FTOC, two parts

What is the difference between the two parts of FTOC, they seem to me to be essentially the same thing:

https://www.math.ucdavis.edu/~kouba/Math21BHWDIRECTORY/MVTFTC.pdf

Solution 1:

No, they are not the same. FTC1 is the big gun: It states that if $f$ is continuous on $[a,b],$ then $f$ has an antiderivative $F,$ namely the function

$$F(x)= \int_a^x f(t)\, dt,\,\, x\in [a,b].$$

Recall that before the student has even seen FTC1, a lot of work has already been done in guaranteeing that the integrals $\int_a^x f(t)\, dt$ even exist (limits of Riemann sums and all that). FTC1 is a crowning acheivement in that it says not only do those integrals exist, the derivative of the function so formed gives us back $f.$

FTC2 is a lesser acheivement. All it says is that if you have any antiderivative $G$ of a continuous function $f$ on $[a,b],$ then $\int_a^b f(t)\,dt = G(b)-G(a).$ In FTC1 we already had an antiderivative, namely the $F$ defined there, which does this. FTC2 simply says any antiderivative will do this. The proof of FTC2 is almost trivial: By the MVT, any two antiderivatives on an interval differ by a constant, and the result follows.