# Taylor polynomials expansion with substitution

I am working on some practice exercises on Taylor Polynomial and came across this problem:

Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$

In the solution provided, the author makes a substitution such that $x=3+h$ and $y=1+k$. I am not sure why he makes this substitution. Also, why not just find the Taylor polynomial for $f(x,y)$, then plug in the values for $x$ and $y$ to solve for $f(x,y)$?

If you could provide some references for reading on this I would appreciate that as well.

In your case, you want to have $h$ go to zero for $x$ and $k$ go to zero for $y$. For this to happen at $x=3$ and $y=1$, you want to use $x=3+h$ and $y = 1+k$.
One reason that you want to see what happens when $h \to 0$ is that $h^2$ (and higher powers) are small compared to $h$, so they can be disregarded when you are seeing what happens. If $h$ does not tend to zero, then $h^2$ and higher powers cannot be disregarded and, in fact, may dominate.