Question $1$

$1\mathrm{.}1.$ Discuss the Newton's method for finding roots of nonlinear equations and describe a scenario where it might not be the most efficient method.

$1\mathrm{.}2.$ Discuss the accuracy and applicability of numerical methods compared to analytical methods in solving structural engineering problems.

$1\mathrm{.}3.$ Write the necessary $R$ commands $($or any programming tool of your choice$)$ and present screenshots to calculate the eigenvalues and eigenvectors of matrix $B$ below:

$B=\left[\begin{array}{cc}5& 2\\ -1& 3\end{array}\right]$

$1\mathrm{.}4.$ Solve the following system of equations using lower$-$ upper $($LU$)$ decomposition, by hand or using any programming tool:

$2x-3y+z=5$

$x+y-4z=-2$

$3x-y+2z=6$

$1\mathrm{.}5$ Provide the $R$ commands $($or any programming tool of your choice$)$ and a screenshot to show the plot, including a title and labels on the horizontal and vertical axes, for the following equation:

$M=12\mathrm{.}5x-0\mathrm{.}75x^3$

$x$ is to be defined from $0$ to $5$ at an interval of $0\mathrm{.}05$

Title: Bending Moment Diagram

Horizontal axis label: Distance along the beam $($m$)$

Vertical axis label: Bending Moment $($kNm$)$

Also determine the value of $x$ where the bending moment is the highest.

$[30$ marks$]$