If a line is a linear approximation to a function

Davon Irwin 2022-07-01 Answered
If a line is a linear approximation to a function in $1$ variable and a hyperplane is the linear approximation to a function in $2$ variables what is the linear approximation to a function in $3$ variables?
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humbast2
Your terminology is slightly off.
The linear approximation to a function of 1 variable is indeed a line.
The linear approximation to a function of 2 variables forms a plane.
The linear approximation to a function of any number of variables is a hyperplane.
Unlike the terms "line" and "plane", a hyperplane does not traditionally have a fixed dimension. Instead, the dimensionality depends on context: in ${\mathbb{R}}^{2}$, a hyperplane has one dimension; in ${\mathbb{R}}^{3}$, it has two; and, in general, a hyperplane has one fewer dimension than the ambient space (a nice term for this is "codimension $1$").
In general, we don't bother to describe lines as "hyperplanes in $2D$ space," mainly because one cannot distinguish between dimension $1$ and codimension $1$ when the ambient space has $2$ dimensions. At the same time, it's not wrong.
The distinction is relevant in $3D$ and above, and so we do describe planes as "hyperplanes in $3D$ space."