4) Order of approximation for numerical integration methods Let f(x) = x?e-. Using u-substitution, find the exact value of f(x)dx. i) For each of the three numerical integration methods above (rectangle, trapezoid, Simpsons), compute the approxi- ii) mate integral using n=10,100,1000,10000. Calculate the error of approximation, and make a table showing n, the approximate integral, and the error of approximation. Explain how each of these three tables shows the expected order of approximation for each of the numerical integration methods. iii) For each method, determine the value of K such that the error behaves like K/nP, where p is the order of approxi- mation for the method. iv) For each method, using your answer from part iii, predict the value of n so that the error is equal to 3.5 x 10-4. Calculate the actual error, using this value of n (or the closest even number to it, for Simpsons method).
4) Order of approximation for numerical integration methods Let f(x) = x?e-. Using u-substitution, find the exact value of f(x)dx. i) For each of the three numerical integration methods above (rectangle, trapezoid, Simpsons), compute the approxi- ii) mate integral using n=10,100,1000,10000. Calculate the error of approximation, and make a table showing n, the approximate integral, and the error of approximation. Explain how each of these three tables shows the expected order of approximation for each of the numerical integration methods. iii) For each method, determine the value of K such that the error behaves like K/nP, where p is the order of approxi- mation for the method. iv) For each method, using your answer from part iii, predict the value of n so that the error is equal to 3.5 x 10-4. Calculate the actual error, using this value of n (or the closest even number to it, for Simpsons method).
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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Transcribed Image Text:4) Order of approximation for numerical integration methods
Let f(x) = 2?e-
Using u-substitution, find the exact value of f f (x)dx.
%3D
i)
For each of the three numerical integration methods above (rectangle, trapezoid, Simpsons), compute the approxi-
ii)
mate integral using n=10,100,1000,10000.
Calculate the error of approximation, and make a table showing n, the approximate integral, and the error of
approximation. Explain how each of these three tables shows the expected order of approximation for each of the
numerical integration methods.
iii) For each method, determine the value of K such that the error behaves like K/nP, where p is the order of approxi-
mation for the method.
iv) For each method, using your answer from part iii, predict the value of n so that the error is equal to 3.5 x 10-4.
Calculate the actual error, using this value of n (or the closest even number to it, for Simpsons method).
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