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 $a_{11}^2-4 a_{20} a_{02}$.

  • $\begingroup$ Your computations will give you $2xy+x+y+1=0 \Leftrightarrow y=-\dfrac{x+1}{2x+1}$ which is an hyperbola. The consequence is that you have a saddle point at the origin. $\endgroup$ – Jean Marie Mar 3 '16 at 15:58
  • $\begingroup$ @JeanMarie: Cool, but I left my answer as is so that the OP could work out the result for himself. $\endgroup$ – Ron Gordon Mar 3 '16 at 15:59
  • $\begingroup$ I understand your point. I think that giving the answer leaves some work to do... Regarding the nature of the conic section, I have found with my students, that they had not the idea to find rather easily the nature of a conic section using the "trick" of expressing one variable as a function (or as 2 functions) of the other variable. The knowledge about conic sections is now very scarce and criteria like the sign of $ a_{11}^2-4 a_{20} a_{02}$ is no longer part of the curricula, in most countries, alas... $\endgroup$ – Jean Marie Mar 3 '16 at 16:31

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