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Examples of functional dependence in real life. Functional dependencies in real processes and phenomena. Multiple Argument Functions

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Task No. 16

Interpretation of a real dependence graph.

Examples of dependency graphs surrounding real processes; reading and interpretation.

This task tests the ability to analyze graphs of functions that describe the dependence of m / y quantities.

Theory.

Definition. 1. Function is law , by which each element value x from some set X a single element is matched y from many.

2. The dependence of the variable y on the variable x is called a function if each value of x corresponds to a single value of y. The variable x is called the independent variable or argument, and the variable y is called the dependent variable. The value of y corresponding to a given value of x is called the value of the function. Write down: y = f (x). The letter f denotes this function, that is, the functional relationship between the variables x and y; f(x) is function value , corresponding to the value argument X. They also say that f(x) is the value of the function at point x . All values that the independent variable takes form domain of a function . All values that the function f (x) takes (for x belonging to its domain of definition) form.

function range

Methods for specifying a function
To specify a function, you must specify a way in which, for each argument value, the corresponding function value can be found. The most common way to specify a function is using the formula y = f (x),

where f (x) is some expression with variable x. In this case, they say that the function is given by a formula or that the function is given analytically.

In practice, the tabular method of specifying a function is often used. With this method, a table is provided indicating the function values for the argument values available in the table.

Function graph. Let the function be given analytically by the formula y = f (x). If on coordinate plane mark all points that have the following property: the abscissa of the point belongs to the domain of definition of the function, and the ordinate is equal to the corresponding value of the function, then.

In practice, to construct a graph of a function, they compile a table of function values for certain values of the argument, plot the corresponding points on the plane and connect the resulting points with a line. In this case, it is assumed that the graph of the function is a smooth line, and the found points quite accurately show the progress of the change in the function. The advantage of a graphic image over a tabular one is its clarity and easy visibility; The disadvantage is the low degree of accuracy. A successful choice of scales is of great practical importance. Using the graph, you can find (approximately) the value of the function for those argument values that are not included in the table.

Functional description of real processes The key to a small mathematical problem The golden rule of mechanics Information boom Star chart Mathematical portraits of proverbs

Functional description of real processes Why are there no animals of any size? Why, for example, are there no elephants three times taller than exist, but of the same proportions? Our answer is this: if the elephant were three times larger, its weight would then increase twenty-seven times the size of a cube, but the cross-sectional area of the bones and, consequently, their strength would only increase nine times the size of a square. The strength of the bones would no longer be enough to withstand the enormously increased weight. Such an elephant would be crushed by its own weight.

The reasoning is based on two strict mathematical dependencies. The first establishes a correspondence between the sizes of similar bodies and their volumes: the volume changes like a cube of size. The second connects the sizes of similar figures and their areas: the area changes like the square of the size. With this expressive example, we want to start a conversation about numerical functions of a numerical argument, which can be used to describe real processes.

The key to a small mathematical problem Note that not every functional dependence can be expressed short formula It is no coincidence that we provide you with the key to the door lock as an example: now it will literally serve as the key to a small mathematical problem to which the conversation about functions leads us. Do you know how to open a door lock with such a key? What happens inside this locksmith -mechanical device when you insert the key into the keyhole and make the required number of turns?

In order for the lock to open, you need to turn the drum in which the hole is made. But this is prevented by pins standing in a close formation inside the well, sliding up and down. Each of the pins must be raised to such a height that their upper ends are flush with the surface of the drum. If they move beyond it, they will enter the slot of the clip located exactly above the correspondence well; if they do not reach the surface of the drum, then the pins located there will slide into the keyhole from the slot in the holder. In both cases, the rotation of the drum will be stopped.

Pins in keyhole lifts the key being pushed into it. In this case, the height of each pin, being added to the height of the key profile at the corresponding point, should add up to the diameter of the drum. Only then will he turn it around. Well, what does the function have to do with it? Moreover, from the point of view of a mathematician, all this mechanics is nothing more than the operation of adding two functions. One of them is the key profile. The other is the line that outlines the top ends of the pins when the lock is locked.

The operation of adding functions consists of adding the value of another to the value of one function at each point from the common domain of their definition. This determines what value the function, called the sum of the two original ones, has at a given point. . The secret of the door lock is that as a result of the addition of two functions, expressed by the profile of the key and the structure of the pins, a constant function is obtained, the constant value of which is equal to the diameter of the drum

The Golden Rule of Mechanics Everything richest family mechanisms surrounding modern man, began once with seven simple machines. The ancients knew the lever, block, wedge, gate, screw, inclined plane and gears. These devices, simple according to modern ideas, multiplied human strength. But no matter how many times you win in strength, the same number of times you lose in distance. So says the golden rule of mechanics, which contains the theory of seven simple machines

The graph shown on this page is a visual expression of the famous rule. The horizontal axis represents the force with which, for example, it is necessary to press on the lever arm in order to raise a given load to a given height. Along the vertical axis is the distance that the point of application of force will travel. A line expressing such a functional relationship is called a hyperbola. The law of inverse proportionality also looks at us from the radio dial. You turn the tuning knob and the needle moves along a scale on which there are two rows of numbers - meters and megahertz, wavelength and frequency. The wavelength increases, the frequency decreases. But take a closer look: with any shift of the arrow, by the same factor the wavelength increases, the frequency decreases by the same amount.

A graph of a hyperbola can be seen on a physicist's laboratory bench demonstrating the phenomena of capillarity. The tripod contains several thin glass tubes, arranged in ascending order of diameter. It is known that in a thin channel the wetting liquid rises higher, the smaller its diameter. Therefore, in the narrowest channel the liquid rose the highest, in another channel, the diameter of which is twice as large, - twice as low, in the third, which is three times thicker than the first, - three times lower, and so on. Now let’s lower into the same liquid a wedge formed by two glass plates closed along a vertical edge. The liquid will rush into the narrow gap between the glasses, as if into a capillary. The height of its rise will be determined by the width of the gap. And it increases evenly as it moves away from the tip of the wedge. Therefore, the free surface of the liquid clearly outlines a hyperbola - a graph of inverse proportionality.

Information boom Nowadays there is a lot of talk about the information boom. The flow of information is overwhelming: they claim that its quantity doubles every ten years. Let us depict this process clearly, in the form of a graph of a certain function.

Let us take the amount of information in a certain year as a unit. Since this value will serve as the beginning of further constructions, we will put it above the origin of coordinates in which the graph will be plotted, along the vertical axis. We will construct a segment twice as large above the unit mark of the horizontal axis, considering that this mark corresponds to the first ten years. We will construct another twice as large segment above the point “two”, corresponding to the second ten, and another twice as large - above the point “three”. Decade after decade - the argument values we have chosen will line up along the horizontal axis in order of uniform increase, according to the law arithmetic progression: one two three four. . . The function values will be deposited above them, doubling each time, according to the law of geometric progression: two, four, eight, sixteen. . .

But what if we look at how the flow of information increased until the year that is considered the initial one? Just as evenly, setting aside unit by unit, we will walk along the x-axis to the left of the origin and over the delayed values of the argument, and we will plot the function values on the graph in descending order - doubling with each step. Now let’s connect all the plotted points with a continuous smooth line - after all, the amount of information grows from decade to decade smoothly, and not in leaps. Before us is a graph of the so-called exponential function.

Star chart How many stars are there in the sky? One of the first to try to accurately answer this question was the ancient Greek astronomer Hipparchus. During his lifetime, a new star appeared in the constellation Scorpio. Hipparchus was shocked: the stars are mortal, they, like people, are born and die. And so that future researchers could monitor the rise and fall of stars, Hipparchus compiled his star catalogue. He counted about a thousand stars and divided them into six groups according to their apparent brightness. Hipparchus called the brightest stars of the first magnitude, noticeably less bright - the second, even less bright - the third, and so on in order of uniform decrease in apparent brightness - to stars barely visible to the naked eye, which were assigned the sixth magnitude.

When scientists had sensitive instruments for measuring light at their disposal, it became possible to accurately determine the brightness of stars. It became possible to compare how well the traditional distribution of stars by apparent brightness, made by eye, corresponds to the data of such measurements. We will summarize estimates of both types on one graph. From each of the six groups into which Hipparchus divided the stars, let’s take one typical representative. Along the vertical axis we will plot the brightness of the star in Hipparchus units, that is, its magnitude, and along the horizontal axis - instrument readings. For the scale unit of the horizontal axis we will take the brilliance of the star “b Tauri”, standing in the middle of the series of representatives of the stellar sun. The marks on the horizontal axis are unevenly spaced. Objective (device) and subjective (eye) characteristics of gloss are not proportional to each other.

With each step along the magnitude scale, the instrument records an increase in brightness not by the same amount, as it might seem, but by about two and a half times. Figuratively speaking, the eye compares light sources by their brilliance, asking the question “how many times?” ”, and not with the question “how much?” ". We note not an absolute, but a relative increase in brightness. And when it seems to us that it is increasing or decreasing evenly, in reality we are walking along its scale with ever more sweeping steps, while covering a truly gigantic range: light sources, the weakest and the most powerful, differ in brilliance in the millions, perceived by the human eye.

Precisely because of the described physiological feature the stars, burning brightly in the night sky, are not visible during the day, drowning in the dazzling brilliance of the sun scattered across the sky. And here and there, the radiance of the stars gives the same addition to the background light. However, in the first case (at night) this addition is large compared to the flickering of the sky, while in the second (day) it constitutes a very insignificant fraction of the sun's brightness (less than a billionth even for the most bright stars). That is why the voice of the soloist, when his singing is picked up by the choir, is drowned in the polyphonic sound. The essence of the functional dependence that we described using the example of vision and hearing is that an increase in the argument by the same number of times always corresponds to the same increment in the function. When the argument changes according to the law of geometric progression, the function changes according to the law of arithmetic progression.

What is the name of the function with which we became acquainted with the starry sky? The ordinates of the selected points on the graph are logarithms of abscissas taken to base 2, 5. This function is called logarithmic

Mathematical portraits of proverbs Modern mathematics knows many functions, and each has its own unique appearance, just as the unique appearance of each of the billions of people living on Earth is unique. However, despite all the differences between one person and another, each person has hands and a head, ears and a mouth. In the same way, the appearance of each function can be imagined as composed of a set of characteristic details. They reveal the basic properties of functions.

Functions are mathematical portraits of stable patterns cognizable by humans. To illustrate the characteristic properties of functions, it seemed natural to us to turn to proverbs. After all, proverbs are also a reflection of stable patterns, verified by the centuries-old experience of the people.

“A horse does not gallop higher than its measure” If we imagine the trajectory of a galloping horse as a graph of some function, then the height of the jumps, in full accordance with the proverb, will be limited from above by some “measure”. This will be the familiar graph of the sine function.

“Over-sowing is worse than under-sowing” The yield only grows for a certain time along with the sowing density, then it decreases, because with excessive density the shoots begin to choke each other. This pattern will become especially clear if you depict it in a graph where the yield is presented as a function of sowing density. The harvest is maximum when the field is sown in moderation. The maximum is highest value function compared to its values at all neighboring points. It is like the top of a mountain, from which all roads lead only downwards, no matter where you step.

“The further into the forest, the more firewood” You can graph how the amount of firewood increases as you move deeper into the forest - from the edge, where everything was collected a long time ago, to the thickets, where the harvester has never set foot. The graph represents the amount of firewood as a function of path. According to the proverb, this function invariably increases. This property of a function is called monotonic increase.

“You can’t spoil porridge with oil.” The quality of porridge can be considered as a function of the amount of oil in it. According to the proverb, this function does not decrease with the addition of oil. It may be increasing, but it may remain at the same level. This kind of function is called monotonically nondecreasing.

The mathematical categories discussed are naturally divided into two groups. Some describe the behavior of a function in the vicinity of certain characteristic points (maximum, minimum, inflection). Others describe the behavior of a function in certain intervals (convexity, concavity, decrease, increase).

Thank you for your attention! As you already noticed, the font was 28+, I did this especially for those who cannot see from the second desk: D

Function is one of the main general scientific and mathematical concepts, expressing the dependence between variable quantities. This is the law according to which each element value , by which each element value x from some set a single element is matched a single element is matched y from many .

The dependence of a variable y on a variable x is called a function if each value of x corresponds to a single value of y. The variable x is called the independent variable or argument, and the variable y is called the dependent variable. The value of y corresponding to a given value of x is called the value of the function.

Write down: y = f (x). The letter f denotes this function, that is, the functional relationship between the variables x and y; f(x) is the value of the function corresponding to the value of the argument x. They also say that f(x) is the value of the function at the point x. All the values that the independent variable takes form the domain of the function. All values that the function f (x) takes (with x belonging to its domain of definition) form the range of values of the function.

function range

To specify a function, you must specify a way in which, for each argument value, the corresponding function value can be found. The most common way to specify a function is using the formula y = f (x),

where f (x) is some expression with variable x. In this case, they say that the function is given by a formula or that the function is given analytically.

Let the function be given analytically by the formula y = f (x). If on the coordinate plane we mark all points that have the following property: the abscissa of the point belongs to the domain of definition of the function, and the ordinate is equal to the corresponding value of the function, then the set of points (x; f (x)) is the graph of the function. In physics and technology, functions are often specified graphically, and sometimes a graph is the only available means of specifying a function. Most often this happens when using recording instruments that automatically record changes in one quantity depending on changes in another. As a result, a line is obtained on the device tape that graphically specifies the function recorded by the device.

The function can also be specified in a table. Let's look at examples functional dependence in real life .

Example 1

The table shows the child’s growth during the first 5 months of life:

Having a table of values for the functional dependence of height on age, you can construct a graph point by point:

Example 2

Here shining example function specified graphically. On the graph you can see the maximum and minimum, fragments of a linear function, smoothing of lines, etc.

Cardiogram - a graph of the heart.

Cardiogram is a recording of human heart contractions, which is carried out using some instrumental method. During contraction, the heart moves within chest, it rotates around its axis from left to right.

The essence of electrography is to record potential differences over time. The curve that shows us these changes is the cardiogram. The device that records this curve is called an electrocardiograph. A cardiac cardiogram shows the excitation of the heart and its contraction. When taking a cardiogram, special electrodes are attached to the human body, thanks to which the device receives the necessary data.

The essence of signal processing this study is to diagnose existing problems in the functioning of the heart muscles, using various analytical methods.

Example 3

The transition of a substance from solid to liquid is called melting. In order for a body to begin to melt, it must be heated to a certain temperature. The temperature at which a substance melts is called the melting point of the substance.

Each substance has its own melting point. For some bodies it is very low, for example, for ice. And some bodies have a very high melting point, for example, iron. Melting a crystalline body is a complex process.

The figure shows the ice melting graph known from physics courses.

The graph shows the dependence of the ice temperature on the time it is heated. The vertical axis represents temperature, and the horizontal axis represents time.

The graph shows that initially the ice temperature was -40 degrees. Then they started heating it up. Over time, the temperature increased to 0 degrees. This temperature is considered the melting point of ice. At this temperature, the ice began to melt, but its temperature stopped increasing, although the ice also continued to be heated. Then, when all the ice melted and turned into liquid, the temperature of the water began to increase again. During melting, the body temperature does not change, since all the incoming energy goes to melting. After heating (the peak of the graph), the liquid began to be cooled, the process went in the opposite direction until it solidified.

Let's consider the problem

The tourists went from the camp site to the lake, spent 2 hours there and returned back. Select a graph that describes the distance traveled versus time:

The correct answer will be A., because for two hours the tourists were on the lake, having reached it, and then returned to the camp again, i.e. to the zero reference point.

Back forward

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Item- mathematics

Chapter- "Project"

Form of conduct– a creative mono project with open coordination.

Goals:

Educational aspect: contribute to the generalization of knowledge in the section “Functional dependencies”, expand students’ mathematical understanding of the function and its application in other sciences and Everyday life; to promote the acquisition of specific mathematical knowledge necessary for application in practical activities, for the study of related disciplines, and for continuing education.
Developmental aspect: develop basic methods mental activity students (the ability to analyze, pose and solve problems), form and develop cognitive interest in the subject, develop speech and the ability to convincingly express thoughts, and contribute to the development of students’ independence.
Educational aspect: to cultivate mutual understanding and tolerance, independence, the ability to present oneself, evaluate oneself and others, and lead a team.
Career guidance aspect: contribute to the creation of conditions for the formation of an individual trajectory for the development of students’ professional interests, the development of professional significant qualities personalities (creative, organizational, oratorical).

Equipment: computer, multimedia projector, screen, Internet.

Decor: presentation, creative works of students.

Wise thoughts:

“The greatness of a man lies in his ability to think.”
B.Pascal
“Mathematics is the language that all exact sciences speak.”
N.I.Lobachevsky

Gold words:

Science and labor produce wonderful shoots.
The more you learn, the stronger you will become.
If you read books, you will know everything.

Lesson structure

Lesson stages	Contents of the stage	Material and technical base
1.Organizational and preparatory	Greetings. Checking the attendance of students for classes and their readiness. Activation of students. Formulation of the topic and objectives of the lesson. Setting a learning task for students.	Score sheets
2. Preparation for active educational and cognitive activities	introduction teachers	Lesson format: Computer Multimedia projector
3. Protection design work	Presentation of projects, presentation of final results. Self-assessment and mutual assessment of performances. Assessing the quality of work performed by all students	Project work of students:
4. Summing up	Assessment creative works	Score sheets

During the classes

Introduction.

P tutor:

Good morning dear friends!

Our highlight, “Experience the World,” is all about features. Their significance is great. Functions and the real world are inseparable. They describe phenomena in nature. They establish patterns, help discover laws that serve humanity.

For this lesson, each of you has completed some work, which we will continue now. Gradually revealing the points of the plan that are covered by such beautiful stars, the leaders will lead us to the final stage of the finest hour. (The plan is written on the board in advance)

Teacher:

Dear Guys!

I have passed the stage of routine preparation for this lesson. You worked hard. The tasks that needed to be solved when performing the design work were the following:

establish the significance of the concept of “functional dependence” in real processes and phenomena and in other sciences;
find information from Internet resources related to the topic of the project;
prepare a display of the final product of your work in the form of presentations;

Teacher. Fundamental questions guiding the project “ Can a function describe everything? How to use math skills in your professional activity?”

And in order to answer these questions, you were given the task to systematize and expand the basic knowledge in the “Function” section, to consider and answer the following problematic questions of the project’s educational topic:

What is the role of function in your profession?
And what is the role of function in real life and in the study of other sciences?

During preparation, it was necessary to consider the following training questions:

How did the feature come about?
What real phenomena does it describe?
How is it used in other sciences and in professional activities?
The role of a sine wave in real life?
The role of function in mathematics?

Teacher. It is up to you to evaluate your work. On each table you have a score sheet, you put a grade on your score sheet, and you must take into account:

relevance of the topic
significance of the development
scope and completeness of development
level of creativity
reasoning of the proposed solutions
quality of the report
volume and depth of knowledge on the topic
answers on questions.

Speakers may be asked questions.

Teacher: The lesson today will be held in an unusual form, the role of teachers will be taken on by your fellow students: Dmitry Mastrenko - group TO-13 and Vitaly Chapaev, group TO-11.

Teacher: It's time to demonstrate what you have done. I wish you success! Good luck! We begin.

The main part of the lesson.

Protection of design work. Presentations. Presentation 1. “History of the creation of the function”

Presentation 2. “Function in mathematics”

Presentation 3. “Sine wave in images”

Presentation 4. “Function in the profession”

Presentation 5 “Functional dependencies in other sciences”

Presentation 6 “Function in Economics”

1 student

(Removing the asterisk, reads the words of Galileo written on it):

“It is the function that is the means of mathematical language that allows us to describe the processes of movement and changes inherent in nature.”

Continues. “A function expresses the relationship between variables. Each field of knowledge: chemistry, physics, biology, sociology, etc. has its own objects of study, establishes the properties and relationships between these objects in the real world.”

2 student

The function first entered mathematics under the name “ variable quantity” in the work of the French mathematician and philosopher Rene Descartes in 1637. A complex, very long path of development of the concept of function. A student of group TO-13, Bigvava Daniel, will tell us what great names this concept is associated with.

(Demonstration of presentation).

1 student

Concept functions- one of the main ones in mathematics.

2 student

You often hear this word in math lessons. You build graphs of functions, study the function, find the largest or smallest value of the function. But to understand all these actions, let's determine what role a function plays in mathematics.

1 student

Now many sciences are using mathematical tools. Such functional dependencies, for example, the age of trees, the development of amoeba, and the development of ferns, are studied by the science of biology.

2 student

1 student

Along with other functions, trigonometric functions occupy an important place. A mathematical image of a sinusoid can be obtained by considering the dependence of solar energy on the angle of incidence on a certain section of the plane.

(Removes the second asterisk and reads on it): “Oh sun! Without you there would be no life in the world.” (After a pause he continues): “Let there be light!”

2 student

Functions help describe the processes of mechanical motion of bodies, celestial and terrestrial. Using them, scientists calculate movement trajectories spaceships and solve many technical problems.

1 student

(Removes the asterisk and reads a poem by A.S. Pushkin)

Oh, how many wonderful discoveries we have

Preparing the age of enlightenment!

2 student (Addressing the speaker),

Please tell me, have you made a small discovery for yourself?

Speaker

Yes, my idea of the function has changed. Yes, and in general, I understood why I need to study mathematics.

1 student

There is a lot of talk now about the information boom. The flow of information is overwhelming: they claim that its quantity doubles every ten years. Let us depict this process clearly, in the form of a graph of a certain function.

2 student

1 student

Let us construct another twice as large segment above the point “two”, corresponding to the second ten, and another twice as large above the point “three”. Decade after decade - the argument values we have chosen will line up along the horizontal axis in order of uniform increase, according to the law of arithmetic progression: one, two, three, four... The function values will be deposited above them, doubling each time - according to the law of geometric progression: two , four, eight, sixteen...

2 student

Now let’s connect all the plotted points with a continuous smooth line - after all, the amount of information grows from decade to decade smoothly, and not in leaps. Here is a graph of an exponential function.

1 student

He takes off the star and reads:

2 student

To date.

1 student

What does this have to do with the function?

2 student

We'll find out now.

1 student

(Removes the last star with the words of M.V. Lomonosov):

An abyss has opened and is full of stars;
The stars have no number, the bottom of the abyss.

2nd teacher:

Astronomers compare the brightness of stars using a logarithmic function. Today several mathematical stars have lit up in our sky.

Rewarding.

4. Summing up.

Teacher:

What did you learn during the project work?

During the project implementation we acquired the following specific skills:

freely navigate diverse functional dependencies;
apply acquired knowledge in practice;
put forward hypotheses;
quickly and accurately select the resources needed for work, search on the Internet;
work in various search engines;
formulate the question precisely;
present research results in the form of presentations;
interpret research results;
draw conclusions;
discuss the results of the study, participate in the discussion.

Teacher: The main goal that we set when starting work on the project, I consider achieved. What do you think? You have a coordinate system on your tables, please build a graph of the function that evaluates all the presentations.

Send your good work in the knowledge base is simple. Use the form below

Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

Posted on http://www.allbest.ru/

Examples of functional dependencies

Functionaladdiction- a form of stable relationship between objective phenomena or quantities reflecting them, in which a change in some phenomena causes a certain quantitative change in others. Objectively F. z. manifests itself in the form of laws and relations that have precise quantitative certainty. They can, in principle, be expressed in the form of equations that combine given quantities or phenomena as a function and an argument. F. z. can characterize the relationship:

1) between the properties and states of material objects and phenomena;

2) between the objects, phenomena themselves, or material systems within the framework of an integral system of a higher order;

3) between objective quantitative laws that are in relation to subordination, depending on their generality and scope;

4) between abstract mathematical quantities, sets, functions or structures, regardless of what they express.

Key to a small math problem

Let us note that not every functional dependence can be expressed in a short formula; it is no coincidence that we provide you with the key to the door lock as an example: now it will literally serve as the key to a small mathematical problem to which the conversation about functions leads us. Do you know how to open a door lock with such a key? What happens inside this locksmith-mechanical device when you insert the key into the keyhole and turn the required number of turns?

The pins in the keyhole are lifted by the key being pushed into it. In this case, the height of each pin, being added to the height of the key profile at the corresponding point, should add up to the diameter of the drum. Only then will he turn it around.

Well, what does the function have to do with it? Moreover, from the point of view of a mathematician, all this mechanics is nothing more than the operation of adding two functions. One of them is the key profile. The other is the line that outlines the top ends of the pins when the lock is locked.

The operation of adding functions consists of adding the value of another to the value of one function at each point from the common domain of their definition.

The golden rule of mechanics

The entire rich family of mechanisms surrounding modern man once began with seven simple machines. The ancients knew the lever, block, wedge, gate, screw, inclined plane and gears. These devices, simple according to modern ideas, multiplied human strength. But no matter how many times you win in strength, the same number of times you lose in distance. So says the golden rule of mechanics, which contains the theory of seven simple machines.

The law of inverse proportionality also looks at us from the radio dial. You turn the tuning knob, and the needle moves along a scale on which there are two rows of numbers - meters and megahertz, wavelength and frequency. The wavelength increases, the frequency decreases. But take a closer look: with any shift of the arrow, by the same factor the wavelength increases, the frequency decreases by the same amount.

Information boom

Star chart

How many stars are there in the sky? One of the first to try to accurately answer this question was the ancient Greek astronomer Hipparchus. During his lifetime, a new star appeared in the constellation Scorpio. Hipparchus was shocked: the stars are mortal, they, like people, are born and die. And so that future researchers could monitor the rise and fall of stars, Hipparchus compiled his star catalogue. He counted about a thousand stars and divided them into six groups according to their apparent brightness. Hipparchus called the brightest stars of the first magnitude, noticeably less bright - the second, even less bright - the third, and so on in order of uniform decrease in apparent brightness - to stars barely visible to the naked eye, which were assigned the sixth magnitude.

When scientists had sensitive instruments for measuring light at their disposal, it became possible to accurately determine the brightness of stars. It has become possible to compare how well the traditional distribution of stars by apparent brightness, made by eye, corresponds to the data of such measurements. We will summarize the estimates of both types on one graph. From each of the six groups into which Hipparchus divided the stars, let’s take one typical representative. Along the vertical axis we will plot the brightness of the star in Hipparchus units, that is, its magnitude, and along the horizontal axis - instrument readings. With each step along the magnitude scale, the instrument records an increase in brightness not by the same amount, as it might seem, but by about two and a half times. Figuratively speaking, the eye compares light sources by brightness, asking the question “how many times?”, and not the question “by how much?”. We note not an absolute, but a relative increase in brightness. And when it seems to us that it is increasing or decreasing evenly, in reality we are walking along its scale with increasingly sweeping steps, while covering a truly gigantic range: the light sources, the weakest and the most powerful, differ in brilliance by a million million times, perceived by the human eye .

It is precisely because of the described physiological feature that the stars, shining brightly in the night sky, are not visible during the day, drowning in the dazzling brilliance of the sun scattered across the sky. And here and there, the radiance of the stars gives the same addition to the background light. However, in the first case (at night) this addition is large compared to the twinkling of the sky, while in the second (day) it constitutes a very insignificant fraction of the solar brightness (less than a billionth even for the brightest stars). That is why the voice of the soloist, when his singing is picked up by the choir, is drowned in the polyphonic sound...

Mathematical portraits of proverbs

Modern mathematics knows many functions, and each has its own unique appearance, just as the unique appearance of each of the billions of people living on Earth is unique. However, despite all the differences between one person and another, each person has hands and a head, ears and a mouth. In the same way, the appearance of each function can be imagined as composed of a set of characteristic details. They reveal the basic properties of functions.

"HighermeasureshorseNotjumps" If we imagine the trajectory of a galloping horse as a graph of some function, then the height of the jumps, in full accordance with the proverb, will be limited from above by some “measure”. This will be the familiar graph of the sine function.

"Reseedingworseunder-sown" The yield only grows with the sowing density for a certain period of time; then it decreases, because if the density is excessive, the shoots begin to choke each other. This pattern will become especially clear if you depict it in a graph where the yield is presented as a function of sowing density. The harvest is maximum when the field is sown in moderation. The maximum is the largest value of a function compared to its values at all neighboring points. It is like the top of a mountain, from which all roads lead only downwards, no matter where you step.

“Start cool, end cool” And“Hot at the start, but soon cooled down”

functional relationship mathematical equation

Both time-dependent functions are increasing. But, as you can see, you can grow in different ways. The slope of one curve is constantly increasing. The growth of the function increases with the growth of the argument. This property of a function is called concavity.

The slope of the other curve invariably decreases. The growth of the function weakens as the argument grows. This property of a function is called convexity.

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Examples of functional dependence in real life. Functional dependencies in real processes and phenomena. Multiple Argument Functions

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Examples of functional dependence in real life. Functional dependencies in real processes and phenomena. Multiple Argument Functions

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