(a) Given an initial approximation x 1 to a root of the equation f (x) 0, explain geometrically, with a diagram, how the second approximation x 2 in Newtons method is obtained (b) Write an expression for x 2 in terms of x 1 , f (x 1 ), and f '(x 1 ) (c) Write an expression for x n 1 in terms of x n...

A given X 1 is initial approximation of the root for equati ...

(a) Given an initial approximation x 1 to a root of the equation f (x) = 0,...

Question:

(a) Given an initial approximation x₁ to a root of the equation f (x) = 0, explain geometrically, with a diagram, how the second approximation x₂ in Newton’s method is obtained.
(b) Write an expression for x₂ in terms of x₁, f (x₁), and f'(x₁).
(c) Write an expression for x_n+1 in terms of x_n, f (x_n), and f'(x_n).
(d) Under what circumstances is Newton’s method likely to fail or to work very slowly?