What is the difference between the two parts of FTOC?

Question

g2joey15 · Accepted Answer

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) d t, 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) d t

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) d t = 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.

What is the difference between the two parts of FTOC?

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