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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 cb (absolute value of the) error in using the trapezoid rule to approximate S f(x) dx cannot a exceed the bound (b-a)³ -M 12n² where M is the maximum value of f"(x)| on the interval [a, b]. (a) Compute E(n) = = [² = de and give your answer as a decimal rounded to three decimal places. (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.

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 cb (absolute value of the) error in using the trapezoid rule to approximate S f(x) dx cannot a exceed the bound (b-a)³ -M 12n² where M is the maximum value of f"(x)| on the interval [a, b]. (a) Compute E(n) = = [² = de and give your answer as a decimal rounded to three decimal places. (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.

Advanced Engineering Mathematics
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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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 f(x) dx cannot (absolute value of the) error in using the trapezoid rule to approximate Tº exceed the bound (b − a)³ 12n² where M is the maximum value of f"(x)| on the interval [a, b]. E(n) = M 1 (a) Compute S de 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. (d) Find the error bound E(n) for approximating f n = 20, and n = 30. That is, find E(10), E(20), and E(30). 1 da 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.
Transcribed Image Text: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 f(x) dx cannot (absolute value of the) error in using the trapezoid rule to approximate Tº exceed the bound (b − a)³ 12n² where M is the maximum value of f"(x)| on the interval [a, b]. E(n) = M 1 (a) Compute S de 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. (d) Find the error bound E(n) for approximating f n = 20, and n = 30. That is, find E(10), E(20), and E(30). 1 da 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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9:08 PM Sun Oct 2 3 of 3 (a) Compute production-gradescope-uploads.s3-us-west-2.amazonaws.com 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 |ƒ"(x)| on the interval [1,7] where f(x) = 1/x. Hint: sketch a graph of f"(x)| and determine what the maximum is. (d) Find the error bound E(n) for approximating [ dx using Trapezoid rule with n = 10, X n = 20, and n = 30. That is, find E(10), E(20), and E(30). (e) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.001. 60%
Transcribed Image Text:9:08 PM Sun Oct 2 3 of 3 (a) Compute production-gradescope-uploads.s3-us-west-2.amazonaws.com 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 |ƒ"(x)| on the interval [1,7] where f(x) = 1/x. Hint: sketch a graph of f"(x)| and determine what the maximum is. (d) Find the error bound E(n) for approximating [ dx using Trapezoid rule with n = 10, X n = 20, and n = 30. That is, find E(10), E(20), and E(30). (e) Find the first value of n which is large enough that the error bound E(n) is smaller than 0.001. 60%
9:08 PM Sun Oct 2 3 of 3 production-gradescope-uploads.s3-us-west-2.amazonaws.com 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 b [º (absolute value of the) error in using the trapezoid rule to approximate exceed the bound (b-a)³ M 12n² where M is the maximum value of |ƒ"(x)| on the interval [a, b]. (a) Compute E(n)= = f(x) dx cannot 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. n = 20, and n = (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 |ƒ"(x)| and determine what the maximum is. (d) Find the error bound E(n) for approximating dx using Trapezoid rule with n = 10, X 30. That is, find E(10), E(20), and E(30). 60%
Transcribed Image Text:9:08 PM Sun Oct 2 3 of 3 production-gradescope-uploads.s3-us-west-2.amazonaws.com 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 b [º (absolute value of the) error in using the trapezoid rule to approximate exceed the bound (b-a)³ M 12n² where M is the maximum value of |ƒ"(x)| on the interval [a, b]. (a) Compute E(n)= = f(x) dx cannot 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. n = 20, and n = (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 |ƒ"(x)| and determine what the maximum is. (d) Find the error bound E(n) for approximating dx using Trapezoid rule with n = 10, X 30. That is, find E(10), E(20), and E(30). 60%
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