# FTC - THE FUNDAMENTAL THEOREM OF CALCULUS The first part of...

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THE FUNDAMENTAL THEOREM OF CALCULUS The first part of the Fundamental Theorem of Calculus describes functions de- fined by an equation of the form g ( x ) = Z x a f ( t ) dt where f is a continuous function on [ a, b ] and x varies between a and b . Example 1. Let g ( x ) = Z x a f ( t ) dt , where a = 1 and f ( t ) = t , find a formula for g ( x ) and calculate g 0 ( x ). Solution g ( x ) = Z x 1 t dt = x 2 2 - 1 2 g 0 ( x ) = d dx ( x 2 2 - 1 2 ) = x Note that g 0 ( x ) = f ( x ). If we assume the existence of an antiderivative of f , i.e. F 0 = f , then g ( x ) = Z x a f ( t ) dt = F ( x ) - F ( a ), and g 0 ( x ) = F 0 ( x ) = f ( x ). Thus, we can get the following theorem. Theorem 1 (The Fundamental Theorem of Calculus 1) . If f is continuous on [ a, b ] , then the function g defined by g ( x ) = Z x a f ( t ) dt is an antiderivative of f , that is, g 0 ( x ) = f ( x ) for a < x < b . Example 2. Find the derivative of the function g ( x ) = Z x 0 e - t 2 dt . Solution Since e - t 2 is continuous, Part 1 of the Fundamental Theorem of Cal- culus gives g 0 ( x ) = e - x 2 . This g ( x ) is called the error function and frequently used in statistics.

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