$($a$)$ The function $cos(x)$ can be approximated by the following series

$cos(x)={\displaystyle \underset{n=0}{\overset{}{}}}(-1{)}^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+$cdots

Using a loop, create line plots of the series expansion of $cos(x)$ against xin$[-,]$ using $n=0,1,$dots,$4$ on the same set of axes. Annotate your plot with axis labels, a title and add a legend with appropriate labels.

$($b$)$ The Sierpinski triangle is a fractal that can be generated using the following steps: starting at a point with co$-$ordinates $(0,0)$

Randomly select one of three vertex points of an equilateral triangle: bottom$-$left vertex $(0,0),$ bottom right vertex $(1,0)$ or top vertex $(\frac{1}{2},\frac{\sqrt[\phantom{2}]{3}}{2}).$

Move the point half the distance from its current position to the selected vertex.

Plot the new position.

Repeat these steps a given number of times.

Use this method to produce a plot of the Sierpinski triangle using $10000$ points with a point marker of size $1.$ Colour the points red if the bottom left vertex was selected, blue if the bottom right vertex was selected or red if the top vertex was selected.

$($c$)($i$)$ Take a selfie and read in the image file using the imread$()$ matplotlib or MATLAB function $($if you don't want to use a selfie you can using the image kitten.jpg on the Moodle area for this module instead$).$ Determine the size of the image in pixels and plot the image.

$($ii$)$ An image can be converted to sepia tone using the following conversion

$R_{\text{s}\text{e}\text{p}\text{i}\text{a}}=$min$(0\mathrm{.}393R+0\mathrm{.}769G+0\mathrm{.}189B,255),$

$G_{\text{s}\text{e}\text{p}\text{i}\text{a}}=$min$(0\mathrm{.}349R+0\mathrm{.}686G+0\mathrm{.}168B,255),$

$B_{\text{s}\text{e}\text{p}\text{i}\text{a}}=$min$(0\mathrm{.}272R+0\mathrm{.}534G+0\mathrm{.}131B,255).$

where $R,G$ and $B$ are the colour values for the original image and $R_{\text{s}\text{e}\text{p}\text{i}\text{a}},G_{\text{s}\text{e}\text{p}\text{i}\text{a}}$ and $B_{\text{s}\text{e}\text{p}\text{i}\text{a}}$ are the colour values for the sepia toned image. Apply this conversion to your image and plot it$.$