# Clarify definition

1. Nov 27, 2004

### AKG

I have a question that asks me to find the Taylor polynomial of degree 2 of the function:

$$f(x, y, z) = (x^2 + 2xy + y^2)e^z$$

at (1, 2, 0). I have Taylor's Theorem given as follows:

If $f : V \to \mathbb{R}$, V is open, $V \subseteq \mathbb{R}^n$, $c \in V$, $h \in \mathbb{R}^n$, f is of class $C^{k + 1}$, and $c + th \in V$ if $0 \leq t \leq 1$, then:

$$f(c + h) = \sum _{l = 0} ^{k} \left ( \sum _{\{\alpha \in \mathbb{Z}_+ ^n : |\alpha | = l\} } \frac{D_1 ^{\alpha _1} \dots D_n ^{\alpha _n}f(c)}{l!}(h_1 ^{\alpha _1}, \dots , h_n ^{\alpha _n})\right ) + \sum _{\{\alpha \in \mathbb{Z}_+ ^n : |\alpha | = k + 1\}}\left ( \int _0 ^1 \frac{(1 - t)^k}{k!}D_1 ^{\alpha _1} \dots D_n ^{\alpha _n} f(c + th)(h_1 ^{\alpha _1}, \dots , h_n ^{\alpha _n})dt\right )$$

Is this ugly thing above the thing I'm supposed to be working with? And if I'm asked for the polynomial of degree 2, then should my value for k be 1 or 2 (or something else)? Thanks.