3. Linearization

Let’s take an example of a sine function.

If we take a closer look at the point $(0,1)$, we can see that it is almost like a straight line.

Linearization

Definition

Given a function $y=f(x)$ the tangent line to the graph of $f$ at the point $x=a$ is given by the equation:

$y=f(a)+f^{\prime }(a)(x-a)$

The linear function whose graph is the tangent line at $x=a$ is called the linearization of $f$ at $a$:

$L(x)=f(a)+f^{\prime }(a)(x-a)$