0.1 Rates of Change
By definition, a function is simply a relation between the points or members of two sets. In this
setting, the output of a function at a single point may be completely unrelated to the output of the
function at other points. However, if a function represents a physical quantity, then it’s reasonable
to believe the outputs of the function for different points in the domain are related (at least in the
realm of classical mechanics). One of the basic goals of calculus is finding and understanding these
relationships, in order to understand the evolution of a system.
Let us suppose we know the position of an object as a function of time, at all times. We know
that there must be at least some relationship between the positions of the object for closely related
times. In other words, the position of the object at a given time is definitely related to the position
of the object 1 second later. We may not be able to find one position given the other, but we know
the distance traveled can only be so great, as macroscopic objects do not teleport. A very natural
and interesting question that arises is as follows: given the position of an ob ject at all times, can
we find its velocity? Roughly speaking, an object’s velocity (or speed) is a measure of how fast it
is moving, or changing positions. Noting that velocity (or speed) is simply the rate of change of
position with respect to time, we certainly should be able to find it.
Definition 0.1.1 (Average Rate of Change).The average rate of change of a function fover an
interval [x0,x1] is the ratio of its change in output (f(x1)−f(x0)) over change in input (x1−x0=h).
Mathematically,
Average Rate of Change = f(x1)−f(x0)
x1−x0
=f(x0+h)−f(x0)
h
The above definition holds in a more general context, but for the moment we’re interested in a
changing position.
Definition 0.1.2 (Velocity and Speed).The velocity of an object is the rate of change of its position
with respect to time. The speed of an object is the magnitude of its velocity.
According to the above definition, velocity describes how fast an object is moving, and in which
direction, whereas speed simply denotes how fast an object is moving. Since velocity is the rate of
change of position with respect to time, average velocity is the average rate of change of position
with respect to time.
How should we interpret average velocity? Let x(t) denote the position of an ob ject D, and t
represent time. The average velocity of an object over an interval [a,b] in time is given by
vab =x(b)−x(a)
b−a=x(a+h)−x(a)
h,
where h=b−a. If another object Eis moving with a constant velocity vab,then over the interval
of time [a, b], Ewill travel a distance of
vab ·h=x(a+h)−x(a)
h
·h=x(a+h)−x(a) = x(b)−x(a),
which is exactly the distance that Dtravels over the interval [a, b]. In other words, the average
velocity of an object over a given interval of time is the unique number such that for any other
object, if the second object travels at a constant velocity of the average velocity for that interval
of time, it will travel the same distance as the original object. The key thing to note here is that
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