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Theorem 1 (Fundamental Theorem of Calculus - Part I). If fis continuous on [a;b], then the function gdeﬁned by: g(x) = Z. x a. f(t)dt axb is continuous on [a;b], differentiable on (a;b) and g0(x) = f(x) Theorem2(Fundamental Theorem of Calculus - Part II). If fis continuous on [a;b], then: Z.Do you have to prove theorems in AP Calculus?
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn. First, we prove the first part of the theorem.How do you prove theorems you learn?
In general, it's always good to require some kind of proof or justification for the theorems you learn. First, we prove the first part of the theorem. Next, we offer some intuition into the correctness of the second part.What is the assumption of continuousity theorem?
The theorem guarantees that if f(x) is continuous, a point c exists in an interval [a, b] such that the value of the function at c is equal to the average value of f(x) over [a, b]. We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section.