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.