Hello.

This is our second encounter with you.

In the first lesson in your life, perhaps you would benefit

this about the general structure of mathematics The structure of the course in general mathematics

location, general mathematics at university What courses in univariate

functions, multivariate functions, linear

In algebra and differential equations

The second of the four main lessons of this course that I had introduced.

Course information about the general structure I gave.

They are completely within the scope input.

In this lesson again to the same extent in here needed to be successful

preparation, prior knowledge of what your to remind you that, I want to introduce.

These are functions of one variable in the issues.

I say again, repeat it.

You know the functions of one variable, We expect you to know the main issues.

But I do not know the issues, know and even the If you have forgotten the issues

that it is a good opportunity in this I want to remind you.

Because we will examine each issue very in variables issues

of functions of one variable, so immediately There are many anti.

And we univariate issues,

function of the main concepts in I'll start by reminding.

In this respect, a chance at a good.

Functions of one variable thoroughly for your practice.

Find out more functions and bi the amount of two and three unknowns

solutions of equations with matrices consists of two main areas about.

The most basic of functions of one variable functions power functions.

receiving a force of x.

Here, looking at the properties of these each other

multiplications, derivatives, integrals We expect you to know them.

If you do not know that here it is also the bi does not need anything more.

A very important function of the exponential function.

Where e is the number.

Euler's number e is found.

This is the infinite number of fractions, but

as much as a seven two commas points.

Here e to x therewith function

derivative, integral, this is very awesome bi function.

There is no other function.

Derivative equal to itself.

Equal to the integral itself.

I got to know it.

This function of the product, as part of algebraic properties.

This function is the inverse function, i.e. coin

which can be thought of as two sides of natural There logarithm function.

Natural logarithm of the inverse of the exponential function function.

So if you receive a number exponent,

if you receive the logarithm of that number Do you find it.

Similarly, the logarithm of a number of

Do it again if you receive an exponential number of the ALSA Do you find it.

Logarithm function in a zero in value of the product here

logarithm of the partition of the logarithm logarithmic derivative, integral

You may have it in mind tutmasam but e must sometimes work.

Two genes associated with the exponential function There are more types of functions.

The first of these hyperbolic functions second trigonometric functions.

It's like the names.

Hyperbolic cosine, hyperbolic sine.

In the next immediately b Trigononometrik

we will see in page to the cosine sine looks like.

Each function itself minus x to x turning

collect himself, take half or Remove half

taking symmetric and anti-symmetric possible to build parts.

e to the x of x minus the change in X, and when you have collected at two

hyperbolic cosine of the difference divided by the hyperbolic sine is achieved.

They have some algebraic properties.

How dividing the sine cosine tangent

If there are hyperbolic tangent here occurs.

This function needs to be from time to time.

It always required.

Once more with trigonometric functions Do tanışıyos since.

The most basic algebraic property cosine sine-squared x is equal to x squared plus one.

This tangent is defined.

There are a number of such algebraic properties.

They must keep in mind that not all

but I needed to know that they exist and enough.

Similarly sine cosine derivative cosine derivative

b minus the sine know the difference need.

Gene least their usual I need to be aware of the existence.

But it's something so simple that anyone you know anyway.

Similarly, the derivative of the sine cosine, sinus

derivative is the minus cosine We know.

Gene tangential derivative of a split cosine x squared.

Quite the opposite of the tangent function is a function of plant-available form.

Or we can say that the tangent tangent increases i.e. the opposite function in a negative sense.

Of its derivatives, their integrals again by heart all

of their presence even if you recall your useful to know.

Do not be surprised when you see it's important to at least.

Euler's formula a miraculous function, perhaps.

Ie the square root of minus one cosine

the basic unit of imaginary number

If you collect e over who is possessed with sinus i x involved.

So this is a complex-valued function, but supremely convenient.

Full wave propagation dynamics such as in problems

vibration of the parts of a machine, water of wave propagation,

sound waves, electromagnetic As the propagation of the waves

that works on all matters that is the function.

Even as a very special human being again There are a surprising feature.

to pi, pi number in the circle as we know it including

ALINC to minus i times the strength of pi'inc one involved.

Refer here to the end, irrational i.e. an infinite number of units consisting of fractions.

PI gene comprised of infinitely many fractions points.

i is the imaginary number.

This combination of all of them minus one started off as a simple number.

Take it as a frame involved.

I got to know them.

I got to know many of them immediately.

But many of them are aware of the existence of to be adequate.

These functions exponentials family forms.

So, exponential function, sine, cosine,

hyperbolic sine, hyperbolic cosine of x forces

their multiplications, divisions, compound functions, inverse

functions such as exponential functions family forms.

Just too many problems with this function can be solved.

There is not enough of these problems, but them at this level

comfortable being able to use our business it will be enough to see.

Yet these two functions of one variable in theorem is called the fundamental theorem.

I.e. b take the integral function

derivative function again if you receive Do you find it.

Again taking the derivative of the integral of the function You can find it again if you receive function.

So this is the opposite of each other derivative integrals shows that operations.

It is integral here, then you derivatives receive

the function itself is like nothing no te is coming.

Here likewise comes hard

hard to be zero, because the derivative is coming.

Yet two functions multiplied by a number of

here is the collection of derivatives As shown derivatives

modeled either multiplied by the number of collection In short spell

Almost everyone knows the base of representation.

This function can work with.

Are the product of gene function.

This multiplication of the first derivative of the derivative hit the second plus the first

multiplied by the second derivative, two similar function of the ratio of derivative of formula.

They need to know almost everyone and I suppose you know the formula.

There is a concept called compound functions.

Function is a function of a function As can show.

When we received it directly from the derivative to the right

is a function of the mixed often.

However, instead of f by u the derivative of u with respect to x

as a derivative of a very simplified There are methods of calculation.

Already f, u is a function.

Directly derivative with respect to x You can not.

Therefore the x by u by u is here

because it is such a chain process given name.

Yet sometimes complex integrals, in the integral

Another variant with a function the product of functions

This integration easily by selecting calculated.

This bi, a second Find out integral feature.

Here with the previous ones and functions derivatives,

with algebraic properties of these functions It was about.

Find out more about some of equations There are also issues that need, but we

two and a maximum of three unknowns

with equations and their matrix We will deal with the relationship.

These need.

Two equations with two unknowns here As a

x and x is a matrix consisting of said two

something we call column matrix multiplication.

This sub-bottom on the right side sequencing appropriate

sequences leads to a matrix representation of the b We know that hungry.

But you do not know your work here given definition.

Similarly, three in three unknowns

Even though they lower down the equation those known unknowns

x is equal to b the same as a separating two unknowns as in

It is possible to edit the structure and it is very simplifies things.

Separated because of unknown and known is the situation.

An important concept for matrices the determinant.

Two binary matrix determinant the product of the number of diagonal.

First diagonal multiplication of those over minus

those over the second diagonal multiplication There are as.

Similarly, also in three by three matrix determinants of a generalization of it.

Again, this three by three matrix columns and on lines

A line according to any any According to the column, making it possible to open.

According to the first row as an example here given.

a a a, a a two, a three and that D E, we think

where the number of a unit that is throwing column and

where a is a one-line bi threw two binary matrix remains.

Two binary matrix determinant We know that the calculation here.

Less number of jobs here, signs, plus, minus, plus comes.

Yet where one or two of milk and

b binary matrix into two rows throwing again remains.

This way of expressing this determinant We expect you to know.

Gene has a feature like this.

If you want to get a thousand thousand located in the matrix.

So these are general to all questions, ko, e, results.

But again, we'll take care of our two binary or three by three matrix.

Here bi saw on the previous page.

These equations A x is equal to b We yazabiliyo.

One solution to this equation to be

the determinant is non-zero is required.

This is important information.

These are just the very basics, but here two

about two and three by three matrix We expect you to know them.

When you do not know the easily you can learn topics.

Following three basic operations for matrices has.

Two rows of the matrix are equal to each other determinants is zero.

Determinants gets hit in a row in a t'yl to t'yl

not multiplied in case the determinant is equal to income.

Another hit in a row t'yl when you add the determinant unchanged.

In the example here, for example in the second row a Number

t multiplied by the number of the first row to the we've added.

It does not change the determinant we see.

Now without this preparation course information we can not do.

But you know, at the beginning to know them all not required.

If a good Bilmedikleriniz opportunity.

But they advise you to postpone I would.

In this lesson you learned as soon as possible 'll supremely comfortable.

For those not essential to know all I'm just saying.

Each topic progresses preparedness information You will be reminded.

So he used about preparation information will be reminded and univariate

the learning of functions you do not know You will get the opportunity.

In fact, forget what you learned in the You will get the opportunity to remember.

Now put on a fashion statement takes the diol young people themselves.

I really do not know what to remember that to learn how to

much time you have to invest a little will save.

Everything will be easier.

If you do not know can too.

But if you know much more easily

Scroll through these issues You can go.

So we can ask the following question.

How will you identify your missing?

For this test you one yourself We have prepared questions.

They consist of the following two pages.

For example, where the force function multiplication, division

I function as listed above on the types of features.

Calculation of some determinants, some Calculation of derivatives such things.

Here again the calculation of derivatives it seems.

Integration into account, so I do not want If y'all just forget it

integration of learning lots of complex bi There is no need.

Because we're doing here integrals There will be a simple integration.

We can provide essential issues.

I also have to remember the following functions I'm waiting.

Sasinüs function showing oscillations bi You know that function.

This two x's or sinus sinuses sinus X,

x divided by two relative to each other How will the situation change?

I expect you to know them thoroughly.

These are some of the things you diameter

what happens when you bury, e, to know expect.

Yet over the exponential function e x and e

what happens when x is minus We expect you to know.

e to the x function of the logarithm of X,, We expect you to know drawings.

They are found almost everywhere is information.

Both of you should go to Enternet bulurs, you'll find.

Functions of one variable in your book they are also almost everywhere.

Now our preparatory courses so we we are finished.

Today it is a natural place to accommodate cutting.

Now you're standing here.

The next time we meet now with vectors

The principles related to our subject began will be.

Goodbye.