A real function of two variable is given by: $f(x,y) =\exp(x+y) \cdot \cos(x-y)$.
The approximating polynomal of $\boldsymbol{2}$nd degree for $\boldsymbol{f(x,y)}$ with converging point $\boldsymbol{(x_0,y_0) =(0,0)}$ called $\boldsymbol{P_2(x,y)}$.
a) Find $P_2(x,y)$
b) Equation $P_2(x,y)=0$ describes a conic section in the $(x,y)$ plane. Give a characteristic of the conic section.
Any help would be great on how to approach this question.
Well, you know that a Taylor series about $(0,0)$ takes the form
$$f(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{mn} x^m y^n $$
$$a_{mn} = \frac1{m! n!}\left [\frac{\partial^{m+n} f}{\partial x^m \partial y^n} \right ]_{x=0,y=0} $$
You want the following $6$ terms: $a_{00}$, $a_{10}$,$a_{01}$,$a_{20}$,$a_{11}$,$a_{02}$.
To figure out which conic section you have, look at the sign of $a_{11}^2-4 a_{20} a_{02}$.