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Section 4.4 The Fundamental Theorem of Calculus Part II – The Second Fundamental Theorem

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Section 4.4 The Fundamental Theorem of Calculus We have defined the integral to be an antiderivative in the First Fundamental Theorem of Calculus as follows: where F(x) is an antiderivative of f(x) and f(x) is continuous on the closed interval [a,b] So, what happens if we try to differentiate an integral?

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Section 4.4 The Fundamental Theorem of Calculus The Second Fundamental Theorem of Calculus says the following: If f is continuous on an open interval I containing a, then, for every x in that interval:

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Section 4.4 The Fundamental Theorem of Calculus Let’s examine what that might mean. Where did the F function go in that statement? More importantly, what happens if we play with the limits. What might the following integrals equal?

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Section 4.4 The Fundamental Theorem of Calculus For the problems below, find F’(x).

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