The function $y=cosx$ can be approximated by the polynomial: $=1-\frac{x^2}{4}+\frac{x^4}{24}-\frac{x^6}{720}+\frac{x^{14}}{40320}+\frac{x^{10}}{3628800},$ provided that $x$ is reasonably close to $o.$ We wish to write a function $M$ file to approximate the function $cosx,$ but we need to take advantage of the periodicity of $cosx$ to reduce the errors.

a$)$ Write a function $M$ file called $cosApprox(x)$ to do the following:

i$)$ Test to see that $x$ is positive, and if it is not, stop the function.

ii$)$ While $x$ is still greater than $2,$ subtract $2$ from $x.$

iii$)$ After this is done, $x$ should be between $o$ and $2.$ Use the above degree $8$ polynomial to estimate $cosx.$

b$)$ Write a program $M$ file that does the following.

i$)$ Set up a vector of $x$ values from o to $2,$ in steps of $0\mathrm{.}1.$

ii$)$ Use a For$-$Loop to go through the vector of $x-$values

iii$)$At each $x-$value call your function $cosApprox$ on that $x-$value. Call the Matlab function $cosx$ and subtract the two to find the error. Divide the actual value of $cosx$ and multiply by $100$ to find the percentage error. Display both the $x-$value and the percentage error. $($Note that percentage error is given by:

$\frac{\text{e}\text{s}\text{t}\text{i}\text{m}\text{a}\text{t}\text{e}\text{d}-\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}100\%.$ Sometimes this will be positive and sometimes it will be negative and this is expected.

iv$)$ Plot the percentage error on a graph