3.7 Rates, velocities 177disastrous loss of meaning, be abbreviated to the forms
(3.72) Average rate = difference quotient =Dy=f(x + Ax) - f(x)
Ox Ax(3.73) Rate = derivative = dx = f'(x).
Of course, we are never required to prove definitions, but these are
important and we must have or acquire an understanding of them and a
feeling that they do (or do not) use words of the English language in a
reasonably appropriate way. Shifting the letters from y and x to x and t,
we see that the definition involving (3.72) shows that if x is a number of
miles and t is a number of hours, then the average rate of change of x
with respect to t is a number Ox of miles divided by a number At of hours
and hence is a number of miles per hour. Some applications of this are
very simple and agree with all primitive ideas about rates. When we are
thinking about a particular automobile journey in which the automobile
moves steadily in one direction along a straight road, we can let x and
f(t) denote the distance (number of miles) traveled during the first t hours
of the trip. We are all accustomed to calculating the average rate over a
given time interval and to calling this average rate an average speed.
Suppose now that an untutored (but not necessarily stupid) individual is
asked how he might, without looking at a perfect speedometer, determine
a number Q which could reasonably be called the speed at a particular
time t. His reply might be lengthy and partially intelligible. He
should, sooner or later, arrive at the idea that the average speed over a
long trip is likely to be a very bad approximation to Q, but that the aver-
age speed over the time interval from t to t + At (or from t + At to t in
case At < 0) should be near Q whenever At is near 0 but At 0 0. We
have learned how to make this idea precise. It is done in the definitions
we are discussing. Similar stories involving other rates (degrees centi-
grade per centimeter, coulombs per second, and dollars per year, for
examples) show that the definitions are sensible and should have swarms
of important applications.
Our simple discussion of the journey of an automobile moving steadily
in one direction along a straight road involved the word "speed" but
carefully avoided the words "velocity" and "acceleration." To appre-
ciate what is coming, we should know some history. The words "speed,"
"velocity," and "acceleration" are very old. A long time ago, say before
the year 1900, they were all numbers (scalars); velocity and acceleration
could be negative but speed never could be. Nowadays, in all enlight-
ened communities, velocities and accelerations are always vectors and
we must learn about them. To get started, we consider the path traced
by a bumblebee (or molecule or rocket or satellite or what not) as it