Greenwood press

RATES
99
The slope of the dotted line represents the
average speed of the car from 12:34 
PM
to
1:57
PM
, which is 79 miles per hour. The
automobile has this rate at three other
locations in this interval based on the
equivalent slopes of the small thick lines
(at the points denoted speeding).

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Besides finding the average rate as a means to describe varying speeds, it is
possible to determine the instantaneous rate of an object using differential calcu-
lus. If a total amount, such as distance or production levels, can be described as
a function, then the rate at any moment can be determined by finding the deriv-
ative of that function. Instead of finding the slope at the endpoints of an interval,
a derivative is the slope of a line tangent to a curve at a particular point.
The slope of the tangent line will describe the speed of the car at a specific
moment in time. For example, in the above figure, a tangent line with a slope of
70 miles per hour is drawn on the curve at 1:34 
PM
, illustrating the speed of the
car at that moment. 
In addition to automobile travel, the motion of falling objects shows variable
rates. Since the earth pulls objects at a rate of 9.8 meters per second squared,
falling objects are constantly accelerating. The position of a penny dropped off of
a 400-meter-tall skyscraper can be represented by the function 
h = –4.9t
2
+ 400,
where
h is the height of the penny above the ground in meters, and t is the time
in seconds the penny is airborne. This function is a parabola. It will not have a
constant slope, which means that the penny will not fall at the same rate towards
the ground. However, the slope of the line tangent to the curve at any time, or the
instantaneous rate, can be predicted by the derivative of this function, which is
h
′
= –9.8t. This means that the penny will be falling at a rate of 9.8 meters per
second after one second, 19.6 meters per second after two seconds, and so on.
According to the position function, 
h = –4.9t
2
+ 400, the penny will reach the
ground at approximately 
t = 9 seconds, where h is equal to 0. According to the
derivative of the position function, the velocity of the penny by the time it hit the
ground would be 
h
′
= –9.8(9) = –88.2 meters per second, fast enough to fall
straight through a person’s body. Hence, you are not likely to be permitted to drop
objects from tall buildings!
Human workforce productivity can have varying rates. In a factory, the work-
ers may be less productive in the early morning because they are tired, and then
reach an optimal work rate later in the morning when they are more awake. Later
in the afternoon, they may become less productive again due to fatigue or bore-
dom. Understanding the varying working rates of employees may help manage-
ment determine an optimal time to take a break or to change work shifts. Know-
ing the change in work rates would provide information to make smart decisions