Use Newton's method to find the number arcsin(1/3) rounded to 14 digits after the decimal point by solving numerically the equation sin(x)=1/3 on the interval [0,pi/6].
1) Determine f(a) and f(b).
2) Find analytically f', f'' and check if f '' is continuous on the chosen interval [a,b].
3) Determine the sign of f' and f '' on [a,b] using their plots.
4) Determine using the plot the upper bound C and the lower bound c for |f'(x)|.
5) Calculate the function g(x).
6) Determine an upper bound N for |f ''(x)| on [0.4,1] using its plot on [0.39,1].
7) Determine the initial point x_0.
8) Run the iterations of the Newton's method until the error condition is satisfied and roundoff the answer.
Just part 8 please. Please show all steps and calculations. Thank you
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