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

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(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 xn+1 in terms of xn, f (xn), and f'(xn).
(d) Under what circumstances is Newton’s method likely to fail or to work very slowly?

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