you get 1 + 1 + 1^2 over 2 factorial plus 1^3 over 3 factorial.

得到 1 + 1^2 / 2! + 1^3 / 3!
You do all that you get 2 point something, something.

所有这些做完你就得到了一个二点几的数
Alright, that's e.

那就是 e
So, this is a very powerful notion.

这是个很强大的概念
If you knew all the derivatives of a function,

只要已知函数的任意阶导数
you can predict its value not only the tiny neighborhood,

可求数值的范围就不再局限于领域内
but over big distance.

而是扩展到一段距离外
If you knew the first derivative and you were not very ambitious.

如果仅知一阶导数而又不需精确计算
Look, the logic is very simple.

看这里的逻辑很清晰
If x is a tiny number x^2 is even tinier,

如果 x 数值很小 x^2 就更小了
x^3 is even tinier.

x^3 还要小一些
We've got no respect for these terms.

这些项对我们没有意义