Calculus - Newton's Method Problem

Question

Find the positive value of x which satisfies x = 4.800sin(x). Give the answer to four decimal places of accuracy.

Answer 1

We're trying to find x → x_∞ that is a root of 4.8sinx – x with 1st derivative 4.8cosx – 1. We might guess a starting x₀ by Taylor series 0 ≈ 4.8(x – x³/6) – x = (3.8 – 0.8x²)x, implying x₀ = √(19/4) ≈ 2.17945. Let's iterate x → x – (4.8sinx – x)/(4.8cosx – 1) to get: 2.17945, 2.64909, 2.57652, 2.57529, 2.57529.