The Fundamental Theorem of Calculus goes like this:
[`d/dx int_a^x f(t) dt = f(x)`]
We'll try to unravel it with this exploration. The gist of the FTC is that differentiation
"undoes" integration; in a sense, they are reverse processes of each other. There are three steps.
- We start with a function `f`, and suppose it depends on the variable `t`. (It
doesn't really matter what the variable is, but we'll want to use `x` later.)
- Fix a value `a`, then consider the function `A(x) = int_a^x f(t) dt`. We imagine
`f` and `a` unchanging, so this is a function of `x`; as `x` changes, the value of `A` changes.
- Finally, we differentiate `A(x)`, and surprise! we get `f(x)`.
Do you think it is intuitive that the problem of finding the area between a function
and the `x`-axis is related to the problem of finding the derivative of a function?