6) a) Let the function f been given by when 0 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P) applies that L(f, P.) >1 – e. b) Use (a) to show that f is integrable and that f, f(x)dx = 1 c) Let g be a decreasing continuous function on [0,1), assuming the values g(0) = 1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following inequalities apply to the lower and the upper Riemann sum. 0 < L(g,P) < U(g, P) < 1 d) Use c) to show the inequalities :)dx < 1 (You can use that a continuous function on [0,1] is integrable)
6) a) Let the function f been given by when 0 0 there is a partition P, of [0,1] such that for the lower Riemann sum L(f, P) applies that L(f, P.) >1 – e. b) Use (a) to show that f is integrable and that f, f(x)dx = 1 c) Let g be a decreasing continuous function on [0,1), assuming the values g(0) = 1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following inequalities apply to the lower and the upper Riemann sum. 0 < L(g,P) < U(g, P) < 1 d) Use c) to show the inequalities :)dx < 1 (You can use that a continuous function on [0,1] is integrable)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
6c
![6)
a) Let the function f been given by
(1,
when 0 < x < 1
f(x) =
(o,
when x = 1
Show that for each e > 0 there is a partition P, of [0,1] such that for the lower
Riemann sum L(f,P.) applies that L(f, P.) > 1 - €.
b) Use (a) to show that f is integrable and that f(x)dx = 1
c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) =
1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following
inequalities apply to the lower and the upper Riemann sum.
0 < L(g, P)< U(g,P) < 1
d) Use c) to show the inequalities
0 <
g(x)dx < 1
(You can use that a continuous function on [0,1] is integrable)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f3f25b-2eb2-4186-b342-f9f487e8326d%2F414410dd-4893-420a-b826-22fa8a18ed6e%2Fjmh53e_processed.png&w=3840&q=75)
Transcribed Image Text:6)
a) Let the function f been given by
(1,
when 0 < x < 1
f(x) =
(o,
when x = 1
Show that for each e > 0 there is a partition P, of [0,1] such that for the lower
Riemann sum L(f,P.) applies that L(f, P.) > 1 - €.
b) Use (a) to show that f is integrable and that f(x)dx = 1
c) Let g be a decreasing continuous function on [0,1], assuming the values g(0) =
1 and g(1) = 0. Show that there is a partition P of [0,1] such that the following
inequalities apply to the lower and the upper Riemann sum.
0 < L(g, P)< U(g,P) < 1
d) Use c) to show the inequalities
0 <
g(x)dx < 1
(You can use that a continuous function on [0,1] is integrable)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
