Newton's method

Question

Use newton's method to find the fixed points of the function where f(x) =x, round three decimals.

tan(x) on x= [pi/2, 3pi/2]

Answer 1

We're trying to solve g(x) = tan(x) - x = 0 with Newton's method of iteration: first iteration x₁ from a starting guess x₀ is x₁ = x₀ - [g(x₀) / g'(x₀)] and so forth until we get convergence. g'(x) = sec²(x) - 1. Looking at the Unit Circle between 90° and 270°, I would guess that x = tan(x) about halfway between 4π/3 and 3π/2, i.e. 17π/12 ≅ 4.45 radians.

x₁ = 4.4506 - (-0.72/13.93) = 4.5022.

x₂ = 4.5022 - (0.185/21.97) = 4.4938.

x₃ = 4.4938 - (0.0073/21.26) = 4.4934. x₄ = 4.4934 also, so the answer is tan(x) = x at x = 4.493 radians.