FTC1 - The Fundamental Theorem of Calculus (Part1) If f is...

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The Fundamental Theorem of Calculus (Part1) If f is continuous on an interval I, then f has an antiderivative on I. In particular, if a is any number in I, then the function F defined by = x a dt t f x F ) ( ) ( is an antiderivative of f on I; that is, F’(x) = f(x) for each x in I, or in an alternative notation = x a x f dt t f dx d ) ( ] ) ( [ Proof: If F(x) is defined for all x in the interval I , then we will prove that F(x) is defined at each x in the interval I . F’(x) = x w x F w F x w - - ) ( ) ( lim Since F’(x) = f(x) for each value x, F(w) = w a dt t f ) ( and F(x) = x a dt t f ) ( = - - w a x a x w dt t f dt t f x w ]) ) ( ) ( [ 1 ( lim By the properties of definite integrals, the integral from a to x is equal to the negative integral from x to a . = + - w a a x x w dt t f dt t f x w ]) ) ( ) ( [ 1 ( lim By the other properties of definite integrals, the integral from
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This note was uploaded on 02/04/2011 for the course MATHEMATIC 33 taught by Professor Qian during the Winter '99 term at Hong Kong Shue Yan.

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