(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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f3f25b-2eb2-4186-b342-f9f487e8326d%2F8c5b40c6-d04c-4342-8f6a-b40ee2855bee%2Fd6byfhp_processed.png&w=3840&q=75)
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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