(Taylor series in two variables is
optional material -- check with your instructor to see if you
need this.)
Partial derivatives allow us to approximate functions just like ordinary derivatives do, only with a contribution from each variable. If $x \approx a$ and $y \approx b$, then we can get a two-variable linear approximation that is analogous to the linear approximation $L(x)$ in one variable $$f(x,y) \approx f(a,b) + f_x(a,b) (x-a) +f_y(a,b) (y-b).$$This is sometimes written as $$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy.$$
And, just as in one dimension, we can use higher derivatives to get a more accurate approximation.
| Taylor Series in two variables: $$f(x,y) = \sum_{n,m=0}^\infty c_{n,m} (x-a)^n (y-b)^m,$$ where $$c_{n,m}= \frac{1}{n!m!}\frac{\partial^{n+m}f}{\partial x^n\partial y^m}(a,b).$$ |
This video explores the Taylor Series in two variables and looks
at an example.