Lagrange and Speed: The Invention of the Highway Speed Tutor

Mathematics in Practice: Beyond the Numbers

Lagrange's Theorem is at the heart of how the highway Speed Tutor works: it mathematically guarantees that if average speed exceeds the limit, at least at one instant the instantaneous speed reached that value. This is a perfect example of how mathematics has practical applications in everyday life.

How many times, leafing through a math textbook, have you asked yourself: "What is the point of what I am studying?" In school, students often find themselves immersed in formulas, concepts, and theorems that are difficult to understand or connect to reality. Many times we do not realise just how useful and practical mathematics can be in everyday life.

Today, we will try to answer the question: "What are derivatives good for in real life?" Of course, derivatives are an advanced topic, typically covered towards the end of high school, but their implications are infinite and span every field of science, from biology to physics. And as we shall see, it is very likely that you have already encountered them, even if unconsciously.

The Fascination of Derivatives: The Backbone of Calculus

Derivatives are one of the fundamental concepts of differential calculus, a branch of mathematical analysis that studies the infinitesimal variations of functions through the use of limits, derivatives, and integrals. Differential calculus is fundamental for studying a function because it allows us to analyse its behaviour within its domain.

The central concept of differential calculus is, precisely, the derivative: it represents the instantaneous rate of change of the function with respect to its independent variable. In other words, it tells us how the function changes as the independent variable varies at a given point. The derivative of a function can also be interpreted geometrically as the slope of the tangent line to the graph of the function at a specific point.

All of this highlights how derivatives are a fundamental tool for understanding the graph of a function, identifying its growth and decline and, consequently, pinpointing its maximum and minimum points.

Since they were introduced, mathematicians of the era spent years in the in-depth study of this branch of mathematics, arriving at the formulation of important theorems about derivatives. Among these is certainly Lagrange's Theorem, named after its discoverer, Joseph Louis Lagrange. Lagrange, an Italian-French mathematician and astronomer, is considered one of the greatest mathematicians of the 18th century. He studied in Turin before being called to the court of Louis XVI and Marie Antoinette in Paris, where he continued his academic career.

Lagrange's Theorem states that if we consider a function that is continuous and differentiable (possibly excluding the endpoints) on a closed and bounded interval [a,b], there exists at least one point c interior to that interval where the following equation holds:

f'(c) = [f(b) - f(a)] / (b - a)

The left side of Lagrange's Theorem equation represents the slope m of the tangent line to the graph of the function at point c. The right side of the equation represents the difference quotient of the function over the considered interval, giving the slope of the line connecting the graph points corresponding to the endpoints of that interval.

From a geometric perspective, this means there exists at least one point at which the tangent line to the graph of the function is parallel to the line passing through the endpoints of the interval under consideration. These two lines therefore have the same slope.

This Lagrange's Theorem, of fundamental importance in differential calculus, proved essential for studying the concepts of instantaneous velocity and instantaneous acceleration in physics. Furthermore, it has contributed significantly to the modelling and analysis of change in many natural and scientific phenomena.

The Safety Tutor: The Evolution of Speed Control

Lagrange's Theorem also finds concrete applications in daily life, for example in speed control while driving a car. If you have never heard of the Safety Tutor, you will probably experience it as soon as you get your driver's license.

The Safety Tutor is a speed control system installed on certain stretches of Italian highways that detects whether a vehicle has violated traffic laws by exceeding the legal speed limit. The system works through the installation of two detection points, called portals, that mark the beginning and end of the monitored stretch. These portals are equipped with infrared sensors that detect the passage of vehicles. Each time a vehicle passes through both portals, the system records the exact date and time of transit. Considering the distance between the two detection points and the time elapsed, the Tutor calculates the vehicle's average speed over that highway stretch. If this average speed exceeds the permitted limit, the system generates a violation notice.

But how can we be sure that the vehicle actually exceeded the limit, given that the only data at our disposal is an average speed? This is where Lagrange's Theorem comes in!

If we interpret our function f(x) as the position-time law of a body in motion, where the variable y represents distance and x the time taken to cover that distance, then the left side of the equation would represent the body's instantaneous velocity at time x, while the difference quotient would express the average velocity over the interval [a,b]. This interpretation of the theorem guarantees the existence of at least one instant within the interval where the vehicle's instantaneous velocity equals the average velocity maintained over the entire highway stretch covered by the Tutor. For this reason, Lagrange's Theorem is also known as the Mean Value Theorem.

Therefore, if the calculated average speed exceeds the permitted limit, there will certainly have been an instant when the vehicle reached that speed!

It is important to emphasise how the hypotheses of a theorem are fundamental for its application. The assumption that the function is continuous and differentiable over the considered highway stretch, for example, assures us that a vehicle proceeds linearly in its motion along the highway stretch [a,b], without being able to turn back at any point, which would inevitably increase the total travel time and decrease the average speed detected by the Tutor.

Changes in Speed Control: An Analysis of the Last 20 Years

The Safety Tutor system has significantly contributed to reducing highway accidents and consequently increasing road safety. Drivers now have to maintain a correct speed for several kilometres, not just near speed cameras.

The highway Tutor calculates the average speed of vehicles over the monitored stretch, so even if a vehicle occasionally exceeds the speed limit, it is not necessarily fined unless the average speed over the stretch exceeds the limit. This may seem unfair, especially when someone receives a fine for a small speed excess, while another driver who stops at a rest area manages to keep the average speed below the limit despite higher peak speeds.

However, the Tutor's goal is not to "catch" anyone who exceeds the speed limit, but rather to ensure safer driving and greater compliance with speed limits for longer periods of time compared to the instant monitoring offered by traditional speed cameras. Thanks to this system, a decrease in highway accidents and an increase in road safety have been recorded, since drivers now must maintain proper speed for several kilometres. To deepen your understanding of derivatives and theorems like Lagrange's, math tutoring offers a structured path.

Indeed, data shows that the Tutor has effectively led to a progressive decrease in both peak speed (by 25%) and average speed (by 15%), reducing mortality by 51% in the first year of installation alone and contributing to a 75% decrease in the mortality rate compared to 1999.

It is important to note that, due to legal disputes between Autostrade Per L'Italia and the patent holder of the Tutor system, this system was deactivated for about two years around 2019. However, this did not lead to an increase in violations or mortality during the deactivation period. On the contrary, one can infer that the Tutor system changed drivers' behaviour, discouraging the tendency to "speed" on highways. Road safety on the highway network has improved over the years, regardless of whether the Tutor was active or not, demonstrating the effective long-term benefit it brought to the community and the importance of applying mathematical concepts to real life.

FAQ

What is Lagrange's Theorem?

Lagrange's Theorem (or Mean Value Theorem) states that for a continuous and differentiable function on an interval, there exists at least one point where the derivative (instantaneous velocity) equals the difference quotient (average velocity). It is fundamental in differential calculus.

How does the highway Speed Tutor work?

The Tutor measures a vehicle's average speed between two portals. If it exceeds the limit, Lagrange's Theorem guarantees that at least at one instant the vehicle reached that speed. The system reduced mortality by 51% in its first year.

Do derivatives have practical applications in real life?

Yes, derivatives have countless applications: from physics (velocity and acceleration) to economics (rates of change), from medicine (diffusion models) to engineering. The highway Speed Tutor is just one of the most tangible examples.

Has the Tutor system been effective in reducing accidents?

Data shows that the Tutor reduced peak speed by 25% and average speed by 15%, with a 75% decrease in mortality compared to 1999. Even after temporary deactivation, driver behaviour remained more cautious.