Hi, need help with this proof. I got a,b,c but don't understand d and e. For B, the L(P,f)=3-epsilon and U(P,f)=3. For c, Theorem 5.2 states .Letf: [a,b]−→R be bounded. Then f∈R(x) on [a,b] iff for each epsilon>0 there is a partition P such that U(P,f)−L(P,f)≤epsilon.so I need help to justify each equality and inequality of d and how to write a proof showing how f is intergrable and prove the value of the integral. Thank you! (d) Precisely justify each equality and inequality in the following statement:223 – € = L(P, f)<f dxf dxf dx < U(P, f) = 3.-(e) Now write a formal proof using the steps above. It should have two distinct parts: the part where youprove that f is integrable, and the part where you prove the value of the integral. Define f : [0, 2] →R by f(x)compute the integral by following the scratchwork below. Then write a formal proof.= 1 for 0

As per the question, we are solving part (d) and part (e). For (d), The function f(x)=1, 0≤x≤12,…

Answered: (d) Precisely justify each equality and…

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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Hi, need help with this proof. I got a,b,c but don't understand d and e. For B, the L(P,f)=3-epsilon and U(P,f)=3. For c, Theorem 5.2 states .Letf: [a,b]−→R be bounded. Then f∈R(x) on [a,b] iff for each epsilon>0 there is a partition P such that U(P,f)−L(P,f)≤epsilon.

so I need help to justify each equality and inequality of d and how to write a proof showing how f is intergrable and prove the value of the integral.

Thank you!

(d) Precisely justify each equality and inequality in the following statement:
2
2
3 – € = L(P, f)<
f dx
f dx
f dx < U(P, f) = 3.
-
(e) Now write a formal proof using the steps above. It should have two distinct parts: the part where you
prove that f is integrable, and the part where you prove the value of the integral.

Define f : [0, 2] →R by f(x)
compute the integral by following the scratchwork below. Then write a formal proof.
= 1 for 0 <x<1 and f(x) = 2 for 1 < x < 2. Show that ƒ € R(x) on [0, 2] and
(a) First sketch a graph of the function and draw in the following partition: P = {0, 1, 1+€, 2} of [0, 2].
(b) Using this partition, compute the upper and lower sums: U(P, f), and L(P, f).
(c) Write out the statement of Theorem 5.2. Can we use it to show f is Riemann integrable?

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