In the last part of our linear approximation theme,
we are going to go with one of
the most famous and most important
differential theorems in calculus.
It's mean value theorem.
Usually, you do not understand that it's so important
because it's vague how to use it in real life.
But the result is still impressive,
and actually is the groundbreaking thing for
the mean calculus theorem which we are going to
study further in the fifth week,
and all the others in fact.
So let's start with a puzzle.
A puzzle here is going to be as follows.
Consider some function f and a segment from a to b,
not infinitesimal change of
argument from a to some point x,
a real segment, and the value here is as follows.
Assume we have a chord between ends of
the graph of this segment.
Can you always find the tangent line
at some point of a segment parallel with this chord?
Well, let us draw some basic illustration forces,
and actually we can see it as a theorem all along.
So this is our function f,
usually drawn in blue.
Here are a chord drawn in dotted dark-red color.
An impressive fact is that, okay,
obviously, the orange tangent line isn't answered here,
but the green is.
Assuming, we actually understand that there is always
some point on this segment
with the parallel tangent line.
Well, why it is so important you ask?
It's an obvious question here.
But let us move to the formulas
to repeat to understand it
because it's actually quite
impressive if you just write it down.
The reason is that the theorem stands that there is
always a point C at this segment with a parallel chord.
What is the parallelity in terms of straight lines?
It is the coinciding of slopes.
For the tangent line,
a slope is an attributive at this point,
which is written in our left part,
is the derivative of f at the point C. At the right part,
we are writing down the slope of the chords, which is,
as you may well understand,
is a fraction of functions change between the ends
of the chord towards the arguments
change as the length of the chord.
What if you rewrite it in a singular form?
It's impressive, because what is written down?
It's written down in an exact equation.
It's not the case of our differentiability definition,
which stands for they are
somehow linear with an arrow which is infinitesimal.
Here, we have another thing,
which is an exact equation.
The change of function is exactly
proportional to the change of argument,
where the proportional coefficient equals to a
derivative at some point without any errors.
The trick here is that this point,
this point c can be very tricky,
and misguiding because we're not to
power of finding it analytically in any case.
But it's nice to understand because if you know that
you're function has bound its derivative, for example,
it does not go higher than five and lower than three,
you know that your function change on
every possible segments is
proportional to arguments change,
with promotional coefficient bounded
by two of these numbers.
You can understand
how much your functions changes on this segment,
and more importantly, the order of change.
Which is nice, and it is a groundbreaking thing,
it is mean calculus theorem,
which stands for integrational theorem,
and all the other [inaudible].
Well, let us just take a moment
to look at this beautiful picture,
and I will see you in the next videos.
We will talk about higher-order derivatives
and all the more complex stuff,
which is also funny.