# Taylor series for function of several variables

• November 5th 2010, 04:49 PM
acevipa
Taylor series for function of several variables
Write down the Taylor series up to degree of five for:

$cos(x+y)$ about the point $(0,0)$

For doing questions like these. Should I just remember the formula for the Taylor expansion of $f$ about the point $x_0=(x_0,y_0)$

$\displaystyle f(x,y)=f(x_0)+\frac{1}{1!}(f_x(x_0)(x-x_0)+f_y(x_0)(y-y_0))+\frac{1}{2!}(f_{xx}(x_0)(x-x_0)^2+2f_{xy}(x_0)(x-x_0)(y-y_0)+f_{yy}(x_0)(y-y_0)^2)+...$

Let $P=(0,0)$

$f=1$ at $P$
$f_x=-sin(x+y)=0$ at $P$
$f_y=-sin(x+y)=0$ at $P$
$f_xx=-cos(x+y)=-1$ at $P$
$f_yy=-cos(x+y)=-1$ at $P$
$f_xy=-cos(x+y)=-1$ at $P$

$cos(x+y)\approx 1+\frac{1}{1!}(0(x-0)+y(y-0))+\frac{1}{2!}(-1(x-0)^2-2(x-0)(y-0)-1(y-0)^2)$

Is there an easier way of doing these questions?
• November 5th 2010, 05:23 PM
Prove It
What you are doing is the easiest method.
• November 5th 2010, 05:29 PM
acevipa
Quote:

Originally Posted by Prove It
What you are doing is the easiest method.

So should I just remember the formula I wrote, or is there a way to derive this?
• November 5th 2010, 05:44 PM
Prove It
The derivation requires about three pages of tricky algebra.
• November 5th 2010, 07:43 PM
acevipa
I've just thought of something:

I know that the expansion of $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...$

So isn't it easier just to do the following:

$1-\frac{(x+y)^2}{2!}+\frac{(x+y)^4}{4!}+...$