Application of the primitive in life. Coursework in mathematics

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 to you
will come."
The term "derivative" - (French derive - behind, behind) was introduced in 1797 by J. Lagrange. He also introduced
modern designations y ", f'.
the designation lim is an abbreviation of the Latin word limes (border, boundary). The term "limit" was introduced by I. Newton.
I. Newton called the derivative a flux, and the function itself - a 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 so
Newton appeared. A. Pogue.
Isaac Newton (1643-1727) one of the founders
differential calculus.
His main work is "Mathematical principles
natural philosophy "-had a colossal
influence on the development of natural science
turning point in the history of natural science.
Newton introduced the concept of a derivative while studying the laws
mechanics, thereby revealing its mechanical
meaning.

What is the derivative of a function?

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

The physical meaning of the derivative.

Velocity is the derivative of distance with respect to time:
v(t) = S′(t)
Acceleration is a derivative
speed over time:
a(t) = v′(t) = S′′(t)

The geometric meaning of the derivative:

Slope of the tangent to the graph
function is equal to the derivative of this function,
calculated at the point of contact.
f′(x) = k = tga

Derivative in electrical engineering:

In our homes, in transport, in factories: it works everywhere
electricity. By electric current is meant
directed movement of free electrically charged
particles.
The quantitative characteristic of the electric current is the force
current.
AT
electric current circuits electric charge changes from
over time according to the law q=q (t). Current strength I is a derivative
charge q over time.
In electrical engineering, AC operation is mainly used.
An electric current that changes with time is called
variables. An AC circuit may contain various
elements: heaters, coils, capacitors.
Obtaining an alternating electric current is based on the law
electromagnetic induction, the formulation of which contains
derivative of the magnetic flux.

Derivative in chemistry:

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

Derivative in geography:

The idea of the sociological model of Thomas Malthus is that population growth
proportional to the population at a given time t through N(t), . Model
Malthus did a good job of describing the US population from 1790 to 1860.
years. This model is no longer valid in most countries.

The integral and its application:

A bit of history:

The history of the concept of the integral goes back to
to the mathematicians of Ancient Greece and Ancient
Rome.
The works of the scientist of Ancient Greece, Eudoxus of Knidos (c. 408-c. 355 BC), are known on
finding volumes of bodies and calculations
areas of plane figures.

Integral calculus became widespread in the 17th century. Scientists:
G. Leibniz (1646-1716) and I. Newton (1643-1727) independently discovered
friend and almost simultaneously the formula, later called the formula
Newton - Leibniz, which we use. That the mathematical formula
brought philosopher and physicist does not surprise anyone, because mathematics is the language in which
nature itself speaks.

Symbol entered
Leibniz (1675). This sign is
change of the Latin letter S
(the first letter of the word sum). The very word integral
invented
J. Bernoulli (1690). It probably comes from
Latin integero, which translates as
restore to its original state.
The limits of integration were already indicated by L. Euler
(1707-1783). In 1697 the name appeared
new branch of mathematics - integral
calculus. It was introduced by Bernoulli.

In mathematical analysis, the integral of a function is called
extension of the concept of sum. The process of finding the integral
is called integration. This process is usually used for
finding such quantities as area, volume, mass, displacement, etc.
when the rate or distribution of changes in this quantity is given
with respect to some other quantity (position, time, etc.).

What is an integral?

The integral is one of the most important concepts of mathematical analysis, which
arises when solving problems of finding the area under the curve, the distance traveled when
uneven motion, the mass of an inhomogeneous body, etc., as well as in the problem of
restoring a function from its derivative

Scientists try everything physical
phenomena to express in the form
mathematical formula. how
only we have a formula, further
already possible with it
count anything. And the integral
is one of the main
tools for working with
functions.

Integration methods:

1.Tabular.
2. Reduction to tabular transformation of the integrand
expressions to sum or difference.
3.Integration using a change of variable (substitution).
4. Integration by parts.

Application of the integral:

◦ Mathematics
◦ Compute S shapes.
◦ Curve arc length.
◦ V bodies on S parallel
sections.
◦ V bodies of revolution, etc.
Physics
Work A variable force.
S - (path) of movement.
Mass calculation.
Calculation of the moment of inertia of the line,
circle, cylinder.
◦ Compute center coordinate
gravity.
◦ Amount of heat, etc.
◦
◦
◦
◦

There is no HTML version of the work yet.

Research topic

Application of integral calculus in planning family expenses

Relevance of the problem

Increasingly, in the social and economic spheres, when calculating the degree of inequality in the distribution of income, mathematics is used, namely, integral calculus. By studying the practical application of the integral, we learn:

How does the integral and calculating the area using the integral help in allocating material costs?

How the integral will help in saving money for vacation.

Target

plan family expenses using integral calculation

Tasks

Learn the geometric meaning of the integral.
Consider methods of integration in the social and economic spheres of life.
Make a forecast of the material costs of the family when repairing an apartment using the integral.
Calculate the volume of energy consumption of the family for a year, taking into account the integral calculation.
Calculate the amount of a savings deposit in Sberbank for vacation.

Hypothesis

integral calculus helps in economical calculations when planning family income and expenses.

Research stages

We studied the geometric meaning of the integral and methods of integration in the social and economic spheres of life.
We calculated the material costs required for the repair of an apartment using the integral.
We calculated the volume of electricity consumption in the apartment and the cost of electricity for the family for a year.
We considered one of the options for collecting family income through deposits in Sberbank using the integral.

Object of study

integral calculus in the social and economic spheres of life.

Methods

Analysis of the literature on the topic "Practical application of the integral calculus"
The study of integration methods in solving problems on the calculation of areas and volumes of figures using the integral.
Analysis of family expenses and incomes using integral calculation.

Working process

Literature review on the topic "Practical application of integral calculus"
Solving a system of problems for calculating the areas and volumes of figures using the integral.
Calculation of family expenses and income using an integral calculation: room renovation, electricity volume, deposits in Sberbank for vacation.

Our results

How does the integral and calculating the volume with the help of the integral help in predicting the volume of electricity consumption?

findings

The economic calculation of the necessary funds for the repair of an apartment can be performed faster and more accurately using an integral calculation.

It is easier and faster to calculate the family's electricity consumption using an integral calculation and Microsoft Office Excel, which means predicting the family's electricity costs for a year.

Profit from deposits in Sberbank can be calculated using an integral calculation, which means planning a family vacation.

List of resources

Printed editions:

Textbook. Algebra and the beginning of analysis 10-11 grade. A.G. Mordkovich. Mnemosyne. M: 2007
Textbook. Algebra and the beginning of analysis 10-11 grade. A. Kolmogorov Enlightenment. M: 2007
Mathematics for sociologists and economists. Akhtyamov A.M. M.: FIZMATLIT, 2004. - 464 p.
Integral calculation. Handbook of Higher Mathematics by M. Ya. Vygodsky, Enlightenment, 2000

The motto of the lesson: “Mathematics is the language that all exact sciences speak” N.I. Lobachevsky

The purpose of the lesson: to generalize students' knowledge on the topic "Integral", "Application of the integral"; to broaden their horizons, knowledge about the possible application of the integral to the calculation of various quantities; consolidate the skills to use the integral to solve applied problems; instill a cognitive interest in mathematics, develop a culture of communication and a culture of mathematical speech; be able to learn to speak to students and teachers.

Type of lesson: iterative-generalizing.

Type of lesson: lesson - defense of the project “Application of the integral”.

Equipment: magnetic board, posters “Application of the integral”, cards with formulas and tasks for independent work.

Lesson plan:

1. Project protection:

from the history of integral calculus;
integral properties;
application of the integral in mathematics;
application of the integral in physics;

2. Solution of exercises.

During the classes

Teacher: A powerful research tool in mathematics, physics, mechanics and other disciplines is a definite integral - one of the basic concepts of mathematical analysis. The geometric meaning of the integral is the area of a curvilinear trapezoid. The physical meaning of the integral is 1) the mass of an inhomogeneous rod with density, 2) the displacement of a point moving in a straight line with a speed over a period of time.

Teacher: The guys in our class did a great job, they picked up tasks where a certain integral is applied. They have a word.

2 student: Properties of the integral

3 student: Application of the integral (table on the magnetic board).

4 student: We consider the use of the integral in mathematics to calculate the area of \u200b\u200bfigures.

The area of any plane figure, considered in a rectangular coordinate system, can be composed of the areas of curvilinear trapezoids adjacent to the axis Oh and axes OU. Area of a curvilinear trapezoid bounded by a curve y = f(x), axis Oh and two straight x=a and x=b, where a x b, f(x) 0 calculated by the formula cm. rice. If the curvilinear trapezoid is adjacent to the axis OU, then its area is calculated by the formula , cm. rice. When calculating the areas of figures, the following cases may arise: a) The figure is located above the Ox axis and is limited by the Ox axis, the curve y \u003d f (x) and two straight lines x \u003d a and x \u003d b. (See. rice.) The area of \u200b\u200bthis figure is found by formula 1 or 2. b) The figure is located under the Ox axis and is limited by the Ox axis, the curve y \u003d f (x) and two straight lines x \u003d a and x \u003d b (see. rice.). The area is found by the formula . c) The figure is located above and below the Ox axis and is limited by the Ox axis, the curve y \u003d f (x) and two straight lines x \u003d a and x \u003d b ( rice.). d) The area is bounded by two intersecting curves y \u003d f (x) and y \u003d (x) ( rice.)

5 student: Solve the problem

x-2y+4=0 and x+y-5+0 and y=0

7 student: An integral widely used in physics. A word to physicists.

1. CALCULATION OF THE PATH TRAVENED BY A POINT

The path traveled by a point during non-uniform movement in a straight line with a variable speed for a time interval from to is calculated by the formula .

Examples:

1. Point movement speed m/s. Find the path traveled by the point in 4 seconds.

Solution: according to the condition, . Hence,

2. Two bodies started moving simultaneously from the same point in the same direction in a straight line. The first body moves with a speed m / s, the second - with a speed v = (4t+5) m/s. How far apart will they be after 5 seconds?

Solution: it is obvious that the desired value is the difference between the distances traveled by the first and second bodies in 5 s:

3. A body is thrown vertically upwards from the surface of the earth with a speed u = (39.2-9.8^) m/s. Find the maximum height of the body.

Solution: the body will reach the highest lifting height at a time t when v = 0, i.e. 39.2- 9.8t = 0, whence I= 4 s. By formula (1), we find

2. CALCULATION OF THE WORK FORCE

The work done by the variable force f(x) when moving along the axis Oh material point from x = a before x=b, is found according to the formula When solving problems for calculating the work of a force, the G y k a law is often used: F=kx, (3) where F - force N; X-absolute elongation of the spring, m, caused by the force F, a k- coefficient of proportionality, N/m.

Example:

1. A spring at rest has a length of 0.2 m. A force of 50 N stretches the spring by 0.01 m. What work must be done to stretch it from 0.22 to 0.32 m?

Solution: using equality (3), we have 50=0.01k, i.e. kK = 5000 N/m. We find the limits of integration: a = 0.22 - 0.2 = 0.02 (m), b=0.32- 0.2 = 0.12(m). Now, according to formula (2), we obtain

3. CALCULATION OF THE WORK PERFORMED WHEN LIFTING THE LOAD

Task. A cylindrical tank with a base radius of 0.5 m and a height of 2 m is filled with water. Calculate the work that needs to be done to pump water out of the tank.

Solution: select a horizontal layer at depth x with height dx ( rice.). The work A that must be done to raise a layer of water of weight P to a height x is equal to Px.

A change in depth x by a small amount dx will cause a change in volume V by dV = pr 2 dx and change in weight Р by * dР = 9807 r 2 dх; in this case, the work performed A will change by the value dА=9807пr 2 xdх. Integrating this equality as x changes from 0 to H, we obtain

4. CALCULATION OF THE LIQUID PRESSURE FORCE

The meaning of strength R liquid pressure on a horizontal platform depends on the depth of immersion X this site, i.e., from the distance of the site to the surface of the liquid.

The pressure force (N) on a horizontal platform is calculated by the formula P = 9807Sx,

where - liquid density, kg/m 3 ; S - site area, m 2; X - platform immersion depth, m

If the area under fluid pressure is not horizontal, then the pressure on it is different at different depths, therefore, the pressure force on the area is a function of the depth of its immersion P(x).

5. ARC LENGTH

Let a flat curve AB(rice.) given by the equation y \u003d f (x) (axb) and f(x) and f ?(x) are continuous functions in the interval [а, b]. Then the differential dl arc length AB is expressed by the formula or , and the arc length AB calculated by formula (4)

where a and b are the values of the independent variable X at points A and B. If the curve is given by the equation x =(y)(with yd) then the length of the arc AB is calculated by the formula (5) where with and d independent variable values at at points BUT and V.

6. CENTER OF MASS

When finding the center of mass, the following rules are used:

1) x coordinate ? the center of mass of the system of material points А 1 , А 2 ,..., А n with masses m 1 , m 2 , ..., m n located on a straight line at points with coordinates x 1 , x 2 , ..., x n , are found by the formula

(*); 2) When calculating the coordinate of the center of mass, any part of the figure can be replaced by a material point, placing it in the center of mass of this part, and assigning to it a mass equal to the mass of the considered part of the figure. Example. Let along the rod-segment [a;b] axis Ox - mass is distributed with density (x), where (x) is a continuous function. Let us show that a) the total mass M of the rod is equal to; b) coordinate of the center of mass x " is equal to .

Let's split the segment [a; b] into n equal parts with points a= x 0< х 1 < х 2 < ... <х n = b (rice.). On each of these n segments, the density can be considered constant for large n and approximately equal to (x k - 1) on the k-th segment (due to the continuity of (x). Then the mass of the k-th segment is approximately equal to and the mass of the entire rod is

The concept of an integral is widely applicable in life. Integrals are used in various fields of science and technology. The main tasks calculated using integrals are tasks for:

1. Finding the volume of the body

2. Finding the center of mass of the body.

Let's consider each of them in more detail. Here and below, to denote a definite integral of some function f(x), with integration limits from a to b, we will use the following notation ∫ a b f(x).

Finding the volume of a body

Consider the following figure. Suppose there is some body whose volume is equal to V. There is also a straight line such that if we take a certain plane perpendicular to this straight line, the cross-sectional area S of this body by this plane will be known.

Each such plane will be perpendicular to the x-axis, and therefore will intersect it at some point x. That is, each point x from the segment will be assigned the number S (x) - the cross-sectional area of \u200b\u200bthe body, the plane passing through this point.

It turns out that some function S(x) will be given on the segment. If this function is continuous on this segment, then the following formula will be valid:

V = ∫ a b S(x)dx.

The proof of this statement is beyond the scope of the school curriculum.

Calculating the center of mass of a body

The center of mass is most often used in physics. For example, there is some body which moves with any speed. But it is inconvenient to consider a large body, and therefore in physics this body is considered as the movement of a point, on the assumption that this point has the same mass as the whole body.

And the task of calculating the center of mass of the body is the main one in this matter. Because the body is large, and which point should be taken as the center of mass? Maybe the one in the middle of the body? Or maybe the closest point to the leading edge? This is where integration comes in.

The following two rules are used to find the center of mass:

1. Coordinate x’ of the center of mass of some system of material points A1, A2,A3, … An with masses m1, m2, m3, … mn, respectively, located on a straight line at points with coordinates x1, x2, x3, … xn is found by the following formula:

x’ = (m1*x1 + ma*x2 + … + mn*xn)/(m1 + m2 + m3 +… + mn)

2. When calculating the coordinates of the center of mass, any part of the figure under consideration can be replaced by a material point, while placing it in the center of mass of this separate part of the figure, and the mass can be taken equal to the mass of this part of the figure.

For example, if a mass of density p(x) is distributed along the rod - a segment of the Ox axis, where p(x) is a continuous function, then the coordinate of the center of mass x' will be equal to.

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