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Find the constants of the quadrature rule
Course:- Mathematics
Reference No.:- EM132308112

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Assignment - Final Exam

Problem 1 - Suppose that

limh→0 f(h) = M, and M = f(h) + K1h2 + K2h4 + K3h6 + · · ·

Find a combination of f(h) and f(h/2) that is a better estimate of M.

Problem 2 - Using the data f(x - h), f(x), f(x + 2h) construct the best possible approximation of f'(x). Also, find the order of a truncation error O(hp).

Problem 3 - Consider the following integral

02ex^2 dx.

(a) Use the Composite Trapezoidal rule to find approximations with n = 1, 2, 4.

(b) Use Romberg integration to compute R3,3. (Hint: use previously obtained approximation).

Problem 4 - Consider the integral

0π/4 ln(cos(x)) dx.

(a) Use Gaussian quadrature with n = 3 to approximate the integral.

(b) Use the Composite Simpson's rule with n = 4 to approximate the integral.

Problem 5 - Consider the following improper integral.

01 sin(x)/x¼ dx

Use composite Simpson's rule with n = 4 to approximate the improper integral.

Problem 6 - Find the constants a, b, c, of the following quadrature rule.

-11 f(x)dx = af(-1) + bf(1) + cf'(-1),

that has degree of precision 2. (Hint: the integration must be exact for any polynomial function with degree up to 2)

Problem 7 - Consider the following initial value problem

y'= ty + 1, 1 ≤ t ≤ 2, y(1) = 2, with h = 0.1

(a) Prove that the initial value problem has a unique solution on t ∈ [1, 2] and y ∈ (-∞, ∞).

(b) Use the Modified Euler method to approximate w1.

(c) Starting values from Modified Euler method, use Adams-Moulton Two-Step Implicit method to find w2, w3.

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 len2308112 5/17/2019 4:06:37 AM This exam contains 10 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You may use any calculator on this exam. You are required to show your work on each problem on this exam.