Exercise $1$

For the function $f(x)=x{e}^{\frac{x}{3}}+0\mathrm{.}5x-0\mathrm{.}5$

a$)$ Use Matlab to plot the function for values of $x$ in the range $-3x3$ Important Notes:

i$.$ Include a copy of your Matlab code in your solution.

ii$.$ Include the plot of the function in your solution.

iii. Upload your $.m$ file on Moodle.

b$)$ Find the root of the above function in the range of values of $x$ provided using the following five $(5)$ methods:

i$.$ Bisection Method

ii$.$ Linear Interpolation Method

iii. Newton's Method

iv$.$ Secant Method

v$.$ Fixed Point Iteration Method

Important Notes for part $($b$)$:

i$.$ Show all your calculations in detail.

ii$.$ Use a simple calculator to perform your calculations.

iii. Perform $4($four$)$ iterations for each method.

c$)$ Write a MatLab program to find the numerical solution of $f(x)$ using the Bisection Method.

Important Note for part $($c$)$:

i$.$ Include a copy of your Matlab code in your solution.

ii$.$ Upload your. $m$ file on Moodle.

PLEASE for the $($B$)$ please show all the working and procedure