# Thread: Taylor series for function of several variables

1. ## Taylor series for function of several variables

Write down the Taylor series up to degree of five for:

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

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

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

Let $\displaystyle P=(0,0)$

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

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

Is there an easier way of doing these questions?

2. What you are doing is the easiest method.

3. 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?

4. The derivation requires about three pages of tricky algebra.

5. I've just thought of something:

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

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

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