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.

Thanks in advance.


It's generally a good policy to "always expand around zero". This means that you want to have variables that go to zero at your point of interest.

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.

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  • $\begingroup$ that helps a lot. Thank you very much. $\endgroup$ – Kj Tada Mar 11 '13 at 21:48

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