Problem Sheet 7 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 29 November 2018 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1
1. The lemniscate is the curve of all points whose polar coordinates satisfy r2 = a2 cos 20 where 0<0<21. (a) Use Mathematica to draw a lemniscate in the case when a = 1. (The PolarPlot command might be useful.) (b) Are there values of 0 that do not give points on the curve? Are there values of 0 that give more than one point on the curve? (c) Show that the leminiscate satisfies the cartesian equation x4 + 2x2y2 + 34 - a2 x2 + a2y2 = 0. 2. Let A be the set of positive real numbers. Let f : A -> A be defined by f(x) = (2x2-x+0.6)/x. Use Mathematica to draw a time series plot for f at starting point 1, for 10 iterations. What do you think happens to f(n) (1) as n > co? No justification required. 3. Let a and b be constants, and let f : R -> R be defined by f (x) = ax + b. (i) Suppose that a # 1. Prove by induction that for any positive integer n, f(n)(x) = anx + 1 = " b. 1-a 1 - an (ii) Suppose that a = 1. Find expressions for f(2) (x) and f(3)(x). Write down (without including a justification) an expression for f(n)(x) that holds for any positive integer n. 4. Let A be the set of positive real numbers. Let f : A -> A be defined by f(x) = x2 - x + 0.5. (i) Find the two fixed points u and u' of f, where u < u'. (ii) Suppose s E R satisfies 0 < < < u. What happens to f(n) (s) as n > ? There is no need to give a rigorous argument, but you should provide a cobweb diagram (use Mathematica) to illustrate your answer. (iii) Now suppose u < s < u. What happens to f(n) (s) as n -> 00? Provide a cobweb diagram to illustrate your answer; no need for any other justification. (iv) Finally, suppose u' < s. What happens to f(n)(s) as n -> 00? Provide a cobweb diagram to illustrate your answer; no need for any other justification. 2