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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.