一..微分中值定理
1. 预备知识(极值点)
![](https://img.haomeiwen.com/i15252533/6e8f9b6f7379527c.png)
2 罗尔定理
![](https://img.haomeiwen.com/i15252533/b20864bd5f322471.png)
3.拉格朗日中值定理
![](https://img.haomeiwen.com/i15252533/02688676f9a6fb38.png)
4.柯西中值定理
![](https://img.haomeiwen.com/i15252533/3a01a23934eda0a0.png)
5.情况分析
![](https://img.haomeiwen.com/i15252533/aee5d76e63b7223c.png)
![](https://img.haomeiwen.com/i15252533/8867df5738f37de5.png)
二.洛必达法则
1.零比零型
![](https://img.haomeiwen.com/i15252533/8fc7f00e40f0b14b.png)
2.无穷大比无穷大型
![](https://img.haomeiwen.com/i15252533/3c04dc07d1b672ea.png)
3.注意事项
![](https://img.haomeiwen.com/i15252533/7b4aaccb36297aec.png)
三.泰勒公式
1.泰勒公式
![](https://img.haomeiwen.com/i15252533/891e329f1824c804.png)
其中:将x0取为0则泰勒公式变为麦克劳林公式
拉格朗日型余项:
![](https://img.haomeiwen.com/i15252533/751cbd64450697a0.png)
2.佩亚诺型余项
![](https://img.haomeiwen.com/i15252533/76db1386cc703624.png)
3.熟记的公式
![](https://img.haomeiwen.com/i15252533/553f970facc8015f.png)
四(a).函数的单调性与极值
(1)增减函数的定义
![](https://img.haomeiwen.com/i15252533/28d4542a4920a056.png)
(2)单调性判定方法
![](https://img.haomeiwen.com/i15252533/ec64df879ba2accd.png)
(3)极值的定义
![](https://img.haomeiwen.com/i15252533/e3281ec1fe2a5d4d.png)
(4)求极值步骤
![](https://img.haomeiwen.com/i15252533/e1c0120a3ee63154.png)
四(b).凹凸性与拐点
(1)凹凸性的定义
![](https://img.haomeiwen.com/i15252533/ee25cb2776fbaf69.png)
(2)凹凸性的判别法
![](https://img.haomeiwen.com/i15252533/030dcf576c502db2.png)
(3)拐点的定义
![](https://img.haomeiwen.com/i15252533/061c9b134ef5bf71.png)
五.函数的最大最小值
函数的极值与两个端点值的大小比较,最大的为函数的最大值
六..函数图形的描绘(了解)
![](https://img.haomeiwen.com/i15252533/c2d8bfd6d318a5f7.png)
七.曲率
(1)弧微分
![](https://img.haomeiwen.com/i15252533/41279bbae5d584d7.png)
(2)曲率
![](https://img.haomeiwen.com/i15252533/232b754023d4e7d5.png)
(3)曲率圆,曲率半径
记忆公式:
![](https://img.haomeiwen.com/i15252533/e5a449de4c81ff10.png)
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