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.