![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/1.png)
01. 概论
圆的面积
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Circle.png)
02. 导数的悖论
微积分之父
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Fathers_of_Calculus.png)
汽车
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/dt.png)
03. 用几何来求导
应用
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Applications.png)
\(x^{2}\)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/2.png)
\(x^{3}\)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/3.png)
\(x^{n}\)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/n.png)
\(\sin(x)\)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Sin.png)
\(\frac{1}{x})\)
04. 直观理解链式法则和乘积法则
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Chain.png)
组合函数 求导
- 加 Sum
- 乘 Product
- 复合 Composition
加法法则
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Sum.png)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Sum_rule.png)
乘积法则
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Product.png)
复合:链式法则
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Composition.png)
05. 指数函数求导
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/e.png)
2的指数 -> e表示
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/ln2.png)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/ln.png)
06. 隐函数求导是怎么回事
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/hide.png)
变化落在函数曲线上
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/1_x.png)
07. 极限
目标
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Goal.png)
导数定义
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Derivative.png)
\(dx\) -> \(h\)
- 明确表示 \(h\) 是一个普通的数,与无穷小无关
- \(dx\) 就是无穷小量
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Limit.png)
\(\frac{0}{0}\)
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/0_0.png)
08. 积分与微积分的基本定理
积分
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Calculus.png)
积分:求导的逆运算
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Inverse.png)
积分
:小量累积
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Integrates.png)
微积分基本定理
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Theorem.png)
原函数 <==> 导数
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Origin.png)
09. 面积与斜率有什么关系
平均值
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Average.png)
09脚注-高阶导数
二阶导数
- 导数的导数
- 变化量的变化量
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Second_derivatives.png)
高阶导数
![](https://2020.iosdevlog.com/2020/03/17/Essence-of-calculus/Higher_orders.png)
10. 泰勒级数
泰勒级数是利用函数某单个点的导数来近似这个点附近函数的值
用多项式 近似
其它函数
- 好计算
- 好求导
- 好积分
- \(c_{0}\):多项式在 x=0 处与 cos(0) 相等
- \(c_{1}\):两者导数一致
- \(c_{2}\):两者二阶导数一致
控制
泰勒多项式
泰勒公式
\(e^{x}\) 泰勒多项式
几何表示
泰勒级数
- 泰勒多项:有限多项
- 泰勒级数:无阶多项
收敛
发散
11. 你在微积分课上学不到的知识
在这里观看完整的“微积分的本质”播放列表:<hhttp://3b1b.co/calculus>