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## Mean value theorem

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# Mean value theorem

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

## Video transcript

Let's see if we
can give ourselves an intuitive understanding
of the mean value theorem. And as we'll see, once you parse
some of the mathematical lingo and notation, it's actually
a quite intuitive theorem. And so let's just think
about some function, f. So let's say I have
some function f. And we know a few things
about this function. We know that it is
continuous over the closed interval between x equals
a and x is equal to b. And so when we put
these brackets here, that just means closed interval. So when I put a
bracket here, that means we're including
the point a. And if I put the bracket on
the right hand side instead of a parentheses,
that means that we are including the point b. And continuous
just means we don't have any gaps or jumps in
the function over this closed interval. Now, let's also assume that
it's differentiable over the open interval
between a and b. So now we're saying,
well, it's OK if it's not
differentiable right at a, or if it's not
differentiable right at b. And differentiable
just means that there's a defined derivative,
that you can actually take the derivative
at those points. So it's differentiable over the
open interval between a and b. So those are the
constraints we're going to put on ourselves
for the mean value theorem. And so let's just try
to visualize this thing. So this is my function,
that's the y-axis. And then this right
over here is the x-axis. And I'm going to--
let's see, x-axis, and let me draw my interval. So that's a, and then
this is b right over here. And so let's say our function
looks something like this. Draw an arbitrary
function right over here, let's say my function
looks something like that. So at this point right over
here, the x value is a, and the y value is f(a). At this point right
over here, the x value is b, and the y value,
of course, is f(b). So all the mean
value theorem tells us is if we take the
average rate of change over the interval,
that at some point the instantaneous rate
of change, at least at some point in
this open interval, the instantaneous
change is going to be the same as
the average change. Now what does that
mean, visually? So let's calculate
the average change. The average change between
point a and point b, well, that's going to be the
slope of the secant line. So that's-- so this
is the secant line. So think about its slope. All the mean value
theorem tells us is that at some point
in this interval, the instant slope
of the tangent line is going to be the same as
the slope of the secant line. And we can see, just visually,
it looks like right over here, the slope of the tangent line
is it looks like the same as the slope of the secant line. It also looks like the
case right over here. The slope of the tangent
line is equal to the slope of the secant line. And it makes intuitive sense. At some point, your
instantaneous slope is going to be the same
as the average slope. Now how would we write
that mathematically? Well, let's calculate
the average slope over this interval. Well, the average slope
over this interval, or the average change, the
slope of the secant line, is going to be our change
in y-- our change in y right over here--
over our change in x. Well, what is our change in y? Our change in y is
f(b) minus f(a), and that's going to be
over our change in x. Over b minus b minus a. I'll do that in that red color. So let's just remind ourselves
what's going on here. So this right over here,
this is the graph of y is equal to f(x). We're saying that the
slope of the secant line, or our average rate of change
over the interval from a to b, is our change in y-- that the
Greek letter delta is just shorthand for change in
y-- over our change in x. Which, of course,
is equal to this. And the mean value
theorem tells us that there exists-- so
if we know these two things about the
function, then there exists some x value
in between a and b. So in the open interval between
a and b, there exists some c. There exists some
c, and we could say it's a member of the open
interval between a and b. Or we could say some c
such that a is less than c, which is less than b. So some c in this interval. So some c in between it
where the instantaneous rate of change at that
x value is the same as the average rate of change. So there exists some c
in this open interval where the average
rate of change is equal to the instantaneous
rate of change at that point. That's all it's saying. And as we saw this diagram right
over here, this could be our c. Or this could be our c as well. So nothing really--
it looks, you would say f is continuous over
a, b, differentiable over-- f is continuous over the closed
interval, differentiable over the open interval, and
you see all this notation. You're like, what
is that telling us? All it's saying is at some
point in the interval, the instantaneous
rate of change is going to be the same as
the average rate of change over the whole interval. In the next video,
we'll try to give you a kind of a real life example
about when that make sense.