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Clarify definition

  1. Nov 27, 2004 #1


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

    [tex]f(x, y, z) = (x^2 + 2xy + y^2)e^z[/tex]

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

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

    [tex]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 )[/tex]

    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.
  2. jcsd
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