Csider applying newtons method to find the rootnbspalpha

Question

1) Consider applying Newton's method to find the root α = 0  of the equation sin (x) = 0. Find the maximal interval (-r, r) for which the Newton method iterates will converge to α for any choice of x₀ in (-r, r).

Hint: Draw a graph of y = sin (x)  and graphically interpret the placement of x₀ and x₁.

2) Use Newton's method to solve the equation:

0.25x2 - xsin (x) - 0.5 cos (2x) + .5 = 0 with P₀ = π/2.

Iterate using Newton's method until the accuracy of 10^-5 is obtained. Explain why the result seems unusual for the Newton's method. Also, solve the equation with P₀ = 5π and P₀ = 10π.