Download Andrew Hartzell and Kent Skiles

Andrew Hartzell and Kent Skiles
Dr. Weijiu Liu
MATH 1591
December 1, 2005
Law Enforcement with the use of the Mean Value Theorem
INTRODUCTION
According to the National Highway Traffic Safety
Administration, motor vehicle crashes cause over 42,000
deaths per year.
Although these accidents occur for
various reasons, over 13,000, or 33 percent, are cause by
excessive speeding.
While only 5 percent of motorists will
admit to regular speeding, the statistics say otherwise.
PROBLEM
If you have ever been driving down a roadway and come
upon a police car parked on the shoulder, you know that
everyone begins slowing down in fear of being pulled over;
yet, when they think they are out of range, they return to
speeding.
The problem is that motorists are in too big of a
hurry and think that speeding is ok as long as you are not
caught.
This needs to be stopped.
We could accomplish
this by stationing law enforcement officers at two mile
intervals along all major roadways or we could use the Mean
Value Theorem.
Stationing law enforcement officials at two
mile intervals would be impractical; therefore, if speeding
is to be thwarted, then we must use the Mean Value Theorem.
TECHNIQUES
The Mean Value Theorem states: If a function ƒ is
continuous on the closed interval [a,b] and differentiable
on the open interval (a,b), then there exists a number c in
(a,b), such that ƒ’(c) = (ƒ(b) – ƒ(a))/(b – a).
In terms of rates of change, the Mean Value Theorem
implies that there must be a point in the open interval
(a,b) at which the instantaneous rate of change is equal to
the average rate of change over the interval [a,b].
This means that we are able to place two law
enforcement officers, equipped with radar, a certain
distance apart and taking into account the distance(in
miles) to be traveled between the two officials and the
time(in hours) it takes to travel this distance, we can
determine whether any particular car was speeding at any
point between the two officials.
Figure 1.
A diagram is shown in
Figure 1 – Officer Diagram
For example, let’s assume that the two officers are
stationed 10 miles apart and the speed limit is 40 miles
per hour.
A motorist drives by law official “a” and is
clocked at 40 miles per hour.
Eight minutes later, the
motorist passes law official “b” and is clocked at 40 miles
per hour again.
At this point, the officers will probably
assume that the motorist has been going the speed limit for
the whole ten miles, but in fact they have not.
Using the Mean Value Theorem, we can determine the
average velocity the motorist traveled between officer “a”
and officer “b”.
In this case, distance, d, is a function of time.
Let’s denote the initial time by t(0); therefore, d(0) is
then equal the distance traveled at time 0, or 0.
Let’s
also denote the secondary time by t(8), since it took the
motorist eight minutes to travel the distance.
Since t is
measure in hours, eight minutes is converted to 2/15 of an
hour.
Now that we have all the numbers ready, all that is
left is to plug them in where necessary and simplify.
When the numbers are all inserted, the equation reads:
(d(2/15) – d(0))/(2/15 – 0).
After simplification, the
equation then reads d(2/15)/(2/15).
Since the motorist
traveled 10 miles in eight minutes, the equation is
simplified to: 10/2/15 which simplifies to 75.
This means
that at some point between law officer “a” and law officer
“b”, the motorist was traveling 75 miles per hour.
ANALYZING AND DISCUSSING
After obtaining these results, they are appalling.
This means that a motorist could speed at any time except
when in range of law enforcement radars and may never be
caught.
This is alarming considering all of the traffic
accidents cause by speeding motorists.
It also means that,
if they chose to, law enforcement officials could implement
this theorem into their jobs.
They could develop it and
use it prevent motorists from speeding even when they are
not directly being clocked by radar.
If this theorem is
implemented into law enforcement’s array of gadgets, then
it would significantly reduce the amount of traffic
accidents by up to 33 percent.
Works Cited
National Highway Traffic Safety Administration.Research –
NHTSA.30 Nov. 2005
<http://www.nhtsa.dot.gov/STSI/State_Info.cfm?Year=2004&Stat
e=AR&Accessible=0>