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:
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 the Intermediate Value Theorem is 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
X1 = Xo - ((Xo)^3 - 4) / (3*(Xo)^2) = 1.5925
X2 = X1 - ((X1)^3 - 4) / (3*(X1)^2)......