- Another proof of the Evaluation Theorem a. Let a = x

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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- Another proof of the Evaluation Theorem
a. Let a = x <xx < x.<x, = b be any partition of
[a, b], and let F be any antiderivative of f. Show that
F(b) – F(a) = E [F(x) – F(x_1)].
=1
b. Apply the Mean Value Theorem to each term to show that
F(x,) - F(x-1) = f(c)(x, - x-1) for some c, in the interval
(x-1, X;). Then show that F(b) - F(a) is a Riemann sum for
f on [a, b].
c. From part (b) and the definition of the definite integral, show that
F(b) – F(a) =
f(x) dx.
Transcribed Image Text:- Another proof of the Evaluation Theorem a. Let a = x <xx < x.<x, = b be any partition of [a, b], and let F be any antiderivative of f. Show that F(b) – F(a) = E [F(x) – F(x_1)]. =1 b. Apply the Mean Value Theorem to each term to show that F(x,) - F(x-1) = f(c)(x, - x-1) for some c, in the interval (x-1, X;). Then show that F(b) - F(a) is a Riemann sum for f on [a, b]. c. From part (b) and the definition of the definite integral, show that F(b) – F(a) = f(x) dx.
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