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Unformatted text preview: Lecture 30 The definite integral (Relevant section from Stewart, Sixth Edition: Section 5.2) The material presented in this lecture closely followed the presentation of Section 5.2 of the text book, pp. 366376. Lecture 31 The Fundamental Theorem of Calculus The material presented in this lecture, including the proofs of FTC I and II, closely followed the presentation of Section 5.3 of the textbook, pp. 379387. This is perhaps one of the most important results of this course. In both of the results stated below (and proved in class), we assume that f ( x ) is a continuous function on R . 1. First define the function g ( x ) = integraldisplay x a f ( t ) dt. (1) Then g ′ ( x ) = f ( x ) . (2) This result is often known as the First Fundamental Theorem of Calculus , or simply “ FTC I .” 2. The above result establishes that the function g ( x ) is an antiderivative of f ( x ). The question is, “Which one?” The answer is that g ( x ) is the antiderivative of f ( x ) for which g...
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This note was uploaded on 09/22/2011 for the course MATH 137 taught by Professor Speziale during the Fall '08 term at Waterloo.
 Fall '08
 SPEZIALE
 Calculus, Fundamental Theorem Of Calculus

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