Newton’s Method can be used to compute reciprocals without performing division. Let c > 0 and set ƒ(x) = x⁻¹ − c.
(a) Show that x − (ƒ(x)/ ƒ(x)) = 2x − cx².
(b) Calculate the first three iterates of Newton’s Method with c = 10.3 and the two initial guesses x₀ = 0.1 and x₀ = 0.5.
(c) Explain graphically why x₀ = 0.5 does not yield a sequence converging to 1/10.3.