in this specific exercise f(x) = x^3 - 4

Newton's method is an algorithm for finding the zeros of a function, you can find more information about it here:

http://en.wikipedia.org/wiki/Newton's_method

it's an iterative method and we need to come up with an initial guess the closer to the zero the better, in this example theIntermediate Value Theoremis used to choose the initial guess, now we have to calculate f '(x)

f ' (x) = 3*x^2

Xn+1 = Xn - ((Xn)^3 - 4) / (3*(Xn)^2)

Xo = 1.5

so:

X1 = Xo - ((Xo)^3 - 4) / (3*(Xo)^2) = 1.5925

X2 = X1 - ((X1)^3 - 4) / (3*(X1)^2)......