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The mean value theorem for integrals is a crucial concept in Calculus, with many real-world applications that many of us use regularly. If you are calculating the average speed or length of something, then you might find the mean value theorem invaluable to your calculations.

Ultimately, the real value of the mean value theorem lies in its ability to prove that something happened without actually seeing it. Whether it’s a speeding vehicle or tracking the flight of a particle in space, the mean value theorem provides answers for the hard-to-track movement of objects.

## What is the Mean Value Theorem?

Based on the first fundamental theorem of calculus, the mean value theorem begins with the average rate of change between two points. Between those two points, it states that there is at least one point between the endpoints whose tangent is parallel to the secant of the endpoints.

A Frenchman named Cauchy proved the modern form of the theorem. One of the most prolific mathematicians of his time, Cauchy proved the mean value theorem as well as many other related theorems, one of which bears his name.

It is also possible for a function to have more than one tangent that is parallel to the secant. The derivative, or slope, of each tangent line, is always parallel to the secant in the mean value theorem.

As an addition to the mean value theorem for integers, there is the mean value theorem for integrals. This theorem allows you to find the average value of the function on at least one point for a continuous function.

Stipulations for this theorem are that it is continuous and differentiable. That means that the line acts as a traditional function, without any odd stops, gaps, drop-offs, or any other non-continuous feature.

It also must be differentiable, which means you can find the slope of a point on the function. For cube roots or the absolute value of x, you cannot find a derivative because they are either undefined or not tangential to the average rate of change.

Floor and ceiling functions also do not have derivatives because they are not continuous functions. Thus the mean value theorem of integers does not apply to them.

## Why Is the Mean Value Theorem for Integers Important?

Like many other theorems and proofs in calculus, the mean value theorem’s value depends on its use in certain situations. Since this theorem is a regular, continuous function, then it can theoretically be of use in a variety of situations. Any instance of a moving object would technically be a constant function situation.

Real-world applications for the mean value theorem are endless, and you’ve probably encountered them either directly or indirectly at some point in your life.

One of the classic examples is that of a couple of police officers tracking your vehicle’s movement at two different points. You are then issued a ticket based on the amount of distance you covered versus the time it took you to complete that distance.

You were not speeding at either point at which the officer clocked your speed. But, they can still use the mean value theorem to prove you did speed at least once between the two officers.

More specifically, consider modern-day toll roads. These roads have cameras that track your license plate, instantaneously clocking your time spent on the road and where and when you exited and entered. Law enforcement could quickly begin to crack down on speeding drivers on these roads, by merely finding the average rate of change between the two points. Drivers could then blame the mean value theorem of integers as the reason for their ticket.

Further use occurs in sports, such as racing. When investigating the speeds of various racing objects, such as horses or race cars, technicians and trainers need to know the performance of horses or race cars at specific points during the race.

Using data obtained throughout the race, individuals can determine how their horse or car was performing at certain times. For horses, this can mean altering training patterns or other variables to improve performance related to results. Race car drivers can use the data to tune equipment in various ways to better utilize the car’s speed.

The critical part of the theorem is that it can prove specific numbers. It can determine the velocity of a speeding car without direct visual evidence, or the growth, length, and myriad other instances where an object or thing changes over time.

## How to Use the Mean Value Theorem

In the real-world, a continuous function could be the rate of growth of bacteria in a culture, where the number of bacteria is a function of time. You could divide the difference in the number of bacteria by time to find out how fast they multiplied.

Applying the mean value theorem to the above situation would allow you to find the exact time where the bacteria multiplied at the same rate as the average speed. This might be useful to researchers in various ways, to determine the characteristics of certain bacteria.

Another more practical situation would be to determine the average speed of a thrown baseball. The distance of the ball thrown is a function of time. Dividing the difference in the length by the time it took for the ball to get from point a to point b would tell you how fast the ball goes.

When the mean value theorem is applied, a coach could analyze at which point the ball achieved the average speed. If the speed was faster before or after the tangential point, then the coach could alter the mechanics or delivery of the player’s throw. This would make for more optimal speed with the throw reaches the batter.

Finally, let’s find the average speed of the vehicle and then at which point during the drive, the car reached a speed equal to the average rate.

- First, find the total distance traveled by the vehicle. To do this, check the odometer before and after driving. Calculate the difference between the two readings.
- Determine the amount of time spent driving the car between those two points.
- Divide the distance by the time. Let’s say it’s 40 mph.
- Now you need to find the point – or points – during which the car was traveling at 40 mph.
- Graph the function.
- The point at which the vehicle traveled 40 mph will show as the highest or lowest point on the slope connecting the beginning of the drive and the end.
- Using the graph, you can then find the exact time at which the car was traveling at 40 mph.

When using the mean value theorem in practical applications like vehicle speed, it is essential to note that the average rate of change is just that – an average. If your vehicle speed is 50 mph, then at some point during your drive you drove over and under 50 mph. Of course, you would hit that speed at least twice at a minimum.

Another exciting application of the mean value theorem is its use in determining the area. When the point at which the tangent line occurs is understood, draw a line from the new point parallel to the x-axis. This line is the top of your rectangle. The bottom is the x-axis. The left side is the y-axis, and the right is the endpoint of your continuous function.

Once this is complete, the area of your rectangle will be the same as the area beneath the curve of your function. One practical application of this instance is determining the exact height of a liquid in a container. If the liquid is suspended or not at rest, then calculating the mean value theorem of integers for the endpoints of the liquid will help you to determine the resting volume.

Determining amounts of liquid or the properties of a substance are just a few of the many applications of the mean value theorem. All fields of science use this theorem, and merely finding the volume of a liquid at rest is just scratching the surface. As sport becomes more science-based, the value of this theorem will only continue to increase.

## Conclusion

While a fundamental calculus theorem may not change your life, it can make your life a tiny bit more manageable. Understanding the movement of an object and the properties within that movement can help you make a variety of educated conclusions.

When working in scientific fields such as physics or biology, the use of the theorem can aid in the research of particles or microscopic organisms. In sports, you can use the theorem to develop a better understanding of fast-moving objects. On the highway, the police can issue more speeding tickets. The mean value theorem of calculus is an invaluable tool for all types of people.