Find the x2 obtained by using Newton's Method to estimate the real root of f(x)=2−x+sin⁡x=0 with x1=5. x2 =

Question

Find the x₂ obtained by using Newton's Method to estimate the real root of f(x)=2−x+sin⁡x=0 with x₁=5.

x₂ = 

Answer 1

The iterations in Newton's method are given by:

x_{n + 1} = x_n - f(x_n) / f'(x_n)

Let f be the function given on R by f(x) = 2 - x + sin(x)

The derivative of f is given on R by:

f'(x) = cos(x) - 1

x₁ = 5

x₂ = x₁ - f(x₁) / f'(x₁)

= 5 - (2 - 5 + sin(5)) / (cos(5) - 1)

= 5 - (sin(5) - 3) / (cos(5) - 1)

aaaand we need the calculator :-)