How to solve an equation with Newton-Raphson method?

Question

f(x) = 1 − x − cos (x²)

a) Show that the equation f(x) = 0 has a root α in the interval 1.4 < α < 1.5.

b) Using x0 = 1.4 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 decimal places.

c) By considering the change of sign of f(x) over an appropriate interval, show that your answer to part b is correct to 3 decimal places.

Answer 1

a) f(1.4) = 1 - 1.4 - cos(1.96) = - 0.0 < 0; f(1.5) = 1 - 1.5 - cos(2.25) = 0.13 > 0. So on interval (1.4, 1.5) f(x) has at least one zero.

b) Let x₀ = 1.4; f'(x) = - 1 + 2xsin(x²); x₁ = 1.4 - (1 -1.4 - cos1.96)/(- 1 + 2.8sin1.96) = 1.4129

x₂ = 1.4129 - (1 - 1.4129 - cos(1.4129²)/(- 1 + 2.8258sin(1.4129²) = 1.4130

c) Because x₂ - x₁ = 0.0001 we can take α = 1.413 with accuracy 0.001