MA1032 - Assignment 2 Lecturer: Dr. Janitha Gunatilake December 1, 2014 1. Consider the function g(x) = e−x . (a) Prove that g is a contraction on G = [ln 1.1, ln 3]. (b) Prove that g : G → G. (c) Show that g(x) has an unique fixed point z ∈ G. (d) Prove that xn+1 = g(xn ) converges to the unique fixed point z ∈ G for any x0 ∈ G. n , x = limn→∞ xn and en = x−xn . Here [an , bn ] with n ≥ 0 denotes the successive 2. Let xn = an +b 2 intervals that arise in the bisection method when it is applied to a continuous function f . (a) Show that |en | ≤ 2−(n+1) (b0 − a0 ). (b) Show that |en | = O(2−n ) as n → ∞. (c) Is it true that |e0 | ≥ |e1 | ≥ . . . ? Explain. (d) Show that |xn − xn+1 | = 2−n−2 (b0 − a0 ). (e) Show that for all n and m, am ≤ bn . (f) Show that x is the unique element in ∩∞ n=0 [an , bn ]. (g) Show that for all n, [an , bn ] ⊃ [an+1 , bn+1 ]. 3. The Van der Waals equation of state for a gas is given by ( 2 ) N p+a (V − N b) = kN T, V where V is the volume occupied by the gas, T is the temperature, p is the pressure, N is the number of molecules contained and k is the Boltzmann constant. a and b are coefficients that depend on the specific gas. For carbon dioxide (CO2 ) a = 0.401P am6 , b = 42.7 × 10−6 m3 . Use bisection method to find the volume occupied by 1000 molecules of CO2 at a temperature T = 300K and a pressure p = 3.5×107 P a, with a tolerance of 10−12 . The Boltzmann constant is k = 1.3806503×10−23 JK −1 . You may use the MATLAB code provided. 1 MA1032 - Assignment 2 α L l1 γ l2 Figure 1: A rod sliding in a corridor. 4. (a) Implement Newton’s method in MATLAB. (b) A corridor has the form indicated in the Figure 1. The maximum length L of a rod that can pass from one extreme to the other sliding on the ground is given by L= l2 l1 + sin(π − γ − α) sin α where α is the solution of the nonlinear equation l2 cos(π − γ − α) cos α − l1 2 = 0. 2 sin (π − γ − α) sin α Compute α by Newton’s method when l2 = 10, l1 = 8 and γ = 3π/5. Use the MATLAB code in (a) with a tolerance 10−6 . Due date: December 12, 2014 Page 2