极限
核心概念
数列极限：当 n 无限增大时，数列 {aₙ} 无限接近常数 A，记 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to\infty}a_n=A"><span></span><span></span></span>
函数极限：<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\to x_0"><span></span><span></span></span>（或 <span class="equation-text" contenteditable="false" data-index="0" data-equation="x\to\infty"><span></span><span></span></span>）时函数值无限接近常数 A，记 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_0}f(x)=A"><span></span><span></span></span>
左极限与右极限：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_0^-}f(x)"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_0^+}f(x)"><span></span><span></span></span> 分别从左右两侧逼近
无穷小与无穷大：极限为 0 的变量为无穷小，绝对值无限增大的变量为无穷大
无穷小的比较：高阶、同阶、等价无穷小，等价替换简化计算
核心定理
极限存在的充要条件：左极限与右极限都存在且相等（函数极限）；单调有界数列必有极限
夹逼定理：若 <span class="equation-text" contenteditable="false" data-index="0" data-equation="g(x)\le f(x)\le h(x)"><span></span><span></span></span> 且 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim g=\lim h=A"><span></span><span></span></span>，则 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim f=A"><span></span><span></span></span>
两个重要极限：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to 0}\frac{\sin x}{x}=1"><span></span><span></span></span>；<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to\infty}(1+\frac{1}{x})^x=e"><span></span><span></span></span>
四则运算法则：加减乘除的极限等于极限的加减乘除（分母极限不为 0）
复合函数极限法则：内层极限存在且外层函数连续时可代入
连续与间断：函数在点连续等价于 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_0}f(x)=f(x_0)"><span></span><span></span></span>；间断点分第一类（可去、跳跃）与第二类（无穷、振荡）
闭区间连续函数性质：有界性、最值定理、介值定理、零点定理
复习要点
求极限常用方法：直接代入、等价无穷小替换、洛必达法则、夹逼、单调有界、泰勒展开
易错：等价无穷小只能在乘除中替换，加减中慎用
导数
核心概念
定义：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x_0)=\lim_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}"><span></span><span></span></span>，即函数在该点的瞬时变化率
几何意义：曲线在点 <span class="equation-text" contenteditable="false" data-index="0" data-equation="(x_0,f(x_0))"><span></span><span></span></span> 处切线的斜率
可导与连续：可导必连续，连续不一定可导
高阶导数：二阶及以上导数，<span class="equation-text" contenteditable="false" data-index="0" data-equation="f''(x)"><span></span><span></span></span> 表示变化率的变化率
隐函数与参数方程求导：隐函数两边对 x 求导；参数方程 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}=\frac{dy/dt}{dx/dt}"><span></span><span></span></span>
核心定理
基本求导公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x^n)'=nx^{n-1}"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\sin x)'=\cos x"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="(e^x)'=e^x"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\ln x)'=\frac{1}{x}"><span></span><span></span></span> 等
四则求导法则：<span class="equation-text" contenteditable="false" data-index="0" data-equation="(u\pm v)'=u'\pm v'"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="(uv)'=u'v+uv'"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\frac{u}{v})'=\frac{u'v-uv'}{v^2}"><span></span><span></span></span>
复合函数求导（链式法则）：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}"><span></span><span></span></span>
反函数求导：<span class="equation-text" contenteditable="false" data-index="0" data-equation="[f^{-1}(y)]'=\frac{1}{f'(x)}"><span></span><span></span></span>
应用
单调性判定：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x)>0"><span></span><span></span></span> 递增，<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x)<0"><span></span><span></span></span> 递减
极值判定：一阶导数变号法、二阶导数法（<span class="equation-text" contenteditable="false" data-index="0" data-equation="f''(x_0)<0"><span></span><span></span></span> 为极大值）
最值求法：比较区间内驻点、不可导点与端点函数值
凹凸性与拐点：由 <span class="equation-text" contenteditable="false" data-index="0" data-equation="f''(x)"><span></span><span></span></span> 的符号判定
渐近线：水平、垂直、斜渐近线的求法
洛必达法则：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{0}{0}"><span></span><span></span></span> 与 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\infty}{\infty}"><span></span><span></span></span> 型未定式求极限
微分
核心概念
定义：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Delta y=A\Delta x+o(\Delta x)"><span></span><span></span></span> 时，<span class="equation-text" contenteditable="false" data-index="0" data-equation="dy=A\Delta x=f'(x)dx"><span></span><span></span></span>，微分是函数增量的线性主部
可微与可导：一元函数可微与可导等价
微分的几何意义：切线纵坐标的增量
核心定理与法则
微分形式不变性：无论 u 是自变量还是中间变量，<span class="equation-text" contenteditable="false" data-index="0" data-equation="dy=f'(u)du"><span></span><span></span></span> 均成立
微分四则法则：<span class="equation-text" contenteditable="false" data-index="0" data-equation="d(u\pm v)=du\pm dv"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="d(uv)=vdu+udv"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="d(\frac{u}{v})=\frac{vdu-udv}{v^2}"><span></span><span></span></span>
一阶微分形式不变性用于近似计算与积分换元
应用
近似计算：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x_0+\Delta x)\approx f(x_0)+f'(x_0)\Delta x"><span></span><span></span></span>
误差估计：由自变量误差估计函数值误差
与导数的关系：导数是微商 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}"><span></span><span></span></span>，微分侧重增量线性近似
积分
不定积分
概念：<span class="equation-text" contenteditable="false" data-index="0" data-equation="F'(x)=f(x)"><span></span><span></span></span> 时，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int f(x)dx=F(x)+C"><span></span><span></span></span>，是求导的逆运算
基本积分公式：幂函数、指数、三角、反三角等基本积分表
换元积分法：第一类（凑微分）、第二类（变量代换，含三角代换）
分部积分法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int u\,dv=uv-\int v\,du"><span></span><span></span></span>，适用于乘积型被积函数
有理函数积分：部分分式分解后逐项积分
定积分
概念：黎曼和的极限 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b f(x)dx=\lim_{\lambda\to 0}\sum f(\xi_i)\Delta x_i"><span></span><span></span></span>
几何意义：曲边梯形的代数面积
牛顿-莱布尼茨公式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b f(x)dx=F(b)-F(a)"><span></span><span></span></span>，把定积分计算转化为原函数求值
定积分性质：线性性、区间可加性、保号性、估值定理、积分中值定理
变上限积分函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Phi(x)=\int_a^x f(t)dt"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Phi'(x)=f(x)"><span></span><span></span></span>
核心定理与方法
积分中值定理：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b f(x)dx=f(\xi)(b-a)"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi\in[a,b]"><span></span><span></span></span>
微积分基本定理：变上限积分函数的导数等于被积函数，是牛顿-莱布尼茨公式的基础
定积分换元与分部：换元需同步更换上下限
应用
面积：平面图形面积（直角坐标、极坐标）
体积：旋转体体积（圆盘法、壳层法）
弧长、旋转曲面面积
物理应用：变力做功、液体压力、质心
微分方程
核心概念
定义：含未知函数及其导数的方程，阶指数最高阶导数的阶数
通解与特解：通解含与阶数相同个数的独立任意常数；由初始条件确定特解
一阶微分方程
可分离变量方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}=f(x)g(y)"><span></span><span></span></span>，分离后两边积分
齐次方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{dy}{dx}=\varphi(\frac{y}{x})"><span></span><span></span></span>，令 <span class="equation-text" contenteditable="false" data-index="0" data-equation="u=\frac{y}{x}"><span></span><span></span></span> 化为可分离变量
一阶线性方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+P(x)y=Q(x)"><span></span><span></span></span>，通解 <span class="equation-text" contenteditable="false" data-index="0" data-equation="y=e^{-\int P dx}\left(\int Q e^{\int P dx}dx+C\right)"><span></span><span></span></span>
伯努利方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y'+P(x)y=Q(x)y^n"><span></span><span></span></span>，令 <span class="equation-text" contenteditable="false" data-index="0" data-equation="z=y^{1-n}"><span></span><span></span></span> 化为一阶线性
全微分方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Pdx+Qdy=0"><span></span><span></span></span> 且 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}"><span></span><span></span></span> 时存在原函数
高阶微分方程
可降阶类型：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y^{(n)}=f(x)"><span></span><span></span></span> 逐次积分；<span class="equation-text" contenteditable="false" data-index="0" data-equation="y''=f(x,y')"><span></span><span></span></span> 令 <span class="equation-text" contenteditable="false" data-index="0" data-equation="p=y'"><span></span><span></span></span>；<span class="equation-text" contenteditable="false" data-index="0" data-equation="y''=f(y,y')"><span></span><span></span></span> 令 <span class="equation-text" contenteditable="false" data-index="0" data-equation="p=y'"><span></span><span></span></span> 并以 y 为自变量
二阶常系数齐次线性方程：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y''+py'+qy=0"><span></span><span></span></span>，由特征方程 <span class="equation-text" contenteditable="false" data-index="0" data-equation="r^2+pr+q=0"><span></span><span></span></span> 的根确定通解形式（两不等实根、重根、共轭复根）
二阶常系数非齐次线性方程：通解 = 齐次通解 + 特解，特解按自由项类型（<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^{\lambda x}P_m(x)"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^{\lambda x}[P\cos\omega x+Q\sin\omega x]"><span></span><span></span></span>）设待定形式
核心定理
解的结构定理：非齐次方程通解等于对应齐次方程通解加非齐次方程一个特解
叠加原理：自由项为若干项之和时，特解可分别求解后相加
存在唯一性定理：满足连续性条件时初值问题解存在且唯一
级数
常数项级数
概念：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum_{n=1}^{\infty}u_n"><span></span><span></span></span> 的部分和数列极限存在则收敛，否则发散
收敛的必要条件：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to\infty}u_n=0"><span></span><span></span></span>（不满足则发散）
几何级数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum q^n"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="|q|<1"><span></span><span></span></span> 收敛，<span class="equation-text" contenteditable="false" data-index="0" data-equation="|q|\ge 1"><span></span><span></span></span> 发散
p 级数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum\frac{1}{n^p}"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="p>1"><span></span><span></span></span> 收敛，<span class="equation-text" contenteditable="false" data-index="0" data-equation="p\le 1"><span></span><span></span></span> 发散
核心判别法
正项级数比较判别法：与已知收敛/发散级数比较（含极限形式）
比值判别法（达朗贝尔）：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim\frac{u_{n+1}}{u_n}=\rho"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho<1"><span></span><span></span></span> 收敛，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho>1"><span></span><span></span></span> 发散
根值判别法（柯西）：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim\sqrt[n]{u_n}=\rho"><span></span><span></span></span>，判定规则同比值法
交错级数莱布尼茨判别法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="u_n"><span></span><span></span></span> 单调递减且趋于 0 时收敛
绝对收敛与条件收敛：绝对收敛必收敛；条件收敛需另判
幂级数
概念：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum a_n(x-x_0)^n"><span></span><span></span></span>，收敛区间以收敛半径 R 为中心
收敛半径求法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho=\lim|\frac{a_{n+1}}{a_n}|"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="R=\frac{1}{\rho}"><span></span><span></span></span>（或由根值法求得）
收敛域：在收敛区间基础上单独检验端点
幂级数性质：在收敛区间内可逐项求导、逐项积分，收敛半径不变
函数展开：泰勒级数、麦克劳林级数，常用展开式（<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^x"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sin x"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\cos x"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln(1+x)"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{1-x}"><span></span><span></span></span>）
傅里叶级数
概念：周期函数展开为三角级数，系数由积分公式确定
狄利克雷收敛定理：满足条件时傅里叶级数收敛于函数（间断点处收敛于左右极限的平均值）
奇偶函数的傅里叶级数：奇函数只有正弦项，偶函数只有余弦项
