<p> Math 448, Dr. Wyels, S’06 Names: ______</p><p>Approximating π; Exploring Roots and Fixed Points</p><p>Work with at least one and no more than two classmates; submit one worksheet per team. (Remember, the goal is learning, not worksheet completion.)</p><p>1. Directions are in the accompanying Maple file; record your approximations and errors here.</p><p>Method used Approximation Relative error Absolute error a) basic arctan(x) b) Newton’s idea c) arctan identity</p><p>Write 1 – 2 sentences in conclusion. (What did you find?) </p><p>2. Exploring the Bisection Method. Go to http://www.as.ysu.edu/~faires/Numerical- Analysis/DiskMaterial/programs/Java/JavaPrograms.htm . Carry out each “experiment” below as well as at least three of your own (record your own) and analyze the output. Function A B TOL No. Iter. outcome x2  2 1 2 .00001 20 x2  2 1 2 .00000001 20 x3  4x2 10 1 2 .00001 20</p><p>Write a summary of what the algorithm does (outcomes and mechanics). </p><p>3. The functions and some directions are in the accompanying Maple file. a) What is the relationship between the x-value at which each of the g functions crosses the line y = x, and the root of f(x)? Math 448, Dr. Wyels, S’06</p><p> b) Write out two or more equations you could solve to find the x-value discussed in a). Use Maple’s solve (or fsolve) command to solve your equations, and give the results. c) How can you recognize a fixed point1 graphically? </p><p> d) Return to the website given in #2. Use Algorithm 2.1 with TOL .00001 to find the root of f. Report the result and the number of iterations required.</p><p> e) Play with Algorithm 2.2 to see if you can find the fixed points of g1, g2, and g3. (Try different initial points and tolerance levels.) Report what you find.</p><p>4. The functions are in the accompanying Maple file. a) Plot all the functions, together with line y = x on the same axes, with both x and y values from 0 to 3. How can you identify which graph is that of y = x? … of f(x)? b) Complete the table, using Algorithms 2.1 and 2.2, with TOL .00001, A = 1, B = 2, 30 iterations, and p0 = 1.5. If Alg’m 2.2 reports “not within tolerance”, indicate this. Bring this to class Wed.!</p><p>Function Root/ fixed point Number of iterations needed/ comments f g1 g2 g3 g4 g5</p><p>1 A number p is a fixed point for a function g if g(p) = p.</p>