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**Unformatted text preview: **Lecture 26 Fundamental Theorem of Calculus For a given function f , we want to examine the function g ( x ) = Z x a f ( t ) d t . Illustrate g (4) and g (8) for the function above. Is g ( x ) increasing or decreasing at those values? Note that g ( x ) increases (decreases) when When will g ( x ) have a relative maximum/minimum? Is g ( x ) increasing by a greater rate when x = 4 or when x = 11 ? The value that seems to determine the rate of change of g at x is: In fact, we have: Fundamental Theorem of Calculus (I): Let f be a continuous function on [ a,b ] . Then the function g defined by g ( x ) = Z x a f ( t ) d t , a x b is continuous on [ a,b ] and differentiable on ( a,b ) and g ( x ) = Thus, g ( x ) is an of f ( x ) on ( a,b ) and therefore: Now, since g ( x ) is ONE antiderivative of f ( x ) , then EVERY antiderivative has the form: So, if F is ANY antiderivative of f , then: F ( b )- F ( a ) = Fundamental Theorem of Calculus (II): Let f be a continuous function on [ a,b...

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