Of course, the value of the average rate that
we computed depends not only on the point identified as P, but
also on the second point Q on the graph.
If we were to chose a different
endpoint for the same calculation, i.e. a different point for Q,
we would get a different average rate of change.
In the next picture, we show what happens when we make the second
point closer and closer to P. Notice that, as Q is chosen closer to P,
the secant lines (shown in blue) have slopes that measure the change
that take place very close to P.
Notice that the sequence of secant lines shown in the previous
picture accumulate around a unique line through the point P. That
line is called the tangent line. It has been drawn here in
red, together with the secant lines, to show their relationship. The
tangent line represents a limiting process in which the average rate
of change is calculated over smaller intervals around P. As before, we say that this function is
differentiable at P, and we call the slope of the tangent line
the derivative at P. Since the derivative is obtained by
measuring the average rate of change close to P, we can think of it as
measuring an instantaneous rate of change.
Here is a way to see this in an interactive fashion. Simply click
somewhere on the graph. You will then see the secant line which you
can then drag towards the basepoint. As you approach the basepoint,
the secant line approaches the tangent line which is shown in red.
There is nothing special about the point P in our example, and we
could as easily have considered any point on the curve as a location
at which the tangent line is to be found. In the graph below, you can
see the tangent line drawn at several different points along the
curve. The slopes of these tangents change from point to point. As we
will later discuss, the behaviour of these slopes are in themselves an
interesting trend (and will form a function of time that will be
called the derivative of the original function.)
Will every point on every graph allow us to compute, in a similar
way, a tangent line? Can you think of examples in which this type
of process might not work? For example, in which a unique line is not
obtained? Think about these questions and then read on.
Let's consider the absolute value function again:
Here we see that the secant lines on the right all have slope 1
while the secant lines on the left all have slope -1. This means that
the secants do not approach a unique line and so we say that the
absolute value function is not differentiable at x = 0.
Is the function whose graph is represented below differentiable at
x = 0? If so, what is the derivative? What is the equation of the