Slope is equal to 80 miles per hour. Or you could say that your average, your average velocity over that hour was 80 miles per hour. And so 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 literally 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'd 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 can imagine, let's say I had to stop to pay, to get 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 wanna get there at that time, and especially because I have to slow down as I approach it, as I approach the toll booth, the only way I could connect these two things, well, let's see, I'm gonna have to, at some point, I'm actually, at this point, I'm actually going faster than the 80 miles per hour. And the mean value theorem just tells us that, look, if 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 of the secant line.