M471/M571 Homework Assignment 1 Due Aug. 29 (Tuesday) 11:59pm 1. Let f (x) = 3(x+1)(x0.5)(x1). Use the Bisection method on the interval [1.25, 2.5] to find p3 . 2. Consider the bisection method starting with the interval [1.5, 3.5]. (a) What is the width of the interval at the nth steps of this method? (b) What is the maximum distance possible between the root p and the midpoint of the interval [an , bn ]? 3. Let f (x) = (x + 2)(x + 1)2 x(x 1)3 (x 2). To which zero of f does the Bisection method converge when applied on the following intervals? (a) [1.5, 2.5] (b) [0.5, 2.4] (c) [0.5, 3] (d) [3, 0.5] n 4. Let pn = an +b , p = limn pn , and en = p pn . Here [an , bn ], with n 1, denotes 2 the successive intervals that arise in the Bisection method when it is applied to a continuous function f . (a) Is it true that |e1 | |e2 | ? Explain. (b) Show that |pn pn+1 | = 2n1 (b1 a1 ). 5. (Programming) Modify the given Bisection method code to (a) find the solution, accurate to within 105 for problem 3x ex = 0 on interval 1x2 (b) find an approximation to 5 correct to within 104 . [Hint: Consider f (x) = x2 5]