Linearization and the Taylor Series

Linearization

Recall that the derivative $df/dx$ of a function $f(x)$ at a point $x_0$ is given as follows:

Assuming $x = x_0+h$, the equation above can be rearranged and written as follows:

where $\vert x -x_0 \vert < \epsilon$ and where $\epsilon \rightarrow 0$. As can be seen, the derivatives $df/dx$ can now be used to approximate any function by a straight line around the point $x$. For multivariate, the linearization above can be generalized as follows:

where $\nabla_xf(x_0)$ ...