(d) Find the error bound E(n) for approximating x S n = 20, and n = 30. That is, find E(10), E(20), and E(30). 1 de using Trapezoid rule with n = 10, (e) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.001.
(d) Find the error bound E(n) for approximating x S n = 20, and n = 30. That is, find E(10), E(20), and E(30). 1 de using Trapezoid rule with n = 10, (e) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.001.
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
Here is the problem but I just need parts d and e, thank you!
![3/3
O
105% +
4. When using a numerical method to approximate the value of an integral, we expect there
to be some discrepancy between the exact value and the value found by our computation.
We cannot (usually) determine exactly what this error is, but it is possible to show that the
(absolute value of the) error in using the trapezoid rule to approximate f(x) dx cannot
Ts
exceed the bound
;
(b-a)³
E(n)
M
12n²
where M is the maximum value of f"(x)| on the interval [a, b].
1
(a) Compute [ dx and give your answer as a decimal rounded to three decimal places.
x
(b) Compute the trapezoid rule estimate of this integral using n = 10. State the error in
using this approximation; i.e., state the difference between the exact value in (a) and
the approximation.
(c) Find the value of M for this problem. That is, find the maximum value of f"(x)| on
the interval [1, 7] where f(x) = 1/x. Hint: sketch a graph of f"(x)] and determine
what the maximum is.
1
(d) Find the error bound E(n) for approximating x
S
n = 20, and n = 30. That is, find E(10), E(20), and E(30).
dx using Trapezoid rule with n = 10,
(e) Find the first value of n which is large enough that the error bound E(n) is smaller than
0.001.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe55cfba9-7bce-45cb-a868-b9c474dd1b77%2Fdc3402df-e1e7-48f0-9cee-3777c0977a6d%2Filptq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3/3
O
105% +
4. When using a numerical method to approximate the value of an integral, we expect there
to be some discrepancy between the exact value and the value found by our computation.
We cannot (usually) determine exactly what this error is, but it is possible to show that the
(absolute value of the) error in using the trapezoid rule to approximate f(x) dx cannot
Ts
exceed the bound
;
(b-a)³
E(n)
M
12n²
where M is the maximum value of f"(x)| on the interval [a, b].
1
(a) Compute [ dx and give your answer as a decimal rounded to three decimal places.
x
(b) Compute the trapezoid rule estimate of this integral using n = 10. State the error in
using this approximation; i.e., state the difference between the exact value in (a) and
the approximation.
(c) Find the value of M for this problem. That is, find the maximum value of f"(x)| on
the interval [1, 7] where f(x) = 1/x. Hint: sketch a graph of f"(x)] and determine
what the maximum is.
1
(d) Find the error bound E(n) for approximating x
S
n = 20, and n = 30. That is, find E(10), E(20), and E(30).
dx using Trapezoid rule with n = 10,
(e) Find the first value of n which is large enough that the error bound E(n) is smaller than
0.001.
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