So far we've mostly studied rates of change of linear functions, where we could relate rate of change to slope. The amazing thing is that the graphs of almost all functions look straight if we zoom in on them enough. By using very small time steps, we can take everything we already know about linear functions and apply them to arbitrary functions.

Suppose we have a function, like $f(x) = \sqrt{x}$, and we want to understand how this function behaves when $x$ is close to 4. We know that $f(4)=\sqrt{4}=2$. It also happens that the rate of change when $x=4$ is $f'(4) = 1/4$. (Later on we'll learn how to compute $f'(4)$. For this slide, we're just going to use this fact.) This means that
When $x \approx 4$, the graph $y=f(x)$ is very, very close to a straight line of slope $1/4$ going through $(4,2)$.

Let's find the equation of that line. Since it has slope $m=1/4$ and goes through $(4,2)$, the point-slope equation for the line is \begin{eqnarray*} y-2 &=& \frac14 (x-4), \qquad \hbox{or equivalently} \cr y & = & 2 + \frac14 (x-4). \end{eqnarray*} This means that for any value of $x$ close to $4$, $f(x)=\sqrt{x}$ is close to $2 + \frac14(x-4)$. Not exactly equal to, since the graph $y=\sqrt{x}$ isn't exactly a straight line, but very, very close.

Let's see how close:

The same idea works for any reasonably smooth function $f(x)$, not just for square roots. If we know the value $f(x_0)$ and rate of change $f'(x_0)$ at a point $x=x_0$, then we can approximate the graph $y=f(x)$ near $x_0$ by the straight line $$ y = f(x_0) + f'(x_0) \, (x-x_0),$$ or equivalently approximate the function $f(x)$ by $$ f(x) \approx f(x_0) + f'(x_0) \, (x-x_0).$$ This is called the microscope equation because it's what we get by zooming in on the point $(x_0, f(x_0))$.

Of course, to make use of the microscope equation you need to know how to compute $f'(x_0)$. That's what we're going to be spending much of the next month learning.