瞬时变化率,本质是极限
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可导→连续;连续不一定可导;
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类似的, 二阶导函数的导函数称为三阶导函数,
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,是二阶微分Initializing MathJax...
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导数研究的是
,是变化率Initializing MathJax...
微分研究的是
的构成(Initializing MathJax...
,dy即线性主部Initializing MathJax...
可微 ⇔ 可导
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以直代曲
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可以通过导数定义公式,一步步推导出来。能记住最好。
设函数
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设
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像剥卷心菜一样,逐层求导
设函数
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设
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若隐函数
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一阶导:将 y 视作 x 函数,对方程
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二阶导:对方程
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注: 求隐函数在定点
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如果遇到对
求导,相当于y对x的复合函数求导,即2yy'Initializing MathJax...
原函数 | 高阶导数 |
---|---|
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考核形式主要借助高阶导,或莱布尼茨公式
设
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一般是
幂函数
×函数
形式,这样只有展开有限项就能得到结果
公式要记住
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形如
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幂指函数
, 处理方法是对数恒等变形
, 于是Initializing MathJax...
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单调增函数
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单调减函数
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增函数
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减函数
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极大值
:f(x);极大值点
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讨论
极值点
和驻点
的关系
稳定点
(驻点);Initializing MathJax...
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隐函数
、参数方程
一般用这条来判断(直接判断不好判断)
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略
略
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特别的, 当
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注: 若求过曲线外一点处的切法线方程, 先设切点
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连续函数凹凸区间的分界点。(拐点两侧的凹凸不同)
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拐点就是
导函数
的极值点
(不全是,还要去判断)
习题:
提到了
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(都是习题,略)
若
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则在开区间
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几何意义:
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若
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则在开区间
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泰勒公式:设函数
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在
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麦克劳林公式
.常见函数的麦克劳林公式:
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泰勒展开的一般原则:
- 上下同阶
- 多退少补(上下不定阶,则保证有x的最低阶数)