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## Using the mean value theorem

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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 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.

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