MATHEMATICAL MODEL - representation of a phenomenon or process studied in concrete scientific knowledge in the language of mathematical concepts. At the same time, a number of properties of the phenomenon under study are supposed to be obtained on the path of studying the actual mathematical characteristics of the model. Construction of M.m. most often dictated by the need to have a quantitative analysis of the studied phenomena and processes, without which, in turn, it is impossible to make experimentally verifiable predictions about their course.

The process of mathematical modeling, as a rule, goes through the following stages. At the first stage, the links between the main parameters of the future M.m. First of all, we are talking about a qualitative analysis of the phenomena under study and the formulation of patterns that link the main objects of research. On this basis, the identification of objects that allow a quantitative description is carried out. The stage ends with the construction of a hypothetical model, in other words, a record in the language of mathematical concepts of qualitative ideas about the relationships between the main objects of the model, which can be quantitatively characterized.

At the second stage, the study of the actual mathematical problems, to which the constructed hypothetical model leads, takes place. The main thing at this stage is to obtain empirically verifiable theoretical consequences (solution of the direct problem) as a result of the mathematical analysis of the model. At the same time, cases are not uncommon when, for the construction and study of M.m. in different areas of concrete scientific knowledge, the same mathematical apparatus is used (for example, differential equations) and mathematical problems of the same type, although very non-trivial in each specific case, arise. In addition, at this stage, the use of high-speed computing technology (computer) becomes of great importance, which makes it possible to obtain an approximate solution of problems, often impossible in the framework of pure mathematics, with a previously unavailable (without the use of a computer) degree of accuracy.

The third stage is characterized by activities to identify the degree of adequacy of the constructed hypothetical M.m. those phenomena and processes for the study of which it was intended. Namely, in the event that all model parameters have been specified, the researchers try to find out how, within the accuracy of observations, their results are consistent with the theoretical consequences of the model. Deviations beyond the accuracy of observations indicate the inadequacy of the model. However, there are often cases when, when building a model, a number of its parameters remain unchanged.

indefinite. Problems in which the parametric characteristics of the model are established in such a way that the theoretical consequences are comparable within the accuracy of observations with the results of empirical tests are called inverse problems.

At the fourth stage, taking into account the identification of the degree of adequacy of the constructed hypothetical model and the emergence of new experimental data on the phenomena under study, the subsequent analysis and modification of the model takes place. Here, the decision taken varies from an unconditional rejection of the applied mathematical tools to the adoption of the constructed model as a foundation for constructing a fundamentally new scientific theory.

The first M.m. appeared in ancient science. So, to model the solar system, the Greek mathematician and astronomer Eudoxus gave each planet four spheres, the combination of the movement of which created a hippopede - a mathematical curve similar to the observed movement of the planet. Since, however, this model could not explain all the observed anomalies in the motion of the planets, it was later replaced by the epicyclic model of Apollonius from Perge. Hipparchus used the latest model in his studies, and then, subjecting it to some modification, Ptolemy. This model, like its predecessors, was based on the belief that the planets make uniform circular motions, the overlap of which explained the apparent irregularities. At the same time, it should be noted that the Copernican model was fundamentally new only in a qualitative sense (but not as M.M.). And only Kepler, based on the observations of Tycho Brahe, built a new M.m. The solar system, proving that the planets move not in circular, but in elliptical orbits.

At present, the most adequate are the MMs constructed to describe mechanical and physical phenomena. On the adequacy of M.m. outside of physics one can, with a few exceptions, speak with a fair amount of caution. Nevertheless, fixing the hypotheticality, and often simply the inadequacy of M.m. in various fields of knowledge, their role in the development of science should not be underestimated. There are frequent cases when even models that are far from adequate to a large extent organized and stimulated further research, along with erroneous conclusions, contained those grains of truth that fully justified the efforts expended on developing these models.

Literature:

Math modeling. M., 1979;

Ruzavin G.I. Mathematization of scientific knowledge. M., 1984;

Tutubalin V.N., Barabasheva Yu.M., Grigoryan A.A., Devyatkova G.N., Uger E.G. Differential equations in ecology: historical and methodological reflection // Problems of the history of natural science and technology. 1997. No. 3.

Dictionary of philosophical terms. Scientific edition of Professor V.G. Kuznetsova. M., INFRA-M, 2007, p. 310-311.

Computers have firmly entered our lives, and there is practically no such area of ​​human activity where computers would not be used. Computers are now widely used in the process of creating and researching new machines, new technological processes and the search for their optimal options; when solving economic problems, when solving problems of planning and managing production at various levels. The creation of large objects in rocketry, aircraft construction, shipbuilding, as well as the design of dams, bridges, etc., is generally impossible without the use of computers.

To use a computer in solving applied problems, first of all, the applied problem must be "translated" into a formal mathematical language, i.e. for a real object, process or system, its mathematical model must be built.

The word "Model" comes from the Latin modus (copy, image, outline). Modeling is the replacement of some object A with another object B. The replaced object A is called the original or the modeling object, and the replacement B is called the model. In other words, a model is an object-replacement of the original object, providing the study of some properties of the original.

The purpose of modeling is to obtain, process, present and use information about objects that interact with each other and the external environment; and the model here acts as a means of knowing the properties and patterns of the behavior of the object.

Mathematical modeling is a means of studying a real object, process or system by replacing them with a mathematical model that is more convenient for experimental research using a computer.

Mathematical modeling is the process of constructing and studying mathematical models of real processes and phenomena. All natural and social sciences that use the mathematical apparatus are essentially engaged in mathematical modeling: they replace the real object with its model and then study the latter. As in the case of any simulation, the mathematical model does not fully describe the phenomenon under study, and questions about the applicability of the results obtained in this way are very meaningful. A mathematical model is a simplified description of reality using mathematical concepts.

A mathematical model expresses the essential features of an object or process in the language of equations and other mathematical means. Strictly speaking, mathematics itself owes its existence to what it tries to reflect, i.e. to model, in their own specific language, the patterns of the surrounding world.

At mathematical modeling the study of the object is carried out by means of a model formulated in the language of mathematics using certain mathematical methods.

The path of mathematical modeling in our time is much more comprehensive than natural modeling. A huge impetus to the development of mathematical modeling was given by the advent of computers, although the method itself was born simultaneously with mathematics thousands of years ago.

Mathematical modeling as such does not always require computer support. Each specialist professionally engaged in mathematical modeling does everything possible for the analytical study of the model. Analytical solutions (i.e., represented by formulas expressing the results of the study through the initial data) are usually more convenient and informative than numerical ones. The possibilities of analytical methods for solving complex mathematical problems, however, are very limited and, as a rule, these methods are much more complicated than numerical ones.

A mathematical model is an approximate representation of real objects, processes or systems, expressed in mathematical terms and retaining the essential features of the original. Mathematical models in a quantitative form, with the help of logical and mathematical constructions, describe the main properties of an object, process or system, its parameters, internal and external connections

All models can be divided into two classes:

1. real,
2. ideal.

In turn, real models can be divided into:

1. natural,
2. physical,
3. mathematical.

Ideal models can be divided into:

1. visual,
2. iconic,
3. mathematical.

Real full-scale models are real objects, processes and systems on which scientific, technical and industrial experiments are performed.

Real physical models are mock-ups, models that reproduce the physical properties of the originals (kinematic, dynamic, hydraulic, thermal, electrical, light models).

Real mathematical are analog, structural, geometric, graphic, digital and cybernetic models.

Ideal visual models are diagrams, maps, drawings, graphs, graphs, analogues, structural and geometric models.

Ideal sign models are symbols, alphabet, programming languages, ordered notation, topological notation, network representation.

Ideal mathematical models are analytical, functional, simulation, combined models.

In the above classification, some models have a double interpretation (for example, analog). All models, except for full-scale ones, can be combined into one class of mental models, since they are the product of man's abstract thinking.

Elements of game theory

In the general case, solving the game is a rather difficult task, and the complexity of the problem and the amount of calculations required for solving increases sharply with increasing . However, these difficulties are not of a fundamental nature and are associated only with a very large volume of calculations, which in a number of cases may turn out to be practically unfeasible. The fundamental side of the method of finding a solution remains for any one and the same.

Let's illustrate this with the example of a game. Let's give it a geometric interpretation - already a spatial one. Our three strategies, we will depict with three points on the plane ; the first lies at the origin (Fig. 1). the second and third - on the axes Oh and OU at distances 1 from the origin.

Axes I-I, II-II and III-III are drawn through the points, perpendicular to the plane . On the I-I axis, the payoffs for the strategy are plotted on the axes II-II and III-III - the payoffs for the strategies. Every enemy strategy will be represented by a plane cutting off on the axes I-I, II-II and III-III, segments equal to the gains

with appropriate strategy and strategy . Having thus constructed all the strategies of the enemy, we will obtain a family of planes over a triangle (Fig. 2).

For this family, it is also possible to construct a lower payoff bound, as we did in the case, and find a point N on this boundary with the maximum height on the plane . This height will be the price of the game.

The frequencies of the strategies in the optimal strategy will be determined by the coordinates (x, y) points N, namely:

However, such a geometric construction, even for the case, is not easy to implement and requires a great investment of time and imagination. In the general case of the game, however, it is transferred to -dimensional space and loses all clarity, although the use of geometric terminology in some cases may be useful. When solving games in practice, it is more convenient to use not geometric analogies, but computational analytical methods, especially since these methods are the only ones suitable for solving problems on computers.

All these methods are essentially reduced to solving the problem by successive trials, but ordering the sequence of trials allows you to build an algorithm that leads to a solution in the most economical way.

Here we briefly dwell on one computational method for solving games - on the so-called "linear programming" method.

To do this, we first give a general statement of the problem of finding a solution to the game . Let the game be given t player strategies BUT and n player strategies AT and the payoff matrix is ​​given

It is required to find a solution to the game, i.e., two optimal mixed strategies of players A and B

where (some of the numbers and can be equal to zero).

Our optimal strategy S*A should provide us with a payoff not less than , for any behavior of the enemy, and a payoff equal to , for his optimal behavior (strategy S*B).Similarly strategy S*B must provide the enemy with a loss no greater than , for any of our behavior and equal to for our optimal behavior (strategy S*A).

The value of the game in this case is unknown to us; we will assume that it is equal to some positive number. Assuming this, we do not violate the generality of reasoning; in order to be > 0, it is obviously sufficient that all elements of the matrix be non-negative. This can always be achieved by adding a sufficiently large positive value L to the elements; in this case, the cost of the game will increase by L, and the solution will not change.

Let us choose our optimal strategy S* A . Then our average payoff for the opponent's strategy will be equal to:

Our optimal strategy S*A has the property that, for any behavior of the opponent, it provides a gain no less than ; therefore, any of the numbers cannot be less than . We get a number of conditions:

(1)

Divide inequalities (1) by a positive value and denote:

Then condition (1) can be written as

(2)

where are non-negative numbers. Because quantities satisfy the condition

We want to make our guaranteed win as high as possible; Obviously, in this case, the right side of equality (3) takes the minimum value.

Thus, the problem of finding a solution to the game is reduced to the following mathematical problem: define non-negative quantities satisfying conditions (2), so that their sum

was minimal.

Usually, when solving problems related to finding extreme values ​​(maximums and minima), the function is differentiated and the derivatives are equated to zero. But such a technique is useless in this case, since the function Ф, which need minimize, is linear, and its derivatives with respect to all arguments are equal to one, i.e., they do not vanish anywhere. Consequently, the maximum of the function is reached somewhere on the border of the region of change of the arguments, which is determined by the requirement of non-negativity of the arguments and conditions (2). The method of finding extreme values ​​using differentiation is also unsuitable in those cases when the maximum of the lower (or minimum of the upper) payoff boundary is determined for the solution of the game, as we did. for example, they did when solving games. Indeed, the lower boundary is made up of sections of straight lines, and the maximum is reached not at the point where the derivative is equal to zero (there is no such point at all), but at the boundary of the interval or at the point of intersection of straight sections.

To solve such problems, which are quite common in practice, a special apparatus has been developed in mathematics. linear programming.

The linear programming problem is posed as follows.

Given a system of linear equations:

(4)

It is required to find non-negative values ​​of quantities satisfying conditions (4) and at the same time minimizing the given homogeneous linear function of quantities (linear form):

It is easy to see that the game theory problem posed above is a particular case of the linear programming problem for

At first glance, it may seem that conditions (2) are not equivalent to conditions (4), since instead of equal signs they contain inequality signs. However, it is easy to get rid of inequality signs by introducing new fictitious non-negative variables and writing conditions (2) in the form:

(5)

The form Ф, which must be minimized, is equal to

The linear programming apparatus allows, by a relatively small number of successive samples, to select the values , satisfying the requirements. For greater clarity, we will demonstrate here the use of this apparatus directly on the material of solving specific games.

Types of mathematical models

Depending on what means, under what conditions and in relation to what objects of cognition, the ability of models to reflect reality is realized, their great diversity arises, and with it - classifications. By generalizing the existing classifications, we single out the basic models according to the applied mathematical apparatus, on the basis of which special models are developed (Figure 8.1).

Figure 8.1 - Formal classification of models

Mathematical models display the studied objects (processes, systems) in the form of explicit functional relationships: algebraic equalities and inequalities, integral and differential, finite-difference and other mathematical expressions (the distribution law of a random variable, regression models, etc.), as well as relations mathematical logic.

Depending on the two fundamental features of building a mathematical model - the type of description of cause-and-effect relationships and their changes over time - there are deterministic and stochastic, static and dynamic models (Figure 8.2).

The purpose of the diagram shown in the figure is to display the following features:

1) mathematical models can be both deterministic and stochastic;

2) deterministic and stochastic models can be both static and dynamic.

The mathematical model is called deterministic (deterministic), if all its parameters and variables are uniquely determined values, and the condition of complete certainty of information is also satisfied. Otherwise, under conditions of information uncertainty, when the parameters and variables of the model are random variables, the model is called stochastic (probabilistic).

Figure 8.2 - Classes of mathematical models

The model is called dynamic if at least one variable changes over time periods, and static if the hypothesis is accepted that the variables do not change over time.

In the simplest case balance models act in the form of a balance equation, where the sum of any receipts is located on the left side, and the expenditure side is also in the form of a sum on the right side. For example, in this form the annual budget of the organization is presented.

On the basis of statistical data, not only balance, but also correlation-regression models can be built.

If the function Y depends not only on the variables x 1 , x 2 , ... x n , but also on other factors, the relationship between Y and x 1 , x 2 , ... x n is inaccurate or correlational, in contrast to the exact or functional relationship. Correlation, for example, in most cases are the connections observed between the output parameters of the OPS and the factors of its internal and external environment (see topic 5).

Correlation-regression models obtained in the study of the influence of a whole complex of factors on the value of a particular feature by using a statistical apparatus. In this case, the task is not only to establish a correlation relationship, but also to express this relationship analytically, that is, to select equations that describe this correlation dependence (regression equation).

To find the numerical value of the parameters of the regression equation, the least squares method is used. The essence of this method is to choose such a line in which the sum of the squared deviations of the ordinates Y of individual points from it would be the smallest.

Correlation-regression models are often used in the study of phenomena when it becomes necessary to establish a relationship between the corresponding characteristics in two or more series. In this case, pairwise and multiple linear regression of the form

y \u003d a 1 x 1 + a 2 x 2 + ... + a n x n + b.

As a result of applying the least squares method, the values ​​of the parameters a or a 1 , a 2 , …, a n and b are set, and then estimates of the approximation accuracy and significance of the resulting regression equation are performed.

In a special group are graph-analytical models . They use different graphics and therefore have good visibility.

Graph theory - one of the theories of discrete mathematics, studies graphs, which are understood as a set of points and lines connecting them. A graph is an independent mathematical object (first introduced by Koenig D.). On the basis of graph theory, tree-like and network models are most often built.

A tree model (tree) is an undirected connected graph that does not contain loops and cycles. An example of such a model is a goal tree.

Network models are widely used in work management. Network models (graphs) reflect the sequence of work and the duration of each work (Figure 8.3).

Figure 8.3 - Network model of work performance

Each line of a network diagram is some kind of work. The number next to it means the duration of its execution.

Network models allow you to find the so-called critical path and optimize the time schedule for the production of work under restrictions on other resources.

Network models can be deterministic and stochastic. In the latter case, the duration of the work is given by the laws of distribution of random variables.

Optimization Models serve to determine the optimal trajectory for the system to achieve the set goal when some restrictions are imposed on the control of its behavior and movement. In this case, optimization models describe various kinds of problems of finding the extremum of some objective function (optimization criterion).

To identify the best way to achieve the goal of management in conditions of limited resources - technical, material, labor and financial - methods of research operations are used. These include methods of mathematical programming (linear and non-linear, integer, dynamic and stochastic programming), analytical and probabilistic-statistical methods, network methods, methods of queuing theory, game theory (theory of conflict situations), etc.

Optimization models are used for volumetric and scheduling, inventory management, distribution of resources and work, replacement, parameterization and standardization of equipment, distribution of commodity supply flows on the transport network and other management tasks.

One of the main achievements of the theory of operations research is the typification of control models and problem solving methods. For example, to solve a transport problem, depending on its dimension, standard methods have been developed - the Vogel method, the potential method, the simplex method. Also, when solving the inventory management problem, depending on its formulation, analytical and probabilistic-statistical methods, methods of dynamic and stochastic programming can be used.

In management, special importance is attached to network methods of planning. These methods made it possible to find a new and very convenient language for describing, modeling and analyzing complex multi-stage works and projects. In operations research, a significant place is given to improving the control of complex systems using the methods of queuing theory (see Section 8.3) and the apparatus of Markov processes.

Models of Markov stochastic processes- a system of differential equations describing the functioning of a system or its processes as a set of ordered states on a certain trajectory of the system's behavior. This class of models is widely used in mathematical modeling of the functioning of complex systems.

Game theory models serve to select the optimal strategy under conditions of limited random information or complete uncertainty.

A game is a mathematical model of a real conflict situation, the resolution of which is carried out according to certain rules, algorithms that describe a certain strategy for the behavior of a person making a decision under conditions of uncertainty.

There are "games with nature" and "games with the enemy". Based on the situation, methods and criteria for evaluating decision-making are determined. So, when “playing with nature”, the following criteria are used: Laplace, maximin (Wald criterion) and minimax, Hurwitz and Savage and a number of other algorithmic rules. In “games with the enemy”, payoff matrices, maximin and minimax criteria, as well as special mathematical transformations are used for making decisions due to the fact that the decision maker is opposed by an unfriendly opponent.

The considered types of mathematical models do not cover all their possible diversity, but only characterize individual types depending on the accepted aspect of the classification. V.A. Kardash made an attempt to create a system for classifying models according to four aspects of detailing (Figure 8.4).

A - models without spatial differentiation of parameters;

B - models with spatial differentiation of parameters

Figure 8.4 - Classification of models according to four aspects of detailing

With the development of computing tools, one of the most common decision-making methods is a business game, which is a numerical experiment with the active participation of a person. There are hundreds of business games. They are used to study a number of problems of management, economics, organization theory, psychology, finance and trade.

What is a mathematical model?

The concept of a mathematical model.

A mathematical model is a very simple concept. And very important. It is mathematical models that connect mathematics and real life.

In simple terms, a mathematical model is a mathematical description of any situation. And that's it. The model can be primitive, it can be super complex. What is the situation, what is the model.)

In any (I repeat - in any!) business, where you need to calculate something and calculate - we are engaged in mathematical modeling. Even if we don't know it.)

P \u003d 2 CB + 3 CM

This record will be the mathematical model of the expenses for our purchases. The model does not take into account the color of the packaging, expiration date, politeness of cashiers, etc. That's why she model, not a real purchase. But the costs, ie. what we need- we'll know for sure. If the model is correct, of course.

It is useful to imagine what a mathematical model is, but this is not enough. The most important thing is to be able to build these models.

Compilation (construction) of a mathematical model of the problem.

To compose a mathematical model means to translate the conditions of the problem into a mathematical form. Those. turn words into an equation, formula, inequality, etc. Moreover, turn it so that this mathematics strictly corresponds to the original text. Otherwise, we will end up with a mathematical model of some other problem unknown to us.)

More specifically, you need

There are an infinite number of tasks in the world. Therefore, to offer clear step-by-step instructions for compiling a mathematical model any tasks are impossible.

But there are three main points that you need to pay attention to.

1. In any task there is a text, oddly enough.) This text, as a rule, has explicit, open information. Numbers, values, etc.

2. In any task there is hidden information. This is a text that assumes the presence of additional knowledge in the head. Without them - nothing. In addition, mathematical information is often hidden behind simple words and ... slips past attention.

3. In any task there must be given communication between data. This connection can be given in clear text (something equals something), or it can be hidden behind simple words. But simple and clear facts are often overlooked. And the model is not compiled in any way.

I must say right away that in order to apply these three points, the problem has to be read (and carefully!) several times. The usual thing.

And now - examples.

Let's start with a simple problem:

Petrovich returned from fishing and proudly presented his catch to his family. Upon closer examination, it turned out that 8 fish come from the northern seas, 20% of all fish come from the southern seas, and not a single one from the local river where Petrovich fished. How many fish did Petrovich buy in the Seafood store?

All these words need to be turned into some kind of equation. To do this, I repeat, establish a mathematical relationship between all the data of the problem.

Where to start? First, we will extract all the data from the task. Let's start in order:

Let's focus on the first point.

What is here explicit mathematical information? 8 fish and 20%. Not a lot, but we don't need a lot.)

Let's pay attention to the second point.

Are looking for covert information. She is here. These are the words: "20% of all fish". Here you need to understand what percentages are and how they are calculated. Otherwise, the task cannot be solved. This is exactly the additional information that should be in the head.

There is also here mathematical information that is completely invisible. it task question: "How many fish did you buy... It's also a number. And without it, no model will be compiled. Therefore, let us denote this number by the letter "X". We do not yet know what x is equal to, but such a designation will be very useful to us. For more information on what to take for x and how to handle it, see the lesson How to solve math problems? Let's write it right away:

x pieces - the total number of fish.

In our problem, southern fish are given as a percentage. We need to translate them into pieces. What for? Then what's in any the task of the model should be in the same sizes. Pieces - so everything is in pieces. If we are given, let's say hours and minutes, we translate everything into one thing - either only hours, or only minutes. It doesn't matter what. It is important to all values ​​were the same.

Back to disclosure. Whoever does not know what a percentage is will never reveal, yes ... And who knows, he will immediately say that the percentages here of the total number of fish are given. We don't know this number. Nothing will come of it!

The total number of fish (in pieces!) is not in vain with the letter "X" designated. It will not work to count the southern fish in pieces, but can we write it down? Like this:

0.2 x pieces - the number of fish from the southern seas.

Now we have downloaded all the information from the task. Both explicit and covert.

Let's pay attention to the third point.

Are looking for mathematical connection between task data. This connection is so simple that many do not notice it... This often happens. Here it is useful to simply write down the collected data in a bunch, and see what's what.

What do we have? There is 8 pieces northern fish, 0.2 x pieces- southern fish and x fish- total. Is it possible to link this data somehow together? Yes Easy! Total number of fish equals sum of southern and northern! Well, who would have thought ...) So we write down:

x = 8 + 0.2x

This will be the equation mathematical model of our problem.

Please note that in this problem we are not asked to fold anything! It was we ourselves, out of our heads, who realized that the sum of the southern and northern fish would give us the total number. The thing is so obvious that it slips past attention. But without this evidence, a mathematical model cannot be compiled. Like this.

Now you can apply all the power of mathematics to solve this equation). This is what the mathematical model was designed for. We solve this linear equation and get the answer.

Answer: x=10

Let's make a mathematical model of another problem:

Petrovich was asked: "How much money do you have?" Petrovich wept and answered: “Yes, just a little bit. If I spend half of all the money, and half of the rest, then I will have only one bag of money left ...” How much money does Petrovich have?

Again, we work point by point.

1. We are looking for explicit information. You won't find it right away! Explicit information is one money bag. There are some other halves... Well, we'll sort it out in the second paragraph.

2. We are looking for hidden information. These are halves. What? Not very clear. Looking for more. There is another issue: "How much money does Petrovich have?" Let's denote the amount of money by the letter "X":

X- all the money

And read the problem again. Already knowing that Petrovich X of money. This is where the halves work! We write down:

0.5 x- half of all money.

The remainder will also be half, i.e. 0.5 x. And half of the half can be written like this:

0.5 0.5 x = 0.25x- half of the remainder.

Now all the hidden information is revealed and recorded.

3. We are looking for a connection between the recorded data. Here you can simply read the sufferings of Petrovich and write them down mathematically):

If I spend half of all the money...

Let's write down this process. All money - X. Half - 0.5 x. To spend is to take away. The phrase becomes:

x - 0.5 x

yes half of the rest...

Subtract another half of the remainder:

x - 0.5 x - 0.25 x

then only one bag of money will remain with me ...

And there is equality! After all the subtractions, one bag of money remains:

x - 0.5 x - 0.25x \u003d 1

Here it is, the mathematical model! This is again a linear equation, we solve, we get:

Question for consideration. Four is what? Ruble, dollar, yuan? And in what units do we have money in the mathematical model? In bags! So four bag Petrovich's money. It's not bad too.)

The tasks are, of course, elementary. This is specifically to capture the essence of drawing up a mathematical model. In some tasks, there may be much more data in which it is easy to get confused. This often happens in the so-called. competency tasks. How to pull mathematical content out of a pile of words and numbers is shown with examples

One more note. In classical school problems (pipes fill the pool, boats are sailing somewhere, etc.), all the data, as a rule, is chosen very carefully. There are two rules:
- there is enough information in the problem to solve it,
- there is no extra information in the task.

This is a hint. If there is some unused value in the mathematical model, think about whether there is an error. If there is not enough data in any way, most likely, not all hidden information has been revealed and recorded.

In competence and other life tasks, these rules are not strictly observed. I don't have a hint. But such problems can also be solved. Unless, of course, practice on the classic.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

First level

Mathematical models at the OGE and the Unified State Examination (2019)

The concept of a mathematical model

Imagine an airplane: wings, fuselage, tail, all this together - a real huge, immense, whole airplane. And you can make a model of an airplane, small, but everything is real, the same wings, etc., but compact. So is the mathematical model. There is a text problem, cumbersome, you can look at it, read it, but not quite understand it, and even more so it is not clear how to solve it. But what if we make a small model of it, a mathematical model, out of a large verbal problem? What does mathematical mean? So, using the rules and laws of mathematical notation, remake the text into a logically correct representation using numbers and arithmetic signs. So, A mathematical model is a representation of a real situation using a mathematical language.

Let's start simple: The number is greater than the number by. We need to write it down without using words, just the language of mathematics. If more by, then it turns out that if we subtract from, then the very difference of these numbers will remain equal. Those. or. Got the gist?

Now it’s more complicated, now there will be a text that you should try to present in the form of a mathematical model, until you read how I will do it, try it yourself! There are four numbers: , and. A product and more products and twice.

What happened?

In the form of a mathematical model, it will look like this:

Those. the product is related to as two to one, but this can be further simplified:

Well, with simple examples, you get the point, I guess. Let's move on to full-fledged tasks in which these mathematical models also need to be solved! Here is the task.

Mathematical model in practice

Task 1

After rain, the water level in the well may rise. The boy measures the time of falling small pebbles into the well and calculates the distance to the water using the formula, where is the distance in meters and is the time of falling in seconds. Before the rain, the time for the fall of the pebbles was s. How much must the water level rise after the rain in order for the measured time to change to s? Express your answer in meters.

Oh God! What formulas, what kind of well, what is happening, what to do? Did I read your mind? Relax, in tasks of this type, conditions are even more terrible, the main thing to remember is that in this task you are interested in formulas and relationships between variables, and what all this means in most cases is not very important. What do you see useful here? I personally see. The principle of solving these problems is as follows: you take all known quantities and substitute them.But sometimes you have to think!

Following my first advice, and substituting all the known ones into the equation, we get:

It was I who substituted the time of the second, and found the height that the stone flew before the rain. And now we need to count after the rain and find the difference!

Now listen to the second advice and think about it, the question clarifies, "how much the water level must rise after rain in order for the measured time to change by s." You need to figure it out right away, soooo, after the rain the water level rises, which means that the time for the stone to fall to the water level is less, and here the ornate phrase “so that the measured time changes” takes on a specific meaning: the fall time does not increase, but is reduced by the specified seconds. This means that in the case of a throw after the rain, we just need to subtract c from the initial time c, and we get the equation for the height that the stone will fly after the rain:

And finally, in order to find how much the water level should rise after the rain, so that the measured time changes by s, you just need to subtract the second from the first height of the fall!

We get the answer: per meter.

As you can see, there is nothing complicated, most importantly, don’t bother too much where such an incomprehensible and sometimes complex equation came from in the conditions and what everything in it means, take my word for it, most of these equations are taken from physics, and there the jungle is worse than in algebra. It sometimes seems to me that these tasks were invented to intimidate the student at the exam with an abundance of complex formulas and terms, and in most cases they require almost no knowledge. Just carefully read the condition and substitute the known values ​​in the formula!

Here is another problem, no longer in physics, but from the world of economic theory, although knowledge of sciences other than mathematics is again not required here.

Task 2

The dependence of the volume of demand (units per month) for the products of a monopoly enterprise on the price (thousand rubles) is given by the formula

The company's monthly revenue (in thousand rubles) is calculated using the formula. Determine the highest price at which the monthly revenue will be at least a thousand rubles. Give the answer in thousand rubles.

Guess what I'll do now? Yeah, I'll start substituting what we know, but, again, you still have to think a little. Let's go from the end, we need to find at which. So, there is, equal to some, we find what else it is equal to, and it is equal, and we will write it down. As you can see, I don’t particularly bother about the meaning of all these quantities, I just look from the conditions, what is equal to what, that’s what you need to do. Let's return to the task, you already have it, but as you remember, from one equation with two variables, none of them can be found, what to do? Yeah, we still have an unused particle in the condition. Here, there are already two equations and two variables, which means that now both variables can be found - great!

Can you solve such a system?

We solve by substitution, we have already expressed it, which means we will substitute it into the first equation and simplify it.

It turns out here is such a quadratic equation: , we solve, the roots are like this, . In the task, it is required to find the highest price at which all the conditions that we took into account when we compiled the system will be met. Oh, it turns out that was the price. Cool, so we found the prices: and. The highest price, you say? Okay, the largest of them, obviously, we write it in response. Well, is it difficult? I think not, and you don’t need to delve into it too much!

And here's a frightening physics for you, or rather, another problem:

Task 3

To determine the effective temperature of stars, the Stefan–Boltzmann law is used, according to which, where is the radiant power of the star, is a constant, is the surface area of ​​the star, and is the temperature. It is known that the surface area of ​​a certain star is equal, and the power of its radiation is equal to W. Find the temperature of this star in degrees Kelvin.

Where is it clear? Yes, the condition says what is equal to what. Previously, I recommended that all unknowns be immediately substituted, but here it is better to first express the unknown sought. Look how simple everything is: there is a formula and they are known in it, and (this is the Greek letter "sigma". In general, physicists love Greek letters, get used to it). The temperature is unknown. Let's express it as a formula. How to do it, I hope you know? Such assignments for the GIA in grade 9 usually give:

Now it remains to substitute numbers instead of letters on the right side and simplify:

Here's the answer: degrees Kelvin! And what a terrible task it was!

We continue to torment problems in physics.

Task 4

The height above the ground of a ball tossed up changes according to the law, where is the height in meters, is the time in seconds that has elapsed since the throw. How many seconds will the ball be at a height of at least three meters?

Those were all the equations, but here it is necessary to determine how much the ball was at a height of at least three meters, which means at a height. What are we going to make? Inequality, yes! We have a function that describes how the ball flies, where is exactly the same height in meters, we need the height. Means

And now you just solve the inequality, most importantly, do not forget to change the inequality sign from more or equal to less or equal when you multiply by both parts of the inequality in order to get rid of the minus in front.

Here are the roots, we build intervals for inequality:

We are interested in the interval where the sign is minus, since the inequality takes negative values ​​there, this is from to both inclusive. And now we turn on the brain and think carefully: for inequality, we used an equation that describes the flight of the ball, it somehow flies along a parabola, i.e. it takes off, reaches a peak and falls, how to understand how long it will be at a height of at least meters? We found 2 turning points, i.e. the moment when it soars above meters and the moment when it reaches the same mark while falling, these two points are expressed in our form in the form of time, i.e. we know at what second of the flight it entered the zone of interest to us (above meters) and into which it left it (fell below the meter mark). How many seconds was he in this zone? It is logical that we take the time of exit from the zone and subtract from it the time of entry into this zone. Accordingly: - so much he was in the zone above meters, this is the answer.

You are so lucky that most of the examples on this topic can be taken from the category of problems in physics, so catch one more, it is the final one, so push yourself, there is very little left!

Task 5

For a heating element of a certain device, the temperature dependence on the operating time was experimentally obtained:

Where is the time in minutes. It is known that at a temperature of the heating element above the device may deteriorate, so it must be turned off. Find the maximum time after the start of work to turn off the device. Express your answer in minutes.

We act according to a well-established scheme, everything that is given, we first write out:

Now we take the formula and equate it to the temperature value to which the device can be heated as much as possible until it burns out, that is:

Now we substitute numbers instead of letters where they are known:

As you can see, the temperature during operation of the device is described by a quadratic equation, which means that it is distributed along a parabola, i.e. the device heats up to a certain temperature, and then cools down. We received answers and, therefore, during and during minutes of heating, the temperature is critical, but between and minutes it is even higher than the limit!

So, you need to turn off the device after a minute.

MATHEMATICAL MODELS. BRIEFLY ABOUT THE MAIN

Most often, mathematical models are used in physics: after all, you probably had to memorize dozens of physical formulas. And the formula is the mathematical representation of the situation.

In the OGE and the Unified State Examination there are tasks just on this topic. In the USE (profile) this is task number 11 (formerly B12). In the OGE - task number 20.

The solution scheme is obvious:

1) From the text of the condition, it is necessary to “isolate” useful information - what we write in physics problems under the word “Given”. This useful information is:

• Formula
• Known physical quantities.

That is, each letter from the formula must be assigned a certain number.

2) Take all the known quantities and substitute them into the formula. The unknown value remains as a letter. Now you just need to solve the equation (usually quite simple), and the answer is ready.

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

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In conclusion...

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