Language: Python

Problem A.4. Finding with Newton's method It is common knowledge that 3.14, but in this exercise you will use Newton's method and knowledge of the sine function to improve an approximation of . Newton's method can be written as a difference equation defined f(In-1) In = In-1-f(n-1) and is used to find roots of f(r), based on the function f(x), the derivative of the function f'(x) and an initial guess Io. Consider the function f(x) = sin(I). Finding the functions derivative f'(x), should be trivial. This function has infinitely many roots, which we know are located at x = km, where k is an integer. This exercise will focus on the root for k = 1, and use Newton's method to approximate 1. For Newton's method to converge towards the correct root, the initial condition, to, needs to be set close to the value of . Set Xo = 3.14 and calculate 11 and 12 following Newton's method. Print out 20, 1 and 22, as well as the value of numpy.pi with 13 decimals. You should print them in a tidy way such that the values are easy to compare. If Newton's method was used correctly, your values for should improve