A **nonlinear function** can be approximated with an **linear function** in a certain operating point. The process of **linearization**, in mathematics, refers to the process of finding a **linear approximation** of a nonlinear function at a given point *(x _{0}, y_{0})*.

For a given nonlinear function, its linear approximation, in an operating point *(x _{0}, y_{0})*, will be the

**tangent line**to the function in that point.

### Linearization – theoretical background

A line is defined by a linear equation as:

where:

*m* – the **slope** of the line

*b* – the vertical offset of the line

The slope *m* of the line can be defined as the tangent function of the angle (*α*) between the line and the horizontal axis:

where *dy* and *dx* are small variations in the coordinates of the line.

Another way of defining a line, is by specifying the **slope** *m* and a **point** (*x _{0}, y_{0}*) through which the line passes. The equation of the line will be:

Replacing equation (2) in (3) gives:

Equation (4) translates into: for a given nonlinear function, its linear approximation in an operating point (*x _{0}, y_{0}*) depends on the derivative of the function in that point.

In order to get a general expression of the linear approximation, we’ll consider a function

*f(x)*and the x-coordinate of the function

*a*. The y-coordinate of the function will be

*f(a)*.

Replacing these in equation (3) gives:

We can now write the general linear approximation *L(x)* of a nonlinear function *f(x)* in a point *a* as:

### Linearization – practical example

Let’s find a linear approximation of the function *f(x)* in the point *a = 1*.

The graphical representation of the function is:

**Step 1**. Calculate *f(a)*

**Step 2**. Calculate the derivative of *f(x)*

**Step 3**. Calculate the slope of the linear approximation *f'(a)*

**Step 4**. Write the equation

*L(x)*of the linear approximation

If we plot the linear approximation *L(x)* on the same graph, we get:

As expected, the linear approximation *L(x)* in the point *(a, f(a))* is tangent to the nonlinear function.

If we consider an interval close to our linearization point *a*, we can see that the results of the linear approximation are very close to the ones of the nonlinear function. For example, let’s plot both nonlinear function *f(x)* and linear approximation *L(x)* between *0.9* and *1.1*.

If we calculate the relative error between the results of the nonlinear and linear functions, we’ll notice small errors, below *0.5 %*. This means that we can use our linear approximation to predict the behavior of the nonlinear function, but only around the linearization point *(a, f(a))*.

The **Scilab instruction** to plot the above graphical representations are: