Use Newton's Method to approximate the zero(s) of the function.

Question

Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) to three decimal places using a graphing utility and compare the results.

f(x) = x³ − cos x

Newtons method -

Graphing -

Answer 1

Newton's method is x_n+1 = x_n - f(x_n)/f'(x_n)

f'(x) = 3x² +sin(x)

x_n+1 = x_n - (x³-cos(x))/(3x²+sin(x)) (The entire RHS evaluated at x = x_n, the last estimate and starting with a guess)

You have to put in reasonable guesses. The method converges quickly, but sometimes it doesn't converge, and sometimes you don't get to all the roots.

Go to Desmos and plot the function. If you click on the points that the function crosses the access, you can read off the x values.

Please consider a tutor. Take care.