Please answer the following fully, answering all the parts

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Please answer the following fully, answering all the parts

Please Answer The Following Fully Answering All The Parts 1 (63.75 KiB) Viewed 32 times

Please Answer The Following Fully Answering All The Parts 2 (259.23 KiB) Viewed 32 times

Question 2 (Unit D2) 30 marks (a) Let f(2)= 2² + 2z - 1. (i) Find the fixed points a and 3 of the function f, and classify them as attracting, repelling or indifferent, identifying any attracting fixed points that are super-attracting. (ii) Prove that the iteration sequence Zn+1 = f(zn), n = 0, 1, 2,..., is conjugate to the iteration sequence Wn+1 = P-1/2 (Wn), n = 0, 1, 2,..... and determine the conjugating function h. (iii) Verify that h(a) and h(3) are the fixed points of P-1/2- (iv) Use Lemma 4.1 on page 143 of Book D to find a 2-cycle of P-1/2. Hence find a 2-cycle of f, and classify it as attracting (possibly super-attracting), repelling or indifferent. (v) Determine four distinct points that are not real and that lie in the keep set K-1/2- (vi) Determine whether or not the points 0 and 3/2 lie in K-1/2- (b) Determine which of the following points lie in the Mandelbrot set. (i) c = 1.2i (ii) c=–0.5+0.5i (c) Prove that the closed circle with centre -1 and radius is a subset of the Mandelbrot set. [5] [3] [2] [5] [3] [3] [3] [3] [3]
Lemma 4.1 Suppose that c-3. Then Pe has a single 2-cycle a₁, a2, where a₁ = -1/2+√√√-³-c, a2=-1-√√-3-c C₂ with multiplier (P²)'(a1) = 4a1a2 = 4(c + 1).