[MUSIC] We actually know more
about tangent lines and let us proceed with understanding of
linear approximations of functions. First of all, let's just look at the very
same equation where you've used earlier. And what does it say? Basically say that in the close
neighborhood of the given point a this or well, this one you can substitute your
function with its linear approximation. In other words, you can move with differential is the
change of a function in the given point. And hope that this arrow
here which is infinitesimal towards function change doesn't
actually cost very much. How far it is? Well, it's kind of tricky and
in order to understand more of it. You need to consider Taylor series and
its accuracy, but for our purposes, we just go in to
understand basic approximations with which can be derived from this
first order approximations here. So let us consider people
just some basic example. For example, let us compute sign of 29 degrees, okay. First of all,
we need to define three basic things here. We need to define function
we're looking at because we're actually looking at some number. We need to write something about f at x. So we are going to write this out. Well, it's kind of obvious here. What is the function which is
referrals in this example. It's sine of x well. Nobody's actually surprised here. So but what we're going to do we
are going to define two points, point a which is a good point. A point, which is most fittest. In other way, we are comfortable to
compute the derivative at this point and the value of function
itself at this point. So what is this a good convenience
point near as value and his argument of the functions that
we are going to compute here. Well, actually let us just skip for
a second and try to bad point here because
bad point is kind of easy. Bad point is point that
we are looking for. It's 29 degrees. But as good point well most of us
that remember school calls off. Metric can easily write this in
down its such activities and social science and faction of 30
degrees are quite famous and overall. We are going to go with it. So now we are having here now
function are good and our bad point. We can calculate easily
the derivative of a function f and then substitute with good point. We can easily calculate as a change of
arguments and we'll get the result here, but we need to be cautious here
because it's kind of tricky. First of all, there is some strange
thing here, which is our degrees. You need to understand some basics
in if we just move forward with our degrees our change of arguments
will be measured in degrees. I know it sounds like a physical
explanation here, but it's kind of useful to understand
if we move forward with degrees our answer will be measured in degrees,
which is kind of not the case. Sine of something is not an angular thing. It's just a matrix thing. It's actually is not measured in anything. It's measured in units because if
you remember from Pythagoras Theorem scientist is fraction between hypotenuse
and as a segment of the triangle. So we need to get rid of degrees and move to some other angular measurements, which is measured in units not
in some physical units here. So it is actually radians and if you do not remember what I've done is
I've got a nice room for you as rules. Basically, it works like this. 180 degrees is pi radius
half a circle is pi. It's quite easy. Well in real life you kind of
expecting that pi is a circle but nothing actually works good
within these odd jokes. So another way if we need to go for
the derivative we need to use transition to
write down some degrees here. So in other words 30 degrees is six parts of semicircle and in our case it is pi divided by 6. Well and well 29 degrees. It's kind of ugly, but
we actually do know that it is 30 degrees minus 1 degree and
1 degree is pi divided by 180. So we are going to go with it and
it's quite nice and easy. So in other words x minus
is a change on function here is minus pi divided
by 180 which is nice. So there are the last thing to do is to
write down our derivative at the point a. So we need to find
the derivative of sine function, which is I hope you remember
cosine function and we need to substitute x with the good
point which is pi divided by 6. And as a result we get okay, this is kind of tricky because
nobody actually remembers it, but it's square root
out of 3 divided by 2. So let us just write the thing down. So what is sine of 29 degrees? It is sine of 30 degrees,
which is one-half plus the derivative
which is square root out of 3 divided by 2 multiplied
by the arguments change, which is minus 5 divided by 180. And I am warning you. Finite change can be negative and
you should not lose this minus. It's crucial here and it's your first
way to just mess with it completely. The change is actually can be negative or
a positive and well actually 0, but that's not our case. So what do we get here? Well, first of all, they need to understand that
it's kind of question of well, not very good accuracy because
well what is sine of 29 degrees? It's one-half plus something and
this something is actually not so great because square root out of
3 divided by 2 is less than 1 and minus pi divided by 180 is close to 1
divided by 60 or something like that. And this is somehow close to well one-half minus 1 divided by 60 which is what kind of loser approximation. And we actually just said that
the value of sine of 29 degrees is close to saying as of 30 degrees which
kind of not a great discovery here. But it gave us a sense how close and we can go with some value here so
it still works. And the last thing I need to point out
that it's actually [INAUDIBLE] and we can now calculate
approximate values in many, many, many,
many bad points using good points. And well,
it's mathematically absolutely correct, which is a national standard and
a nice trick. [SOUND]