Linearization

Linearization is the idea of approximating a continuous function around a point with a line.
For a continuous function \(f(x)\), we linearize \(f(x)\) around point \(a\) as
\[f(x)\approx f(a) + f'(a)(x-a).\]

More interesting things happen in higher dimensions:
\[f(\mathbf{x}) \approx f(\mathbf{a}) + \left. \frac{\partial f(\mathbf{x})}{\partial x_1}\right|_{\mathbf{a}} + \left. \frac{\partial f(\mathbf{x})}{\partial x_2}\right|_{\mathbf{a}} + \ldots \]
or
\[f(\mathbf{x}) \approx f(\mathbf{a}) + \left. \nabla f \right|_{\mathbf{a}}  \cdot ({\mathbf{x}} - {\mathbf{a}}) .\]