| 196 |
linear approximation to the function
$f:\mathbb{R}^n\rightarrow \mathbb{R}$
at
$\vect{x}=\vect{a}=\left(a_1,\ldots,a_n\right)$
|
\[
\begin{align*}
L(x_1, \ldots, x_n) =
&f(a_1,\ldots,a_n)\\
&+ \left[
\frac{\partial f}{\partial x_1}
(a_1,\ldots,a_n)
\right]
(x_1-a_1)\\
&+ \cdots+
\left[
\frac{\partial f}{\partial x_n}
(a_1,\ldots,a_n)
\right]
(x_n-a_n)
\end{align*}
\]
\[
\begin{align*}
L\left(\vect{x}\right) =
f\left(\vect{a}\right)
&+ \left[
\frac{\partial}{\partial x_1}
f\left(\vect{a}\right)
\right]
\left(x_1-a_1\right)\\
&+ \cdots+
\left[
\frac{\partial}{\partial x_n}
f\left(\vect{a}\right)
\right]
\left(x_n-a_n\right)
\end{align*}
\]
|
Alex Notes