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# Mean value theorem application

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.B (LO)
,
FUN‑1.B.1 (EK)

## 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 now your honor you're on the Turnpike and this is a toll booth at Point a and you get your tote you get a you reach it at exactly 1 p.m. and the highways computers and stuff register that let's say you have some type of a one of those devices so that when you pay the toll it just knows who you are and it read it takes your money from an account someplace so it sees that you got there at exactly you got there at exactly 1 p.m. and then let's say that there's and let's say you get off of the off of the toll toll highway the Turnpike let's say you get off of it at point B and you get there at exactly 2 p.m. 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 80 miles and let's say that the speed limit on this stretch of highway is 55 miles per hour speed limit speed limit is 55 miles per hour so the question is 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 its you know 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 use for gets used for position very often so let's say s is our position and let's see this is T for time 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 one time one to time two time to I'm not really drawing the axes completely at scale see let me let me just assume that there's a gap here just because I don't want to I don't want to actually want to don't 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 two hours and so at time equal one you're at where you're right over here the let's say this this position is we'll just call that s of one and a time to a time to your this position right over here you're right over there and so though your position is s of s of 2 you're 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 our change in position which is equal to s of 2 s of 2 minus s of 1 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 as 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 80 miles and then what is our change in time over our change in time well that's going to be 2 minus 1 2 minus 1 which is just going to be one hour over one hour or we could say that the slope of the line that connects these two points let me do that in another color the slope of that's the same color the slope of this line right over here the slope right over here is 80 miles per hour 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 big very controversial the authorities already 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 mile at least frankly 80 miles per hour and it would have been very hard to disprove because your your position as a function of time is definitely continuous and differentiable over that interval is continuous you're not 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 you know I challenge anyone try to connect these two points with a continuous and differentiable curve where at some point the instantaneous velocity the the slope of the tangent line is not the same thing as the slope of this line it's impossible that's the mean value theorem tells us it's impossible so let me just draw so we can imagine let's say you know I I had to stop to pay to get to kind of register where I am on the on the highway then I start to accelerate a little bit so right now my 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 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 this same as the slope of the secant line so that point right over there that point looks like that right over there and so if this is time C and that looks like it's like at around 115 this the mean value theorem says that at some point there exist some time where your where s prime of C is equal to this average velocity is equal to 80 miles per hour and doesn't look like that's the only one it looks like this one over here this could also be this could also be a candidate for C