Q$1$ Consider the nonlinear equation

$2xcos(2x)-(x-2{)}^2=0$

$($a$)$ Write the general term $x_n$ of the Newton's iteration sequence with an arbitrary

initial value $x_0$ for the given equation.

$($b$)$ Write a Python program which computes the iterations of the Newton's method

starting with $x_0=3\mathrm{.}63$ until $|x_{n+1}-x_n|<{10}^{-4}$ and prints the computed terms of the

sequence $\{x_n\}$ and the value of $n$ for which $|x_{n+1}-x_n|<{10}^{-4}$ holds.

$($c$)$ Return your solution file containing the solution of part $($a$),$ the Python code and

the output of your Python code.

Q$2$ Consider the definite integral

${\displaystyle \underset{0}{\overset{6}{}}}\frac{dx}{x^4+2\mathrm{.}63}$

$($a$)$ Write the Composite Simpson's rule formula with $2M$ subintervals.

$($b$)$ Write a Python program which computes the approximate value with the Com$-$

posite Simpson's rule formula written in part $($a$),$ for $M=1,2,3,4,5,6,7,8$ and prints

the results as

$M=k,Sk=$ computed value, for $k=1,$dots,$8.$