一,利用導數求函數的單調性
- 導數與函數單調性的關系
<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" />
- 用導數求函數單調性的步驟
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-3.png" width="500" />
- 舉例
<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" />
- 應用單調性進行證明
<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,極值定理
- 極大值:函數在x0某一個鄰域 左側單調遞增(左導數>0),右側單調遞減(右導數<0),x0為極大值點;
- 極小值:函數在x0某一個鄰域 左側單調遞減(左導數<0),右側單調遞增(右導數>0),x0為極小值點;
<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" />
- 求函數極值點的方法一:一階導數法
<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" />
- 求函數極值點的方法二:二階導數法, 注意是在駐點的基礎上求二階導數
<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" />
- 盡量使用方法二求極值,但是如果如果使用方法二極值不存在,那么就要使用第一種方法繼續求:
<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、函數的最值
- 函數在給定區間內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" />
- 函數的可能的最值點可能是 駐點(函數的導數=0的點)或導數不存在(使導數無意義的點) 或 函數兩個端點 其中的一點;
<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" />
上圖中的最值點就是駐點,使導數不存在的點不存在;
- 實際問題的最值就是極值;
<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、曲線的凸凹性
- 定義
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-24png" width="500" />
- 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. 函數曲線拐點的判定
<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、曲線的漸進線, 水平漸進線用極限求,鉛直漸進線是使得函數分母趨向于0,分子不為0的點求;
- 水平漸進線
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-daoshu-user-30.png" width="500" />
- 鉛直漸進線
<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" />