Mathematical analysis for teapots. Lesson 1

understand multiplier

Bezlich- Tse sukupnіst deyakyh ob'єktіv. How can you multiply? First, kіntseі chi neskіnchenni. For example, the nameless sirniks at the box are a lot of kіntsev, you can take them and frighten them. The quantity of food on the beach is much more important, but, in principle, it is possible. І tsya kіlkіst manifests itself as a final number. So the faceless pishchinok on the beach is also amazing. And the axis of the impersonal point is directly tse impersonal indistinct. So it’s easier, it’s straight on its own, unskinned, and you can put skilki on it forever. An impersonal point in a straight line is also unskinned. That is why, theoretically, the point can be quite small. Obviously, it is physically impossible to draw a point, with a difference, for example, less than the size of an atom, but, from the point of view of mathematics, a point cannot be measured. Її expansion is close to zero. And what about going out, like adding a number to zero? That's right, inconsistency. I want an impersonal point on the straight line and on the edge of the right inconsistency, but not the same. Bezlich - it's not a bunch of things there, but a collection of objects. I vvazhayutsya less than those multiplied, yakі avenge absolutely the same objects. If in one multiplicity to avenge those objects, which is another multiplicity, but plus one more "livy" object, then it is no longer equal multiplier.

Let's look at an example. Let us have two multiplies. First - the succession of all points on a straight line. The other is the confluence of the short points on the straight line. Why are they not jealous? In a first step, the breeze that straight line can not be overwhelmed. Todі stinks are definitely not equal, oskіlki vindicate their absolutely different points. If the stench is overwhelmed, then they have more than one hot point. Everything else is also different. And how can the arms lie on a straight line? Todі all points in vіrіzka є th points of the straight line. Not all points of a straight line are points of a cut. So in this way, multiplicity cannot be equal (equal).

The skin multiplier is set by the rule, as it is unambiguously determined, to lie down the element to the tsієї multiplier chi nі. What can be the rules? For example, like a faceless kіtsev, you can stupidly overthrow this object. You can set the range. For example, let's use numbers from 1 to 10. Tse will be the same kіnceve impersonal, but we will not overdo it, but we will formulate the rule. But the inconsistency, for example, must be the numbers, more than 10. great number- if you didn’t name the number, zavzhd є number plus 1.

As a rule, plurals are denoted by great letters of the Latin alphabet A, B, C and so on. This impersonality is composed of specific elements and you want to set it with a list of these elements, we can put it in the shape of the bow, for example A=(a, b, c, d). If a is an element of the multiplier A, then write it like this: a Î A. If a is not an element of the multiplier A, then write a Ï A. One of the important multipliers is the multiplier N of all natural numbers N=(1,2,3,...,) . Isnuye is also special, so called an empty lot, so as not to avenge the native element. An empty multiplier is indicated by the symbol Æ .

Designation 1 (designation of the equality of multiples). Bezlich BUTі B рівні, as if the stench is formed from the quiet elements themselves, so, as xÎ A follows x Î B i backwards, from x Î B turns x Î A.

Formally, the equality of two sets is written as follows:

(A = B) := " x (( x Î A ) Û (x Î B )),

Tse means what for any object xÎ A and xÎ B are equal.

Here " - The quantifier of heat (" xread like "for skin x").

Appointment 2 (appointment submultiple). Bezlich BUTє multiplicity of impersonality At yakscho be-yak X what to lay faceless BUT, lie faceless Art. Formally, you can look at the virazu:

(A Ì B) := " x((x Î A) Þ (x Î B))

Yakscho A Ì B, ale A ¹ B, then A is a clear submultiplier Art. Like a butt, you can restore it straight and straight. If the arms lie on a straight line, then a rich point is a multiple point on a straight line. But another example. Anonymous integers, yakі dilyatsya without excess by 3, є submultiple anonymous integers.

Respect. An empty multiplier is a submultiple, be it a multiplier.

Operations on multiplies

Over the impersonal, the following operations are possible:

United. The essence of this operation is to combine two multiples into one, to avenge the elements of the skin from multiples that are combined. Formally, it looks like this:

C=AÈ B:= {x:x Î A or xÎ B}

butt. Virіshimo nerіvnіst | 2 x+ 3 | > 7.

From a new standpoint or unevenness 2x+3 >7, for 2x+3≥0 then x>2

or unevenness 2x+3<-7, для 2x+3 <0, тогда x<-5.

An impersonal solution to this unevenness is the union of multiples (-∞,-5) È (2, ∞).

Let's revise. Porahuemo meaning viraz | 2 x+ 3 | for how many points to lie and not to lie in a given range:

Like Bachimo, everything is written correctly (the chervonim signifies the near-cordon ranges).

Peretin. Peretin is the operation of creating a new multiplier in two, in order to avenge the elements, as if to enter into offense and multiplier. To depict the exact value, let's notice that we have two multiplies points on the plane, and the figure A and figure B itself.

Formally, the multiplication operation is written as follows:

C=A Ç B:= (x: x Î A i x О B )

butt. Let us have an impersonal C=A Ç B = {5,6,7}

Vіdnіmannya. Vіdnіmannya mnozhin - tse vinyatok z vіdnіmaєtsya impersonal quiet elements, yakі mіstjatsya vіdnіmaєtsya i vіdchituvachi:

Formally, the multiples are written as follows:

A\B:={x:x Î A and xÏ B}

butt. Let us have an impersonal A = (1,2,3,4,5,6,7), B = (5,6,7,8,9,10). Todi C=A\ B = { 1,2,3,4}

Addition. Additional - this is a unary operation (an operation on two, and on one without a person). This operation is the result of the given multiplier from the same universal multiplier (multiplier, as it includes all other multipliers).

A := (x:x О U and x П A) = U \ A

Graphically, you can depict it at a glance:

Symmetrical difference. On the vіdmіnu vіd zvichaynoї raznitsi at symmetrical raznіtіnі many elements are left only those that are present either in one or another plurality. But, speaking in simple words, it is created from two multitudes, and then those elements are turned off from it, as if they are in this and in the other multitude:

Mathematically, it can be expressed like this:

A D B:= (A/B) È ( B&A) = (A È B) \ (A Ç B)

The power of operations over multiples.

The purpose of the association is that the network of many people is screaming, that the operation of the network of that association may be so powerful:

1. Commutativity.

A È B=BÈ A
AÇ B=BÇ A

1. Associativity.

(A È B) È C=AÈ ( B È C)
(A Ç B) Ç C=AÇ ( B Ç C)

For the quiet, who wants to learn to know the boundaries in this article, it is peaceful about it. Do not delve into theory, start to give її at lectures and lectures. So the "tedious theory" can be outlined in your zoshites. If you don’t know anything, then you can read the assistants taken in the library of the initial mortgage or on other Internet resources.

Later, it is more important to understand the distance between the students in the course of higher mathematics, especially if you are familiar with the integral calculations and understand the relationship between the boundary and the integral. Which materials will be looked at, simple applications, as well as ways of their perfection.

Apply solution

What does it look like: $ \bigg [\frac(0)(0) \bigg ] $

Directly between the remaining two butts to inconsistency and discernible insignificance: $ \bigg [\frac(\infty)(\infty) \bigg ] $

Algorithm for calculating limits

Otzhe, let's briefly pіdіb'єmo pіdіb'єmo pіdіb'єmo pіdіb'іmok razіbrannym buttovі sladimo algorithm razvyazannya between:

1. Submit a point x viraz, next after the sign of the boundary. If you go out the same number, or inconsistency, then the boundary is clear. In a different way, it is possible to have insignificance: "zero divided by zero" or "inconsistency of division by inconsistency" and let's move on to the next paragraphs of the instruction.
2. In order to use the insignificance of "zero divided by zero", it is necessary to lay out a number book and a banner into multipliers. Speed ​​it up. Submit a point x at the viraz, what should be under the sign of the boundary.
3. Like insignificance "inconsistency of division into inconsistency", the same is to be blamed both in the numeral book, and in the banner x of the greatest world. We'll soon ixi. We represent the value of xx z-pid between in viraz, which is lost.

In these articles, you learned the basics of perfection between, which are often quoted in the course of Mathematical Analysis. Obviously, not all the types of tasks that the examiners are proposing, but the simplest ones. In the upcoming articles, we’ll talk about other types of tasks, but first I need to learn the lesson, so that the distance will collapse. It is negotiable that work, that is the root, the world, that there are infinitely small equivalent functions, miraculous boundaries, Lopital's rule.

If you don't know how to cross the line on your own, then don't panic. We are here to help!

The category Mathematical analysis is selected without cost online video lessons on this topic. Mathematical analysis is the sum total of the divisions of mathematics, yakі are engaged in the development of functions and their zagalnen methods of differential and integral calculation. These include: functional analysis, including the theory of the Lebesgue integral, complex analysis (TFKT), which shows functions, assignments on the complex plane, the theory of series and rich integrals, non-standard analysis, which develops infinitely small and infinitely large numbers, vector analyses. calculation. The development of mathematical analysis with video lessons will be as colorful as the beginnings, so the knowledge of mathematicians. Video lessons from the heading Mathematical analysis You can marvel without cost at any hour. Additional materials have been added to some video lessons on mathematical analysis, which can be taken advantage of. Good luck!

A lot of materials: 12
Materials shown: 1-10

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Introduction to analysis. Tension of multiples

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Between the functions at the point - appointment, application

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Convenience of state rows - an example of how to know the area of ​​concession, doslidzhennya

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What is the first

In this video, I will tell you about the first, like a close relative of the dead. Tell the truth about her, and so you already know everything, like looking at my previous video, and we only had to put specks over the i. Primary - tse "batkivska" function for traveling. Know the first - tse means the answer to the question: and whose child? If you see a donka, then we can know my mother. Previously, we navpak joked on the donka on the instructions of my mother. At once mi robimo turnaround - vіd...

Geometric zm_st pokhіdnoy

In this video, I'll tell you about the geometric meaning of the opposite. You know that the geometrical zmist is similar to the fact that it is similar to such a smart dot - it’s the same. I say “mayzhe” to that which is similar to the old tangent of the kuta badly dot. Can vvazhat, scho that nahіdna frail dotichї - the stench is closely tied. If it’s a big one, then it’s big, and the function of this point is sharply growing. If it’s a small one, then it’s ugly ...

What is a function in mathematics

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Between functions on inconsistency - purpose, butt

Lesson "Between functions on inconsistency - purpose, apply" assignments to nutrition about those that are so bordering on inconsistency. The majority of elementary functions are assigned a certain amount of great value to the argument. And here it is important to know the behavior of the function on inconsistency. One of the elements of achieving such a behavior is the importance of inter-functions on inconsistency. Although inconsistency is not a number, and їy does not show the same point on the number line, the designated boundary on ...

Heaps of terrible formulas, helpers of great mathematics, yakі vіdkrієsh і vіdrazu zakriєsh, bolіsnі nіsnі pіsіnі prіshennya it would be more simple zavdannya .... This situation is not uncommon, especially if the mathematician's assistant has recovered from the distant 11th grade. And in the meantime, at the VNZ, the main plans of a variety of specialties are transferred to the education of everyone who loves mathematics. And in this situation, you often see yourself with the same teapot in front of a heap of stingy mathematical gibberish. Moreover, the situation is similar for any subject, especially for the cycle of natural sciences.

What work? For a full-time student, everything is much simpler, obviously, obviously, the subject is not neglected. You can consult with the vikladach, classmates, and simply write off from the sus_da in the party. To create a new teapot in the world of mathematics for such an expansion of the session through.

And how does a person study at the correspondence department of the VNZ, and even a mathematician, mildly seeming, is unlikely to be needed in the future? Until then, there is no time to be busy. So it’s out, in the majority of vipadkiv so, but no one skasovuvav vykonannya control work and folded іspitu (mostly, written). With control robots, everything is simpler in general mathematics, chi is a teapot, chi is not a teapot - you can learn the control robot from mathematics. For example, I have. І s іnshih subjects tezh you can learn. Already not here. Ale vikonannya that building for the review of the control work has not yet been brought to a reserved entry in the book of books. Often buvay, scho vitvir mystetstva, vykonaniya on zamovlennya, it is necessary to protect, and explain why this formula sings out of these letters. In addition, check to sleep, and there already happen to be virishuvati vyznachniki, between and pokhіdnі INDEPENDENTLY. Yakshcho, obviously, the vikladach does not accept valuable gifts, like a hired good-natured person outside the walls of the audience.

Allow me, I will give you a more respectful pleasure. In the halls, іspitah z exactі і natural sciences Remember, please. Most of the time, the rozumovyh processes are just fighting the vikladach, I have a flutter, if the students-part-time students gagged 5-6 times. Remember, one young man gave the control robot 4 times, and after the skin reissue, he turned up to me for a cost-free guarantee consultation. I’m sorry, I’ve mentioned that the deputy of the letter “pi” had written the letter “pe” in the case of the vidpovidi wine, for which there were severe sanctions on the side of the reviewer. A student to NAVIT NOT WANTING TO DEPRECIATE into the task, as if he rewrote a lot

You can be a real teapot at all mathematics, but it’s better to know that the constants are better than zero. To that, as you seem to be foolishness on elementary nutrition, then the greatness of the fact that the VNZ will end for you is great. It’s more convenient to put yourself up to that student, who WANTS to learn about the subject, to the one who, high and mildly, even if you try to be virishiti, explain or bring it up. І tse assertion є fair for all disciplines. To that, it is necessary to give the position “I don’t know anything, I don’t understand anything.”

Another important joy is to give lectures, to teach them not too much. I already guessed about it on the main side of the site Mathematics for correspondence students. There is no sense to repeat, why is it important, read there.

Otzhe, scho robiti, as if on the nose of the hall, sleeps with great mathematics, but in deplorable terms - the camp of a full, or rather seemingly, empty teapot?

One of the options is to hire a tutor. With the largest base of tutors you can get acquainted with (more importantly, Moscow) or (more importantly, St. Petersburg). You can find a tutor at your place behind the click system, or you can look at the local advertising newspapers. The price for the services of a tutor can vary from 400 and more rubles per year depending on the qualification of the tutor. Note that cheap does not mean bad, especially since your mathematical background is bad. At the same hour for 2-3K rubles, you take a LOT. No one takes such pennies for free, and no one pays such pennies for nothing ;-). The only important moment is to ask a tutor with specialized pedagogical knowledge. And it’s true, we don’t go for legal help to the dentist.

In the rest of the hour, the service of online tutors is gaining popularity. Vіn duzhe zruchny, if it is necessary to solve one or two tasks in terms, explore topics or get ready for sleep. Insane perevagou є tsіni, yakі at kіlka razіv nizhchi, nizh at offline tutor + saving an hour on travel, which is especially relevant for mechantsіv megacities.

In the course of higher mathematics, it is even more important to master certain speeches without a tutor, it is necessary for you to "lively" explain.

Tim is not less, for rich types of managers, you can develop them independently, and the meta of this division of the site - teach you how to write about types, apply that task, so that you can practically learn about experiences. Moreover, for the low task, “zhorstky” algorithms are developed, deducing the correct solution from the start “nowhere to go”. I, in the world of my knowledge, I will try to help you, there is more pedagogical education and knowledge of work in these specialties.

Let's do some mathematical gibberish. It's okay, just like a teapot, more mathematics - it's really simple and accessible.

And rozpochati requires repetition of the school mathematics course. Repetition is the mother of torment.

First of all, you proceed to the development of my methodical materials, and then proceed to the development of any materials from higher mathematics, I really recommend that you read the lower studies.

In order to successfully solve the problem of great mathematics, it is NECESSARY:

BUY A MICRO CALCULATOR.

3 programs - Excel (optional choice!). Manual for "teapots" I got from the library.

Chi є? Already good.

– Type of permutation of addendums - the amount does not change: .
And the axis of the same difference of speech:

Rearranging "ix" and "four" is simply not possible. At the same time, we create the cult letter "iks", as in mathematics it designates an unknown or changing value.

– Type of permutation of multipliers - tvir does not change: .
With a meanness, such a trick can’t be passed, and two different fractions and a permutation of the numeral with a banner can’t do without naslіdkіv.
It is also assumed that the plural sign (“dots”) is most often not used to write:,

– Guessing the rules for opening the bow:
- here the signs of the dodanks do not change
- And here they are changing on the floor.
І for plural:

Vzagali, enough memory, scho TWO MINUS GIVE A PLUS, a THREE MINUS - GIVE MINUS. І, try not to get lost in the course of solving problems with higher mathematics (already often, and pardon pardon).

– Let's guess, there are similar dodankiv, You are guilty of good understanding of this diyu:

– Guess what the steps are:

, , , .

Steps are only a few words.

–Guessing that fractions can be shortened: (Shorted by 2), (Shorted by five), (Shortened by).

– Guessing diї with fractions:

and even more important is the rule of bringing fractions to a double banner:

Likewise, apply little minds, marvel at school mentors.
Without which you will be TIGHT.

PORADA: in most mathematics, it is more common to carry out the calculations in PRIVATE REGULAR AND IRREGULAR FRACTIONS, so there will be scary fractions for kshtalt. The axis of this piece DOES NOT REQUIRED to be presented at a glance, i, more, it is NOT REQUIRED to add a number to the banner on the calculator, taking into account 4.334552102.

EXCLUDED from the rules є kіntseva vіdpovіd zavdannya, even if it's better to write down abo.

– Rivnyannia. The new one has a part that right part. For example:

You can transfer whether it’s dodanok to another part, changing into a new sign:
Let's transfer, for example, the warehouses in the left part:

Abo is right:

Mezhі zavdat all students, yakі vyvchayut mathematics, chimalo trouble. In order to cross the line, sometimes you have to stop a lot of cunning and choose from rich methods of developing the very same, which is a pidide for a particular butt.

In this article, we can’t help you to understand the boundaries of your abilities and touch the boundaries of control, but we will try to improve your knowledge: how to understand the boundaries of your mathematics? Rozuminnya to come with the knowledge, then at once we will bring a few report examples of versification between the explanations.

Understanding inter-mathematics

First food: what did you go for between that between what? You can talk about inter-numerical sequences and functions. We have to understand the inter-functions; Ale on the back - a scorched line between:

Admissible, є pevna changeable value. That the value of the process of change is inexorably approaching before the next day a , then a - Between values.

For singing in the interval function f(x)=y the boundary is called such a number A , what is the right function for X , sho pragne to the singing point a . Krapka a lie within the interval for which the function is assigned.

It sounds cumbersome, but it’s even easier to write down:

Lim- in English limit- Mezha.

It is also a geometric explanation of the designation of the boundary, but here we are not liztimemo in theory, the shards of us are more practical, the lower is the theoretical side of nutrition. When we say what X It doesn’t have any meaning, it means that the change doesn’t take on the meaning of the number, but it’s getting closer to the new one.

Let's take a concrete example. Zavdannya - know the boundary.

To write such a butt, imagine the meaning x=3 at the function. We take:

Until then, to tell you about basic operations on matrices, read the following article on this topic.

At the butts X you can pragnet be of any significance. Tse mozhe buti be-like the number of chi neskіchennіst. Axis butt, if X pragne inconsistency:

Intuitively realized that the larger the number of the bannerman, the less important the function is. So, for the unbound growth X meaning 1/x will change and approach zero.

Like a bachimo, to cross the line, it is necessary to simply put on the function a value, some kind of pragmatic X . However, the simplest one. Often perebuvannya between negarazd is more obvious. At the borders there are inconsistencies to the type 0/0 or inconsistency / inconsistency . What is the job of such vipadkas? Go to the tricks!

Insignificance in between

Insignificance to the mind inconsistency / inconsistency

Let me go between:

If we try to introduce inconsistency in a function, then we take inconsistency like a numeralist, and a bannerman. It was necessary to say that the manifestation of such insignificance has a singing element of mysticism: it is necessary to remember how it is possible to transform the function in such a manner that the insignificance is gone. At our point of view, we divide the number and banner into X at the senior level. What can you see?

From what I have already looked at, I will butt we know that the members, like revenge on the banner of x, will bend zero. Some solutions between:

For razkrittya non-insignificance type inconsistency / inconsistency divide the numeral and banner on X at the highest level.

Before speech! For our readers at once a discount of 10% on some kind of robot

Another kind of insignificance: 0/0

How to start, substitution of the value of the function x=-1 yes 0 at the numeral and the bannerman. Take a look at the three more respectfully and remember that we have a square equal to the number book. We know the root and write it down:

We quickly take it away:

Also, as you see, it sticks to the insignificance of the type 0/0 - Lay out the number book and the banner on the multipliers.

So that it would be easier for you to write it down, we will set up a table for the boundaries of the current functions:

Lopital's rule in the borders

Another straining way allows us to use the insignificance of both types. What is the essence of the method?

As a boundary between insignificance, I’ll take a look at the numeral and bannerman until that hour, until the insignificance is not known.

Initially, Lopital's rule looks like this:

important moment : the border, in the yakіy zamіst of the number book and the bannerman, to stand like the number book and the bannerman, you can use it.

And now - a real example:

In manifestness, type non-insignificance 0/0 . Let's take a look at the date of the numeral and banner:

Voila, invisibility is dressed up elegantly.

Let's hope that you can put this information into practice and know the evidence for nutrition "how to overcome the difference between other mathematics." It is necessary to calculate between sequences, or between functions in points, and there is no time for a robot to use the word “zom”, go back to a professional student service for a quick report and solutions.