Math 448, Dr. Wyels, S’06 Names: ______

Approximating π; Exploring Roots and Fixed Points

Work with at least one and no more than two classmates; submit one worksheet per team. (Remember, the goal is learning, not worksheet completion.)

1. Directions are in the accompanying Maple file; record your approximations and errors here.

Method used Approximation Relative error Absolute error a) basic arctan(x) b) Newton’s idea c) arctan identity

Write 1 – 2 sentences in conclusion. (What did you find?)

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

Write a summary of what the algorithm does (outcomes and mechanics).

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

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?

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.

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.

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.!

Function Root/ fixed point Number of iterations needed/ comments f g1 g2 g3 g4 g5

1 A number p is a fixed point for a function g if g(p) = p.