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[music] How does information about the
derivative connect back to information

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about the original function.
There is a technical tool that connects

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this differential information just to the
value of the function and that tool is the

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mean value theorum.
Well here's a statement of the mean value

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theorem.
Suppose that f is continuous on the closed

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interval between a and b, and it's
differentiable on the open interval a b.

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So that's the setup.
F is a pretty nice function.

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Then there exist some point in between a
and b, so a point in the interval a b.

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So that the derivative of the function at
the point c, is equal to this.

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Which is really calculating the slope
between the point a f of a.

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And b f of b.
That statement seems really complicated.

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Let's try to boil that down to a statement
we can really believe in.

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Well, here's one interpretation.
If that function is giving you position

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and the input to that function is time, so
that the derivative of that function is

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your velocity, then that formula is saying
that your average velocity is achieved, at

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some point instantaneously.
Right?

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That fraction, that difference ratio, is
calculating the average velocity.

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And instantaneous velocity is the
derivative.

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And the mean value theorem is saying that
those are equal at some point in between.

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Let's go back now to the official
statement of the mean value theorem.

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So here again, is the statement of the
mean value theorem, right, a nice enough

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function that there's some point in the
middle so that the derivative, your

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instantaneous velocity is equal to this.
Which is, you know, if you like

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calculating your average velocity.
We can take a look at this from a graph as

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well and might help even more.
All right here's the graph of just some

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function that I've made up.
And I've picked points a and b and it's

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certainly a nice enough function.
I mean it's a continuous function a

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differentiable function, and I've picked
points a and b.

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And here's f of a and f of b and then I've
drawn this red line that connects the

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point a f of a and b f of b.
And the slope of that line is exactly what

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this quantity calculates.
All right, so I'll label this as the slope

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of that secant line.
Now, if asserting the existence of some

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point in between, where the derivative has
the same value as the slope of this secant

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line you know, it just sort of looks like,
from this picture, maybe that point is

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here.
And, yeah, it looks like if I draw a

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tangent line to the curve at that point,
the slope of that tangent line, which is

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the derivative at the point c, is the same
as the slope of that red secant line.

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Right?
So, in the statement of the mean value

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theorem, this f prime of c, that's the
slope of the tangent line.

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To the graph of the function at the point
c, and this statement is asserting that

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the slope of tangent line at some point in
between is equal to the slope of the

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secent line between a f of a, and b f of
b.

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The mean-value theorem is often told as a
story about somebody driving a car.

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Well here's the story.
At noon, you're in some city A, and at 1

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p.m.
You're driving your car and you've arrived

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in a city B, which is 100 miles away from
city A.

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You're driving your car and you've arrived
in a city B, which is 100 miles away from

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city A.
Now what does the mean-value theorem say?

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Well, the Mean Value Theorem tells you
that at some point in between, the

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derivative is equal to the slope of the
secant line.

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Which in this story means, that at some
point during your journey, your

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speedometer.
Said 100 miles per hour.

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Your speedometer's reporting your
instantaneous speed, right?

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That's this, the derivative of your
position with respect to time, at some

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point in between.
And I'm claiming that at some point it

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said 100 miles per hour, and that's
because your average speed was 100 miles

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per hour, right?
In one hour you traveled 100 miles and

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that's exactly what this difference
quotient is calculating.

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So at some point along your journey, you
must have exceeded the speed limit.

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Alright, so all this is very interesting,
the mean value theorem, really great but

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what is this used for, how does the mean
value theorem actually help us understand

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anything about a function once we know
something about its derivative.

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Well, here's one very important
application; it's a theorem.

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You've got some function f and it's
differentiable on some open interval and

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on that interval the derivative is
identically zero, so the derivative is

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just zero no matter what I plug in.
Then that function is constant on that

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interval.
So it's a really exciting result, because

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it's relating information about the
derivative back to information about the

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value of the function.
It's saying that if the derivative is

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equal to 0, that function only outputs one
value.

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It's a constant function.
So let's prove this statement using the

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most valuable theorem, the MVT, or the
mean value theorem.

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Well here we go.
So I want to prove this and what I'm going

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to do is I'm going to pick two points,
I'll call them a and b, in the interval.

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And since f is differentiable on the whole
interval and I've picked a and b inside

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that interval, this is actually enough to
guarantee that the hypotheses, the Mean

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Value Theorem, applied.
The function's continuous on the closed

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interval and differentiable on the open
interval a b.

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Okay, now that means value theorem
applies.

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So f of b minus f of a over b minus a is
equal to the derivative of f at some

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mystery point c in between a and b.
But no matter where I evaluate the

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derivative the assumption here is that the
derivative is identically 0.

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So that is equal to 0.
This means that no matter which a and b I

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pick f of b minus f of a over b minus a is
equal to 0.

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Now how can a fraction be equal to 0?
The only way the fraction is equal to 0 is

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if the numerator's equal to 0.
So that means that f of b minus f of a is

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equal to 0.
And now, if I just add f of a to both

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sides, I conclude that f of b is equal to
f of a.

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What I'm really saying here, is that no
matter which a and which b I pick, f of a

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is equal to f of b.
So, f, is a constant function, right?

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Any two output values are the same so
there must only be one output value.

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Which is exactly what it means to say the
function is constant.