Let 1 (x) x and define a sequence of functions by n 1 (x) 1 2 ( n (x) x n (x)) For example, 2 (x) 1 2 (x 1) Use a computer algebra system to compute n (x) for n 3, 4, 5 and plot y n (x) together with y x for x 0 What do you notice ...

and With fix x and f2x x 1 we calculate 11 22 f3x f4x x 1 1x 6x 1 4x 1 X x1 X x ...

Let 1 (x) = x and define a sequence of functions by n+1 (x) =

Question:

Let ƒ₁(x) = x and define a sequence of functions by ƒ_n+1(x) = 1/2 (ƒ_n(x) + x/ƒ_n(x)). For example,ƒ₂(x) = 1/2 (x + 1). Use a computer algebra system to compute ƒ_n(x) for n = 3, 4, 5 and plot y = ƒ_n(x) together with y = √x for x ≥ 0. What do you notice?