If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Mean value theorem application

Even if a cop never spots you while you are speeding, he can still infer when you must have been speeding... Created by Sal Khan.

## Want to join the conversation?

• So does that mean that if I were to speed, then stop my car and wait a while. (Theoretically ) would I evade the law?
(74 votes)
• In a way, yes, the computer/camera system uses a distance and a time. By stopping your car somewhere, you are increasing the time between point A and point B. Therefore lowering your average speed, in the eyes in the computer. Although, in reality, highway patrol would probably ticket you for speeding (via radar), but that's a whole other topic.
(71 votes)
• I still don't really see the full purpose of the mean value theorem here because if the speed limit is 55 MILES PER HOUR and the distance between the two points is 80 miles, then wouldn't it be kind of obvious that the person must have exceeded the speed limit if they went from one point to the other in just an hour?
(42 votes)
• Exactly. That is the simplified version of what the mean value theorem is exactly telling you.

Is it obvious? Sure.
But it can be mathematically proved using the mean value theorem, and that's why it is of importance.
(64 votes)
• I think there is a simpler way to tackle this problem.
Can not we say without bringing the notion of mean value theorem in mind that if the average speed is 80 mph, the maximum speed that the driver achieved during the journey is equal to or more than 80 mph. Therefore the maximum allowed speed limit (55 mph) must have been crossed. And this does not require the knowledge of nature of the function (continuity and differentiability).
Is there mean value theorem underlying in inferring the problem this way?
If so then we are using mean value theorem without knowing it !!
(24 votes)
• Yes, you are right in all your affirmations. Knowing that the average speed was 80mph is enough to know that at some point the speed was equal to 80mph, and that is the essence of the mean value theorem, in a very day to day situation.
(20 votes)
• Why doesn't Sal do a video on the proof of Mean Value Theorem?
(4 votes)
• There are probably several reasons for this. First, Sal is great but he isn't perfect! It takes time to create videos. I'm sure there are many ideas he or others at KA would love to implement but don't have the time to implement. Also, relatedly, KA focuses on the concepts usually taught in any given class. In my own calculus class, the proof of the MVT was not taught nor was it in the book (unless it appeared significantly later). If this is common, then the majority of students would not need a video covering it. As time is limited, Sal probably considers what videos to make based on what would be the most useful to the most people.
(33 votes)
• Did we really need the MVT for this, isn't it just common sense? There's no way you can travel 80mph at any given time under 80mph in erm... less than an hour. Ya know?
(7 votes)
• Yes, it's just common sense.
But if you want to learn the Mean Value Theorem why not start with an easy example where you'll also be able to understand the intuition of the problem.
(12 votes)
• why mr. khan took a gap in x-axis?
(4 votes)
• The time axis represent a continuous increase in t values from the moment of observation which you can put at t = 0 or, as has been done in this case, start your observation at a later instant of time such as 1 PM. Since in this example we are interested in the travel time between 1 PM and 2 PM. We are not focusing on where the car was at 8 AM or 11 AM or 12 PM or even PM. The gap here shows those instants of time before the clock struck 1 PM and we started observing the car which was the position S(1).
(7 votes)
• why did Sal make that gap/hole on the x-axis.
(3 votes)
• Sal made the hole on the x-axis in order to show that the graph is not to scale. When looking at the distance between 1st hour and the 2nd hour on the x-axis, it is not the same length as the distance between the 0th hour and the 1st hour. Adding the gap on the x-axis shows us that part of it was omitted when drawing the graph.
(4 votes)
• So if you were given a ticket for this, and you argued that you only went at the average speed for a short period of time, they would be right to argue that you went over the average speed?
(1 vote)
• Correct, they would be right to assume that you went either 80mph for the full hour, or more probably, went both faster and slower than 80 over the course of the trip.
(7 votes)
• What if it isnt continuous?
(3 votes)
• What is the difference between position and distance or displacement?
(1 vote)
• Suppose you're going from home to school.
Your home is the STARTING POSITION and the school the ENDING POSITION. School is 30m away from home (That means the straight line that connects the two positions is 30 meters long).
You take a path and walk a distance of 35m to get to your destination.

Your total displacement is 30 meters and the total distance travelled is 35 m.

Position refers to single points (the location of a body)
Displacement is a vector and measures the change in position.
Distance is the length traveled

What happens if you go back home taking the same path?
Now, the total distance travelled is 70m and the total displacement is 0, because you ended in the same position you started (The difference positions is 0).
(2 votes)

## Video transcript

You may think that the mean value theorem is just this arcane theorem that shows up in calculus classes. But what we will see in this video is that it has actually been used-- at least implicitly used-- to give people speeding tickets. So let's think of an example. So let's say that this is a toll booth, right here. You're on the turnpike, and this is a toll booth at point A. And you get your toll-- you reach it at exactly 1:00 PM, and then the highway's computers and stuff register that. Let's say you have some type of-- one of those devices so that when you pay the toll it just knows who you are and it registers-- it takes your money from an account someplace. So it sees that you got there at exactly 1:00 PM. And then, let's say that you get off of the toll highway, the turnpike. Let's say you get off of it at point B, and you get there exactly 2:00 PM. I'm making these numbers very easy to work with. And let's say that they are 80 miles apart. So this distance right over here is 80 miles. And let's say that the speed limit on this stretch of highway is 55 miles per hour. So the question is, can the authorities prove that you went over the speed limit? Well, let's just graph this. I think you know where this is going. So let's graph it. So let's say this right over here is our position. So I'll call that the s-axis, s for position. And that's going to be in miles. And s is, obviously, s doesn't really stand for position. But p, you know, it kind of looks like rho for density. And d we use for differentials for distance or displacement. So s is what gets used for position very often. So let's say s is our position. And let's see, this is t for time. And let's say this is in hours. And let's see, we care about the interval from time going from time 1 to time 2. I'm not really drawing the axes completely at scale. Would you let me just assume that there's a gap here just because I don't actually want to make you think that I'm drawing it completely at scale. Because I really want to focus on this part of the interval. So this is time equals to 2 hours. And so at time equal 1, you're right over here. And let's say this position is, we'll just call that s of 1. And at time 2, you're at this position right over here. You're right over there. And so your position is s of 2. You're at that coordinate right over there. And that's all we know. That's all we know. Well, we know a few other things. We know what our change in time is, it's 2 minus 1. And we know what our change in position is. We know that our change in position, which is equal to s of 2 minus s of 1, is equal to 80 miles. The change in position is 80 miles. So let me write that, and we'll just for simplicity assume it was a straight highway. So our change in distance is the same as our change in position, same as change in displacement. So this is 80 miles. And then what is our change in time? Over our change in time, well that's going to be 2 minus 1. Which is just going to be 1 hour. Or we could say that the slope of the line that connects these two points-- let me do that in another color-- that's the same color-- the slope of this line right over here is 80 miles per hour. Slope is equal to 80 miles per hour. Or you could say that your average velocity over that hour was 80 miles per hour. And what the authorities could do in a court of law, and I've never heard a mathematical theorem cited like this, but they could. And I remember reading about this about 10 years ago, and it was very controversial. The authorities said look, over this interval, your average velocity was clearly 80 miles per hour. So at some point in that hour-- and they could have cited, they could have said by the mean value theorem-- at some point in that hour, you must have been going at exactly 80 miles, at least, frankly, 80 miles per hour. And it would have been very hard to disprove because your position as a function of time is definitely continuous and differentiable over that interval. It's continuous, you're not just getting teleported from one place to another. That would be a pretty amazing car. And it is also differentiable. You always have a well defined velocity. And so I challenge anyone. Try to connect these two points with a continuous and differentiable curve, where at some point the instantaneous velocity, the slope of the tangent line, is not the same thing as the slope of this line. It's impossible. The mean value theorem tells us it's impossible. So let me just draw. So we could imagine. Say I had to stop to pay, to kind of register where I am on the highway, then I start to accelerate a little bit. So right now, my instantaneous velocity is less than my average velocity. I'm accelerating. The slope of the tangent line. But if I want to get there at that time, and especially because I have to slow down as I approach it, as I approach the tollbooth. The only way I could connect these two things-- well let's see, I'm going to have to-- at some point, at this point, I'm actually going faster than the 80 miles per hour. And the mean value theorem just tells us, that look, that this function is continuous and differentiable over this interval. Continuous over the closed interval. Differentiable over the open interval. That there's at least one point in the open interval, which it calls c, so there's at least one point where your instantaneous rate of change, where the slope of the tangent line, is the same as the slope as the secant line. So that point right over there, that point looks like that right over there. And so if this is time c, that looks like it's like at around 1:15, this-- the mean value theorem says that at some point, there exists some time where s prime of c is equal to this average velocity, is equal to 80 miles per hour. And it doesn't look like that's the only one. It looks like this one over here, this could also be a candidate for c.