You've seen Taylor series for functions of 1 variable. For a function satisfying the appropriate conditions, we have
is the remainder term:
z is a number between c and x. The Remainder Term gives the error that occurs in approximating with the degree Taylor polynomial.
There is a similar formula for functions of several variables. To make the notation a little better, I'll define higher-order differentials as follows. Let .
And so on. Here's Taylor's formula for functions of several variables. With more variables, it's more complicated and technical; try to see the resemblance between the formula here and the one for functions of one variable.
Theorem. Suppose , where U is an open set in . Suppose f has continuous partial derivatives at all points of U through order . Let , where and the segment from c to x is contained in U. Then for some point z on the segment from c to x,
Example. Write out the Taylor expansion through terms of degree 2 for a function of 2 variables .
Let's say we're expanding at a point . Then
Example. For a function ,
Write out the Taylor expansion of f at through terms of degree 2.
Example. Construct the Taylor series through the order for at .
Example. Let . Use a -order Taylor approximation to approximate .
I'll use a Taylor expansion at , since it's the closest "nice" point to .
The series is
Bruce Ikenaga's Home Page
Copyright 2018 by Bruce Ikenaga