Application of derivatives in real life. Abstract application of derivative

06.08.2020

Presentation on the topic: “Application of derivatives in science and in life” Performed by a student of the PkhI-17 group Anastasia Dolzhenkova Information from the history of the appearance of the derivative: The slogan of many mathematicians of the 17th century. was: “Move forward, and faith in the correctness of the results will come to you.” The term “derivative” - (French derive - behind, behind) was introduced in 1797 by J. Lagrange. He also introduced modern notation y" , f'. the designation lim is an abbreviation of the Latin word limes (interface, border). The term “limit” was introduced by I. Newton. I. Newton called the derivative fluxion, and the function itself - fluent. G. Leibniz spoke about the differential relation and denoted the derivative as follows: Lagrange Joseph Louis (1736-1813) French mathematician and mechanic Newton:

“This world was shrouded in deep darkness. Let there be light! And then Newton appeared." A. Pogue.

Isaac Newton (1643-1727) one of the creators of differential calculus.

His main work, “Mathematical Principles of Natural Philosophy,” had a tremendous impact on the development of natural science and became a turning point in the history of natural science.

Newton introduced the concept of derivative while studying the laws of mechanics, thereby revealing its mechanical meaning.

What is the derivative of a function?

The derivative of a function at a given point is the limit of the ratio of the increment of the function at this point to the increment of the argument, when the increment of the argument tends to zero.

Physical meaning of derivative.

Speed is the time derivative of the path:

v(t) = S′(t)

Acceleration is the derivative of speed with respect to time:

a(t) = v′(t) = S′′(t)

Geometric meaning of the derivative: The angular coefficient of a tangent to the graph of a function is equal to the derivative of this function, calculated at the point of tangency. f′(x) = k = tga Derivative in electrical engineering:

In our homes, in transport, in factories: electric current works everywhere. Electric current is understood as the directed movement of free electrically charged particles.

A quantitative characteristic of electric current is current strength.

In an electric current circuit, the electric charge changes over time according to the law q=q (t). Current strength I is the derivative of charge q with respect to time.

Electrical engineering mainly uses alternating current.

An electric current that changes over time is called alternating. An AC circuit may contain various elements: heaters, coils, capacitors.

The production of alternating electric current is based on the law of electromagnetic induction, the formulation of which contains the derivative of the magnetic flux.

Derivative in chemistry:

And in chemistry, differential calculus has found wide application for constructing mathematical models of chemical reactions and subsequent description of their properties.
Chemistry is the science of substances, of chemical transformations of substances.
Chemistry studies the patterns of various reactions.
The rate of a chemical reaction is the change in the concentration of reactants per unit time.
Since the reaction rate v changes continuously during the process, it is usually expressed derivative concentrations of reactants over time.

Derivative in geography:

The idea behind Thomas Malthus's sociological model is that population growth is proportional to the number of people at a given time t through N(t), . Malthus's model worked well for describing the population of the United States from 1790 to 1860. Nowadays this model does not work in most countries.

Integral and its application: A little history: The history of the concept of integral goes back to the mathematicians of Ancient Greece and Ancient Rome. The works of the scientist of Ancient Greece - Eudoxus of Cnidus (c.408-c.355 BC) on finding the volumes of bodies and calculating the areas of plane figures are known. Integral calculus became widespread in the 17th century. Scientists: G. Leibniz (1646-1716) and I. Newton (1643-1727) discovered independently of each other and almost simultaneously a formula, later called the Newton-Leibniz formula, which we use. The fact that a philosopher and physicist derived a mathematical formula does not surprise anyone, because mathematics is the language spoken by nature itself. Integral calculus became widespread in the 17th century. Scientists: G. Leibniz (1646-1716) and I. Newton (1643-1727) discovered independently of each other and almost simultaneously a formula, later called the Newton-Leibniz formula, which we use. The fact that a philosopher and physicist derived a mathematical formula does not surprise anyone, because mathematics is the language spoken by nature itself. Character entered Character entered Leibniz (1675). This sign is a modification of the Latin letter S (the first letter of the word sum). The word itself integral invented J. Bernoulli (1690). It probably comes from the Latin integero, which is translated as bringing to a previous state, restoring. The limits of integration were already indicated by L. Euler (1707-1783). In 1697, the name of a new branch of mathematics appeared - integral calculus. It was introduced by Bernoulli. In mathematical analysis, the integral of a function is an extension of the concept of sum. The process of finding the integral is called integration. This process is usually used to find quantities such as area, volume, mass, displacement, etc., when the rate or distribution of changes of this quantity in relation to some other quantity (position, time, etc.) is given. What is an integral? Integral is one of the most important concepts mathematical analysis, which arises when solving problems of finding the area under a curve, the distance traveled during uneven motion, the mass of an inhomogeneous body, etc., as well as in the problem of restoring functions according to her derivative Scientists try to express all physical phenomena in the form of a mathematical formula. Once we have a formula, we can then use it to calculate anything. And the integral is one of the main tools for working with functions. Scientists try to express all physical phenomena in the form of a mathematical formula. Once we have a formula, we can then use it to calculate anything. And the integral is one of the main tools for working with functions. Integration methods:

Tabular.
Reduction to a table by converting the integrand into a sum or difference.
Integration using change of variable (substitution).
Integration by parts.

Application of the integral:

Mathematics
Calculations of S figures.
Length of the arc of the curve.
V bodies on S parallel sections.
V bodies of rotation, etc.

Physics
Work A of variable force.
S – (path) of movement.
Mass calculation.
Calculation of the moment of inertia of a line, circle, cylinder.
Calculation of the coordinates of the center of gravity.
Amount of heat, etc.

Ministry of Education of the Saratov Region

State Autonomous Professional Educational Institution of the Saratov Region "Engels Polytechnic"

APPLICATION OF DERIVATIVE IN VARIOUS FIELDS OF SCIENCE

Performed: Verbitskaya Elena Vyacheslavovna

mathematics teacher at GAPOU SO

"Engels Polytechnic"

Introduction

The role of mathematics in various fields of natural science is very great. No wonder they say “Mathematics is the queen of sciences, physics is her right hand, chemistry is her left.”

The subject of the study is derivative.

The leading goal is to show the significance of the derivative not only in mathematics, but also in other sciences, its importance in modern life.

Differential calculus is a description of the world around us, done in mathematical language. The derivative helps us successfully solve not only mathematical problems, but also practical problems in various fields of science and technology.

The derivative of a function is used wherever there is an uneven process: uneven mechanical movement, alternating current, chemical reactions and radioactive decay of a substance, etc.

Key and thematic questions of this essay:

1. History of the derivative.

2. Why study derivatives of functions?

3. Where are derivatives used?

4. Application of derivatives in physics, chemistry, biology and other sciences.

I decided to write a paper on the topic “Application of derivatives in various fields of science” because I think this topic is very interesting, useful and relevant.

In my work, I will talk about the application of differentiation in various fields of science, such as chemistry, physics, biology, geography, etc. After all, all sciences are inextricably linked, which is very clearly seen in the example of the topic I am considering.

Application of derivatives in various fields of science

From the high school algebra course, we already know that the derivative is the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists.

The act of finding a derivative is called differentiating it, and a function that has a derivative at a point x is called differentiable at that point. A function that is differentiable at each point of an interval is said to be differentiable in that interval.

The honor of discovering the fundamental laws of mathematical analysis belongs to the English physicist and mathematician Isaac Newton and the German mathematician, physicist, and philosopher Leibniz.

Newton introduced the concept of derivative while studying the laws of mechanics, thereby revealing its mechanical meaning.

Physical meaning of the derivative: the derivative of the function y = f (x) at the point x 0 is the rate of change of the function f (x) at the point x 0.

Leibniz came to the concept of derivative by solving the problem of drawing a tangent to a derivative line, thereby explaining its geometric meaning.

The geometric meaning of the derivative is that the derivative function at the point x 0 is equal to the slope of the tangent to the graph of the function drawn at the point with the abscissa x 0 .

The term derivative and modern designations y ", f" were introduced by J. Lagrange in 1797.

The 19th century Russian mathematician Panfutiy Lvovich Chebyshev said that “of particular importance are those methods of science that make it possible to solve a problem common to all practical human activity, for example, how to dispose of one’s means to achieve the greatest benefit.”

Representatives of a variety of specialties have to deal with such tasks nowadays:

Technological engineers try to organize production in such a way that as many products as possible are produced;

Designers are trying to develop a device for a spacecraft so that the mass of the device is minimal;

Economists try to plan the plant’s connections with sources of raw materials so that transport costs are minimal.

When studying any topic, students have a question: “Why do we need this?” If the answer satisfies curiosity, then we can talk about the students’ interest. The answer for the topic "Derivative" can be obtained by knowing where derivatives of functions are used.

To answer this question, we can list some disciplines and their sections in which derivatives are used.

Derivative in algebra:

1. Tangent to the graph of a function

Tangent to the graph of a function f, differentiable at the point x o, is a straight line passing through the point (x o; f(x о)) and having a slope f′(x o).

y = f(x o) + f′(x о) (x – x о)

2. Search for intervals of increasing and decreasing functions

Function y=f(x) increases over the interval X, if for any and the inequality . In other words, a larger value of the argument corresponds to a larger value of the function.

Function y=f(x) decreases on the interval X, if for any and the inequality . In other words, a larger value of the argument corresponds to a smaller value of the function.

3. Search for extremum points of the function

The point is called maximum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the maximum point is called maximum of the function and denote .

The point is called minimum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the minimum point is called minimum function and denote .

The neighborhood of a point is understood as the interval , where is a sufficiently small positive number.

The minimum and maximum points are called extremum points , and the values of the function corresponding to the extremum points are called extrema of the function .

4. Finding the intervals of convexity and concavity of a function

convex, if the graph of this function within the interval lies no higher than any of its tangents (Fig. 1).

The graph of a function differentiable on the interval is on this interval concave, if the graph of this function within the interval lies not lower than any of its tangents (Fig. 2).

The inflection point of the graph of a function is the point separating the intervals of convexity and concavity.

5. Finding bending points of a function

Derivative in physics:

1. Velocity as a derivative of path

2. Acceleration as a derivative of speed a =

3. Decay rate of radioactive elements = - λN

And also in physics, the derivative is used to calculate:

Velocities of a material point

Instantaneous speed as the physical meaning of the derivative

Instantaneous AC current value

Instantaneous value of EMF of electromagnetic induction

Maximum power

Derivative in chemistry:

And in chemistry, differential calculus has found wide application for constructing mathematical models of chemical reactions and subsequent description of their properties.

A derivative in chemistry is used to determine a very important thing - the rate of a chemical reaction, one of the decisive factors that must be taken into account in many areas of scientific and industrial activity. V (t) = p ‘(t)

Derivative in biology:

A population is a collection of individuals of a given species, occupying a certain area of territory within the species’ range, freely interbreeding and partially or completely isolated from other populations, and is also an elementary unit of evolution.

Derivative in geography:

1. Some meanings in seismography

2. Features of the electromagnetic field of the earth

3. Radioactivity of nuclear-geophysical indicators

4.Many meanings in economic geography

5. Derive a formula for calculating the population in a territory at time t.

y'= k y

The idea of Thomas Malthus's sociological model is that population growth is proportional to the number of people at a given time t through N(t). Malthus' model worked well to describe the population of the United States from 1790 to 1860. This model is no longer valid in most countries.

Derivative in electrical engineering:

In our homes, in transport, in factories: electric current works everywhere. Electric current is understood as the directed movement of free electrically charged particles.

A quantitative characteristic of electric current is current strength.

In an electric current circuit, the electric charge changes over time according to the law q=q (t). Current strength I is the derivative of charge q with respect to time.

Electrical engineering mainly uses alternating current.

An electric current that changes over time is called alternating. An AC circuit may contain various elements: heaters, coils, capacitors.

The production of alternating electric current is based on the law of electromagnetic induction, the formulation of which contains the derivative of the magnetic flux.

Derivative in economics:

Economics is the basis of life, and differential calculus, an apparatus for economic analysis, occupies an important place in it. The basic task of economic analysis is to study the relationships of economic quantities in the form of functions.

The derivative in economics solves important issues:

1. In what direction will state income change with an increase in taxes or with the introduction of customs duties?

2. Will the company's revenue increase or decrease if the price of its products increases?

To solve these questions, it is necessary to construct connection functions of the input variables, which are then studied by methods of differential calculus.

Also, using the extremum of the function (derivative) in the economy, you can find the highest labor productivity, maximum profit, maximum output and minimum costs.

CONCLUSION: derivative is successfully used in solving various applied problems in science, technology and life

As can be seen from the above, the use of the derivative of a function is very diverse, not only in the study of mathematics, but also in other disciplines. Therefore, we can conclude that studying the topic: “Derivative of a function” will have its application in other topics and subjects.

We were convinced of the importance of studying the topic “Derivative”, its role in the study of processes in science and technology, the possibility of constructing mathematical models based on real events, and solving important problems.

“Music can uplift or soothe the soul,
Painting is pleasing to the eye,
Poetry is to awaken feelings,
Philosophy is to satisfy the needs of the mind,
Engineering is to improve the material side of people's lives,
A mathematics can achieve all these goals.”

That's what the American mathematician said Maurice Kline.

Bibliography:

1. Bogomolov N.V., Samoilenko I.I. Mathematics. - M.: Yurayt, 2015.

2. Grigoriev V.P., Dubinsky Yu.A., Elements of higher mathematics. - M.: Academy, 2014.

3. Bavrin I.I. Fundamentals of higher mathematics. - M.: Higher School, 2013.

4. Bogomolov N.V. Practical lessons in mathematics. - M.: Higher School, 2013.

5. Bogomolov N.V. Collection of problems in mathematics. - M.: Bustard, 2013.

6. Rybnikov K.A. History of mathematics, Moscow University Publishing House, M, 1960.

7. Vinogradov Yu.N., Gomola A.I., Potapov V.I., Sokolova E.V. – M.: Publishing Center “Academy”, 2010

8. Bashmakov M.I. Mathematics: algebra and principles of mathematical analysis, geometry. – M.: Publishing Center “Academy”, 2016

Periodic sources:

Newspapers and magazines: “Mathematics”, “Open Lesson”

Use of Internet resources and electronic libraries.

Back forward

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Lesson type: integrated.

The purpose of the lesson: study some aspects of the application of derivatives in various fields of physics, chemistry, and biology.

Tasks: broadening the horizons and cognitive activity of students, developing logical thinking and the ability to apply their knowledge.

Technical support: interactive board; computer and disk.

DURING THE CLASSES

I. Organizational moment

II. Setting a lesson goal

– I would like to conduct a lesson under the motto of Alexei Nikolaevich Krylov, a Soviet mathematician and shipbuilder: “Theory without practice is dead or useless, practice without theory is impossible or detrimental.”

– Let’s review the basic concepts and answer the questions:

– Tell me the basic definition of a derivative?
– What do you know about the derivative (properties, theorems)?
– Do you know any examples of problems using derivatives in physics, mathematics and biology?

Consideration of the basic definition of a derivative and its rationale (answer to the first question):

Derivative – one of the fundamental concepts of mathematics. The ability to solve problems using derivatives requires good knowledge of theoretical material and the ability to conduct research in various situations.

Therefore, today in the lesson we will consolidate and systematize the acquired knowledge, review and evaluate the work of each group and, using the example of some problems, we will show how to use the derivative to solve other problems and non-standard problems using the derivative.

III. Explanation of new material

1. Instantaneous power is the derivative of work with respect to time:

W = lim ΔA/Δt ΔA – job change.

2. If a body rotates around an axis, then the angle of rotation is a function of time t
Then the angular velocity is equal to:

W = lim Δφ/Δt = φ׳(t) Δ t → 0

3. Current strength is a derivative Ι = lim Δg/Δt = g′, Where g– positive electric charge transferred through the cross section of the conductor during time Δt.

4. Let ΔQ– the amount of heat required to change the temperature in Δt time, then lim ΔQ/Δt = Q′ = C – specific heat.

5. Problem about the rate of a chemical reaction

m(t) – m(t0) – amount of substance that reacts over time t0 before t

V= lim Δm/Δt = m Δt → 0

6. Let m be the mass of the radioactive substance. Radioactive decay rate: V = lim Δm/Δt = m׳(t) Δt→0

In differentiated form, the law of radioactive decay has the form: dN/dt = – λN, Where N– number of nuclei that have not decayed time t.

Integrating this expression, we get: dN/N = – λdt ∫dN/N = – λ∫dt lnN = – λt + c, c = const at t = 0 number of radioactive nuclei N = N0, from here we have: ln N0 = const, hence

n N = – λt + ln N0.

Potentiating this expression we get:

– the law of radioactive decay, where N0– number of cores at a time t0 = 0, N– number of nuclei that have not decayed during time t.

7. According to Newton’s heat transfer equation, the heat flow rate dQ/dt is directly proportional to the window area S and the temperature difference ΔT between the inner and outer glass and inversely proportional to its thickness d:

dQ/dt =A S/d ΔT

8. The phenomenon of Diffusion is the process of establishing an equilibrium distribution

Within phases of concentration. Diffusion goes to the side, leveling the concentrations.

m = D Δc/Δx c – concentration
m = D c׳x x – coordinate, D – diffusion coefficient

Fick's Law:

9. It was known that an electric field excites either electric charges or a magnetic field, which has a single source - electric current. James Clark Maxwell introduced one amendment to the laws of electromagnetism discovered before him: a magnetic field also arises when the electric field changes. A seemingly small amendment had enormous consequences: a completely new physical object appeared, albeit only at the tip of the pen - an electromagnetic wave. Maxwell masterfully, unlike Faraday, who thought its existence was possible, derived the equation for the electric field:

∂E/∂x = M∂B/Mo ∂t Mo = const t

A change in the electric field causes the appearance of a magnetic field at any point in space; in other words, the rate of change of the electric field determines the magnitude of the magnetic field. Under greater electric current, there is a greater magnetic field.

IV. Consolidation of what has been learned

– You and I studied the derivative and its properties. I would like to read Gilbert’s philosophical statement: “Every person has a certain outlook. When this horizon narrows to the infinitesimal, it turns into a point. Then the person says that this is his point of view.”
Let's try to measure the point of view on the application of the derivative!

The plot of "Leaf"(use of derivative in biology, physics, life)

Let's consider the fall as an uneven movement depending on time.

So: S = S(t) V = S′(t) = x′(t), a = V′(t) = S″(t)

(Theoretical survey: mechanical meaning of derivative).

1. Problem solving

Solve problems yourself.

2. F = ma F = mV′ F = mS″

Let us write down Porton’s II law, and taking into account the mechanical meaning of the derivative, we rewrite it in the form: F = mV′ F = mS″

The plot of "Wolves, Gophers"

Let's return to the equations: Consider the differential equations of exponential growth and decrease: F = ma F = mV" F = mS""
The solution of many problems in physics, technical biology and social sciences comes down to the problem of finding functions f"(x) = kf(x), satisfying the differential equation, where k = const .

Human Formula

A person is as many times larger than an atom as he is smaller than a star:

It follows that
This is the formula that determines man’s place in the universe. In accordance with it, the size of a person represents the average proportionality of a star and an atom.

I would like to end the lesson with the words of Lobachevsky: “There is not a single area of mathematics, no matter how abstract it may be, that someday will not be applicable to the phenomena of the real world.”

V. Solution of numbers from the collection:

Independent problem solving on the board, collective analysis of problem solutions:

№ 1 Find the speed of movement of a material point at the end of the 3rd second, if the movement of the point is given by the equation s = t^2 –11t + 30.

№ 2 The point moves rectilinearly according to the law s = 6t – t^2. At what moment will its speed be zero?

№ 3 Two bodies move rectilinearly: one according to the law s = t^3 – t^2 – 27t, the other according to the law s = t^2 + 1. Determine the moment when the velocities of these bodies turn out to be equal.

№ 4 For a car moving at a speed of 30 m/s, the braking distance is determined by the formula s(t) = 30t-16t^2, where s(t) is the distance in meters, t is the braking time in seconds. How long does it take to brake until the car comes to a complete stop? How far will the car travel from the start of braking until it comes to a complete stop?

№5 A body weighing 8 kg moves rectilinearly according to the law s = 2t^2+ 3t – 1. Find the kinetic energy of the body (mv^2/2) 3 seconds after the start of movement.

Solution: Let's find the speed of movement of the body at any moment of time:
V = ds / dt = 4t + 3
Let's calculate the speed of the body at time t = 3:
V t=3 = 4 * 3 + 3=15 (m/s).
Let us determine the kinetic energy of the body at time t = 3:
mv2/2 = 8 – 15^2 /2 = 900 (J).

№6 Find the kinetic energy of the body 4 s after the start of movement, if its mass is 25 kg, and the law of motion has the form s = 3t^2- 1.

№7 A body whose mass is 30 kg moves rectilinearly according to the law s = 4t^2 + t. Prove that the motion of a body occurs under the influence of a constant force.
Solution: We have s" = 8t + 1, s" = 8. Therefore, a(t) = 8 (m/s^2), i.e., with a given law of motion, the body moves with a constant acceleration of 8 m/s^2. Further, since the mass of the body is constant (30 kg), then, according to Newton’s second law, the force acting on it F = ma = 30 * 8 = 240 (H) is also a constant value.

№8 A body weighing 3 kg moves rectilinearly according to the law s(t) = t^3 – 3t^2 + 2. Find the force acting on the body at time t = 4s.

№9 A material point moves according to the law s = 2t^3 – 6t^2 + 4t. Find its acceleration at the end of the 3rd second.

VI. Application of derivative in mathematics:

A derivative in mathematics shows a numerical expression of the degree of change of a quantity located at the same point under the influence of various conditions.

The derivative formula dates back to the 15th century. The great Italian mathematician Tartagli, considering and developing the question of how much the flight range of a projectile depends on the inclination of the gun, applies it in his works.

The derivative formula is often found in the works of famous mathematicians of the 17th century. It was used by Newton and Leibniz.

The famous scientist Galileo Galilei devotes an entire treatise on the role of derivatives in mathematics. Then the derivative and various presentations with its application began to be found in the works of Descartes, the French mathematician Roberval and the Englishman Gregory. Great contributions to the study of the derivative were made by such minds as L'Hopital, Bernoulli, Langrange and others.

1. Plot a graph and examine the function:

Solution to this problem:

A moment of relaxation

VII. Application of derivative in physics:

When studying certain processes and phenomena, the task of determining the speed of these processes often arises. Its solution leads to the concept of derivative, which is the basic concept of differential calculus.

The method of differential calculus was created in the 17th and 18th centuries. The names of two great mathematicians – I. Newton and G.V. – are associated with the emergence of this method. Leibniz.

Newton came to the discovery of differential calculus when solving problems about the speed of motion of a material point at a given moment in time (instantaneous speed).

In physics, the derivative is used mainly to calculate the largest or smallest values of any quantity.

Problem solving:

№1 Potential energy U the field of a particle in which there is another, exactly the same particle has the form: U = a/r 2 – b/r, Where a And b- positive constants, r- distance between particles. Find: a) value r0 corresponding to the equilibrium position of the particle; b) find out whether this situation is stable; V) Fmax the value of the force of attraction; d) draw approximate dependence graphs U(r) And F(r).

Solution to this problem: To determine r0 corresponding to the equilibrium position of the particle we study f = U(r) to the extreme.

Using the connection between the potential energy of the field

U And F, Then F = – dU/dr, we get F = – dU/dr = – (2a/r3+ b/r2) = 0; wherein r = r0; 2a/r3 = b/r2 => r0 = 2a/b; We determine stable or unstable equilibrium by the sign of the second derivative:
d2U/dr02= dF/dr0 = – 6a/r02 + 2b/r03 = – 6a/(2a/b)4 + 2b/(2a/b)3 = (– b4/8a3)< 0 ;
stable equilibrium.

For determining Fmax attraction, I examine the function for extrema: F = 2a/r3 - b/r2;
dF/dr = –6a/r4 + 2b/ r3 = 0; at r = r1 = 3a/b; substituting, I will get Fmax = 2a/r31 - b/r31 = – b3/27a2; U(r) = 0; at r = a/b; U(r)min at r = 2, a/b = r0;F = 0; F(r)max at r = r1 = 3a/b;
Answer: F(r)max at r = r1 = 3a/b;

№2 Circuit with external resistance R = 0.9 Ohm powered by battery from k = 36 identical sources, each of which has an emf E=2 IN and internal resistance r0 = 0.4 Ohm. Battery include n groups connected in parallel, and each of them contains m series connected batteries. At what values m, n the maximum will be obtained J in the external R.

Solution to this problem:

When connecting batteries in series E gr = m* E ; rgr = r0*m;
and when connecting identical rbat = r0m/n; E baht = m* E,
According to Ohm's law J = m E /(R+ r0m/n) = m E n/(nR + r0m)
Because k is the total number of batteries, then k = mn;
J = k E /(nR + r0m) = k E /(nR + kr0/n);
To find the condition under which J current in the circuit maximum I study the function J = J(n) to the extremum by taking the derivative with respect to n and equating it to zero.
J'n – (k E (R - kr0/n2))/ (nR + kr0/n)2 = 0;
n2 = kr/R
n = √kr/R = √3.6*0.4/0.9 = 4;
m = k/n = 36/4 = 9;
wherein Jmax = k E /(nR + mr0) = 36*2/(4*0.9 + 9*0.4) = 10 A;

Answer: n = 4, m = 9.

№3 Platform mass M begins to move to the right under the influence of a constant force F. Sand is poured onto her from a stationary bunker. The loading speed is constant and equal to µ kg/s. Neglecting friction, find the dependence on the acceleration time of the platform during loading. Determine acceleration a1 platform in case the sand is not poured onto the platform, but pours out from the filled one through a hole in its bottom at a constant speed µ kg/s.

Solution to this problem: Let us first consider the case when sand is poured onto the platform
The motion of the platform – sand system can be described using Newton’s second law:
dP/dt = FΣ
P– impulse of the platform-sand system, FΣ– force acting on the platform-sand system.
If through p to denote the momentum of the platform, we can write: dp/dt = F
Let us find the change in the momentum of the platform over an infinitesimal period of time Δ t: Δ p = (M + µ(t + Δ t))(u +Δ u) – (M + µt)u = FΔ t;
Where u– platform speed.
Opening the brackets and making abbreviations we get:
Dp = µuΔ t+MΔ u+ Δ µut +Δ µuΔ t = FΔ t
Divide by Δt and go to the limit Δ t→ 0
Mdu/dt + µtdu/dt + µu= F or d[(M + µt)u]/dt = F
This equation can be integrated by considering the initial speed of the platform to be zero: (M + µt)u = Ft.
Hence: u = Ft/(M + µt)
Then, platform acceleration: a = du/dt = (F(M + µt) – Ftµ)/(M + µt) 2 = FM / (M + µt) 2

Consider the case when sand spills out of a filled platform.
Change in momentum over a short period of time:
Δ p = (M – µ(t +Δ t))(u+Δ u) +Δ µtu – (M – µt)u = FΔ t
Term Δ µtu is the impulse of the amount of sand that poured out of the platform during time Δ t. Then:
Δ p = MΔ u – µtΔ u –Δ µtΔ u = FΔ t
Divide by Δ t and move on to the limit Δ t→ 0
(M – µt)du/dt = F
Or a1= du/dt= F/(M – µt)

Answer: a = FM / (M + µt) 2 , a1= F/(M – µt)

VIII.Independent work:

Find derivatives of functions:

The straight line y = 2x is tangent to the function: y = x 3 + 5x 2 + 9x + 3. Find the abscissa of the point of tangency.

IX. Summing up the lesson:

– What questions was the lesson devoted to?
– What did you learn in the lesson?
– What theoretical facts were summarized in the lesson?
– Which tasks considered turned out to be the most difficult? Why?

Bibliography:

Amelkin V.V., Sadovsky A.P. Mathematical models and differential equations. – Minsk: Higher School, 1982. – 272 p.
Amelkin V.V. Differential equations in applications. M.: Science. Main editorial office of physical and mathematical literature, 1987. – 160 p.
Erugin N.P. A book for reading on the general course of differential equations. – Minsk: Science and Technology, 1979. – 744 p.
.Magazine "Potential" November 2007 No. 11
“Algebra and principles of analysis” 11th grade S.M. Nikolsky, M.K. Potapov and others.
“Algebra and mathematical analysis” N.Ya. Vilenkin et al.
"Mathematics" V.T. Lisichkin, I.L. Soloveichik, 1991

We are studying derivative. Is it really that important in life? “Differential calculus is a description of the world around us, performed in mathematical language. The derivative helps us successfully solve not only mathematical problems, but also practical problems in various fields of science and technology.”

Concept in the language of chemistry Designation Concept in the language of mathematics Quantity of substance at time t 0 p = p(t 0) Function Time interval t = t– t 0 Increment of argument Change in quantity of substance p= p(t 0 + t) – p(t 0) Increment of the function Average rate of the chemical reaction p/t Ratio of the increment of the function to the increment of the argument V (t) = p (t) Solution:

Solution: Concept in the language of biology Designation Concept in the language of mathematics Number at time t 1 x = x(t) Function Time interval t = t 2 – t 1 Argument increment Change in population size x = x(t 2) – x(t 1) Increment of a function Rate of change in population size x/t Ratio of increment of function to increment of argument Relative increment at a given moment Lim x/t t 0 Derivative P = x (t)

Algorithm for finding the derivative (for the function y=f(x)) Fix the value of x, find f(x). Give the argument x an increment of Dx, (move x+Dx to a new point), find f(x+Dx). Find the increment of the function: Dy= f(x+Dx)-f(x) Compose the ratio of the increment of the function to the increment of the argument Calculate the limit of this ratio (this limit is f `(x).)

Project activities in mathematics lessons

Project topic: Application of derivative

Participants: 1st year students of State Educational Institution "SKSiS"

Fundamental Question : How to measure speed speed?

Problematic issues

Who worked on the issue of “differentiation”?

How is the derivative used when studying a function?

How does a derivative help biologists and chemists?

What problems in physics are solved using derivatives?

How is derivative used in economics?

What is the connection between derivatives and geography?

Target: Studying the use of the derivative to solve problems in principles of analysis, physics, economics, biology, chemistry and geography; deepening and expanding knowledge on the topic “Derivative”.

Tasks:

Find information about the history of the derivative, study it and systematize it.

Study of functions for monotonicity, extrema, convexity-concavity using the derivative.

Selection of problems from different branches of biology that can be solved using derivatives

Find out what processes the derivative regulates in geography. Consider geography problems that can be solved using derivatives

Select problems from different branches of physics that can be solved using derivatives.

Select economic problems that can be solved using derivatives.

Consider the application of the rules for calculating the derivative to solving practical problems with economic content.

“I warn you to beware of throwing away dx - this is a mistake that is often made and which prevents progress.”

G.W. Leibniz

Using derivatives to solve problems requires students to think unconventionally. It should be noted that knowledge of non-standard methods and techniques for solving problems contributes to the development of new, unconventional thinking, which can also be successfully applied in other areas of human activity (economics, physics, chemistry, biology, etc.). This proves the relevance of this work. When working on a project, certain stages of student activity must be observed. Each of them contributes to the formation of personal qualities.

Preparatory stage

At this stage, the students and I are immersed in the project: the activity is motivated, the topic, problem and goals are determined. The topic of the project should not only be close and interesting, but also accessible to the student. In terms of time, this stage of the project is the shortest, but it is very important for achieving the expected results.The conversation takes place during the introductory presentation; updating existing knowledge on the topic, discussing the general project plan, planning work on the project. Determining the direction of searching for information in different sources.

The topic “Derivative” is one of the most important sections of the course in mathematical analysis, since this concept is fundamental in differential calculus and serves as the starting point for constructing integral calculus. But often, when students encounter this concept for the first time, they do not understand why they need to study it. They don't see the practical application of this topic. Therefore, this project “Application of Derivative” is aimed at making students find out why they need to study derivative, where they can use the knowledge related to derivative in life as well as in other subjects.

Planning stage and organization of activities.

At this stage, we define groups according to areas of activity, highlighting the goals and objectives of each group. Suggested topics for selecting groups:

Group 1 – “Historical information of differential calculus”;

Group 2 – “Geometric meaning of derivatives”

Group 4 - “Application of derivatives in solving physical problems”;

Group 3 – “Finding the best solution in applied, including socio-economic, problems”

Group 4 – “Application of derivatives in chemistry and biology”

Group 5 - “Application of derivatives in solving problems with geographical content.”

The group included students with different educational abilities. Each group was given the task to analyze the selected topic and find information. The work of the groups is planned: responsibilities are distributed among students, sources of information, methods of collecting and analyzing information, ways of presenting the results of activities are determined (in our case - presentations and booklets).

Search stage.

At this stage, there is a search and collection of information on your chosen topic, and the solution of intermediate problems. Analysis and synthesis of the collected material. A written presentation of the results and intermediate control by the teacher of the results obtained. Consultations were held on the programs PowerPoint, Publisher, Word, for students who had problems in practical work to formalize the results. Formulation of conclusions.

Stage of presenting results, report.

The presentation stage is necessary to complete the work, to analyze what has been done, to self-assess and be assessed by others, and to demonstrate the results. The form for presenting the results in our project: an oral report with a demonstration of materials presented in the form of a presentation, booklet, or abstract.

Evaluation of results, reflection

One of the final stages of working on a project is evaluating the results and reflection. The project is defended in class or in a group activity.

The appendices present student work prepared as part of project activities in the form of presentations and a booklet.

When assessing students’ work on a project, the content is taken into account (completeness of the topic, presentation of aspects of the topic, presentation of the strategy for solving the problem, logic of presentation of information, use of various resources), the degree of independent work of the group (coordinated work in the group, distribution of roles in the group, author’s originality), design of the presentation product (grammar, suitable vocabulary, absence of spelling errors and typos), defense (quality of the report, volume and depth of knowledge on the topic, culture of speech, demeanor in front of the audience, answers to questions).