Solution

First, let's define our curve:

Our point of tangency:

So, for a given Δx, the slope of the secant line is:

This works great until , at which time bad things happen...  So, a way to avoid this is to use the following trick:

Thus, the equation of the secant line is:

This gets the basic idea across, but it jumps around a lot.  The way to fix this is to use the PlotRange option:

Let's spice it up a little more with different colors:

Another cool trick is to plot all of these together on the same axes: