Average rates of change

A simple illustrative example of rates of change is the speed of a moving object. An object moving at aconstant speed travels a distance that is proportional to the time. For example, a car moving at 50 kilometres per hour (km/hr) travels 50 km in 1 hr, 100 km in 2 hr, 150 km in 3 hr, and so on. Agraph of the distance traveled against the time elapsed looks like a straightline whose slope, or gradient, yields the speed (seefigure).

Constant speeds pose no particular problems—in the example above, any time interval yields the same speed—but variable speeds are less straightforward. Nevertheless, a similar approach can be used to calculate the average speed of an object traveling at varying speeds: simply divide the total distance traveled by the time taken totraverse it. Thus, a car that takes 2 hr to travel 100 km moves with an average speed of 50 km/hr. However, it may not travel at the same speed for the entire period. It may slow down, stop, or even go backward for parts of the time, provided that during other parts it speeds up enough to cover the total distance of 100 km. Thus, average speeds—certainly if the average is taken over long intervals of time—do not tell us the actual speed at any given moment.

Instantaneous rates of change

In fact, it is not so easy to make sense of the concept of “speed at a given moment.” How long is a moment?Zeno of Elea, a Greek philosopher who flourished about 450bce, pointed out in one of his celebratedparadoxes that a moving arrow, at any instant of time, is fixed. During zero time it must travel zero distance. Another way to say this is that the instantaneous speed of a moving object cannot be calculated by dividing the distance that it travels in zero time by the time that it takes to travel that distance. This calculation leads to a fraction,0/0, that does not possess any well-defined meaning. Normally, a fraction indicates a specific quotient. For example,6/3 means 2, the number that, when multiplied by 3, yields 6. Similarly,0/0 should mean the number that, when multiplied by 0,yields 0. But any number multiplied by 0 yields 0. In principle, then,0/0 can take any value whatsoever, and in practice it is best considered meaningless.

Despite these arguments, there is a strong feeling that a moving object does move at a well-defined speed at each instant. Passengers know when a car is traveling faster or slower. So the meaninglessness of0/0 is by no means the end of the story. Various mathematicians—both before and after Newton and Leibniz—argued that good approximations to the instantaneous speed can be obtained by finding the average speed over short intervals of time. If a car travels 5 metres in one second, then its average speed is 18 km/hr, and, unless the speed is varying wildly, its instantaneous speed must be close to 18 km/hr. A shorter time period can be used to refine the estimate further.

If a mathematical formula is available for the total distance traveled in a given time, then this idea can be turned into a formal calculation. For example, suppose that after timet seconds an object travels a distancet² metres. (Similar formulas occur for bodies falling freely under gravity, so this is a reasonable choice.) To determine the object’s instantaneous speed after precisely one second, its average speed over successively shorter time intervals will be calculated.

To start the calculation, observe that between timet = 1 andt = 1.1 the distance traveled is 1.1² − 1 = 0.21. The average speed over that interval is therefore 0.21/0.1 = 2.1 metres per second. For a finer approximation, the distance traveled between timest = 1 andt = 1.01 is 1.01² − 1 = 0.0201, and the average speed is 0.0201/0.01 = 2.01 metres per second.

The table displays successively finer approximations to the average speed after one second. It is clear that the smaller the interval of time, the closer the average speed is to 2 metres per second. The structure of the entire table points very compellingly to an exactvalue for the instantaneous speed—namely, 2 metres per second. Unfortunately, 2 cannot be found anywhere in the table. However far it is extended, every entry in the table looks like 2.000…0001, with perhaps a huge number of zeros, but always with a 1 on the end. Neither is there the option of choosing a time interval of 0, because then the distance traveled is also 0, which leads back to the meaningless fraction0/0.

Formal definition of thederivative

More generally, suppose an arbitrary time intervalh starts from the timet = 1. Then the distance traveled is (1 +h)² −1², which simplifies to give 2h +h². The time taken ish. Therefore, the average speed over that time interval is (2h +h²)/h, which equals 2 +h, providedh ≠ 0. Obviously, ash approaches zero, this average speed approaches 2. Therefore, the definition of instantaneous speed is satisfied by the value 2 and only that value. What has not beendone here—indeed, what the whole procedure deliberately avoids—is toseth equal to 0. As BishopGeorge Berkeley pointed out in the 18th century, to replace (2h +h²)/h by 2 +h, one must assumeh is not zero, and that is what the rigorous definition of a limit achieves.

Even more generally, suppose the calculation starts from an arbitrary timet instead of a fixedt = 1. Then the distance traveled is (t +h)² −t², which simplifies to 2th +h². The time taken is againh. Therefore, the average speed over that time interval is (2th +h²)/h, or 2t +h. Obviously, ash approaches zero, this average speed approaches the limit 2t.

This procedure is so important that it is given a special name: the derivative oft² is 2t, and this result is obtained bydifferentiatingt² with respect tot.

One can now go even further and replacet² by any otherfunctionf of time. The distance traveled between timest andt +h isf(t +h) −f(t). The time taken ish. So the average speed is(f(t +h) −f(t))/h. (3) If (3) tends to a limit ash tends to zero, then that limit is defined as the derivative off(t), writtenf′(t). Another common notation for the derivative isdf/dt, symbolizing small change inf divided by small change int. A function is differentiable att if its derivative exists for that specific value oft. It is differentiable if the derivative exists for allt for whichf(t) is defined. A differentiable function must be continuous, but the converse is false. (Indeed, in 1872 Weierstrass produced the first example of a continuous function that cannot bedifferentiated at any point—a function now known as a nowhere differentiable function.) Table 2 lists the derivatives of a small number of elementary functions. It also lists theirintegrals, described below.