记号
<br><span class="equation-text" data-index="0" data-equation="f(u,v):f'_1(u,v)=f_u(u,v)=f_u;f'_2=f_v';f''_{12}=f_{uv};f''_{21}=f_{vu}" contenteditable="false"><span></span><span></span></span>
全微分形式不变性<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x};P(x,y),Q(x,y)分别为x,y的偏导" contenteditable="false"><span></span><span></span></span>
题型<br>
求偏导数
定积分/重积分<br>
一元函数定积分<br>
概念与定义
<br><span class="equation-text" data-index="0" data-equation="\int_{a}^{b} f(x)\, \mathrm{d}x" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="= \begin{matrix} \lim_{\lambda \to \ 0} \sum_{i=1}^{n} f(\xi_i)\Delta x_i \end{matrix}" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="2" data-equation="=I" contenteditable="false"><span></span><span></span></span>
和式极限表达式<br><span class="equation-text" data-index="0" data-equation="c\lim \frac{1}{n}\sum_{i=1}^nf(\frac{i}{n})" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\Delta x_i=\frac{1}{n};\xi_i=\frac{i}{n}" contenteditable="false"><span></span><span></span></span>
微积分公式<br><span class="equation-text" data-index="0" data-equation="\int_{a}^{b} f(x)\, \mathrm{d}x=F(b)-F(a)=[F(x)]^b_a" contenteditable="false"><span></span><span></span></span><br>
性质
<br><span class="equation-text" data-index="0" data-equation="\int_a^b [\alpha f(x)+\beta g(x)]dx=\alpha\int_a^b f(x)dx+\beta\int_a^b g(x) dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^b g(t)dx=\int_a^c g(t)dx+\int_c^b g(t)dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^b Cdx=C(b-a)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f(x)\leq g(x)则\int_a^b f(x)dx\leq\int_a^b g(x)dx" contenteditable="false"><span></span><span></span></span>
介值定理<br><span class="equation-text" data-index="0" data-equation="m\xi\leq \int_a^bf(x)dx \leq M\xi" contenteditable="false"><span></span><span></span></span>
定积分中值定理<br><span class="equation-text" data-index="0" data-equation="F(b)-F(a)=\int_a^b f(x)dx=f(\xi)(b-a)=F'(\xi)(b-a)" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="(a\leq \xi\leq b)" contenteditable="false"><span></span><span></span></span>
补充
积分上下限相等为0<br><span class="equation-text" data-index="0" data-equation="\int_a^af(x)dx=0" contenteditable="false"><span></span><span></span></span>
上下限相反，值取反<br><span class="equation-text" data-index="0" data-equation="\int_a^b f(x)dx=-\int_b^a f(x)dx" contenteditable="false"><span></span><span></span></span>
积分变量无关性<br><span class="equation-text" data-index="0" data-equation="\pi\int_0^\pi \frac{\sin t}{\pi -t}dt-\int_0^\pi \frac{x\sin x}{\pi -x}dx=\int_0^\pi \sin xdx=2" contenteditable="false"><span></span><span></span></span><br>
定积分存在定理<br>(可积性判断)<br>
必要条件<br>
可积必有界<br><span class="equation-text" data-index="0" data-equation="可积则f(x)在[a,b]上有界" contenteditable="false"><span></span><span></span></span><br>
充分条件<br>
闭区间连续函数必然可积<br><span class="equation-text" data-index="0" data-equation="f(x)在[a,b]上连续" contenteditable="false"><span></span><span></span></span>
闭区间单调函数必然可积<br><span class="equation-text" data-index="0" data-equation="f(x)在区间[a,b]上单调" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="有界，且有有限个间断点(无穷除外)" contenteditable="false"><span></span><span></span></span>
计算方法<br>
分部积分
【连接原函数与导数的桥梁】<br><span class="equation-text" data-index="0" data-equation="\int_a^{b} u(x)v'(x)dx=[u(x)v(x)]^b_a-\int_a^{b} u'(x)v(x)dx" contenteditable="false"><span></span><span></span></span>
换元法<br>
条件<br>
<br><span class="equation-text" data-index="0" data-equation="\varphi(t)在[\alpha,\beta]上有连续导数，且值域为R=[a,b]" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\varphi(\alpha)=a,\varphi(\beta)=b" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^b f(x)dx=\int_{\alpha}^{\beta}f[\varphi(t)]\varphi'(t)dt" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="x的积分域=t积分域对应函数\varphi(t)的值域" contenteditable="false"><span></span><span></span></span>
题型总结<br>
三角
<br><span class="equation-text" data-index="0" data-equation="\int_a^b dx/[(x-1)^4\sqrt{(x-1)^2-1}]=\int \frac{\sec t\tan t}{\sec^4t \tan t}dt" contenteditable="false"><span></span><span></span></span>
指数/对数<br>
<br><span class="equation-text" data-index="0" data-equation="\int_0^{+\infty} \frac{e^{-x}}{1+e^{-x}}dx=-\ln (1+e^{-x})|_0^{+\infty}=\ln 2" contenteditable="false"><span></span><span></span></span>
倒数
<br><span class="equation-text" data-index="0" data-equation="\int_0^{+\infty} \frac{x^2}{1+x^4}dx=\int_0^{+\infty}\frac{1}{1+t^4}dt【t=1/x】" contenteditable="false"><span></span><span></span></span>
结论公式<br>
三角函数<br>
<br><span class="equation-text" data-index="0" data-equation="\int_0^\pi xf(sinx)dx=\frac{\pi}{2}\int_0^\pi f(sinx)dx=\frac{\pi}{2}\int_0^\pi f(cosx)dx" contenteditable="false"><span></span><span></span></span>
【奇偶】点火公式<br><span class="equation-text" data-index="0" data-equation="\int_0^{\frac{\pi}{2}}\sin^nxdx=\int_0^{\frac{\pi}{2}}\cos^nxdx=\begin{cases} \frac{n-1}{n} * \frac{n-3}{n-2}*...*\frac{1}{2}*\frac{\pi}{2} ,n为偶\\ \frac{n-1}{n} * \frac{n-3}{n-2}*...*\frac{2}{3} ,n>1 且奇\end{cases}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_0^a\cos^2xdx=\int_0^a\frac{1}{2}(\cos 2x+1)dx" contenteditable="false"><span></span><span></span></span>
周期函数
<br><span class="equation-text" data-index="0" data-equation="\int_a^{a+T} f(x)dx=\int_0^T f(x)dx" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{a+nT} f(x)dx=n\int_0^T f(x)dx" contenteditable="false"><span></span><span></span></span>
区间再现<br>
<br><span class="equation-text" data-index="0" data-equation="\int_0^\frac{\pi}{2} f(sinx)dx=\int_0^\frac{\pi}{2} f(cosx)dx\\" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{\pi}{4}=\int_0^{\frac{\pi}{2}} \frac{\sin ^px}{\sin^px+\cos^px}dx=\int_0^{\frac{\pi}{2}} \frac{\cos ^pt}{\cos^pt+\sin^pt}dt【t=\frac{\pi}{2}-x】" contenteditable="false"><span></span><span></span></span><br>
利用性质<br>
对称性
区间再现<br>
奇偶性<br>
周期性
常见定积分
<span class="equation-text" data-index="0" data-equation="\int_{-\infty}^{+\infty} e^{-t^2}dt=\sqrt{\pi}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(\int_{-\infty}^{+\infty} e^{-t^2}dt)^2=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-(x^2+y^2)}dxdy=\pi" contenteditable="false"><span></span><span></span></span>
变上限积分函数<br>
公式<br>
可积必有原函数<br><span class="equation-text" data-index="0" data-equation="\phi(x)=\int_a^x f(t)dt" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^x f(t)dt=-\int_x^a f(t)dt" contenteditable="false"><span></span><span></span></span>
基本公式<br><span class="equation-text" data-index="0" data-equation="\phi'(x)=\frac{d}{dx}\int_a^x f(t)dt" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="=f(x)(a\leq x\leq b)" contenteditable="false"><span></span><span></span></span>
统一性<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dx}\int_0^x tf(t)dt=\frac{d}{dx}\int_0^x g(t)dt=g(t)=xf(x)" contenteditable="false"><span></span><span></span></span>
【一元】复合变上限求导<br><span class="equation-text" data-index="0" data-equation="\int_{\varphi(x)}^{\psi(x)}f(x)dx=\varphi'(x)f(\varphi(x))-\psi'(x)f(\psi(x))" contenteditable="false"><span></span><span></span></span>
【多元】变上限求导<br><span class="equation-text" data-index="0" data-equation="\int_{\varphi(x)}^{\psi(x)}f(x,t)dx=\int_{\varphi(x)}^{\psi(x)}\frac{\partial f(x,t)}{\partial x}dt+\varphi'(x)f(x,\varphi(x))-\psi'(x)f(x,\psi(x))" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(\int_a^bf(x,t)dt)'=\int_a^b\frac{\partial f(x,t)}{\partial x}dt" contenteditable="false"><span></span><span></span></span>
常见换元法<br><span class="equation-text" data-index="0" data-equation="\int_0^x f(x-t)dt=\int_0^x f(u)du" contenteditable="false"><span></span><span></span></span>
积分中值定理<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow 0}\int_0^x f(t)dt=xf(\xi)" contenteditable="false"><span></span><span></span></span>
变上限积分等价代换<br><span class="equation-text" data-index="0" data-equation="f(x)和g(x)在x=0某邻域U_0连续，且\lim_{x\rightarrow 0}f(x)/g(x)=1,则\int_0^x f(t)dt\sim \int_0^x g(t)dt" contenteditable="false"><span></span><span></span></span><br>
结论<br>
连续性<br><span class="equation-text" data-index="0" data-equation="f(x)上[a,b]可积，则\int_a^xf(t)dt在[a,b]上连续" contenteditable="false"><span></span><span></span></span><br>
可导性<br><span class="equation-text" data-index="0" data-equation="f(x)在[a,b]上除某点外可积，则在点除F(x)性质由f(x)确定" contenteditable="false"><span></span><span></span></span><br>
奇偶性<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)为奇函数，则\int_a^xf(t)dt为偶函数" contenteditable="false"><span></span><span></span></span>
偶函数奇偶性只在[0，x]区间上体现<br><span class="equation-text" data-index="0" data-equation="f(x)为奇函数，则\int_a^xf(t)dt为偶函数" contenteditable="false"><span></span><span></span></span>
反常积分
无穷限反常积分【三类】
定义
<br><span class="equation-text" data-index="0" data-equation="\int_{a}^{+\infty}f(x)dx=\lim_{t\rightarrow +\infty} \int_a^{t} f(x)dx" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\int_{-\infty}^bf(x)dx=\lim_{t\rightarrow -\infty} \int_t^{b} f(x)dx" contenteditable="false"><span></span><span></span></span>
敛散判别
比较判别法<br><span class="equation-text" data-index="0" data-equation="0\leq f(x)\leq g(x),[a,+\infty)连续" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}g(x)dx收敛，\int_a^{+\infty}f(x)dx收敛" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}f(x)dx发散，\int_a^{+\infty}g(x)dx发散" contenteditable="false"><span></span><span></span></span>
比较法的极限形式<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow +\infty}\frac{f(x)}{g(x)}=\lambda,[a,+\infty)非负连续" contenteditable="false"><span></span><span></span></span>
λ ≠ 0<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}f(x)dx与\int_a^{+\infty}g(x)dx同敛散" contenteditable="false"><span></span><span></span></span><br>
λ = 0<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}g(x)dx收敛,则\int_a^{+\infty}f(x)dx收敛" contenteditable="false"><span></span><span></span></span><br>
λ = +∞<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}g(x)dx发散,则\int_a^{+\infty}f(x)dx发散" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}\frac{1}{x^p}dx,\begin{cases} p>1,收敛\\p\leq 1,发散\end{cases}(a>0)" contenteditable="false"><span></span><span></span></span>
性质
同一函数正负无穷区间的极限，必须两项<font color="#B71C1C">同时收敛时</font>和才收敛<br><span class="equation-text" data-index="0" data-equation="\int_{-\infty}^{+\infty} f(x)dx=\int_{-\infty}^{0} f(x)dx+\int_{0}^{+\infty} f(x)dx" contenteditable="false"><span></span><span></span></span>
不同函数的反常积分的发散的和，不能够确定【例如:f(x)+g(x)=0】<br><span class="equation-text" data-index="0" data-equation="\int_{a}^{+\infty} f(x)dx,\int_{a}^{+\infty} g(x)dx发散，则\int_{a}^{+\infty} [f(x)+g(x)]dx敛散性不能确定" contenteditable="false"><span></span><span></span></span>
无界函数反常积分<br>
瑕点<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)在点a的任一邻域内都无界，则a称为f(x)的瑕点" contenteditable="false"><span></span><span></span></span>
定义
f(x) 在 (a,b] 上连续，点 a 为 f(x) 的瑕点<br><span class="equation-text" data-index="0" data-equation="\int_a^bf(x)dx=\lim_{t\rightarrow b^-} \int_a^t f(x)dx" contenteditable="false"><span></span><span></span></span>
f(x) 在 [a,b) 上连续，点 b 为 f(x) 的瑕点<br><span class="equation-text" data-index="0" data-equation="\int_a^bf(x)dx=\lim_{t\rightarrow a^+} \int_a^t f(x)dx" contenteditable="false"><span></span><span></span></span>
f(x) 在 [a,b] 上除点 c (a<c<b) 外连续，点c为f(x)瑕点<br><span class="equation-text" data-index="0" data-equation="\int_a^bf(x)dx= \int_a^c f(x)dx+\int_c^a f(x)dx" contenteditable="false"><span></span><span></span></span>
敛散判别<br>
比较判别法<br><span class="equation-text" data-index="0" data-equation="0\leq f(x)\leq g(x)，x=a为f(x)和g(x)的瑕点" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}g(x)dx收敛，\int_a^{+\infty}f(x)dx收敛" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{+\infty}f(x)dx发散，\int_a^{+\infty}g(x)dx发散" contenteditable="false"><span></span><span></span></span>
比较法的极限形式<br><span class="equation-text" data-index="0" data-equation="\lim_{x\rightarrow +\infty}\frac{f(x)}{g(x)}=\lambda,[a,+\infty)非负连续" contenteditable="false"><span></span><span></span></span>
λ ≠ 0<br><span class="equation-text" data-index="0" data-equation="\\int_a^{b}f(x)dx与\int_a^{b}g(x)dx同敛散" contenteditable="false"><span></span><span></span></span><br>
λ = 0<br><span class="equation-text" data-index="0" data-equation="\int_a^{b}g(x)dx收敛,则\int_a^{b}f(x)dx收敛" contenteditable="false"><span></span><span></span></span><br>
λ = +∞<br><span class="equation-text" data-index="0" data-equation="\int_a^{b}g(x)dx发散,则\int_a^{b}f(x)dx发散" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{b}\frac{1}{(x-a)^p}dx,\begin{cases} p<1,收敛\\p\geq 1,发散\end{cases}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_a^{b}\frac{1}{(b-x)^p}dx,\begin{cases} p<1,收敛\\p\geq 1,发散\end{cases}" contenteditable="false"><span></span><span></span></span>
性质
<span class="equation-text" data-index="0" data-equation="f(x)为奇函数，则\int_{-\infty}^{+\infty} f(x)dx=0 【错误】" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\lim_{R \rightarrow +\infty}\int_{-R}^{+R} f(x)dx=0, 则\int_{-\infty}^{+\infty} f(x)dx 收敛【错】" contenteditable="false"><span></span><span></span></span>
常见反常积分
<span class="equation-text" data-index="0" data-equation="1/x 同敛散" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^{+\infty} \frac{1}{\sqrt{1-x^2}}" contenteditable="false"><span></span><span></span></span>
应用
几何运用
平面图形面积
旋转体体积
一元定积分
区域D由y=f(x)和直线x=a,x=b以及x围成<br><span class="equation-text" data-index="0" data-equation="绕x轴旋转V_x=2\pi\iint_Dyd\sigma=2\pi\int_a^bdx\int_0^{f(x)}ydy=\pi\int_a^bf^2(x)dx=\pi\int_a^b r^2dx" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="绕y轴旋转V_x=2\pi\iint_Dxd\sigma=2\pi\int_a^bdx\int_0^{f(x)}xdy=\pi\int_a^bxf^2(x)dx" contenteditable="false"><span></span><span></span></span>
二重积分·
通用公式<br><span class="equation-text" data-index="0" data-equation="V=2\pi \iint_D r(x,y)d\sigma" contenteditable="false"><span></span><span></span></span>
沿直线 <span class="equation-text" data-index="0" data-equation="L: ax+by+c=0" contenteditable="false"><span></span><span></span></span> 旋转<br><span class="equation-text" data-index="1" data-equation="r(x,y) = \frac{|ax+by+c|}{\sqrt{a^2+b^2}}" contenteditable="false"><span></span><span></span></span><br>
旋转体侧面积
<span class="equation-text" data-index="0" data-equation="\int 2\pi" contenteditable="false"><span></span><span></span></span> d弧长<br><span class="equation-text" data-index="1" data-equation="S=2\pi \int_a^b f(x)\sqrt{1+f'^2(x)}dx" contenteditable="false"><span></span><span></span></span><br>
曲线弧长<br>
物理运用<br>
变力沿直线做工<br><span class="equation-text" data-index="0" data-equation="dW=Fdr \rightarrow \int dW=W=\int Fdr" contenteditable="false"><span></span><span></span></span>
库仑力【距离平方成反比】<br><span class="equation-text" data-index="0" data-equation="F=k\frac{q}{r^2}" contenteditable="false"><span></span><span></span></span>
水压【压强与深度成正比】<br><span class="equation-text" data-index="0" data-equation="P=pA=\rho gh*A" contenteditable="false"><span></span><span></span></span>
引力【距离平方成反比】<br><span class="equation-text" data-index="0" data-equation="F=G\frac{m_1m_2}{r^2}" contenteditable="false"><span></span><span></span></span>
考场题型
定积分比大小<br>
<span class="equation-text" data-index="0" data-equation="\int_0^\frac{\pi}{2} \frac{\sin x}{x} dx>\int_0^\frac{\pi}{2} f(\frac{\pi}{2}) dx=\int_0^\frac{\pi}{2} \frac{2}{\pi} dx=1" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_0^\frac{\pi}{4} \frac{\tan x}{x} dx<\int_0^\frac{\pi}{4} f(\frac{\pi}{4}) dx=\int_0^\frac{\pi}{2} \frac{4}{\pi} dx=1" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_{-\frac{\pi}{2}}^\frac{\pi}{2} \frac{(1+x)^2}{1+x^2} dx=\int_{-\frac{\pi}{2}}^\frac{\pi}{2} \frac{1+x^2+2x}{1+x^2} dx=\int_{-\frac{\pi}{2}}^\frac{\pi}{2} (1+0) dx=\pi" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="f(x)=g(x)+\int f(x) 型" contenteditable="false"><span></span><span></span></span>
未知变上限求原函数【同时求导】<br><span class="equation-text" data-index="0" data-equation="f(x)=g(x)\pm \int_0^xf(x)dx" contenteditable="false"><span></span><span></span></span>
未知不定积分求原函数【同时积分】<br><span class="equation-text" data-index="0" data-equation="f(x)=g(x)\pm \int_a^bf(x)dx" contenteditable="false"><span></span><span></span></span>
多元函数重积分
二重积分
几何意义
曲顶柱体的体积 <br><span class="equation-text" data-index="0" data-equation="面积\times 高" contenteditable="false"><span></span><span></span></span><br>
薄片的质量<br><span class="equation-text" data-index="0" data-equation="面积\times 密度" contenteditable="false"><span></span><span></span></span><br>
定义
<span class="equation-text" data-index="0" data-equation="\iint_D f(x,y)d\sigma = \lim_{\lambda\rightarrow 0}\sum_{i=1}^n f(\xi_i,\eta_i)\Delta \sigma_i" contenteditable="false"><span></span><span></span></span>
坐标系
直角坐标
点线面计算逻辑
积分顺序<br>
先 y 后 x<br><span class="equation-text" data-index="0" data-equation="\iint_D f(x,y)d\sigma = \int_a^bdx\int_{y_1(x)}^{y_2(x)}f(x,y)dy" contenteditable="false"><span></span><span></span></span>
先 x 后 y<br><span class="equation-text" data-index="0" data-equation="\iint_D f(x,y)d\sigma = \int_a^bdy\int_{x_1(y)}^{x_2(y)}f(x,y)dx" contenteditable="false"><span></span><span></span></span><br>
面积元素
<br><span class="equation-text" data-index="0" data-equation="d\sigma=dxdy[直角坐标系]" contenteditable="false"><span></span><span></span></span>
正方形面积<br><span class="equation-text" data-index="0" data-equation="\Delta\sigma_i=\Delta x_i · \Delta y_i=dxdy" contenteditable="false"><span></span><span></span></span>
极坐标
常见极坐标计算被积函数<br><span class="equation-text" data-index="0" data-equation="f(\sqrt{x^2+y^2}),f(\frac{y}{x}),,f(\frac{x}{y})" contenteditable="false"><span></span><span></span></span>
面积元素
<br><span class="equation-text" data-index="0" data-equation="d\sigma=\rho d\rho d\theta[极坐标系]" contenteditable="false"><span></span><span></span></span>
扇环面积<br><span class="equation-text" data-index="0" data-equation="\Delta\sigma_i=\frac{1}{2}(\rho_i+\Delta \rho_i)^2 · \Delta \theta-\frac{1}{2}(\rho_i)^2 · \Delta \theta=\rho_i · \Delta \rho_i · \Delta \theta_i" contenteditable="false"><span></span><span></span></span>
适合极坐标积分域<br>
<span class="equation-text" data-index="0" data-equation="x^2+y^2\leq R^2[圆]\\ r^2\leq x^2+y^2\leq R^2[环]\\x^2+y^2\leq2ax[偏心圆]\\x^2+y^2\leq 2ay" contenteditable="false"><span></span><span></span></span>
运算性质
二重积分中值定理<br>
<br><span class="equation-text" data-index="0" data-equation="\iint_D f(x,y)d\sigma=f(\xi,\eta)\sigma" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\iint_Df(x,y)dxdy=f(\xi,\eta)S" contenteditable="false"><span></span><span></span></span>
交换相等<br>
奇偶性<br>
积分域 D 关于 y 轴对称<br><span class="equation-text" data-index="0" data-equation="\iint_Df(x,y)d\sigma =\begin{cases}2\iint_{D_x\geq 0} f(x,y)d\sigma, f(-x,y)=f(x,y)[偶]\\0,f(-x,y)=-f(x,y)[奇]\end{cases}" contenteditable="false"><span></span><span></span></span><br>
积分域关于 x = a 对称【奇偶平移】<br><span class="equation-text" data-index="0" data-equation="\int_D x dx = \int_D [(x-a)+a]dx=\int_D adx" contenteditable="false"><span></span><span></span></span><br>
积分域 D 关于 x 轴对称<br><span class="equation-text" data-index="0" data-equation="\iint_Df(x,y)d\sigma =\begin{cases}2\iint_{D_y\geq 0} f(x,y)d\sigma, f(x,y)=f(x,-y)[偶]\\0,f(x,-y)=-f(x,y)[奇]\end{cases}" contenteditable="false"><span></span><span></span></span><br>
对称性<br>
积分变量记号无关<br><span class="equation-text" data-index="0" data-equation="\iint_{D(x,y)}f(x,y)d\sigma =\iint_{D(y,x)}f(y,x)d\sigma" contenteditable="false"><span></span><span></span></span>
沿直线 y=x 对称：变量对调积分制不变<br><span class="equation-text" data-index="0" data-equation="\iint_{D}f(x,y)d\sigma =\iint_{D}f(y,x)d\sigma" contenteditable="false"><span></span><span></span></span>
微分形式不变性
<br><span class="equation-text" data-index="0" data-equation="\iint_Df(x,y)dxdy=\iint_Df(u,v)dudv" contenteditable="false"><span></span><span></span></span>
基本运算
同积分区域可线性运算<br><span class="equation-text" data-index="0" data-equation="\iint_D [\alpha f(x,y)+\beta g(x,y)]d\sigma=\alpha\iint_D f(x,y)d\sigma+\beta\iint_D g(x,y) d\sigma" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\iint_D g(t)d\sigma=\iint_{D1} g(t)d\sigma+\iint_{D2} g(t)d\sigma" contenteditable="false"><span></span><span></span></span>
不等式性质<br><span class="equation-text" data-index="0" data-equation="f(x,y)\leq g(x,y)则\iint_D f(x,y)d\sigma\leq\iint_D g(x,y)d\sigma" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="m\xi\leq \iint_Df(x,y)d\sigma \leq M\xi" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\iint_D |f(x,y)d\sigma|\leq \iint_D |f(x,y)|d\sigma" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\iint_D Cd\sigma=C\iint_D d\sigma" contenteditable="false"><span></span><span></span></span>
二重可拆积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果\iint_Df(x,y)dxdy被积函数f(x,y)=f_1(x) f_2(y)\\ 且积分区域D=\{(x,y)|a\leq x \leq b ,c\leq y\leq d\} \\则有 \iint f_1(x)f_2(y)dxdy=[\int_a^b f_1(x)dx][\int_c^d f_2(y)dy]"><span></span><span></span></span>
题型<br>
累次积分交换次序<br>
步骤
画域
重新定限<br>
常见题型
极坐标→直角坐标<br>
<span class="equation-text" data-index="0" data-equation="直线 x+y=a" contenteditable="false"><span></span><span></span></span>
极坐标系重积分<br><span class="equation-text" data-index="0" data-equation="\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}d\theta\int_{\frac{1}{cos\theta+sin\theta}}^{\frac{2}{cos\theta+sin\theta}}dr" contenteditable="false"><span></span><span></span></span>
极坐标系表示<br><span class="equation-text" data-index="0" data-equation="\begin{cases} r= \frac{a}{\cos \theta + \sin \theta}\\x=\frac{a\sin \theta}{\cos \theta + \sin \theta}\\ y=\frac{a\cos \theta}{\cos \theta + \sin \theta} \\ x+y = a \end{cases}" contenteditable="false"><span></span><span></span></span><br>
偏心圆
<span class="equation-text" data-index="0" data-equation="圆心 (a,0)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int d\theta\int_0^{2a\sin\theta}f(rcos\theta,r\sin \theta)rdr" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\begin{cases} r= 2a\sin\theta,r^2=2ar\sin \theta=2ay\\r^2=x^2+y^2,x^2+y^2=2ay\\ (y-a)^2+x^2 = a\end{cases}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="圆心 (0,a)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}d\theta\int_0^{2acos\theta}f(rcos\theta,r\sin \theta)rdr" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\begin{cases} r= 2a\cos \theta\\x=2a\cos^2 \theta \\ y=2a\sin \theta\cos \theta=a\sin2\theta \\x-a=a(\cos ^2\theta-\sin^2\theta)=a\cos 2\theta\\ (x-a)^2+y^2 = a\end{cases}" contenteditable="false"><span></span><span></span></span>
直角坐标 <span class="equation-text" data-index="0" data-equation="\rightarrow" contenteditable="false"><span></span><span></span></span> 极坐标<br>
偏心圆<br>
图形<br>
<br>
公式
极坐标：<span class="equation-text" data-index="0" data-equation="\int_0^{\frac{\pi}{2}}d\theta\int_0^{2a\cos \theta}f(r)rdr" contenteditable="false"><span></span><span></span></span><br>直角坐标：<span class="equation-text" data-index="1" data-equation="(x-a)^2+y^2=a^2 \iff y^2=2ax-x^2" contenteditable="false"><span></span><span></span></span><br>
累次积分上限函数求导<br>
<br><span class="equation-text" data-index="0" data-equation="\int_0^xdu\int_0^{u^2}f(t,u)dt=\int_0^{x^2}f(t,x)dt" contenteditable="false"><span></span><span></span></span>
二重积分计算<br>
参数方程 积分<br>
<span class="equation-text" data-index="0" data-equation="\begin{cases} x=a(1-\sin t)\\ y=a(1-\cos t)\end{cases},计算\iint_D y^2d\sigma" contenteditable="false"><span></span><span></span></span>
全平面 抽象函数<br>
不等式<br>
平移法
轮换对称性<br>
去绝对值
利用区间对称型
<span class="equation-text" data-index="0" data-equation="f(x,y)=g(x,y)+\iint f(x,y) 型" contenteditable="false"><span></span><span></span></span>
未知变上限求原函数【同时求导】<br><span class="equation-text" data-index="0" data-equation="f(x)=g(x)\pm \int_0^xf(x)dx" contenteditable="false"><span></span><span></span></span>
未知不定积分求原函数【同时积分】<br><span class="equation-text" data-index="0" data-equation="f(x)=g(x)\pm \int_a^bf(x)dx" contenteditable="false"><span></span><span></span></span>
二重变上限积分<br>
<span class="equation-text" data-index="0" data-equation="\int_0^1 dx\int_0^{f(x)} g(t)dt" contenteditable="false"><span></span><span></span></span>
三重积分
几何意义
三维物体的质量<br><span class="equation-text" data-index="0" data-equation="体积\times 密度" contenteditable="false"><span></span><span></span></span><br>
坐标系<br>
直角坐标
核心：化为三个单次积分<br>
由此可见n维，即化为n个单次积分<br>
点线面体计算法<br>
在二重积分的薄片【面】基础上，增加对z轴积分【体】<br>
积分域的选择和积分顺序与二重积分原理一致<br>
体积元素
<br><span class="equation-text" data-index="0" data-equation="dv=dxdydz[直角坐标系]" contenteditable="false"><span></span><span></span></span>
立方体<br><span class="equation-text" data-index="0" data-equation="\Delta v_i=\Delta x_i \Delta y_i \Delta z=dxdydz" contenteditable="false"><span></span><span></span></span>
三化二/先二后一<br>
球体<br>
柱面坐标<br>
坐标元素<br><span class="equation-text" data-index="0" data-equation="\rho,\theta,z" contenteditable="false"><span></span><span></span></span>
分别对应 以 z 轴为轴的圆柱面[半径]，过z轴半平面[从 x 到 y 的旋转角度]，与 xOy 平面平行平面[高度]
取值范围<br><span class="equation-text" data-index="0" data-equation="0\leq\rho \leq +\infty，0\leq\theta \leq 2 \pi,-\infty\leq z \leq +\infty" contenteditable="false"><span></span><span></span></span>
右手为正
直角坐标参数方程<br>
<br><span class="equation-text" data-index="0" data-equation="\begin{cases} x=\rho cos\theta \\ y=\rho sin\theta \\z=z \end{cases}" contenteditable="false"><span></span><span></span></span>
常见被积函数形式<br><span class="equation-text" data-index="0" data-equation="f(z)g(\sqrt{x^2+y^2})" contenteditable="false"><span></span><span></span></span><br>
体积元素
扇环体<br><span class="equation-text" data-index="0" data-equation="dv =\rho d\rho d\theta dz[柱面坐标]" contenteditable="false"><span></span><span></span></span>
球面坐标系
<br><span class="equation-text" data-index="0" data-equation="r,\varphi,\theta" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="半径，与z轴夹角，从x到y轴旋转角度" contenteditable="false"><span></span><span></span></span>
参数方程<br>
<br><span class="equation-text" data-index="0" data-equation="\begin{cases} x=rsin\varphi cos\theta \\ y=rsin\varphi sin\theta \\z=rcos\varphi \end{cases}" contenteditable="false"><span></span><span></span></span>
常见被积函数形式<br><span class="equation-text" data-index="0" data-equation="f(x^2+y^2+z^2)=f(r)" contenteditable="false"><span></span><span></span></span><br>
体积元素<br>
<br><span class="equation-text" data-index="0" data-equation="dv = r^2sin\varphi drd\varphi d\theta[球面坐标]" contenteditable="false"><span></span><span></span></span>
图解<br>
计算时注意利用相似三角形求 r3 长度<br>
由上图可得<br><span class="equation-text" data-index="0" data-equation="r_1=\sqrt{dx^2+dy^2}, dx= r_1\cos \theta,dy= r_1 \sin \theta\\r_2=dz, r_3【球半径】=\sqrt{r_2^2+r_1^2}\\ r_1= r_3\sin \varphi,r_2=r_3\cos \varphi" contenteditable="false"><span></span><span></span></span>
应用 [元素法的使用推广]
曲面面积
质心
转动惯量
引力
题型<br>
含参变量的积分
坐标系计算
直角坐标（先二后一）<br>
柱坐标<br>
球坐标
常见图形<br>
图形 <span class="equation-text" data-index="0" data-equation="|x|+|y|=1" contenteditable="false"><span></span><span></span></span><br>
星形线<br>
参数方程<br><span class="equation-text" data-index="0" data-equation="\begin{cases}x =a\cos^3t\\y=a\sin^3t\end{cases}" contenteditable="false"><span></span><span></span></span>
图形<br>
<br>
摆线<br>
参数方程<br><span class="equation-text" data-index="0" data-equation="\begin{cases}x =a(t-\sin t)\\y=a(1-\cos t)\end{cases}" contenteditable="false"><span></span><span></span></span>
图形
<br>
向量代数<br>
向量<br>
概念
自由向量<br>
负向量<br>
零向量
角平分线向量
归一化处理<br><span class="equation-text" data-index="0" data-equation="\frac{\overrightarrow b}{|\overrightarrow a|}=\frac{\overrightarrow a}{|\overrightarrow a|}" contenteditable="false"><span></span><span></span></span><br>
向量夹角
范围<br><span class="equation-text" data-index="0" data-equation="[0,\pi]" contenteditable="false"><span></span><span></span></span>
向量的大小【模】<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow a|" contenteditable="false"><span></span><span></span></span><br>
模长为1<br><span class="equation-text" data-index="0" data-equation="单位向量\frac{\overrightarrow a}{|\overrightarrow a|}=\overrightarrow e_a" contenteditable="false"><span></span><span></span></span>
模长为0<br>零向量，方向任意<br>
向量间关系<br>
向量共线/平行
充要条件<br><span class="equation-text" data-index="0" data-equation="\overrightarrow a \neq 0,且存在唯一实数\lambda,使得 \lambda \overrightarrow b=\overrightarrow a" contenteditable="false"><span></span><span></span></span><br>
向量共面
三向量共面<br>混合积为0<br>
向量相等
大小相等方向相同<br><span class="equation-text" data-index="0" data-equation="\overrightarrow a=\overrightarrow b" contenteditable="false"><span></span><span></span></span>
向量运算
线性运算
向量加减<br>
线性代数<br>
线性组合
线性相关
<br><span class="equation-text" data-index="0" data-equation="存在k(k\neq0)使，k\alpha=0" contenteditable="false"><span></span><span></span></span>
向量组的线性相关
线性无关
线性表示(出)
<br><span class="equation-text" data-index="0" data-equation="存在k\alpha(a_1,...,a_n)=\beta" contenteditable="false"><span></span><span></span></span>
向量表示
向量组表示
向量组等价
三角形不等式
共线时相等<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow a+\overrightarrow b| \leq |\overrightarrow a|+|\overrightarrow b|" contenteditable="false"><span></span><span></span></span>
相反时相等<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow a-\overrightarrow b| \leq |\overrightarrow a|+|\overrightarrow b|" contenteditable="false"><span></span><span></span></span>
性质<br>
交换律
结合律<br>
向量数乘
结合律
分配律
空间直角坐标系
坐标轴<br>
三个两两垂直的单位向量<br>
建立规则
右手规则
卦限<br>
坐标分解式<br>
(x,y,z)<br><span class="equation-text" data-index="0" data-equation="\overrightarrow r=\overrightarrow {OM} = x\overrightarrow i+y\overrightarrow j+z \overrightarrow k" contenteditable="false"><span></span><span></span></span>
线性运算<br>
求解线性方程组
求证等腰三角形
求出空间两点距离
方向角与方向余弦
方向角<br>
向量转换<br><span class="equation-text" data-index="0" data-equation="(cos\alpha,cos\beta,cos\gamma)=\frac{1}{|\overrightarrow r|}(\overrightarrow x,\overrightarrow y,\overrightarrow z)=\overrightarrow e_r=\frac{\overrightarrow r}{|\overrightarrow r|}；cos\alpha = \frac{\overrightarrow x}{|\overrightarrow r|}" contenteditable="false"><span></span><span></span></span>
模长公式<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow x|^2+|\overrightarrow y|^2+|\overrightarrow z|^2=|\overrightarrow r|^2" contenteditable="false"><span></span><span></span></span>
性质【恒等式】<br><span class="equation-text" data-index="0" data-equation="cos^2\alpha+cos^2\beta+cos^2\gamma=1" contenteditable="false"><span></span><span></span></span>
向量表示<br><span class="equation-text" data-index="0" data-equation="注: \overrightarrow r=\overrightarrow {OM}=(x,y,z)" contenteditable="false"><span></span><span></span></span>
轴上的投影<br>
数量积/点乘->内积<br>
公式<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} ·\overrightarrow{b}=|\overrightarrow{a} ||\overrightarrow{b}|cos\alpha" contenteditable="false"><span></span><span></span></span>
代数表示
<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} =\{x_0,x_1,...,x_n\},\overrightarrow{b}= \{y_0,y_1,...,y_n\}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} ·\overrightarrow{b}=x_0y_0+...+x_ny_n" contenteditable="false"><span></span><span></span></span>
运算规律
交换律<br>
分配率
几何表示<br>
<b>b</b>向量在<b>a</b>向量方向投影的模长与<b>a</b>向量模长的乘积
几何应用
求模长<br>
<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow{a}|=\sqrt{[\overrightarrow{a} ·\overrightarrow{a}]}=\sqrt{x_0^2+...+x_n^2}" contenteditable="false"><span></span><span></span></span>
求夹角
<br><span class="equation-text" data-index="0" data-equation="cos\alpha=\frac{\overrightarrow{a} ·\overrightarrow{b}}{|\overrightarrow{a} |·|\overrightarrow{b}|}=\frac{x_0y_0+..+x_ny_n}{\sqrt{x_0^2+..+x_n^2}\sqrt{y_0^2+..+y_n^2}}" contenteditable="false"><span></span><span></span></span>
判断垂直<br>
<b>b</b>向量在<b>a</b>向量的分量为0【反之同理】<br><span class="equation-text" data-index="0" data-equation="cos\alpha=0" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation=",\overrightarrow{a} \perp \overrightarrow{b}\Rightarrow \overrightarrow{a}· \overrightarrow{b}=0" contenteditable="false"><span></span><span></span></span>
向量积/叉乘->外积
运算规律
交换异号<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} \times \overrightarrow{b}=-\overrightarrow{b} \times \overrightarrow{a}" contenteditable="false"><span></span><span></span></span>
分配率<br>
代数表示<br>
对应向量行列式运算<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} \times \overrightarrow{b}" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="=\begin{vmatrix} \overrightarrow i & .. &\overrightarrow z \\ a_x & ... & a_z \\ b_x & ... & b_z \end{vmatrix}" contenteditable="false"><span></span><span></span></span>
几何表示<br>
向量积乘积结果仍为向量
模长<br><span class="equation-text" data-index="0" data-equation="|\overrightarrow{a} \times\overrightarrow{b}|=|\overrightarrow{a} ||\overrightarrow{b}|sin\alpha" contenteditable="false"><span></span><span></span></span>
几何应用
求向量<b>a,b</b>临边的平行四边形面积<br><span class="equation-text" data-index="0" data-equation="S=|\overrightarrow{a} \times\overrightarrow{b}|=|\overrightarrow{a} ||\overrightarrow{b}|sin\alpha" contenteditable="false"><span></span><span></span></span><br>
判断向量平行<br><span class="equation-text" data-index="0" data-equation="a//b\iff a \times b =0" contenteditable="false"><span></span><span></span></span><br>
求垂直于<b>a,b</b>向量的平面法向量
叉乘向量方向<br>
右手法则<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{a} \times\overrightarrow{b}, 四指由a方向开始，指向b,大拇指为叉乘结果方向" contenteditable="false"><span></span><span></span></span><br>
混合积<br>
定义<br><span class="equation-text" data-index="0" data-equation="(\overrightarrow{a} \times\overrightarrow{b})·\overrightarrow{c}" contenteditable="false"><span></span><span></span></span>
代数表示
<br><span class="equation-text" data-index="0" data-equation="\begin{vmatrix} a_x & ... & a_z \\ b_x & ... & b_z \\ c_x & ... & c_z \end{vmatrix}" contenteditable="false"><span></span><span></span></span>
运算规律
轮换对称性<br><span class="equation-text" data-index="0" data-equation="[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]=[\overrightarrow{b}\overrightarrow{c}\overrightarrow{a}]=[\overrightarrow{c}\overrightarrow{a}\overrightarrow{b}]" contenteditable="false"><span></span><span></span></span><br>
调换变号<br><span class="equation-text" data-index="0" data-equation="(\overrightarrow{a}\overrightarrow{b}\overrightarrow{c})=-(\overrightarrow{a}\overrightarrow{c}\overrightarrow{b})" contenteditable="false"><span></span><span></span></span><br>
几何应用
平行六面体体积<br><span class="equation-text" data-index="0" data-equation="|(\overrightarrow{a}\overrightarrow{b}\overrightarrow{c})|" contenteditable="false"><span></span><span></span></span>
三向量共面<br><span class="equation-text" data-index="0" data-equation="|[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]|=0" contenteditable="false"><span></span><span></span></span><br>
空间解析几何
空间曲面方程<br>
曲面方程
概念<br>
<br><span class="equation-text" data-index="0" data-equation="如果\\1)曲面S上任意一点坐标满足与三元方程组F(x,y,z)=0\\2)不在曲面S上的点都不满足方程F(x,y,z)=0\\则\\F(x,y,z)=0为曲面S的方程" contenteditable="false"><span></span><span></span></span>
相关问题<br>
已知一曲面作为点的几何轨迹时，建立曲面方程<br>
已知坐标x,y,z之间的一个方程，研究该曲面形状<br>
常见曲面形状
球面
<br><span class="equation-text" data-index="0" data-equation="(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2" contenteditable="false"><span></span><span></span></span>
旋转曲面<br>
绕坐标轴旋转<br>
L绕y轴旋转<br><span class="equation-text" data-index="0" data-equation="f(y,\pm\sqrt{x^2+z^2}) =0" contenteditable="false"><span></span><span></span></span>
L绕z轴旋转<br><span class="equation-text" data-index="0" data-equation="f(\pm\sqrt{x^2+y^2},z)=0" contenteditable="false"><span></span><span></span></span>
柱面
母线平行于z轴<br><span class="equation-text" data-index="0" data-equation="准线方程联立消去z,得H(x,y)=0" contenteditable="false"><span></span><span></span></span><br>
二次曲面【常考】<br>
圆锥面<br><span class="equation-text" data-index="0" data-equation="x^2+y^2=z^2" contenteditable="false"><span></span><span></span></span><br>
椭球面<br><span class="equation-text" data-index="0" data-equation="\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1" contenteditable="false"><span></span><span></span></span><br>
球面<br><span class="equation-text" data-index="0" data-equation="x^2+y^2+z^2=R^2" contenteditable="false"><span></span><span></span></span><br>
旋转抛物面<br><span class="equation-text" data-index="0" data-equation="z=x^2+y^2" contenteditable="false"><span></span><span></span></span><br>
旋转双叶双曲面【了解】<br><span class="equation-text" data-index="0" data-equation="\frac{x^2}{a^2} - \frac{y^2+z^2}{c^2}=1" contenteditable="false"><span></span><span></span></span>
旋转单叶双曲面【了解】<br><span class="equation-text" data-index="0" data-equation="\frac{x^2+y^2}{a^2} - \frac{z^2}{c^2}=1" contenteditable="false"><span></span><span></span></span><br>
马鞍面
柱面<br>
构成<br>
母线
准线
二次曲面
椭圆曲面
椭球面
单叶双曲面
双叶双曲面
椭圆抛物面
双曲抛物面[马鞍面]
平面方程
点法式方程[点+法向量]<br>
相关问题形式<br><span class="equation-text" data-index="0" data-equation="已知法向量\overrightarrow n=(a,b,c),以及点M_0(x_0,y_0,z_0)则平面方程为?" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="hint:\overrightarrow n*\overrightarrow {M_0M}=0[平面内某向量与法向量垂直]" contenteditable="false"><span></span><span></span></span>
一般<b>方程</b><br>
形式<br>
方程<br><span class="equation-text" data-index="0" data-equation="Ax+By+Cz+D=0;D=-Ax_0-By_0-Cz_0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A(x-x_0)+B(y-y_0)+C(z-z_0)=0 [三元一次方程]" contenteditable="false"><span></span><span></span></span>
与点法式联系<br>
<br><span class="equation-text" data-index="0" data-equation="法向量\overrightarrow n=(A,B,C),平面内向量法向量\overrightarrow {M_0M}=(x-x_0,y-y_0,z-z_0)" contenteditable="false"><span></span><span></span></span>
值与特性特点联系<br>
齐次方程零解<br><span class="equation-text" data-index="0" data-equation="D=0：过原点平面" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A=0[即x为任意值]:法向量\overrightarrow n=(0,B,C)垂直于x轴，即平行于x轴平面(B=0或C=0同理)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="当A=B=0时，即法向量\overrightarrow n=(0,0,C), 则平面垂直与z轴" contenteditable="false"><span></span><span></span></span>
相关问题形式
<br><span class="equation-text" data-index="0" data-equation="已知平面过x轴，和某点M(x_0,y_0,z_0)求平面方程" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="注：过x轴表示，平面内包含x轴[必然过原点]" contenteditable="false"><span></span><span></span></span>
截距式方程
<br><span class="equation-text" data-index="0" data-equation="1）已知平面内三点P(a,0,0).Q(0,b,0),R(0,0,c)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="2）得A=\frac{D}{a},B=\frac{D}{b},C=\frac{D}{c}，再带入一般方程形式同除以D" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="3）\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1" contenteditable="false"><span></span><span></span></span>
平面束方程
对称式方程平面束<br><span class="equation-text" data-index="0" data-equation="(x-x_0)/{m}=(y-y_0)/n=(z-z_0)/p\\ [(x-x_0)/{m}-(y-y_0)/n]+\lambda[(y-y_0)/n-(z-z_0)/p]=0" contenteditable="false"><span></span><span></span></span><br>
一般方程平面束<br><span class="equation-text" 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}\\A_1x+B_1y+C_1z+D_1+\lambda(A_1x+B_1y+C_1z+D_2)=0" contenteditable="false"><span></span><span></span></span><br>
三点式
<br><span class="equation-text" data-index="0" data-equation="已知平面内三坐标点，求平面方程?" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="hint:如何构造法向量?利用向量叉乘?" contenteditable="false"><span></span><span></span></span>
空间曲线方程
曲线方程
一般方程<br>
<span class="equation-text" data-index="0" data-equation="可以看作两个曲面的交线,则满足方程组\\\begin{cases} F(x,y,z)=0, \\ G(x,y,z)=0. \end{cases}" contenteditable="false"><span></span><span></span></span>
参数方程
<br><span class="equation-text" data-index="0" data-equation="\\ \begin{cases} x=x(t), \\ y=y(t) \\ z=z(t) \end{cases}" contenteditable="false"><span></span><span></span></span>
切线方向<br>
利用方程曲面的<b>法向量叉积</b>求出其切线法向量<br>
参数方程直接求出三个变量的导数<br>
直线方程
一般方程[两个平面一般方程]<br>
<br><span class="equation-text" 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}" contenteditable="false"><span></span><span></span></span>
方向向量<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{n_1}\times\overrightarrow{n_2}=\overrightarrow{s}" contenteditable="false"><span></span><span></span></span><br>
对称式方程[点向式]
基本形式<br><span class="equation-text" data-index="0" data-equation="(x-x_0)/{m}=(y-y_0)/n=(z-z_0)/p" contenteditable="false"><span></span><span></span></span><br>
方向向量<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{s}=(m,n,p)" contenteditable="false"><span></span><span></span></span><br>
方向余弦
参数方程<br>
基本形式<br><span class="equation-text" data-index="0" data-equation="\\\begin{cases} x=mt+x_0, \\ y=nt+y_0, \\ z=pt+z_0 \end{cases}" contenteditable="false"><span></span><span></span></span>
方向向量<br><span class="equation-text" data-index="0" data-equation="\overrightarrow{s}=(m,n,p)" contenteditable="false"><span></span><span></span></span><br>
空间曲线投影<br>
在xOy面上投影<br><span class="equation-text" data-index="0" data-equation="\Gamma =\begin{cases} F(x,y,z)=0, \\ G(x,y,z)=0. \end{cases}，消去z得 \begin{cases}H(x,y)=0\\z=0 \end{cases}" contenteditable="false"><span></span><span></span></span>
空间夹角<br>
平面与平面夹角<br>
范围<br><span class="equation-text" data-index="0" data-equation="[0,\frac{\pi}{2}]" contenteditable="false"><span></span><span></span></span><br>
夹角余弦[两法向量夹角]<br><span class="equation-text" data-index="0" data-equation="cos\theta = \frac{|A_1A_2+B_1 B_2+C_1 C_2|}{\sqrt{A_1^2+ B_1^2 + C_1^2}\sqrt{A_2^2+ B_2^2 + C_2^2}} [绝对值]" contenteditable="false"><span></span><span></span></span><br>
平面关系<br>
平行【法向量对应成比例】<br><span class="equation-text" data-index="0" data-equation="\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}" contenteditable="false"><span></span><span></span></span><br>
垂直【法向量数量积为0】<br><span class="equation-text" data-index="0" data-equation="A_1A_2+B_1B_2+C_1C_2=0" contenteditable="false"><span></span><span></span></span><br>
直线与直线夹角
范围<br><span class="equation-text" data-index="0" data-equation="[0,\frac{\pi}{2}]" contenteditable="false"><span></span><span></span></span><br>
夹角余弦[两方向向量夹角]<br><span class="equation-text" data-index="0" data-equation="cos\theta = \frac{|n_1n_2+m_1 m_2+p_1 p_2|}{\sqrt{n_1^2+ m_1^2 + p_1^2}\sqrt{n_2^2+ m_2^2 + p_2^2}} [绝对值]" contenteditable="false"><span></span><span></span></span><br>
直线与平面夹角<br>
范围<br><span class="equation-text" data-index="0" data-equation="[0,\frac{\pi}{2}]" contenteditable="false"><span></span><span></span></span><br>
夹角余弦[方向向量与法向量夹角]<br><span class="equation-text" data-index="0" data-equation="sin\varphi=cos\theta = \frac{|n_1A_1+m_1 B_1+p_1 C_1|}{\sqrt{A_1^2+ B_1^2 + C_1^2}\sqrt{n_2^2+ m_2^2 + p_2^2}} [绝对值]" contenteditable="false"><span></span><span></span></span><br>
距离公式<br>
点到直线<br><span class="equation-text" data-index="0" data-equation="P(x_0,y_0,z_0)到直线Ax+By+C=0" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="方向向量\overrightarrow \alpha=(A,B,C)\\任取一点P_1=(x_1,y_1,z_1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="距离d=|\frac{\overrightarrow{P_1P_0}\overrightarrow \times\alpha}{ |\overrightarrow \alpha|}|" contenteditable="false"><span></span><span></span></span>
点到平面<br><span class="equation-text" data-index="0" data-equation="P(x_0,y_0,z_0)到Ax+By+Cz+D=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="法向量\overrightarrow n=(A,B,C);任取一点P_1=(x_1,y_1,z_1)\\ \overrightarrow{P_1P_0}[斜边]与法向量[底边]夹角余弦:|cos\theta| =|\frac{\overrightarrow{P_1P_0}\overrightarrow n}{|\overrightarrow{P_1P_0}| |\overrightarrow n|}|" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="d=|\overrightarrow{P_1P_0}|*cos\theta=|\frac{\overrightarrow{P_1P_0}\overrightarrow n}{ |\overrightarrow n|}|=\frac{(\overrightarrow {OP_0}-\overrightarrow {OP_1})\overrightarrow n}{|\overrightarrow n|}=\frac{|F(P_0)|}{|\overrightarrow n|}=\frac{|Ax_0+By_0+Cz+D|}{\sqrt{A^2+B^2+C^2}}" contenteditable="false"><span></span><span></span></span>
二维空间<br><span class="equation-text" data-index="0" data-equation="d=\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}" contenteditable="false"><span></span><span></span></span>
三维空间<br><span class="equation-text" data-index="0" data-equation="\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}=\frac{|F(P_0)|}{|\overrightarrow n|}" contenteditable="false"><span></span><span></span></span><br>
平行平面<br><span class="equation-text" data-index="0" data-equation="Ax+By+Cz+D_0=0\\Ax+By+Cz+D_1=0" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="d=\frac{(r2-r1)\overrightarrow n}{|\overrightarrow n|}=\frac{|D_1-D_0|}{\sqrt{A^2+B^2+C^2}}" contenteditable="false"><span></span><span></span></span>
两直线之间<br><span class="equation-text" data-index="0" data-equation="s_1,s_2为直线L_1,L_2方向向量，\overrightarrow{AB}为两直线各取一点连线" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="d=|s_1s_2\overrightarrow{AB}|/|s_1\times s_2|" contenteditable="false"><span></span><span></span></span>
方向导数与梯度
方向导数<br>
定义<br><span class="equation-text" data-index="0" data-equation="\frac{\partial f}{\partial l}|_{(x_0,y_0)}=\lim_{t\rightarrow 0+}[f(x_0+t\cos \alpha,y_0+t\cos \beta)-f(x_0,y_0)]/t" contenteditable="false"><span></span><span></span></span>
二元函数方向导数计算<br><span class="equation-text" data-index="0" data-equation="\frac{\partial f}{\partial l}|_{(x_0,y_0)}=f_x(x_0,y_0)\cos \alpha + f_y(x_0,y_0)\cos \beta（\cos \alpha，\cos \beta 为方向余弦）" contenteditable="false"><span></span><span></span></span><br>
三元函数方向导数计算<br><span class="equation-text" data-index="0" data-equation="\frac{\partial f}{\partial l}|_{(x_0,y_0,z_0)}=f_x(x_0,y_0)\cos \alpha + f_y(x_0,y_0)\cos \beta+f_x(x_0,y_0)\cos \gamma" contenteditable="false"><span></span><span></span></span><br>
方向余弦<br><span class="equation-text" data-index="0" data-equation="\cos \alpha=\frac{x}{\sqrt{x^2+y^2+z^2}},\cos \beta=\frac{y}{\sqrt{x^2+y^2+z^2}},\cos \gamma=\frac{z}{\sqrt{x^2+y^2+z^2}}" contenteditable="false"><span></span><span></span></span><br>
梯度
定义<br><span class="equation-text" data-index="0" data-equation="\nabla f(x_0,y_0)=f_x(x_0,y_0)i+f_y(x_0,y_0)j" contenteditable="false"><span></span><span></span></span><br>
关系
等式<br><span class="equation-text" data-index="0" data-equation="\frac{\partial f}{\partial l}|_{(x_0,y_0)}=f_x(x_0,y_0)\cos \alpha + f_y(x_0,y_0)\cos \beta=\nabla f(x_0,y_0)\cdot e_l=|\nabla f(x_0,y_0)|\cos \theta" contenteditable="false"><span></span><span></span></span>
θ=0<br>f(x,y)增加最快，该方向导数达到最大值，即为梯度的模<br>
θ=pi<br>方向与梯度方向相反，函数减少最快，达到最小值<br>
θ=pi/2<br>方向与梯度方向正交时，函数变化率为0<br>
多元微分学及运用
一元向量值函数<br>
参数方程<br><span class="equation-text" data-index="0" data-equation="\Gamma =\begin{cases} x=\varphi(t) \\ y=\psi(t), \\ z=\omega(t) \end{cases} \\" contenteditable="false"><span></span><span></span></span><br>
空间曲线参数方程及其向量形式<br><span class="equation-text" data-index="0" data-equation="r=xi+yj+zk,f(t)=\varphi(t)i+\psi(t)j+\omega(t)k;r=f(t)" contenteditable="false"><span></span><span></span></span>
导向量<br>
<br><span class="equation-text" data-index="0" data-equation="f'(t_0)=\frac{dr}{dt}|_{t=t_0}=\varphi'(t)i+\psi'(t)j+\omega'(t)k;" contenteditable="false"><span></span><span></span></span>
向量复合求导法则<br>
<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dt}[u(t)\cdot v(t)]=u'(t)\cdot v(t)+u(t)\cdot v'(t)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{d}{dt}[u(t)\times v(t)]=u'(t)\times v(t)+u(t)\times v'(t)" contenteditable="false"><span></span><span></span></span>
曲切平面[法线]<br>
法向量
F(x,y,z)=0<br><span class="equation-text" data-index="0" data-equation="n=\{F_x,F_y,F_z\}" contenteditable="false"><span></span><span></span></span><br>
z=f(x,y)<br><span class="equation-text" data-index="0" data-equation="n=\{f_x,f_y,f_z\}" contenteditable="false"><span></span><span></span></span><br>
曲线切线[法平面]<br>
曲线法线
与切线垂直的直线<br>
切线向量<br>
参数方程切向量<br><span class="equation-text" data-index="0" data-equation="f(t_0)'=(\varphi'(t_0),\psi'(t_0),\omega'(t_0))" contenteditable="false"><span></span><span></span></span><br>
一般方程切向量<br><span class="equation-text" data-index="0" data-equation="\tau=\overrightarrow n_1\times \overrightarrow n_2" contenteditable="false"><span></span><span></span></span><br>
法平面方程<br>
<br><span class="equation-text" data-index="0" data-equation="(x-x_0){\varphi'(t_0)}+(y-y_0)\psi'(t_0)+(z-z_0)\omega'(t_0)=0" contenteditable="false"><span></span><span></span></span>
多元积分学及运用<br>
曲线积分
概念<br>
平面单连通区域<br>不含"洞"的区域<br>
复连通区域<br>含"洞"的区域<br>
弧长曲线积分 (第一类)<br>
类型<br>
封闭曲线弧长积分<br><span class="equation-text" data-index="0" data-equation="\oint_{L} f(x,y)ds" contenteditable="false"><span></span><span></span></span>
平面曲线<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\lim_{\lambda \rightarrow 0}\sum_{i=1}^n f(\xi_i,\eta_i)\Delta s_i" contenteditable="false"><span></span><span></span></span><br>
空间曲线<br><span class="equation-text" data-index="0" data-equation="\int_\Gamma f(x,y,z)ds=\lim_{\lambda \rightarrow 0}\sum_{i=1}^n f(\xi_i,\eta_i,\zeta_i)\Delta s_i" contenteditable="false"><span></span><span></span></span><br>
性质
可拆性<br><span class="equation-text" data-index="0" data-equation="\int_a^b [\alpha f(x)+\beta g(x)]dx=\alpha\int_a^b f(x)dx+\beta\int_a^b g(x) dx" contenteditable="false"><span></span><span></span></span>
可加性<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\int_{L_1}f(x,y)ds+\int_{L_2}f(x,y)ds" contenteditable="false"><span></span><span></span></span>
不等式<br><span class="equation-text" data-index="0" data-equation="f(x)\leq g(x)则\int_L f(x)ds\leq\int_L g(x)ds" contenteditable="false"><span></span><span></span></span>
绝对值不等式<br><span class="equation-text" data-index="0" data-equation="|\int_Lf(x)ds|\leq \int_L|f(x,y)|ds" contenteditable="false"><span></span><span></span></span>
<font color="#B71C1C">积分路径无关</font><br><span class="equation-text" data-index="0" data-equation="\int_{L\overset{\frown} {AB}}f(x,y)ds=\int_{L\overset{\frown} {BA}}f(x,y)ds" contenteditable="false"><span></span><span></span></span><br>
奇偶对称性<br>
积分曲线 L 关于 y 轴对称<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\begin{cases}2\int_{L：x\geq 0} f(x,y)ds,f(x,y)=f(-x,y)\\0,f(-x,y)=-f(x,y)\end{cases}" contenteditable="false"><span></span><span></span></span><br>
计算
参数方程<br><span class="equation-text" data-index="0" data-equation="\begin{cases} x=\varphi(t) \\ y=\psi(t) \\ z=\omega(t)\end{cases}" contenteditable="false"><span></span><span></span></span>
平面曲线参数方程<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\int_\alpha^\beta f[\varphi(t),\psi(t)]\sqrt{\varphi'^2(t)+\psi'^2(t)}dt" contenteditable="false"><span></span><span></span></span>
空间曲线参数方程<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\int_\alpha^\beta f[\varphi(t),\psi(t)]\sqrt{\varphi'^2(t)+\psi'^2(t)+\omega'^2(t)}dt" contenteditable="false"><span></span><span></span></span><br>
极坐标系<br>
平面极坐标系<br><span class="equation-text" data-index="0" data-equation="\int_Lf(x,y)ds=\int_\alpha^\beta f(r\cos \theta,r\sin \theta)\sqrt{r^2+r'^2}d\theta【ds=\sqrt{(d\rho)^2+(\rho d\theta)^2};\rho=\rho(\theta)】" contenteditable="false"><span></span><span></span></span><br>
坐标曲线积分 (第二类)<br>
定义
<span class="equation-text" data-index="0" data-equation="\int_\Gamma P(x,y,z)dx+ Q(x,y,z)dy+ R(x,y,z)dz" contenteditable="false"><span></span><span></span></span>
性质<br>
<font color="#B71C1C">积分路径方向相关</font><br><span class="equation-text" data-index="0" data-equation="\int_{L\overset{\frown} {AB}}Pdx+Qdy=-\int_{L\overset{\frown} {BA}}Pdx+Qdy" contenteditable="false"><span></span><span></span></span><br>
计算<br>
平面曲线<br>
直接法
<span class="equation-text" data-index="0" data-equation="\int_LP(x,y)dx+Q(x,y)dy=\int_\alpha^\beta \{P[\varphi(t),\psi(t)]\varphi'(t)+Q[\varphi(t),\psi(t)]\psi'(t)\}dt" contenteditable="false"><span></span><span></span></span>
格林公式
<span class="equation-text" data-index="0" data-equation="\\ \iint_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy=\oint_L Pdx+Qdy【L为D取正向的边界曲线】" contenteditable="false"><span></span><span></span></span>
补线使用格林公式<br>
路径无关线积分<br>
判定<br>
<span class="equation-text" data-index="0" data-equation="\int_{L}Pdx+Qdy" contenteditable="false"><span></span><span></span></span> 与路径无关
闭曲线积分为0<br><span class="equation-text" data-index="0" data-equation="\oint_{L_1+L_2^-}Pdx+Qdy=0" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\oint_{L}Pdx+Qdy=0" contenteditable="false"><span></span><span></span></span>，L为D中任一分段光滑闭曲线
两偏导相等<br><span class="equation-text" data-index="0" data-equation="\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}，\forall(x,y)\in D" contenteditable="false"><span></span><span></span></span><br>
原函数微分<br><span class="equation-text" data-index="0" data-equation="P(x,y)dx+Q(x,y)dy=dF(x,y)" contenteditable="false"><span></span><span></span></span>
计算<br>
该换路径计算<br>
利用原函数计算<br><span class="equation-text" data-index="0" data-equation="\int_{(x_1,y_1)}^{(x_2,y_2)}Pdx+Qdy=F(x_2,y_2)-F(x_1,y_1)" contenteditable="false"><span></span><span></span></span>
空间曲线<br>
直接法
<span class="equation-text" data-index="0" data-equation="\int_\Gamma P(x,y,z)dx+ Q(x,y,z)dy+ R(x,y,z)dz" contenteditable="false"><span></span><span></span></span>
斯托克斯公式<br>
<span class="equation-text" data-index="0" data-equation="\oint_LP(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz=\iint\begin{vmatrix}\cos\alpha &\cos \beta &\cos \gamma\\\partial/\partial x & \partial/\partial y & \partial/\partial z\\P&Q&R\end{vmatrix}ds" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="=\iint(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial x})dydz+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})dzdx+(\frac{\partial Q}{\partial x}-\frac{\partial R}{\partial y})dxdy" contenteditable="false"><span></span><span></span></span>
两类曲线积分联系<br>
平面曲线<br><span class="equation-text" data-index="0" data-equation="\int_L Pdx+Qdy=\int_L(P\cos \alpha+Q\cos\beta)ds" contenteditable="false"><span></span><span></span></span>
空间曲线<br><span class="equation-text" data-index="0" data-equation="\int_\Gamma Pdx+Qdy+Rdz=\int_\Gamma(P\cos \alpha+Q\cos\beta+R\cos \gamma)ds" contenteditable="false"><span></span><span></span></span><br>
曲面积分
面积曲面积分<br>
定义<br><span class="equation-text" data-index="0" data-equation="\iint f(x,y,z)dS=\lim_{\lambda\rightarrow 0}\sum_{i=1}^n f(\xi_i,\eta_i,\zeta_i)(\Delta S_i)" contenteditable="false"><span></span><span></span></span><br>
性质【与积分曲面<font color="#B71C1C">方向无关</font>】<br><span class="equation-text" data-index="0" data-equation="\iint_{\Sigma}f(x,y,z)dS=\iint_{-\Sigma}f(x,y,z)dS" contenteditable="false"><span></span><span></span></span><br>
公式<br><span class="equation-text" data-index="0" data-equation="\iint_{\Sigma}f(x,y,z)dS=\iint_{D_{xy}}f[x,y,z(x,y)]\sqrt{1+z_x'^2+z_y'^2}dxdy" contenteditable="false"><span></span><span></span></span><br>
化简条件<br>
奇偶性<br>
对称性<br>
计算步骤<br>
坐标曲面积分
定义<br><span class="equation-text" data-index="0" data-equation="\iint R(x,y,z)dxdy=\lim_{\lambda\rightarrow 0}\sum_{i=1}^n R(\xi_i,\eta_i,\zeta_i)(\Delta S_i)_{xy}" contenteditable="false"><span></span><span></span></span>
性质【与积分曲面方向有关】<br><span class="equation-text" data-index="0" data-equation="\iint_{\Sigma}Pdydz+Qdzdx+Rdxdy=-\iint_{-\Sigma}Pdydz+Qdzdx+Rdxdy" contenteditable="false"><span></span><span></span></span><br>
计算
直接法<br><span class="equation-text" data-index="0" data-equation="\iint_\Sigma R(x,y,z)dxdy=\pm \iint_{D_{xy}} R[x,y,z(x,y)]dxdy" contenteditable="false"><span></span><span></span></span><br>
高斯公式<br><span class="equation-text" data-index="0" data-equation="\oiint_{\Sigma}Pdydz+Qdzdx+Rdxdy=\iiint(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z})dv" contenteditable="false"><span></span><span></span></span><br>
真题
2021, (二), 14<br>
两类面积分的联系<br>
<br><span class="equation-text" data-index="0" data-equation="\iint_{\Sigma} Pdydz+Qdxdz+Rdxdy= \iint_{\Sigma} (P\cos \alpha+Q\cos\beta+R\cos \gamma)ds" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\cos \alpha=\frac{-Z_x}{\sqrt{1+Z_x'^2+Z_y'^2}},\cos \beta=\frac{-Z_y}{\sqrt{1+Z_x'^2+Z_y'^2}},\cos \gamma=\frac{1}{\sqrt{1+Z_x'^2+Z_y'^2}}" contenteditable="false"><span></span><span></span></span>
沿任意闭曲面积分为零<br>
充要条件<br><span class="equation-text" data-index="0" data-equation="\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}=0" contenteditable="false"><span></span><span></span></span><br>
多元积分应用
质量<br>
平面域 <br><span class="equation-text" data-index="0" data-equation="m = \iint_D \rho(x,y) d\sigma" contenteditable="false"><span></span><span></span></span><br>
空间体 <br><span class="equation-text" data-index="0" data-equation="m=\iiint_\Omega \rho(x,y,z)dv" contenteditable="false"><span></span><span></span></span><br>
曲线段<br> <span class="equation-text" data-index="0" data-equation="m=\int_C f(x,y,z)ds" contenteditable="false"><span></span><span></span></span><br>
曲面片<br> <span class="equation-text" data-index="0" data-equation="m=\iint_\Sigma\rho(x,y,z) dS" contenteditable="false"><span></span><span></span></span><br>
质心
平面域 <br><span class="equation-text" data-index="0" data-equation="\bar x = \iint_D x\rho(x,y) d\sigma/m" contenteditable="false"><span></span><span></span></span><br>
空间体 <br><span class="equation-text" data-index="0" data-equation="\bar x=\iiint_\Omega x\rho(x,y,z)dv/m" contenteditable="false"><span></span><span></span></span><br>
曲线段<br> <span class="equation-text" data-index="0" data-equation="\bar x=\int_C xf(x,y,z)ds/m" contenteditable="false"><span></span><span></span></span><br>
曲面片<br> <span class="equation-text" data-index="0" data-equation="\bar x=\iint_\Sigma\rho(x,y,z) dS/m" contenteditable="false"><span></span><span></span></span><br>
形心<br>
转动惯量<br>
平面域 <br><span class="equation-text" data-index="0" data-equation="I_x = \iint_D y^2\rho(x,y) d\sigma" contenteditable="false"><span></span><span></span></span><br>
空间体 <br><span class="equation-text" data-index="0" data-equation="I_x=\iiint_\Omega (y^2+z^2)\rho(x,y,z)dv" contenteditable="false"><span></span><span></span></span><br>
曲线段<br> <span class="equation-text" data-index="0" data-equation="I_x=\int_C (y^2+z^2)f(x,y,z)ds" contenteditable="false"><span></span><span></span></span><br>
曲面片<br> <span class="equation-text" data-index="0" data-equation="I_x=\iint_\Sigma (y^2+z^2)\rho(x,y,z) dS" contenteditable="false"><span></span><span></span></span><br>
变力作功<br>
力：<span class="equation-text" data-index="0" data-equation="F=Pi+Qj+Rk" contenteditable="false"><span></span><span></span></span>
功：<span class="equation-text" data-index="0" data-equation="W=\int_{\overset{\frown} {AB} }Pdx+Qdy+Rdz" contenteditable="false"><span></span><span></span></span><br>
通量/流量<br>
通量：<span class="equation-text" data-index="0" data-equation="\Phi=\iint_{ \Sigma}Pdx+Qdy+Rdz" contenteditable="false"><span></span><span></span></span><br>
场论初步
梯度<br>
<span class="equation-text" data-index="0" data-equation="\bold {grad}u=\nabla u=\frac{\partial u}{\partial x}i+\frac{\partial u}{\partial y}j+\frac{\partial u}{\partial z}k" contenteditable="false"><span></span><span></span></span>
散度<br>
<br><span class="equation-text" data-index="0" data-equation="\bold{divA}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}" contenteditable="false"><span></span><span></span></span>
旋度
<br><span class="equation-text" data-index="0" data-equation="\bold {rotA}=\nabla \times A=\begin{vmatrix}i &j&k\\\partial/\partial x & \partial/\partial y & \partial/\partial z\\P&Q&R\end{vmatrix}" contenteditable="false"><span></span><span></span></span>
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