M371 - Alexiades
Problem Set 2:   Roots

1. Do 3 iterations of bisection by hand on  f(x) = x6 - 2 on [0,2].
   Give an upper bound for the error.
 
2. Consider the function    F(x) = x6 - 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 x0, x1 the two (integer) endpoints.