导数的应用

一，利用导数求函数的单调性

• 导数与函数单调性的关系

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-1.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-2.png" width="500" />

• 用导数求函数单调性的步骤

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-3.png" width="500" />

• 举例

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-4.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-5.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-6.png" width="500" />

• 应用单调性进行证明

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-7.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-8.png" width="500" />

二，极值定理

• 极大值：函数在x0某一个邻域 左侧单调递增（左导数>0）,右侧单调递减（右导数<0），x0为极大值点；
• 极小值：函数在x0某一个邻域 左侧单调递减（左导数<0）,右侧单调递增（右导数>0），x0为极小值点；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-9.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-10.png" width="500" />

• 求函数极值点的方法一：一阶导数法

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-11.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-12.png" width="500" />

• 求函数极值点的方法二：二阶导数法, 注意是在驻点的基础上求二阶导数

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-13.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-14.png" width="500" />

• 尽量使用方法二求极值，但是如果如果使用方法二极值不存在，那么就要使用第一种方法继续求：

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-15.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-16.png" width="500" />

三、函数的最值

• 1. 函数在给定区间内的最大值和最小值求法

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-17.png" width="500" />

• 2. 函数的可能的最值点可能是　驻点（函数的导数＝０的点）或导数不存在（使导数无意义的点） 或　函数两个端点　其中的一点；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-18.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-19.png" width="500" />　　
上图中的最值点就是驻点，使导数不存在的点不存在；

• 3. 实际问题的最值就是极值；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-20.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-21.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-22.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-23.png" width="500" />

四、曲线的凸凹性

• 1. 定义

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-24png" width="500" />

• 2. 凹凸型的判定

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-25png" width="500" />

• 3.　函数曲线拐点的判定

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-26png" width="500" />

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-27png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-28png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-29png" width="500" />

四、曲线的渐进线, 水平渐进线用极限求，铅直渐进线是使得函数分母趋向于０，分子不为０的点求；

• 1. 水平渐进线

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-30.png" width="500" />

• 2. 铅直渐进线

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-31.png" width="500" />

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-32.png" width="500" />