Taylor series to approximate a nonlinear problem to a linear problem

Question

How do I convert the function $f(x,y) = xy$ (area of a rectangle) to a linear function using Taylor Series? My attempt at this is as follows (please tell me if I am right/wrong:

$L(x,y) = f(a,b) + df/dx(a,b)(x-a) + df/dy(a,b)(y-b)$

Answer 1

Yes, that's the linear Taylor polynomial at $(a,b)$, but why didn't you compute the function values?

$$\begin{align} L(x,y)&=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\\ &=ab+b(x-a)+a(y-b)\\ &=bx+ay-ab \end{align}$$

EDIT

We have $$\begin{align}f(x,y)&=xy,\\ f_x(x,y)&=y,\\ f_x(a,b)&=b\end{align}$$ Similarly, $$f_y(a,b)=a$$

Answer 2

This is very correct at the first order just as you want.

So, if you need to make the problem more linear, this is the formula to be used. Since the problem is now bilinear, you could shorten the notations.