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Unformatted text preview: 24. The Fundamental Theorem of Calculus P. K. Lamm 11/15/11 (22:13) p. 1 / 5 Lecture Notes: 24. The Fundamental Theorem of Calculus These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. The Fundamental Theorem of Calculus, Part 1 In this set of lecture notes we consider the Fundamental Theorem of Calculus, taken in two parts. The first part is another statement of the inverse relationship between differentiation and integration, only this time using a definite integral. Theorem (The Fundamental Theorem of Calculus, Part 1) : If f is continuous on [ a,b ], then the function F defined on [ a,b ] by F ( x ) = Z x a f ( t ) dt, x [ a,b ] has a derivative at every point of [ a,b ]. Further, F ( x ) = d dx Z x a f ( t ) dt satisfies F ( x ) = f ( x ) , x [ a,b ] . Proof: Using the definition of the derivative of F , F ( x ) = lim h F ( x + h )- F ( x ) h = lim h 1 h " Z x + h a f ( t ) dt- Z x a f ( t ) dt # = lim h 1 h " Z x a f ( t ) dt + Z x + h x f ( t ) dt !...
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