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256 11 Discrete Dynamical Systems X|> 149. However, taking xo very small will make xvery large. After that, it is also apparent from the graph that . decreases monotoncly and quickly to 149. Similarly, starting with xo <0, the same procedure follows from reflection of the whole picture in the y-axis--the sequence converges rapidly to -V149. The point X0 =0 is a nonstarter because f'(0) - 0. Exercises for Section 11.2 A Show that f(x) +*+ 1 has exactly one real rool, Uso Newton's method to approximate it to eight decimal places. Show your error estimates. B. The equation sinx=x/2 has exactly one positive solution. Use Newton's method to approxi- mate it to eight decimal places. Show your error estimates. C. (a) Set up Newton's method for computing cube roots. (b) Show by band that V2 1.25 <0.01. (c) Compule 2 to cight decimal places. 1. Let S(x) = (v2)* for X E R. Sketch y = f(x) and y = x on the same graph. Given X CB. define a sequence by Xn+1 = S(x) for n>0. (a) Find all fixed points off. (b) Show that the sequence is monotone (c) II (..) converges to a number x*, prove that f(x") - x (d) for which E R does the sequence (x) converge, and what is the limit? E. Find the largest critical point of f(x) = sin(1/x) to four decimal places. F. Find the minimum value of f(x) = (logx)2 +* on (0w) to four decimal places. G. Apply Newton's method to find be root of f(x) - (-) Start with any point xu / r, and computer. Explain what went wrong here. H. Modified Newton's method. With the same stup us for Newton's method. show that the sequence Xn+1 = x - forn > 0 converges lux SO) 1. Before computers had high-precision division, the following algorithms were used. Notice that they involve only multiplication (a) Let a > 0. Show that Newton's formula for solving 1/x = a yields the iteration X.+1 2x, an (b) Suppose tbat x = (1-C)/a for some c < 1. Derive the formula for X. (c) Do the same analysis for the iteration scheme 1 = *(1 + (1 -axX1 + (1 -ax.))) Explain why this is a superior algorithm J. Let (x) = x/3 (a) Set up Newton's method for this function. (b) If0<x<1/V6, then Xn+1>2xn. (C) Show that if X>1/V6. then Xx+1 > *a+ (d) Hence show that Newton's method never works unless Xo = 0. However, given e > 0, there will be an N 50 large that X-1 -X. <for N (c) Sketch hand try to explain this nasty behaviour. K. Three towns are situated around the shore of a circulur lake of radius 1 km. The largest town. Alphaville, claims one-half of the area of the lake as its territory. The town mathematician is charged with computing the radius of a circle from the town bull (which is right on the shore) that will cut off half the area of the lake. Computer to seven decimal places. )