Linearization use tangent space to approximate a function
It can be seen as first-degree Taylor expansion.
Function with Multiple Inputs
See also: gradient > Gradient as Best Linear Approximation
Given a function
In one and two dimensions, we have tangent line and tangent plane, respectively. Visualizing higher dimensions is more challenging.
Example: take
and estimate at Answer: first:
. Then , , and and only is non-zero at : . Therefore, And so The actual value of is