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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>