All right. If the derivative is small, it's not changing, maybe want to take a larger step, but let's not worry about that all right?

好，如果导数很小的话，函数就基本没什么变化,可能我们就想把步子迈大一点儿了,但是别为这个担心？

麻省理工公开课 - 计算机科学及编程导论课程节选

What's a function, what's a derivative, what's a second derivative, how to take derivatives of elementary functions, how to do elementary integrals.

什么是函数,什么是导数,什么是二阶导数,如何对初等函数求导,如何进行初等积分

耶鲁公开课 - 基础物理课程节选

One way to think about this intuitively if the derivative is very large the function is changing quickly, and therefore we want to take small steps.

关于这个方法很直观的一点想法是,如果导数非常大,函数也就变化的非常快，因此我们想一小步一小步的来。

麻省理工公开课 - 计算机科学及编程导论课程节选

I tell you something about the second derivative of a function and ask you what is the function.

我告诉你一个函数的二阶导数,然后问你这个函数是什么

耶鲁公开课 - 基础物理课程节选

Normally, I will give you a function and tell you to take any number of derivatives.

通常情况下,我会给你一个函数,然后让你求任意阶的导数

耶鲁公开课 - 基础物理课程节选

Of course, you can take a function and take derivatives any number of times.

当然,你可以随意拿一个函数,对它求任意阶的导数

耶鲁公开课 - 基础物理课程节选

Then you're supposed to know derivatives of simple functions like sines and cosines.

你应该知道一些简单函数的导数,比如正弦函数和余弦函数

耶鲁公开课 - 基础物理课程节选

This function will have the right second derivative.

这个函数会得出正确的二阶导数

耶鲁公开课 - 基础物理课程节选

In other words, your job is to guess a function whose second derivative is a, and this is called integration, which is the opposite of differentiation, and integration is just guessing.

换言之,你的任务是要猜出一个二阶导数为a的函数,这就是积分,和微分恰恰相反,积分就是猜

耶鲁公开课 - 基础物理课程节选

Mathematically complete problem is that you can find the function x of t by saying that the second derivative of the function is equal to -k over m times the function.

这个数学问题就是,你可以求出函数 x,只要令函数的二阶导数,等于 -k 除以 m 再乘以函数

耶鲁公开课 - 基础物理课程节选

Then, to find the meaning of b, we take one derivative of this, dx/dt, that's velocity as a function of time, and if you took the derivative of this guy, you will find as at+b. That's the velocity of the object.

接下来,为了弄清b的含义,我们取它的一阶导数,dx/dt,得到速度作为时间的函数,如果你对它求导的话,你会得到at+b,这就是物体的速度

耶鲁公开课 - 基础物理课程节选