**M371 - Alexiades**
Problem Set 2: Roots

1. Do 3 iterations of bisection by hand on ** f(x) = x**^{6} - 2 on [0,2].
Give an upper bound for the error.
2. Consider the function ** F(x) = x**^{6} - x - 1.
We want to find its positive root(s).
a. Prove there is only one positive root.
b. What would be a good interval [a,b] for bisection (with integer endpoints) ?
c. Estimate how many bisection iterations it would take to find the root to
accuracy of 0.1 and of 0.001.
d. Use bisection, by hand (on a calculator!) to find the root to an accuracy
of 0.1 (it should not take more than 5 iterations).
e. Use Newton-Raphson, by hand (on a calculator!) to find the root to an accuracy
of 0.1, with initial guess one of the endpoints from above.
e. Use Secant method, by hand (on a calculator!) to find the root to an accuracy
of 0.1, with initial guesses x_{0}, x_{1} the two (integer) endpoints.