Rates of change and the slope of a curve

To see another way in which the derivative appears, let's go back to our earlier discussion about making measurements. Recall that we looked at a graph that describes the result of some scientific observation (the measurement of the value of the variable y at different times t).

By displaying these measurements in the form of a graph, we can start to ask questions about how changes in y are occurring and assign more concrete meanings to the notions of rates of change. It is much more convenient to do this on a graph than a table of values.

Average rate of change

In the figure below, we have identified a point P on the graph, and a second point, also on the graph which will serve as example. Note that a straight line has been drawn, connecting these two points. By how much has the value of y changed between the two points? Over what period of time has this change occurred? The ratio of these two values, i.e. (change in y)/(change in t) is called the average rate of change in y. Average rate of change = (change in y)/(change in t) The average rate of change is an important quantity which we can discuss without a graph. For instance, economists are interested in the average rate of change in unemployment over the past year. However, once we draw our data points on a graph as above, we have an appealing geometrical interpretation of the average rate of change. Notice that the average rate of change is a slope; namely, it is the slope of a line which we call the secant line joining P and Q. In other words, we can look at this concept from two different angles---one shows us a rate of change and the other the slope of a line. We will often move back and forth between these two perspectives.

Instantaneous rates of change

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.) Questions: 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. Questions: Is the function whose graph is represented below differentiable at x = 0? If so, what is the derivative? What is the equation of the tangent line?