Fundamental Theorem of Calculus

# Fundamental Theorem of Calculus - Fundamental Theorem of...

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Fundamental Theorem of Calculus, Part II Suppose is a continuous function on [ a,b ] and also suppose that is any anti-derivative for . Then, Proof First let and then we know from Part I of the Fundamental Theorem of Calculus that and so is an anti-derivative of on [ a , b ]. Further suppose that is any anti-derivative of on [ a , b ] that we want to choose. So, this means that we must have, Then, by Fact 2 in the Mean Value Theorem section we know that and can differ by no more than an additive constant on . In other words for we have, Now because and are continuous on [ a , b ], if we take the limit of this as and we can see that this also holds if and . So, for we know that . Let’s use this and the definition of to do the following.

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Note that in the last step we used the fact that the variable used in the integral does not matter and so we could change the t ’s to x ’s. Average Function Value The average value of a function over the interval [ a,b ] is given by, Proof We know that the average value of
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Fundamental Theorem of Calculus - Fundamental Theorem of...

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