7) Approximating Roots (Newton's Method)
Consider the function f(x)
> f:=x->x^3-2*x-1;
> plot(f(x),x=-3..3,-5..5);
The newton method formula is given by the recurrence formula
> newton:=x->x-f(x)/D(f)(x);
As is needed near the root guess
> evalf(newton(-.5));
> evalf(newton(%));
> evalf(newton(%));
> evalf(newton(%));
> evalf(newton(%));