Linear Approximations

As we have seen on previous pages, the derivative is recording how a function changes over a very small scale. This page introduces one important fact that will be used repeatedly in this course.

Tangent Line Approximation

If $ f(x) $ is differentiable at $ x_0 $ , then for small values of $ h $ , the following is a good approximation:
$f(x_0 + h) \cong f(x_0) + f^\prime(x_0) h$

Questions:

These questions give two different reasons as to why the tangent line approximation works.

Write an expression for the average rate of change of the function $ f $ between $ x_0 $ and $ x_0 + h $ .
Give an argument as to why
$\frac{f(x_0 + h) - f(x_0)}{h} \cong f^\prime(x_0)$
for small values of $ h $ .
Justify the tangent line approximation.

We know that the tangent line to the graph of $ y = f(x) $ passes through the point $ (x_0,f(x_0)) $ . We also know that the derivative $f^\prime(x_0)$ is the slope of the tangent line to the graph at $ x_0 $ .
Use this information to find the equation of the tangent line. (Do you need a review? )
Remember that in the Linear Nature of Graphs we saw how the graph of the function near $
(x_0,f(x_0)) $ essentially looks like this tangent line. Here is the demonstration again with the tangent line drawn in red.

Use this observation to argue that the tangent line approximation holds for small values of $ h $ .

The tangent line approximation is fundamental for it underlies every application of the derivative. Basically, it is telling us how to approximate any function, which could be very complicated, by a linear function, which is very easy to work with. We'll see how to use it as we proceed. For now, you should try to understand what it is saying and why it works. Let's consider an example so that we might understand it better.

An Example

Earlier, we considered the function $
f(x) = x^2 $ and saw that $f^\prime(1) = 2$ . Now consider the point $ x_0 = 1 $ . The tangent line approximation then says that

$\begin{eqnarray*} f(x_0 + h) & \cong & f(x_0) + f^\prime(x_0) h \\ (1 + h)^2 & \cong & 1 + 2h \\ 1 + 2h + h^2 & \cong & 1 + 2h \end{eqnarray*}$
We can see that this is a pretty good approximation. The error that is made from the tangent line is simply the term $ h^2 $ and we know that when $ h $ is small, then $ h^2 $ is small as well.

The power of this approximation is not so apparent from this example---namely because the original function $ f(x) = x^2 $ is fairly simple. However, if $ f(x) $ were a more complicated function or a function which we did not yet know, it would be very convenient to approximate it by a linear function $f(x_0) + f^\prime(x_0)h$ . This kind of reasoning will be important as we explore derivatives further.