[SOLVED] MA 322 - Lab Quiz 1

1. (2 points) 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).

• How many iterations are required? Does the iteration count match the expectations,based on our convergence analysis?
• Convert this into a fixed point iteration and find the approximated value of theroot of f with altol = 10⁻⁸.

2. (3 points) 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?
   • Bisection method.
   • Secant method.
3. (2 points) Draw the graph of a function having the following properties:
   • The function has exactly two fixed points.
   • Give two choices of the initial guess x₀ and y₀ such that the corresponding sequences {x_n} and {y_n} have the properties that {x_n} converges to one of the fixed point and the sequence {y_n} goes away and diverges. Point out the first three terms of both the sequences on the graph.
4. (3 points) 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

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