What we have done with average velocities for the displacement function in the previous section applies in much more general settings. We can move in the direction of generality by abstracting what we did to a more purely mathematical context. In the above context, the velocity measures the rate of change of the displacement. There are three concepts to keep in mind. Let f(t) be the position function of a particle during a time interval .

1.

the net displacement during the time interval [c,d] is simply the value of f(d) minus the value of f(c) - it tells how far the particle has moved (and in what direction) during the time interval.

2.

the average velocity of the particle over the time interval . It is the average rate of change of the position of the particle during the time interval, and is given by the familiar expression:

3.

the instantaneous velocity of the particle at a specific time c. It is computed by calculating the average velocities of the particle over smaller and smaller time intervals containing c (and is the ``limiting value'' of these average velocities as the length of the time intervals shrinks to zero.)

In our more abstract mathematical context, we will talk of functions, and the three concepts above are abstracted as quantities which say something about the growth of the value of the function as the independent variable varies over its domain. Now, just let f(t) be a function defined on some interval .

1.

The net change (or net growth) of the function over the time interval [c,d] is simply the difference of function values

	net change = f(d) - f(c)

2.

The average growth rate (or average rate of change) of the function over the interval [c,d] is given by the usual ``rise over run'' formula:

Notice that we can assign a geometrical meaning to this expression: it is the slope of the line connecting the points (c, f(c)) and (d, f(d)) on the graph of f.

3.

The instantaneous growth rate (or instantaneous rate of change) of the function at a specific value of t, say t = c. It is computed by calculating the average growth rates of the function over smaller and smaller t-intervals containing c (and is the ``limiting value'' of these average growth rates as the length of the t-intervals shrinks to zero). The notation mathematicians use for this concept is

(where h stands in for d - c in the expressions we were using before - Verify this). Notice how this fits in with the Maple expressions we used in Exercise 2.3 above.

Thinking back to what we did in relating the instantaneous velocity to the average velocity, and noting that average velocity can be estimated from the slope of a line on a distance vs. time graph, you can probably get a good sense of the derivative as the slope of a curve (i.e. a function) evaluated at a point on the curve. If you wanted to check this notion, you follow the prescription of the limit expression we just wrote down earlier. Namely, evaluate the slope over an interval. Then, reduce the size of the interval, find the slope, reduce the interval again and find the slope, etc. As you continue that process to smaller and smaller intervals, you get a more and more accurate estimate of the slope at a point somewhere inside the interval. The derivative is defined for the interval of zero size, i.e. the interval includes only the point for which you want the slope. Since you're probably already familiar with the mechanics for how to find a derivative given a function, try out the limit prescription for your favorite function using this Java applet.

Make use of Maple to review the concept of instantaneous rate of change in Exercise 2-4.

Maple can compute instantaneous growth rates (also known as derivatives). There are two ways, depending on whether you want to compute the derivative of a function such as the f you have defined, or just of an expression, which is what you get when you type f(t). For functions, the statement D(f); (or D(f,t) when the definition of f contains other names you are using as parameters and Maple needs to be told which is the actual variable) will return the definition of a function which gives the instantaneous growth rate (i.e., the derivative) of f; for an expression, the statement diff(f(t),t) (here the ``,t'' is obligatory) does the same thing.

Notation:

Just as Maple has two different notations for derivatives, mathematics has (at least) two standard notations for derivatives. Sometimes, the derivative of f(t) is denoted f ´ (t) (read ``f-prime of t''), and other times it is denoted by the fraction which is supposed to be vaguely reminiscent of .

Be sure you understand the distinction between expressions and functions in Maple, and the (subtle!) difference between D and diff. Exercises 2-5 and 2-6 will help clear up your understanding.

The preceding problem should indicate that computing average growth rates over small enough intervals is, for all practical purposes, just as good as computing derivatives. Furthermore, the definition we have given of derivative leaves a lot to be desired (it is essentially the same as the one Newton gave, and the loose ends he left over took more than 200 years to tie up), and seems much more complicated than computing average rates of change. So why work with derivatives at all?

As it turns out, there are pragmatic, theoretical and philosophical reasons for working with derivatives. To begin with, we have seen that average growth rates over small intervals are effectively as good as derivatives when the intervals are ``small enough''. But how small is that? It turns out not just to be a matter of taste, but to depend in an essential way on the function whose growth rate we are trying to measure - it can even depend over which values of the variable we are trying to measure it. For example, consider the function f(t) = sin(t² ).

Do exercise 2-7.

The second reason the derivative works better than average growth rates over small intervals is that, after all is said and done, derivatives are easier to compute. As you will learn (or have learned) in your regular calculus course, there are algorithmic rules for computing derivatives of practically any function that can be written down. Maple knows all these rules (that's how diff and D work), but they are still good to have memorized, because they provide some insight into the behavior of functions that arise in applications.

Another reason derivatives are better is that calculating average growth rates over extremely small intervals is fraught with serious numerical pitfalls. For instance, consider the problem of finding the growth rate of the function f(x) = cos(x) for x = 0.1 (radians). Maple reports that the values of cos(0.1) and cos(0.10001) differ only in the sixth decimal place (they are 0.995004 and 0.995003 respectively). So to get a good growth rate estimate, we have to be able to compute cosines to many significant figures. It's just not worth the effort when we know that the derivative of cos(x) is -sin(x) (verify this using Maple) and computing -sin(0.1) to a few significant figures is a breeze.

Finally (for the moment), derivatives provide a powerful theoretical tool for studying quantities that change. The following section is meant to provide some idea of this.