Suppose $$f(x,y) = x^2 + xy + y^3$$
I have to find the Taylor series for this function about the point $(1,-1)$. How should one solve this?
Now usually, I'm used to solving questions that ask you to find the Taylor polynomial of a certain degree of a function near a given point. But I feel like this one is different. It doesn't ask you to find the taylor polynomial of a certain degree. It just asks you to find the taylor series about a given point. I suppose there's a different way to do this.
EDIT: Here's a new example:
f(x,y) = 1/(2 + xy^2)
Suppose you had to find the taylor series for this function about a certain point, say (0,0).
In this case, calculating every partial derivative would be quite painstaking. Is it possible to find the taylor series for this function without calculating all the partial derivatives?
Best Answer
Since $f$ is a polynomial it is for each point $(x_0,y_0)$ its own Taylor expansion at $(x_0,y_0)$ in disguise. Given $(x_0,y_0):=(1,-1)$ write $x:=1+\xi$, $y:=-1+\eta$ and obtain $$\eqalign{\hat f(\xi,\eta)&=f(1+\xi,-1+\eta)=(1+\xi)^2+(1+\xi)(-1+\eta)+(-1+\eta)^3\cr &=-1+(\xi+4\eta)+(\xi^2+\xi\eta-3\eta^2)+\eta^3\ .\cr}$$ Of course you can rewrite that in the form $$f(x,y)=-1+\bigl((x-1)+4(y+1)\bigr)+\bigl((x-1)^2+(x-1)(y+1)-3(y-1)^2\bigr)+(y+1)^3\ .$$