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Assignment - Final Exam
Problem 1 - Suppose that
lim_{h→0} f(h) = M, and M = f(h) + K_{1}h^{2} + K_{2}h^{4} + K_{3}h^{6} + · · ·
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(h^{p}).
Problem 3 - Consider the following integral
_{0}∫^{2}e^{x^2} dx.
(a) Use the Composite Trapezoidal rule to find approximations with n = 1, 2, 4.
(b) Use Romberg integration to compute R_{3,3}. (Hint: use previously obtained approximation).
Problem 4 - Consider the integral
_{0}∫^{π/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.
_{0}∫^{1} sin(x)/x^{¼} 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.
_{-1}∫^{1} 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'= t_{y} + 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 w_{1}.
(c) Starting values from Modified Euler method, use Adams-Moulton Two-Step Implicit method to find w_{2}, w_{3}.