1. Apply bisection method to find the root of the function
√ f(x) = x − 1.1
Starting from the interval [0,2], with altol = 10−8 ( absolute error tolerance).
(a) How many iterations are required? Does the iteration count match the expectations,based on our convergence analysis?
(b) Convert this into a fixed point iteration and find the approximated value of theroot of f with altol = 10−8.
2. The function f(x) = tan(πx)−6 has a zero at (1/π)arctan6 ≈ 0.447431543. Let x0 = 0 and x1 = 0.48, and use ten iterations for each of the following methods to approximate this root. Which method is most successful and why?
(a) Bisection method.
(b) Secant method.
3. Draw the graph of a function having the following properties:
(a) The function has exactly two fixed points.
(b) Give two choices of the initial guess x0 and y0 such that the corresponding sequences {xn} and {yn} have the properties that {xn} converges to one of the fixed point and the sequence {yn} goes away and diverges. Point out the first three terms of both the sequences on the graph.
4. Use the Euler’s methods and Runge-Kutta methods of order 2 and 4 to solve the IVP
with initial condition y(0) = 1. Compare the solutions for along with the exact solution y(x) = 3exp(−x/2) + x − 2
