浙江省专升本-一元函数微分学
2022-04-20 12:46:07 0 举报AI智能生成
浙江省专升本-一元函数微分学
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大纲/内容
导数(瞬时变化率)
导数定义
导数定义式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x_0)=\lim_{\nabla\rightarrow {0}}\frac{\nabla y}{\nabla x}=\lim_{\nabla\rightarrow {0}} \frac{f(x_0+\nabla x )-f(x_0)}{\nabla x}(导数的第一定义式)=\lim_{x\rightarrow {x_0}} \frac{f(x)-f(x_0)}{x-x_0}，记为y^{'}|_x=x_0,\frac{dy}{dx} |_x=x_0或\frac{df(x)}{dx} |_x=x_0"><span></span><span></span></span>
其中<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x_0)=\lim_{x\rightarrow {x_0}} \frac{f(x)-f(x_0)}{x-x_0}(导数第二定义式)"><span></span><span></span></span>是求分段函数分段点导数值式子
导数第一定义式推广式(只能作为计算技巧不能作为可导依据)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="lim_{\nabla\rightarrow {0}} \frac{f(x_0+a\nabla x )-f(x_0+b\nabla x)}{c\nabla x}=\frac{a-b}{c}f^{'}(x_o)"><span></span><span></span></span>
导数一般化式子
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x)=lim_{\nabla\rightarrow {0}} \frac{f(x+\nabla x )-f(x)}{\nabla x}"><span></span><span></span></span>,即表示可以求任意点处的导数值
注
函数在点<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>的附近有定义，否则导数不存在
定义式中<span class="equation-text" contenteditable="false" data-index="0" data-equation="\nabla x"><span></span><span></span></span>趋向于0可正、可负、但不为0，<span class="equation-text" contenteditable="false" data-index="1" data-equation="\nabla y "><span></span><span></span></span>可为0
导数反应了因变量随自变量变化快慢程度；比值<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\nabla y}{\nabla x} "><span></span><span></span></span>为函数关于自变量的平均变化率；变化率的问题可以转化为导数问题
顺势变化率与导数是同一概念的两个名词，也就意味着<span class="equation-text" contenteditable="false" data-index="0" data-equation="t_0时的瞬时速度为v(t_0)=S^{'}(t_0)"><span></span><span></span></span>
导数的含义
图解
含义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="K_{PQ}=\frac{\nabla y}{\nabla x}=\frac{f(x_0+\nabla x)-f(x_0)}{\nabla x}"><span></span><span></span></span>
当<span class="equation-text" contenteditable="false" data-index="0" data-equation="\nabla x \rightarrow 0"><span></span><span></span></span>时，割线PQ就会变成切线，那么上式取极限就是曲线在点P处的切线斜率
<span class="equation-text" contenteditable="false" data-index="0" data-equation="K=\lim_{\nabla x \rightarrow 0}\frac{\nabla y}{\nabla x}=\lim_{\nabla x \rightarrow 0}\frac{f(x_0+\nabla x)-f(x_0)}{\nabla x}"><span></span><span></span></span>
导函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="if y=f(x)在开区间(a,b)内每点都有导数，此时对于每一个x\in(a,b)都有一个确定的导数f^{'}(x)，称这个函数f^{'}(x)为y=f(x)的"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="导数f^{'}(x)=y^{'}=\lim_{\nabla\rightarrow {0}}\frac{\nabla y}{\nabla x}=\lim_{\nabla\rightarrow {0}} \frac{f(x_0+\nabla x )-f(x_0)}{\nabla x}"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="if y=f(x)在开区间(a,b)内每点都有导数，称为y=f(x)在开区间(a,b)内可导"><span></span><span></span></span>
导数与导函数称为导数，区分：求一个函数的导数，就求导函数，求一个函数在给定点的导数，就是求导函数值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求导数的时候只需要将x换x_0即可：f^{'}(x)=lim_{\nabla\rightarrow {0}} \frac{f(x+\nabla x )-f(x)}{\nabla x}"><span></span><span></span></span>
根据导数定义求函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>的导数一般方法
求函数的平均该变量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\nabla y =f(x+\nabla x)-f(x)"><span></span><span></span></span>
求平均变化率
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\nabla y}{\nabla x}=\frac{f(x+\nabla x )-f(x)}{\nabla x}"><span></span><span></span></span>
取极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y^{'}=\lim_{\nabla x \rightarrow 0}\frac{\nabla y}{\nabla x}"><span></span><span></span></span>
导数和连续的关系
定理1
如果函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>点可导，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=f(x)"><span></span><span></span></span> 在x_0点连续
可导必定连续，连续不一定可导，不连续一定不可导
左右导数(单侧导数)
左导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f_-^{'}(x_0)=\lim_{\nabla x \rightarrow 0^-}\frac{\nabla y}{\nabla x}=lim_{\nabla\rightarrow {0^-}} \frac{f(x_0+\nabla x )-f(x_0)}{\nabla x}"><span></span><span></span></span>
右导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f_+^{'}(x_0)=\lim_{\nabla x \rightarrow 0^+}\frac{\nabla y}{\nabla x}=lim_{\nabla\rightarrow {0^+}} \frac{f(x_0+\nabla x )-f(x_0)}{\nabla x}"><span></span><span></span></span>
注
即使左右导数存在也不一定<i>相等</i>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x_0)存在\Leftrightarrow f_-^{'}(x_0)=f_+^{'}(x_0)"><span></span><span></span></span>
函数在一点处左右导数存在，即使不相等也可能在这点连续
常见不连续
分式(分母为0的点)不连续
绝对值(绝对值为0的点)不可导
<span class="equation-text" contenteditable="false" data-index="0" data-equation="函数(x-x_0)|x-x_0|在x=x_0处可导"><span></span><span></span></span>
导数的几何意义
含义
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)在x_0处的导数f^{'}(x_0)，表示曲线y=f(x)在点(x_0,f(x_0))处的切线斜率tan\alpha,即f^{'}(x_0)=tan\alpha"><span></span><span></span></span>
切线方程<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x_0,y_0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y-f(x_0)=f^{'}(x_0)(x-x_0)"><span></span><span></span></span>
法线方程<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x_0,y_0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y-f(x_0)=-\frac{1}{f^{'}(x_0)}(x-x_0)"><span></span><span></span></span>
求导数的方法
函数的求导法则
定理1
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=u(x)\pm v(x)\rightarrow f^{'}(x_0)=u^{'}(x_0)\pm v^{'}(x_0)"><span></span><span></span></span>
注
定理可以推广到有限个可导函数上去
简单记<span class="equation-text" contenteditable="false" data-index="0" data-equation="u(x)\pm v(x)=u^{'}(x)\pm v^{'}(x)"><span></span><span></span></span>
定理2
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=u(x) v(x)\rightarrow f^{'}(x_0)=u^{'}(x_0)v(x_0)+ u(x_0)v^{'}(x_0)"><span></span><span></span></span>
注
若取<span class="equation-text" contenteditable="false" data-index="0" data-equation="v(x)=c为常数，则有: (cx)^{'}=cu^{'}"><span></span><span></span></span>
定理可以推广到有限个可导函数的乘积上去,如<span class="equation-text" contenteditable="false" data-index="0" data-equation="(u(x)v(x)w(x))^{'}=u^{'}(x)v(x)w(x)+ u(x)v^{'}(x)w(x)+u(x)v(x)w^{'}(x)"><span></span><span></span></span>
简记<span class="equation-text" contenteditable="false" data-index="0" data-equation="(u(x)v(x))^{'}=u^{'}(x)v(x)+ u(x)v^{'}(x)"><span></span><span></span></span>
定理3
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=\frac{u(x)}{v(x)}\rightarrow f^{'}(x_0)=\frac{u^{'}(x_0)v(x_0)-u(x_)v^{'}(x_0))}{v^2(x_0)}"><span></span><span></span></span>
注
由定理2可推<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=u(x)\cdot \frac{1}{v(x)}"><span></span><span></span></span>
简记<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\frac{u}{v})^{'}=\frac{u^{'}v-uv^{'}}{v^2}(v\neq0)"><span></span><span></span></span>
常用初等求导公式
复合函数、反函数的导数法则
反函数的导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x_0)=\frac{1}{\phi^{'}(y_0)}"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\rightarrow x_0 \Leftrightarrow y \rightarrow y_0 ，因为\phi(y)在y_0点附近连续，严格单调"><span></span><span></span></span>
视若<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>为任意，并用<span class="equation-text" contenteditable="false" data-index="1" data-equation="x代替"><span></span><span></span></span>，使用<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{'}(x)=\frac{1}{\phi^{'}(y)}"><span></span><span></span></span>
求<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)的反函数x=f^{-1}(y)的导数步骤"><span></span><span></span></span>
求出<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x)"><span></span><span></span></span>
反函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=f^{-1}(y)"><span></span><span></span></span>的导数为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{1}{f^{'}(x)}"><span></span><span></span></span>
将<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{f^{'}(x)}"><span></span><span></span></span>中的<span class="equation-text" contenteditable="false" data-index="1" data-equation="x"><span></span><span></span></span>用<span class="equation-text" contenteditable="false" data-index="2" data-equation="x=f^{-1}(y)"><span></span><span></span></span>换，或将所有关于x的式子用x替换即可
复合函数的求导法则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="u=\phi(x)在x=x_0点可导，且y=f(u)在u=u_{0}=\phi(x_0)也可导，y=f(\phi(x))在x=x_0点可导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[f(\phi(x))]^{'}=f^{'}(u_0)\phi^{'}(x_0)"><span></span><span></span></span>
注
若视<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>为任意，并用<span class="equation-text" contenteditable="false" data-index="1" data-equation="x"><span></span><span></span></span>代替得
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[f(\phi(x))]^{'}=f^{'}(\phi(x))\phi^{'}(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(\phi (x))与[f(\phi(x))]^{'}不同，前者对u=\phi(x)求导，后者是对变量x求导"><span></span><span></span></span>
复合函数求导可以推广到有限个函数复合的复合函数上去如
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[f(\phi(h(x)))]^{'}=f^{'}(\phi(h(x)))\cdot \phi^{'}(h(x))\cdot h^{'}(x)"><span></span><span></span></span>
特殊复合函数
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="u(x)^{v(x)}"><span></span><span></span></span>
求导方法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[u(x)^{v(x)}]^{'}=u(x)^{v(x)}[v^{'}(x)ln(u(x))+v(x)\frac{u^{'}(x)}{u(x)}]"><span></span><span></span></span>
高阶导数(n阶导，数学归纳法)
定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{''}(x_0)=\lim_{x\rightarrow x_0} \frac{f^{'}(x)-f^{'}(x_0)}{x-x_0},记作f^{(n)}(x_0),y^{{(n)}}|_{x=x_0}或\frac{d^{n}y}{dx^{n}}|_{x=x_0}"><span></span><span></span></span>
二阶和二阶以上的导数统称为高阶导数
莱布尼茨公式
对于<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=uv"><span></span><span></span></span>求高阶导，则可以一直求导下去，计算结果与二项式<span class="equation-text" contenteditable="false" data-index="1" data-equation="(u+v)^n展开式相似，通过数学归纳法，(uv)^{(n)}=\sum_{k=0}^nC_{n}^{k}u^{(n-k)}v^{(k)},其中u^{(0)}=u,v^{(0)}=v"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_{n}^{k}=\frac{A_{n}^{k}}{A_{k}^{k}},A_{n}^{k}=n\cdot(n-1)\cdots(n-k+1)=\frac{n!}{(n-k)!},C_{n}^{n}=C_{n}^{0}=1,C_{n}^{k}=\frac{n!}{k!(n-k)!},1!=0!=1"><span></span><span></span></span>
莱布尼茨公式中，<span class="equation-text" contenteditable="false" data-index="0" data-equation="u,v"><span></span><span></span></span>的地位对等，所以可以互换
常用高阶导公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x^{m})^n=m(m-1)(m-2)\cdots(m-n+1)x^{m-n}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x^m)^n=m!,m=n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x^m)^n=0,m<n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a^x)^{(n)}=a^x(lna)^n(a>0,a\neq1),特别的(e^x)^{(n)}=e^x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(e^{ax+b})^{(n)}=a^n\cdot e^{ax+b}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[sin(ax+b)]^{(n)}=a^nsin(ax+b+n\cdot\frac{\pi}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[cos(ax+b)]^{(n)}=a^ncos(ax+b+n\cdot\frac{\pi}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[\frac{1}{(ax+b)^m}]^{(n)}=\frac{(-1)^nm(m+1)(m+2)\cdots(m+n-1)a^n}{(ax+b)^{(m+n)}},(a\neq0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[ln(ax+b)]^{(n)}=(-1)^{(n-1)} \cdot \frac{(n-1)!a^n}{(ax+b)^n} "><span></span><span></span></span>
注
不是任何函数所有高阶导都存在
<span class="equation-text" contenteditable="false" data-index="0" data-equation="由二阶导数f^{''}(x)可定义三阶导数f^{'''}(x),由三阶导数f^{'''}(x)可定义四阶导数f^{(4)}(x),可由n-1阶导数定义n阶导数f^{n}(x)"><span></span><span></span></span>
当导数阶数大于3阶时候，现成<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{(n)}(x)"><span></span><span></span></span>
对于正整数米函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^{n}"><span></span><span></span></span>,每求一次导，其幂次降低1，第n阶级导数为一常数，大于n阶的导数都等于0
求高阶导数步骤
将原来的函数化为简单的初等函数的线性组合
求出一阶、二阶、三阶等导数
由第二步做出归纳总结
隐函数及由参数方程所确定的函数的导数
隐函数
显函数和隐函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=...显函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x),x\in I,则有:在I上F(x,f(x))=0隐函数"><span></span><span></span></span>
将隐函数转化为显函数，叫做隐函数的显化
注
不是任一<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x,f(x))=0"><span></span><span></span></span>都能确定出隐函数
隐函数不一定可以显化
隐函数求导方法步骤
方程两端同时对x求导数，注意把y当作复合函数求导的中间变量来看待
从求导后的方程中解出<span class="equation-text" contenteditable="false" data-index="0" data-equation="y^{'}"><span></span><span></span></span>
隐函数求导允许其结果中含有<span class="equation-text" contenteditable="false" data-index="0" data-equation="y"><span></span><span></span></span>
适合对数求导法题型
幂指函数
函数是由几个初等函数经过乘、除、开方构成的
参数方程所确定的函数的导数
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=\phi(t) \\y=\psi(t) \\\end{cases},t\in[\alpha,\beta],在[\alpha,\beta]上取一点t的值，则对应的曲线L上一点(x,y)。当t取遍[\alpha,\beta]上所有值，对应的点(x,y)遍构成曲线L"><span></span><span></span></span>
特殊的情况<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=Rcost \\y=Rsint \\\end{cases},t\in[0,2\pi],可化为圆的标准方程 \Leftrightarrow x^2+y^2=R^2,\begin{cases}x=x_0+rcost \\y=y_0+rsint \\\end{cases},t\in[0,2\pi], \Leftrightarrow (x-x_0)^2+(y-y_0)^2=r^2"><span></span><span></span></span>
链式求导法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{d^2y}{dx^2}=\frac{d}{dt}(\frac{dy}{dx})\cdot\frac{1}{\frac{dx}{dt}}=\frac{\psi^{''}(t)\phi^{'}(t)-\psi^{'}(t)\phi^{''}(t)}{(\phi^{'}(t))^3}"><span></span><span></span></span>
分段函数求导
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=\begin{cases}\phi(x),x>0 \\0,x=0 \\\psi(x),x<0\end{cases}"><span></span><span></span></span>
求导方法
各个区间段内的导数的求法就是普通的求导无异(各段内直接求)，要注意的是分段点处的导数一定要用导数定义求(分段点定义求)。
若分段函数在分段点两侧表达式不同，则要求分别求其左右导数，当且仅当左右导数存在且相等时，函数在分段点的导数存在
含绝对值符号的导数，先去掉绝对值符号，然后再做判断或求解
极限、可导、可微、连续之间的关系
微分
微分的几何含义
定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)在x_0的某个邻域内有定义，当给定x_0一个增量\nabla x时，函数y相应地得到增量\nabla y=f(x_0+\nabla x) -f(x_0).若存在常数A使得"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\nabla y=A\cdot \nabla x+o(\nabla)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="其中：A与\nabla x无关，o(\nabla x)是比\nabla x较高阶的无穷小量，即\lim_{\nabla x \rightarrow 0} \frac{o(\nabla x)}{\nabla x}=0,称为f(x)在x_0处的微分"><span></span><span></span></span>
导数与微分的等价关系
定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在x_0处可微\Leftrightarrow f(x)在x_0处可导，且:f^{'}(x_0)=A,dy|_{x=x_0}=f^{'}(x_0)dx"><span></span><span></span></span>
注
一元函数可导与可微等价
微分形式不变性
对于<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(u)"><span></span><span></span></span>,不论u是自变量，还是中间变量，函数的微分dy都具有相同形式即<span class="equation-text" contenteditable="false" data-index="1" data-equation="dy=f^{'}(u)du"><span></span><span></span></span>
微分的运算法则
微分与增量的关系
当<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x_0)\neq0"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{\nabla \rightarrow 0} \frac{\nabla y}{dy}=\lim_{\nabla \rightarrow 0} \frac{\nabla y}{f^{'}(x_0)\nabla x}=1"><span></span><span></span></span>
在此条件下，当<span class="equation-text" contenteditable="false" data-index="0" data-equation="|\nabla x| 很小时，有\nabla y \approx dy"><span></span><span></span></span>
注
当<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=x时，"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="dy=dx=x^{'}\nabla x=\nabla x,\nabla y=\nabla x,故dy=\nabla y"><span></span><span></span></span>
导数的应用
微分中值定理
费马定理<span class="equation-text" contenteditable="false" data-index="0" data-equation="(Fermat)"><span></span><span></span></span>定理(中值定理基础)
使用条件
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>的某邻域内有定义
在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0处可导"><span></span><span></span></span>
结论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对\forall x \in U(x_0)有f(x)\leq f(x_0)(或f(x) \geq f(x_0)),则f^{'}(x_0)=0"><span></span><span></span></span>
注
通常称为一阶导数为零的点为驻点(或稳定点，临界点)
罗尔<span class="equation-text" contenteditable="false" data-index="0" data-equation="(Rolle)"><span></span><span></span></span>中值定理(拉格朗日中值定理的特殊情况)
使用条件
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在[a,b]上连续"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在(a,b)内可导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(a)=f(b)"><span></span><span></span></span>
结论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\exist \xi\in (a,b),s.t. f^{'}(\xi)=0"><span></span><span></span></span>
几何含义
图解
含义
在每一点都可导的一段连续曲线上，如果曲线的两端的端点高度相等，则至少存在一条水平切线
注
习惯上把结论中的<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi"><span></span><span></span></span>称为中值，罗尔定理三个结论是充分而不必要的
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi不一定唯一，可能有有一个，几个甚至无限多个"><span></span><span></span></span>
利用罗尔定理可以判断函数的导函数的零点个数，判断方程根的个数
题目结论
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=(x-a)(x-b)(x-c)\cdots(x-n)"><span></span><span></span></span>若方程<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)=0"><span></span><span></span></span>有n个根，则方程<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{'}(x)=0"><span></span><span></span></span>有<span class="equation-text" contenteditable="false" data-index="3" data-equation="(n-1)"><span></span><span></span></span>个根，且方程的根数位于原方程的两根之间
拉格朗日<span class="equation-text" contenteditable="false" data-index="0" data-equation="(Largrange)"><span></span><span></span></span>中值定理(柯西中值定理的特殊情况)
使用条件
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在[a,b]上连续"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在(a,b)内可导"><span></span><span></span></span>
结论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\exist \xi\in (a,b),s.t. f^{'}(\xi)=\frac{f(b)-f(a)}{b-a}"><span></span><span></span></span>
几何含义
图解
含义
在满足拉格朗日中值定理条件的曲线<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>上至少存在一点<span class="equation-text" contenteditable="false" data-index="1" data-equation="P(\xi,f(\xi))"><span></span><span></span></span>,该曲线在该点处的切线平行于曲线两端点的连线
注
罗尔定理时拉格朗日中值定理<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(a)=f(b)"><span></span><span></span></span>时的特例
拉格朗日中值定理的两个条件彼此相关，互不独立，<span class="equation-text" contenteditable="false" data-index="0" data-equation="因为f(x)在(a,b)内可导可以推出f(x)在(a,b)内连续，但反之不成立"><span></span><span></span></span>
拉格朗日定理通常用于证明不等式
拉格朗日中值定理证明一个具体的不等式时关键时找到一个函数，一般都是一些我们熟悉的初等函数，再找到一个区间[b,a],再在[b,a]上应用拉格朗日中值定理，找到关于<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi"><span></span><span></span></span>的式子，最后利用<span class="equation-text" contenteditable="false" data-index="1" data-equation="\xi"><span></span><span></span></span>的范围建立一个不等式，将<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{'}(\xi)"><span></span><span></span></span>代入这个不等式即可证得原不等式，不过一定要将由<span class="equation-text" contenteditable="false" data-index="3" data-equation="\xi"><span></span><span></span></span>的范围建立的不等式找准
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在区间I上可导且f^{'}(x)=0\Leftrightarrow f(x)为I上的常值函数"><span></span><span></span></span>
这个推论常常用来证明一个函数恒等于某个常数的命题
证明某一个函数恒等于某个常数的一般步骤
对<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>求导得<span class="equation-text" contenteditable="false" data-index="1" data-equation="f^{'}(x)"><span></span><span></span></span>，再证<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{'}(x)=0"><span></span><span></span></span>
由<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1)得f(x)为一个常量函数，令f(x)=C"><span></span><span></span></span>
某一点<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=x_0"><span></span><span></span></span>一般取<span class="equation-text" contenteditable="false" data-index="1" data-equation="x=0"><span></span><span></span></span>代入<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)=C"><span></span><span></span></span>，得到常数<span class="equation-text" contenteditable="false" data-index="3" data-equation="C"><span></span><span></span></span>的值，即得证
函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)和故g(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上可导且<span class="equation-text" contenteditable="false" data-index="2" data-equation="f^{'}(x)=g^{'}(x)\Rightarrow f(x)=g(x)+c,x\in I"><span></span><span></span></span>
柯西<span class="equation-text" contenteditable="false" data-index="0" data-equation="(Cauchy)"><span></span><span></span></span>中值定理
使用条件
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x),g(x)在[a,b]上连续"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x),g(x)在(a,b)内可导"><span></span><span></span></span>
对于任意<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\in(a,b),g(x)\neq 0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="g(a)\neq g(b)"><span></span><span></span></span>
结论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\exist \xi\in (a,b),s.t. \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{'}(\xi)}{g^{'}(\xi)}"><span></span><span></span></span>
几何含义
图解
含义
一个光滑的线段上，一定存在一个点，这个点的切线跟这个线段首尾两点所在的直线平行，参数方程的形式
泰勒<span class="equation-text" contenteditable="false" data-index="0" data-equation="(Taylor)"><span></span><span></span></span>中值定理
带佩亚诺余项的泰勒公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(x_0)+f^{'}(x_0)(x-x_0)+\frac{1}{2!}f^{''}(x_0)(x-x_0)^2+\cdots +\frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n+o((x-x_0)^n)"><span></span><span></span></span>
其中形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="R_n=o((x-x_0)^n)余项称为佩亚诺余项"><span></span><span></span></span>
泰勒公式在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0=0时的(带有佩亚诺余项)的麦克劳林公式"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(0)+f^{'}(0)x+\frac{1}{2!}f^{''}(0)x^2+\cdots +\frac{1}{n!}f^{(n)}(0)x^n+o(x^n)"><span></span><span></span></span>
带拉格朗日余项的泰勒公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(x_0)+f^{'}(x_0)(x-x_0)+\frac{1}{2!}f^{''}(x_0)(x-x_0)^2+\cdots +\frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n+\frac{1}{(n+1)!} f^{(n+1)}(\xi)(x-x_0)^{(n+1)}(\xi 介于x_0与x之间)"><span></span><span></span></span>
其中形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="R_n=\frac{1}{(n+1)!} f^{(n+1)}(\xi)(x-x_0)^{(n+1)}(\xi 介于x_0与x之间)余项称为拉格朗日余项"><span></span><span></span></span>
泰勒公式在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0=0"><span></span><span></span></span>时的(带有拉格朗日余项)的麦克劳林公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(0)+f^{'}(0)x+\frac{1}{2!}f^{''}(0)x^2+\cdots +\frac{1}{n!}f^{(n)}(0)x^n+\frac{1}{(n+1)!} f^{(n+1)}(\xi)x^{(n+1)}(\xi 介于x_0与x之间)"><span></span><span></span></span>
如果函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在含有<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_0"><span></span><span></span></span>的某个开区间<span class="equation-text" contenteditable="false" data-index="2" data-equation="(a,b)"><span></span><span></span></span>内具有直到<span class="equation-text" contenteditable="false" data-index="3" data-equation="(n+1)阶导数,当x\in(a,b)内时，f(x)可以表示为(x-x_0)的一个n次多项式与一个余项R_n(x)之和"><span></span><span></span></span>
辅助函数构造方法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="构造\frac{f^{(n+1))}(x)}{f^{(n)}(x)}，使其只差一阶导，适合使用ln构造法"><span></span><span></span></span>
出现<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{'}(x)+g(x)f(x)=0,使用公式法F(x)=f(x)\cdot e ^{\int g(x)dx}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\cdots"><span></span><span></span></span>
定理之间的实际含义
图解
结论
罗尔定理是拉格朗日定理的特殊情况<br><br>拉格朗日中值定理又是柯西定理的特殊情况<br><br>其实还有个费马引理作基础（函数的可导极值点为驻点，导数为0）<br><br>三大中值定理建立导数与原函数的关系，辅以拉式泰勒研究函数局部性态
(L'Hospital)洛必达法则（参照第一章）
单调性于凹凸性
单调性
单调函数和严格单调函数
单调函数
单调函数在端点处则可取等号
包括自变量不同是函数值相同情况
包括常函数
严格单调函数
不包括端点，其定义域的两端只能是＞号或者＜号
不能是常函数
定理1
设函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，在<span class="equation-text" contenteditable="false" data-index="2" data-equation="(a,b)"><span></span><span></span></span>内可导
如果在<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b)"><span></span><span></span></span>内<span class="equation-text" contenteditable="false" data-index="1" data-equation="f^{'}(x)>0,"><span></span><span></span></span>那么函数<span class="equation-text" contenteditable="false" data-index="2" data-equation="y=f(x)在[a,b]"><span></span><span></span></span>上单调增加
如果在<span class="equation-text" data-index="0" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内<span class="equation-text" data-index="1" data-equation="f^{'}(x)<0," contenteditable="false"><span></span><span></span></span>那么函数<span class="equation-text" data-index="2" data-equation="y=f(x)在[a,b]" contenteditable="false"><span></span><span></span></span>上单调减少
注
将判定法中的闭区间换成其他各种区间，包括无穷区间，结论依然成立
定理的证明应用到<span class="equation-text" contenteditable="false" data-index="0" data-equation="Lagrange"><span></span><span></span></span>中值定理
在函数的单调区间的分界点处的导数有些为零，有些不存在
在整个定义域上不单调的函数，我们可以取导数为零和导数不存在的点来划分函数的单调区间
如果函数在某个区间内的有限个点处导数值为零，而在其余各点均为正（或负），那么在该区间上函数是单调增加（或减少）的
用函数的单调性来证明不等式是一个重要的方法
利用导数来求函数单调区间的步骤
先求出函数的定义域
对函数求导，令导函数为零，求出驻点(驻点是指横坐标)和不可导点
以(2)中求出的点作为隔断点将定义域分隔成若干区间（不包括分隔点是因为严格单调）
求各个区间的导函数，根据导函数符号判断函数的单调性，若导数大于零，则函数在该区间上单调递增；若导函数小于零，则函数在该区间上单调递减，若到函数恒等于零，则函数在该区间上为一常数。
凹凸性
曲线的凹凸性定义
图解
含义
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上连续，如果对<span class="equation-text" contenteditable="false" data-index="2" data-equation="I"><span></span><span></span></span>上任意两点<span class="equation-text" contenteditable="false" data-index="3" data-equation="x_1,x_2"><span></span><span></span></span>恒有<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(\frac{x_1+x_2}{2})<\frac{f(x_1)+f(x_2)}{2},那么称f(x)在I上的图像是(向下)凹的(凹弧)；"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="5" data-equation="f(\frac{x_1+x_2}{2})>\frac{f(x_1)+f(x_2)}{2},那么称f(x)在I上的图像是(向上)凸的(凸弧)；"><span></span><span></span></span>
函数曲线凹凸性的判定
设函数<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="1" data-equation="[a,b]" contenteditable="false"><span></span><span></span></span>上连续，在<span class="equation-text" data-index="2" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内具有一阶和二阶导数，那么（记忆：大凹小凸）
如果在<span class="equation-text" data-index="0" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内<span class="equation-text" data-index="1" data-equation="f^{''}(x)>0," contenteditable="false"><span></span><span></span></span>则函数<span class="equation-text" data-index="2" data-equation="f(x)在[a,b]" contenteditable="false"><span></span><span></span></span>上的图形是凹的
如果在<span class="equation-text" data-index="0" data-equation="(a,b)" contenteditable="false"><span></span><span></span></span>内<span class="equation-text" data-index="1" data-equation="f^{''}(x)<0," contenteditable="false"><span></span><span></span></span>则函数<span class="equation-text" data-index="2" data-equation="f(x)在[a,b]" contenteditable="false"><span></span><span></span></span>上的图形是凸的
拐点及其求法
定义：(拐点)曲线的凹，凸的分界点称为拐点(拐点是一个点)
判定：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{''}(x)的符号是用来判定曲线凹凸性的一个方法，若f^{''}(x_0)=0,且x=x_0的两侧f^{''}(x)的符号异号，则点(x_0,f(x_0))就是拐点"><span></span><span></span></span>
求法步骤
求<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{''}(x)"><span></span><span></span></span>
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^{''}(x)=0,"><span></span><span></span></span>解出其在<span class="equation-text" contenteditable="false" data-index="1" data-equation="(a,b)"><span></span><span></span></span>内的实根
对于每一个<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>,检查<span class="equation-text" contenteditable="false" data-index="1" data-equation="f^{''}(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_0"><span></span><span></span></span>左右邻近两侧的符号，如果<span class="equation-text" contenteditable="false" data-index="3" data-equation="f^{''}(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="4" data-equation="x_0"><span></span><span></span></span>的左、右邻近两侧分别保持一定的符号，那么当两侧的符号相反的时，点<span class="equation-text" contenteditable="false" data-index="5" data-equation="(x_0,f(x_0))"><span></span><span></span></span>是拐点，当两侧的符号相同的时，点<span class="equation-text" data-index="6" data-equation="(x_0,f(x_0))" contenteditable="false"><span></span><span></span></span>不是拐点
注
在求解拐点时，除了二阶导数为零的点，还应考虑导数不存在的点
拐点处的切线必在拐点处穿过曲线
函数的极值与最值
函数的极值
极值的定义
极大值：若存在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>的某个邻域<span class="equation-text" contenteditable="false" data-index="1" data-equation="U^\circ(x_0)"><span></span><span></span></span>使得任意<span class="equation-text" contenteditable="false" data-index="2" data-equation="x\in U^\circ (x_0)"><span></span><span></span></span>有<span class="equation-text" contenteditable="false" data-index="3" data-equation="f(x)<f(x_0)"><span></span><span></span></span>,称<span class="equation-text" contenteditable="false" data-index="4" data-equation="f(x_0)"><span></span><span></span></span>为<span class="equation-text" contenteditable="false" data-index="5" data-equation="f(x)"><span></span><span></span></span>的一个极大值，而<span class="equation-text" contenteditable="false" data-index="6" data-equation="x_0(指的是横坐标)"><span></span><span></span></span>称为<span class="equation-text" contenteditable="false" data-index="7" data-equation="f(x)"><span></span><span></span></span>极大值点
极小值：若存在<span class="equation-text" data-index="0" data-equation="x_0" contenteditable="false"><span></span><span></span></span>的某个邻域<span class="equation-text" data-index="1" data-equation="U^\circ(x_0)" contenteditable="false"><span></span><span></span></span>使得任意<span class="equation-text" data-index="2" data-equation="x\in U^\circ (x_0)" contenteditable="false"><span></span><span></span></span>有<span class="equation-text" data-index="3" data-equation="f(x)>(x_0)" contenteditable="false"><span></span><span></span></span>,称<span class="equation-text" data-index="4" data-equation="f(x_0)" contenteditable="false"><span></span><span></span></span>为<span class="equation-text" data-index="5" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>的一个极小值，而<span class="equation-text" data-index="6" data-equation="x_0(指的是横坐标)" contenteditable="false"><span></span><span></span></span>称为<span class="equation-text" data-index="7" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>极小值点
极值
极大值和极小值统称为极值
注
极值具有局部性
极值的判别
费马定理可知，可导函数在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>处取得极值的必要条件是<span class="equation-text" contenteditable="false" data-index="1" data-equation="f^{'}(x_0)=0"><span></span><span></span></span>
注
费马定理说明可导函数的极值只能在其驻点取到，即<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0"><span></span><span></span></span>是驻点只是可导函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>在点<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_0"><span></span><span></span></span>取得极值的必要条件，而不是充分条件。
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在点x_0处取得极值，则f(x)在点x_0必须有定义，且x_0不为区间端点"><span></span><span></span></span>
定理1(极值的第一充分条件)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="前提：设f在x_0处连续，在某邻域U^\circ(x_0,\delta)内可导，则"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\in U^\circ_-(x_0,\delta)时f^{'}(x)\leq 0 ,当x\in U^\circ_+(x_0,\delta)时f^{'}(x)\geq 0,则f在点x_0处取得极小值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\in U^\circ_+(x_0,\delta)时f^{'}(x)\geq 0 ,当x\in U^\circ_-(x_0,\delta)时f^{'}(x)\leq 0,则f在点x_0处取得极大值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f^{'}(x)在U^\circ(x_0,\delta)内不变号，则f(x)在x_0点处无极值"><span></span><span></span></span>
图解
注
驻点、不可导点是可能的极值点
计算极值点和极值的步骤
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求出导数f^{'}(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="令f^{'}(x)=0,求得f(x)的全部驻点和不可导点"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="考察f^{'}(x)的符号在每个驻点和不可导点的左、右邻近的情形，以便确定该点是否是极值点，进一步确定对应的"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="函数值是极大值还是极小值"><span></span><span></span></span>
求出各极值点处的函数值，就得到函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>的全部极值
定理2(极值的第二充分条件，可以更好的判断极值)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="前提：设f在x_0的某邻域U^\circ(x_0,\delta)一阶可导，在x=x_0处二阶可导，且f^{'}(x_0)=0，f^{{''}}(x_0)\neq0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f^{''}(x_0)<0，则f在x_0处取得极大值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f^{''}(x_0)>0，则f在x_0处取得极小值"><span></span><span></span></span>
最大值与最小值
如果函数的最值是在区间内部的点取到，则此最值必定是极值点
求最值的方法
方法一
<span class="equation-text" contenteditable="false" data-index="0" data-equation="前提：f(x)在[a,b]上连续，在(a,b)内可导，且至多有有限个驻点，f(x)在闭区间[a,b]上的最值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求出函数f(x)在驻点、不可导点、区间I得端点处的函数值(即找出所有的可能的最值点)"><span></span><span></span></span>
进行比较，最大的就是最大值，最小的就是最小值
方法二(特殊情况)
前提：确定开区间上函数的极值必是最值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="目标函数f(x)在所讨论的区间I(开或闭、有限或无限)内处处可微"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在区间I内部只有一个驻点x_0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在驻点x_0取得极值"><span></span><span></span></span>
函数图像描绘
函数作图的一般步骤
<span class="equation-text" contenteditable="false" data-index="0" data-equation="确定函数y=f(x)的定义域，求出函数的f^{'}(x)和f^{''}(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求出方程f^{'}(x)=0和f^{''}(x)=0的全部实根，用这些根同函数的间断点火导数不存在的点"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="把函数的定义域划分成几个部分区间"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="确定在这些部分区间内f^{'}(x)和f^{''}(x)的符号，并由此确定函数图形的升降和凹凸，极值点和拐点"><span></span><span></span></span>
确定函数图形的水平、铅直渐进线以及其他趋势变化
<span class="equation-text" contenteditable="false" data-index="0" data-equation="描出与方程f^{'}(x)=0和f^{''}(x)=0的根对应的曲线上的点，有时还需补充一些点，再综合前四步讨论结果画出函数的图形"><span></span><span></span></span>
极值点，最值点，驻点，零点，间断点，拐点
极值点，最值点，驻点，零点，间断点都指的是横坐标x<br>拐点指的是(x，y)坐标
补充性质
奇偶函数、周期函数的导数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="可导偶函数的导函数为奇函数，特别地,f(x)为偶函数，且f^{'}(0)存在，则f^{'}(0)=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="可导奇函数的到导函数为偶函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="可导周期函数的到导函数仍为周期函数，特别地，设f(x+T)=f(x),f^{'}(x_0)存在，则f^{'}(x_0+T)=f^{'}(x_0)"><span></span><span></span></span>
含有绝对值函数的可导性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x_0)=0,f^{'}(x)存在，则|f(x)|在x_0处可导\Leftrightarrow f^{'}(x_0)=0"><span></span><span></span></span>
题型
确定方程的根数（数形结合）
单调性+零点+极值
判断出定义域，将所给式子移到一侧,并设<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x)=..."><span></span><span></span></span>
求出<span class="equation-text" contenteditable="false" data-index="0" data-equation="F^{'}(x)=0的点，以及不可导点(在定义域之中的，不在的舍弃)"><span></span><span></span></span>
列表得出单调性
得出极大值或极小值，并写出在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x"><span></span><span></span></span>何处取得
将得出的<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0处点代入到F(x)分情况大于0，小于0，等于0讨论，算出定义域边界的F(x)的值，结合零点+单调性+极值，最后综上所述分开讨论结果"><span></span><span></span></span>
单调性+零点
判断出定义域，将所给式子移到一侧,并设<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x)=..."><span></span><span></span></span>
求出<span class="equation-text" contenteditable="false" data-index="0" data-equation="F^{'}(x)=0的点"><span></span><span></span></span>
列表得出单调性
算出定义域端点的值或极限
运用零点定理+单调性进行判断
利用单调性证明不等式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)=左-右"><span></span><span></span></span>
求导判断单调性
验证最大(最小)值
得出结论
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