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(a) Let f be the real function on the interval [0, 1] given by when x = 0 I when 0 < x < 1. f(x) = 0, π, Show that for every & > 0 there exists a partition P, such that U(f, Pe) - L(f, P₂) < ɛ, where U (f, Pc) and L(f, P.) are the upper and lower Riemann sums for the partition. Use this to determine if f is Riemann integrable. (b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n > 2. Show that g is Riemann integrable on the entire interval [0, 1]. (Hint: Let € > 0 be given and let M > 0 be a constant such that g(x)| ≤ M for all x = [0, 1]. Choose n ≥ 2 so that

(a) Let f be the real function on the interval [0, 1] given by when x = 0 I when 0 < x < 1. f(x) = 0, π, Show that for every & > 0 there exists a partition P, such that U(f, Pe) - L(f, P₂) < ɛ, where U (f, Pc) and L(f, P.) are the upper and lower Riemann sums for the partition. Use this to determine if f is Riemann integrable. (b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n > 2. Show that g is Riemann integrable on the entire interval [0, 1]. (Hint: Let € > 0 be given and let M > 0 be a constant such that g(x)| ≤ M for all x = [0, 1]. Choose n ≥ 2 so that

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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Find the Maclaurin series for the following function. Specify the convergence interval

(a) Let f be the real function on the interval [0, 1] given by when x = 0 when 0 < x≤ 1. f(x): Show that for every > 0 there exists a partition P such that U(f, Pe) - L(f, Pe) < &, where U (f, P) and L(f, P.) are the upper and lower Riemann sums for the partition. Use this to determine if f is Riemann integrable. 0, π, (b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n > 2. Show that g is Riemann integrable on the entire interval [0, 1]. (Hint: Let & > 0 be given and let M > 0 be a constant such that g(x)| ≤ M for all x = [0, 1]. Choose n ≥ 2 so that and note that 2€ sup{g(x): x = [0, 1/n]} - inf{g(x) : x = [0,1/n]} < 3 Now use that g is Riemann integrable on [1/n, 1] to find a suitable partition of [0, 1].) (c) Leth be a continuous function on [0, 1] and let g be the function given by when x = 0 0, h(x), when 0<x ≤ 1. Show that g is Riemann integrable on [0, 1]. g(x) = (Hint: Use the result from (b).)
Transcribed Image Text:(a) Let f be the real function on the interval [0, 1] given by when x = 0 when 0 < x≤ 1. f(x): Show that for every > 0 there exists a partition P such that U(f, Pe) - L(f, Pe) < &, where U (f, P) and L(f, P.) are the upper and lower Riemann sums for the partition. Use this to determine if f is Riemann integrable. 0, π, (b) Let g be a bounded function on [0, 1] and assume that the restriction of g to the interval [1/n, 1] is Riemann integrable for every n > 2. Show that g is Riemann integrable on the entire interval [0, 1]. (Hint: Let & > 0 be given and let M > 0 be a constant such that g(x)| ≤ M for all x = [0, 1]. Choose n ≥ 2 so that and note that 2€ sup{g(x): x = [0, 1/n]} - inf{g(x) : x = [0,1/n]} < 3 Now use that g is Riemann integrable on [1/n, 1] to find a suitable partition of [0, 1].) (c) Leth be a continuous function on [0, 1] and let g be the function given by when x = 0 0, h(x), when 0<x ≤ 1. Show that g is Riemann integrable on [0, 1]. g(x) = (Hint: Use the result from (b).)
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