In this section we’re going to take a look at an application
not of derivatives but of the tangent line to a function. Of course, to get the tangent line we do need
to take derivatives, so in some way this is an application of derivatives as
well.

Given a function, ,
we can find its tangent at . The equation of the tangent line, which we’ll
call for this discussion, is,

Take a look at the following graph of a function and its
tangent line.

From this graph we can see that near the tangent line and the function have nearly
the same graph. On occasion we will use
the tangent line, ,
as an approximation to the function, ,
near . In these cases we call the tangent line the **linear approximation** to the function at
.

So, why would we do this?
Let’s take a look at an example.

Linear approximations do a very good job of approximating
values of as long as we stay “near” . However, the farther away from we get the worse the approximation is liable
to be. The main problem here is that how
near we need to stay to in order to get a good approximation will
depend upon both the function we’re using and the value of that we’re using. Also, there will often be no easy way of
predicting how far away from we can get and still have a “good”
approximation.

Let’s take a look at another example that is actually used
fairly heavily in some places.

This is actually a somewhat important linear
approximation. In optics this linear
approximation is often used to simplify formulas. This linear approximation is also used to
help describe the motion of a pendulum and vibrations in a string.