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 f(x) dx cannot exceed the bound E(n) = (b-a)3³ 12n² -M where M is the maximum value of f"(x)| on the interval [a, b]. (a) Compute √™1² s sin(x) dr. (b) Compute the trapezoid rule estimate of this integral using n = 4 to at least 4 decimal places. (c) State the error in using this approximation; i.e., state the difference between the exact value in (a) and the approximation in (b) to at least 4 decimal places. (d) Find the value of M for this problem. That is, find the maximum value of f"(z)| on the interval [0, where f(x)= sin(z). You may do this by sketching a graph of f"(z)| and determine what the maximum is, or by using calculus techniques. (e) Find the error bound E(n) for approximating * sin(z) dr using Trapezoid rule with n = 4, n= 10, and n = 20. That is, find E(4), E(10), and E(20) to at least 4 decimal places. (f) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.0001.
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 f(x) dx cannot exceed the bound E(n) = (b-a)3³ 12n² -M where M is the maximum value of f"(x)| on the interval [a, b]. (a) Compute √™1² s sin(x) dr. (b) Compute the trapezoid rule estimate of this integral using n = 4 to at least 4 decimal places. (c) State the error in using this approximation; i.e., state the difference between the exact value in (a) and the approximation in (b) to at least 4 decimal places. (d) Find the value of M for this problem. That is, find the maximum value of f"(z)| on the interval [0, where f(x)= sin(z). You may do this by sketching a graph of f"(z)| and determine what the maximum is, or by using calculus techniques. (e) Find the error bound E(n) for approximating * sin(z) dr using Trapezoid rule with n = 4, n= 10, and n = 20. That is, find E(4), E(10), and E(20) to at least 4 decimal places. (f) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.0001.
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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Please solve d-f
![7. 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
exceed the bound
·T
E(n) =
(b-a)³
12²
-M
where M is the maximum value of f"(x) on the interval [a, b].
(a) Compute
100%
π/2
sin(x) dr.
(b) Compute the trapezoid rule estimate of this integral using n = 4 to at least 4 decimal
places.
(c) State the error in using this approximation; i.e., state the difference between the exact
value in (a) and the approximation in (b) to at least 4 decimal places.
(d) Find the value M for this problem. That is, find the maximum value of f"(z)| on
the interval [0,] where f(x) = sin(r). You may do this by sketching a graph of f"(x)|
and determine what the maximum is, or by using calculus techniques.
/2
(e) Find the error bound E(n) for approximating
[/² s
n = 4, n= 10, and n = 20. That is, find E(4), E(10), and E(20) to at least 4 decimal
places.
sin(z) dr using Trapezoid rule with
(f) Find the first value of n which is large enough that the error bound E(n) is smaller than
0.0001](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe55cfba9-7bce-45cb-a868-b9c474dd1b77%2F613196a8-0b65-473c-a881-f7967a69751e%2Frbgpfr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. 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
exceed the bound
·T
E(n) =
(b-a)³
12²
-M
where M is the maximum value of f"(x) on the interval [a, b].
(a) Compute
100%
π/2
sin(x) dr.
(b) Compute the trapezoid rule estimate of this integral using n = 4 to at least 4 decimal
places.
(c) State the error in using this approximation; i.e., state the difference between the exact
value in (a) and the approximation in (b) to at least 4 decimal places.
(d) Find the value M for this problem. That is, find the maximum value of f"(z)| on
the interval [0,] where f(x) = sin(r). You may do this by sketching a graph of f"(x)|
and determine what the maximum is, or by using calculus techniques.
/2
(e) Find the error bound E(n) for approximating
[/² s
n = 4, n= 10, and n = 20. That is, find E(4), E(10), and E(20) to at least 4 decimal
places.
sin(z) dr using Trapezoid rule with
(f) Find the first value of n which is large enough that the error bound E(n) is smaller than
0.0001
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