2. Let S(f, P) denote the upper Darboux sum of f with partition P. Prove that if Q is a refinement of P, then S(f, Q) ≤ S(f, P).
2. Let S(f, P) denote the upper Darboux sum of f with partition P. Prove that if Q is a refinement of P, then S(f, Q) ≤ S(f, P).
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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2 please
![1. State the definition of a Riemann sum and the definition of what it means
for a function to be Riemann integrable.
2. Let S(f, P) denote the upper Darboux sum of f with partition P. Prove
that if Qis a refinement of P, then S(f, Q) ≤ S(f, P).
3. Let f: [0, 1] →→ R be defined by
if x ≤ 1/1/2
<
{:
if x > 1/1.
Let > 0. Prove that there exists a partition, P, of [0, 1] with
S(f, P) s(f, P) < €.
f(x)
What can you conclude about f?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F1030ab0c-f5ac-433e-96dc-9dc52c85edf0%2Fgvv1f2_processed.png&w=3840&q=75)
Transcribed Image Text:1. State the definition of a Riemann sum and the definition of what it means
for a function to be Riemann integrable.
2. Let S(f, P) denote the upper Darboux sum of f with partition P. Prove
that if Qis a refinement of P, then S(f, Q) ≤ S(f, P).
3. Let f: [0, 1] →→ R be defined by
if x ≤ 1/1/2
<
{:
if x > 1/1.
Let > 0. Prove that there exists a partition, P, of [0, 1] with
S(f, P) s(f, P) < €.
f(x)
What can you conclude about f?
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