Q $2.$ Let $q$ be the function defined by

$q(x,y)=$xycos$(3x^2{y}^2)$

$($i$)$ Use the Taylor series for the one$-$variable cosine function, $cos(x),$ to show that the value of

$q(h,k)$ can be approximated by

$q(h,k)$~~$-\frac{9}{2}{h}^5{k}^5+hk$

$($ii$)$ Show that this polynomial in $h$ and $k$ is exactly the two$-$variable order $15$ Taylor polynomial

for $q$ expanded about $(0,0).$ Show this by directly calculating the definition of the Taylor

polynomial given by

$q(h,k)$~~${\displaystyle \underset{n=0}{\overset{15}{}}}\frac{1}{n!}(D^{n}q)(0,0),$

where the differential operator $D$ is given by $D=h\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}+k\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}.$

$10$ for code that fulfills the requirements

$5$ for explanatory comments

You will not be able to carry out many of caluculcations of the partial derivatives needed for $D^n$ in

part $($ii$)$ by hand as these expressions quickly get very complicated due to the many applications of the

product and chain rules. So instead you should do this calculation using a computer program written

in a suitable programming language. For example, use the diff command from the Matlab symbolic

toolbox to write a program to carry out the calculations and evaluations of $D^{n}q$ needed for the Taylor

polynomial approximation. Include a printout of the Matlab commands and relevant results as part of

your submission and comment on the output. Alternatively you may use SageMath, Python or other

similar software for these computations.