Explanation Proof: Let, f be a continuous function on [ a , b ] and F be any anti-derivatives of f . So, the definite integral from x = a to x = b defined by a limit of Reimann sum is, ∫ a b f ( x ) d x = lim n → ∞ ∑ k = 1 n f ( x • k ) Δ x = lim n → ∞ ∑ k = 1 n F ( x • k ) Δ x Where, for each n , Δ x = b − a n , x k = a + k Δ x and x • k is a point on the sub interval [ x k − 1 , x k ] . Now applying the mean value Theorem to F on [ a , b ] = [ x k − 1 , x k ] such that there exists some point [ a , b ] = [ x k − 1 , x k ] such that, F ′ ( c k ) = F ( x k ) − F ( x k − 1 ) x k − x k − 1 Substituting the above expression for F ′ ( x • k ) in the Reimann sum, ∫ a b f ( x ) d x = lim n → ∞ ∑ k = 1 n F ( x • k ) Δ x = lim n → ∞ ∑ k = 1 n ( F ( x k ) − F ( x k − 1 ) x

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.5, Problem 77P

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To prove: the fundamental theorem of calculus in your own words.

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Chapter 4.5, Problem 76P

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To prove: the fundamental theorem of calculus in your own words.

Solution Summary: The author explains the fundamental theorem of calculus in their own words. f is a continuous function on left[a,bright] and F be any anti-derivative

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