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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.
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