What is the difference between the two parts of FTOC?

Sarai Davenport 2022-06-28 Answered
What is the difference between the two parts of FTOC?
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Answers (1)

g2joey15
Answered 2022-06-29 Author has 21 answers
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 ) = a x f ( t ) d t , x [ a , b ] .Recall that before the student has even seen FTC1, a lot of work has already been done in guaranteeing that the integrals 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 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.
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