$1.($a$)$ Describe the methodology of solving the system of nonlinear equations

using the Newton$-$Raphson method.

$($b$)$ Write a computer program $($using either MATLAB or PYTHON$)$ to

implement it $($considering number of equations and number of variables

both as $n).$

$2.($a$)$ Mention the expression$/$formula for numerical differentiation $($based on

the central difference scheme$)$ against each of the $1^{st},2^{nd},3^{rd},$ and $4^{th}$ order

derivatives.

$($b$)$ Evaluate the $1^{st},2^{nd},$ and $3^{rd}$ order derivatives of the following function

numerically at a point $x=2$ :

$f(x)=25x+3x^2+2x^4$

Further, evaluate the accuracy w$.$r$.$t$.$ the exact values of derivatives.

$3.($a$)$ A function is provided as mentioned below:

$f(x)=0\mathrm{.}2+25x+3x^2+2x^4$

Evaluate the integral $(I)$ numerically in a range between $0$ and $2.$

$I={\displaystyle \underset{0}{\overset{2}{}}}f(x)dx$

$($b$)$ Write a computer program $($using either MATLAB or PYTHON$)$ to

implement it $($considering number of discretization as $n).$

A simply supported beam of a length $(L)$ of $5$ m is subjected to a uniformly

distributed load as $10k\frac{N}{m}.$ Describe the solution for finding the

deflections at $\frac{L}{4},\frac{L}{2},$ and $3\frac{L}{4}$ numerically $($using the Finite Difference

Method$).$ Note: Complete numerical evaluation is not needed.