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

2. 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?

3. 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:

4. 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?

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