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Homework 7 Problem 1 1) Determine the nodes and weights for Gaussian formula of the form Z 1 x4 f (x) ≈ A0 f (x0 ) + A1 f (x1 ). −1 2) Determine the composite rule for a pseudocode). Rb a x4 f (x)dx using two-point Gaussian formula (write Problem 2 Rb Write a program for the calculation of a f (x)dx using Simpson’s rule and trapezoidal rule. Find the convergence rate of these methods for f (x) = cos(x), a = 0, b = π/2 using n equally spaced points with n = 2j + 1, with j = 3, 4, 5, 6, 7, 8, 9, 10. Problem 3 The vibration of a mass m connected to a nonlinear spring of stiffness k(x), is governed by d2 d2 x + k(x) = 0 or x + F (x) = 0, dt2 dt2 where F (x) = k(x)/m and x(t) is the displacement of the mass at time t. a) Show that by integrating the equation we have Z x 1 dξ qR t= √ , x0 2 0 F (η)dη ξ m where the initial displacement and the initial velocity of the mass are assumed to be x0 and 2 d d ( dt x)2 = 2 dtd 2 x). 0 respectively (you can use the identity dx b) Assuming that F (x) = x5 and x0 = 1, find the value of t corresponding to x = 1(= x0 ) using a numerical method (i.e. a period of the oscillation). Problem 4 Show how to use Richardson extrapolation to obtain the accuracy O(h5 ), O(h7 ) and O(h9 ) if L = φ(h) + a1 h + a3 h3 + a5 h5 + . . . . 1