Reference no: EM132308112

Assignment - Final Exam

Problem 1 - Suppose that

lim_h→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(h^p).

Problem 3 - Consider the following integral

₀∫²e^{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 w₁.

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