The calculus is a branch of calculus that studies the derivative, differentials, and their use in the study of a function.

History of appearance

Differential calculus emerged as an independent discipline in the second half of the 17th century, thanks to the work of Newton and Leibniz, who formulated the basic provisions in the calculus of differentials and noticed the connection between integration and differentiation. Since that moment, the discipline has developed along with the calculus of integrals, thus forming the basis of mathematical analysis. The appearance of these calculus opened a new modern period in the mathematical world and caused the emergence of new disciplines in science. It also expanded the possibility of applying mathematical science in natural science and technology.

Basic concepts

The differential calculus is based on the fundamental concepts of mathematics. They are: continuity, function and limit. After a while, they took on a modern look, thanks to integral and differential calculus.

Process of creation

Formation of differential calculus in the form of applied, and then scientific method occurred before the emergence of philosophical theory, which was created by Nicholas of Cusa. His works are considered an evolutionary development from the judgments of ancient science. Despite the fact that the philosopher himself was not a mathematician, his contribution to the development of mathematical science is undeniable. Kuzansky was one of the first to leave the consideration of arithmetic as the most accurate field of science, putting the mathematics of that time in doubt.

For ancient mathematicians, the unit was a universal criterion, while the philosopher proposed infinity as a new measure instead of the exact number. In this regard, the representation of precision in mathematical science is inverted. Scientific knowledge, according to him, is divided into rational and intellectual. The second is more accurate, according to the scientist, since the first gives only an approximate result.

Idea

The main idea and concept in differential calculus is related to a function in small neighborhoods of certain points. To do this, it is necessary to create a mathematical apparatus for studying a function whose behavior in a small neighborhood of the established points is close to the behavior of a polynomial or a linear function. This is based on the definition of derivative and differential.

The appearance was caused big number problems from the natural sciences and mathematics, which led to finding the values ​​of limits of the same type.

One of the main tasks that are given as an example, starting from high school, is to determine the speed of a point moving along a straight line and construct a tangent line to this curve. The differential is related to this, since it is possible to approximate the function in a small neighborhood of the considered point of the linear function.

Compared to the concept of the derivative of a function of a real variable, the definition of differentials simply passes over to a function of a general nature, in particular, to the representation of one Euclidean space onto another.

Derivative

Let the point move in the direction of the Oy axis, for the time we take x, which is counted from a certain beginning of the moment. Such a movement can be described by the function y=f(x), which is assigned to each time moment x of the coordinate of the point being moved. In mechanics, this function is called the law of motion. The main characteristic of movement, especially uneven, is When a point moves along the Oy axis according to the law of mechanics, then at a random time moment x it acquires the coordinate f (x). At the time moment x + Δx, where Δx denotes the increment of time, its coordinate will be f(x + Δx). This is how the formula Δy \u003d f (x + Δx) - f (x) is formed, which is called the increment of the function. It represents the path traveled by the point in time from x to x + Δx.

In connection with the occurrence of this speed at the moment of time, a derivative is introduced. In an arbitrary function, the derivative at a fixed point is called the limit (provided that it exists). It can be designated by certain symbols:

f'(x), y', ý, df/dx, dy/dx, Df(x).

The process of calculating the derivative is called differentiation.

Differential calculus of a function of several variables

This method of calculus is used in the study of a function with several variables. In the presence of two variables x and y, the partial derivative with respect to x at point A is called the derivative of this function with respect to x with fixed y.

It can be represented by the following symbols:

f'(x)(x,y), u'(x), ∂u/∂x or ∂f(x,y)'/∂x.

Required Skills

To successfully study and be able to solve diffuses, skills in integration and differentiation are required. To make it easier to understand differential equations, you should have a good understanding of the topic of the derivative and It also does not hurt to learn how to look for the derivative of an implicitly given function. This is due to the fact that in the process of studying it will often be necessary to use integrals and differentiation.

Types of differential equations

In almost all tests related to there are 3 types of equations: homogeneous, with separable variables, linear inhomogeneous.

There are also rarer varieties of equations: with total differentials, Bernoulli's equations, and others.

Solution Basics

First you need to remember the algebraic equations from the school course. They contain variables and numbers. To solve an ordinary equation, you need to find a set of numbers that satisfy a given condition. As a rule, such equations had one root, and to check the correctness, one had only to substitute this value for the unknown.

The differential equation is similar to this. In general, such a first-order equation includes:

• independent variable.
• The derivative of the first function.
• function or dependent variable.

In some cases, one of the unknowns, x or y, may be missing, but this is not so important, since the presence of the first derivative, without higher order derivatives, is necessary for the solution and the differential calculus to be correct.

To solve a differential equation means to find the set of all functions that fit a given expression. Such a set of functions is often called the general solution of the differential equation.

Integral calculus

Integral calculus is one of the branches of mathematical analysis that studies the concept of an integral, properties and methods for its calculation.

Often, the calculation of the integral occurs when calculating the area of ​​a curvilinear figure. This area means the limit to which the area of ​​a polygon inscribed in a given figure tends with a gradual increase in its side, while these sides can be made less than any previously specified arbitrary small value.

The main idea in calculating the area of ​​an arbitrary geometric figure is to calculate the area of ​​a rectangle, that is, to prove that its area is equal to the product of length and width. When it comes to geometry, all constructions are made using a ruler and a compass, and then the ratio of length to width is a rational value. When calculating the area of ​​a right triangle, you can determine that if you put the same triangle next to it, then a rectangle is formed. In a parallelogram, the area is calculated by a similar, but slightly more complicated method, through a rectangle and a triangle. In polygons, the area is calculated through the triangles included in it.

When determining the mercy of an arbitrary curve, this method will not work. If you break it into single squares, then there will be unfilled places. In this case, one tries to use two covers, with rectangles on top and bottom, as a result, those include the graph of the function and do not. The method of partitioning into these rectangles remains important here. Also, if we take divisions that are increasingly decreasing, then the area above and below must converge at a certain value.

You should return to the method of division into rectangles. There are two popular methods.

Riemann formalized the definition of the integral, created by Leibniz and Newton, as the area of ​​a subgraph. In this case, figures were considered, consisting of a certain number of vertical rectangles and obtained by dividing the segment. When, as the partition decreases, there is a limit to which the area of ​​a similar figure reduces, this limit is called the Riemann integral of a function on a given interval.

The second method is the construction of the Lebesgue integral, which consists in the fact that for the place of dividing the defined area into parts of the integrand and then compiling the integral sum from the values ​​obtained in these parts, its range of values ​​is divided into intervals, and then summed up with the corresponding measures of the inverse images of these integrals.

Modern benefits

One of the main manuals for the study of differential and integral calculus was written by Fikhtengolts - "Course of differential and integral calculus". His textbook is a fundamental guide to the study of mathematical analysis, which has gone through many editions and translations into other languages. Created for university students and long time applied in many educational institutions as one of the main teaching aids. Gives theoretical data and practical skills. First published in 1948.

Function research algorithm

To investigate a function by methods of differential calculus, it is necessary to follow the already given algorithm:

1. Find the scope of the function.
2. Find the roots of the given equation.
3. Calculate extremes. To do this, calculate the derivative and the points where it equals zero.
4. Substitute the resulting value into the equation.

Varieties of differential equations

DE of the first order (otherwise, differential calculus of one variable) and their types:

• Separated variable equation: f(y)dy=g(x)dx.
• The simplest equations, or differential calculus of a function of one variable, having the formula: y"=f(x).
• Linear inhomogeneous DE of the first order: y"+P(x)y=Q(x).
• Bernoulli's differential equation: y"+P(x)y=Q(x)y a .
• Equation with total differentials: P(x,y)dx+Q(x,y)dy=0.

Second order differential equations and their types:

• Linear homogeneous differential equation of the second order with constant values ​​of the coefficient: y n +py"+qy=0 p, q belongs to R.
• Linear inhomogeneous differential equation of the second order with a constant value of the coefficients: y n +py"+qy=f(x).
• Linear homogeneous differential equation: y n +p(x)y"+q(x)y=0, and inhomogeneous second order equation: y n +p(x)y"+q(x)y=f(x).

Higher order differential equations and their types:

• Differential equation allowing lower order: F(x,y (k) ,y (k+1) ,..,y (n) =0.
• The linear equation of higher order is homogeneous: y (n) +f (n-1) y (n-1) +...+f 1 y"+f 0 y=0, and inhomogeneous: y (n) +f (n-1) y (n-1) +...+f 1 y"+f 0 y=f(x).

Stages of solving a problem with a differential equation

With the help of remote control, not only mathematical or physical questions are solved, but also various problems from biology, economics, sociology and other things. Despite the wide variety of topics, one should adhere to a single logical sequence when solving such problems:

1. Compilation of DU. One of the most difficult steps that requires maximum precision, since any mistake will lead to completely wrong results. All factors influencing the process should be taken into account and the initial conditions should be determined. It should also be based on facts and logical conclusions.
2. Solution of the formulated equation. This process is simpler than the first point, since it requires only strict mathematical calculations.
3. Analysis and evaluation of the obtained results. The derived solution should be evaluated to establish the practical and theoretical value of the result.

An example of the use of differential equations in medicine

The use of remote control in the field of medicine is encountered in the construction of an epidemiological mathematical model. At the same time, one should not forget that these equations are also found in biology and chemistry, which are close to medicine, because the study of different biological populations plays an important role in it. chemical processes in the human body.

In the above example of an epidemic, one can consider the spread of an infection in an isolated society. Inhabitants are divided into three types:

• Infected, number x(t), consisting of individuals, carriers of the infection, each of which is contagious (the incubation period is short).
• The second species includes susceptible individuals y(t) that can become infected through contact with infected individuals.
• The third species includes immune individuals z(t), which are immune or have died due to disease.

The number of individuals is constant, accounting for births, natural deaths and migration is not taken into account. It will be based on two hypotheses.

The percentage of incidence at a certain time point is x(t)y(t) (based on the assumption that the number of cases is proportional to the number of intersections between sick and susceptible representatives, which in the first approximation will be proportional to x(t)y(t)), in Therefore, the number of sick people increases, and the number of susceptible people decreases at a rate that is calculated by the formula ax(t)y(t) (a > 0).

The number of immune individuals that have acquired immunity or died increases at a rate that is proportional to the number of diseased, bx(t) (b > 0).

As a result, it is possible to draw up a system of equations taking into account all three indicators and draw conclusions based on it.

Example of use in economics

The differential calculus is often used in economic analysis. The main task in economic analysis is the study of quantities from the economy, which are written in the form of a function. This is used when solving problems such as changes in income immediately after an increase in taxes, introduction of duties, changes in company revenue when the cost of production changes, in what proportion can retired workers be replaced with new equipment. To solve such questions, it is required to construct a connection function from the input variables, which are then studied using the differential calculus.

In the economic sphere, it is often necessary to find the most optimal indicators: maximum labor productivity, the highest income, the lowest costs, and so on. Each such indicator is a function of one or more arguments. For example, production can be viewed as a function of labor and capital inputs. In this regard, finding a suitable value can be reduced to finding the maximum or minimum of a function from one or more variables.

Problems of this kind create a class of extremal problems in the economic field, the solution of which requires differential calculus. When an economic indicator needs to be minimized or maximized as a function of another indicator, then at the point of maximum, the ratio of the increment of the function to the arguments will tend to zero if the increment of the argument tends to zero. Otherwise, when such a ratio tends to some positive or negative value, the specified point is not suitable, because by increasing or decreasing the argument, you can change the dependent value in the required direction. In the terminology of differential calculus, this will mean that the required condition for the maximum of a function is the zero value of its derivative.

In economics, there are often tasks to find the extremum of a function with several variables, because economic indicators are made up of many factors. Such questions are well studied in the theory of functions of several variables, applying the methods of differential calculation. Such problems include not only maximized and minimized functions, but also constraints. Such questions are related to mathematical programming, and they are solved with the help of specially developed methods, also based on this branch of science.

Among the methods of differential calculus used in economics, an important section is marginal analysis. In the economic sphere, this term refers to a set of methods for studying variable indicators and results when changing the volume of creation, consumption, based on the analysis of their marginal indicators. The limiting indicator is the derivative or partial derivatives with several variables.

The differential calculus of several variables is an important topic in the field of mathematical analysis. For a detailed study, you can use various study guides for higher educational institutions. One of the most famous was created by Fikhtengolts - "Course of differential and integral calculus". As the name implies, skills in working with integrals are of considerable importance for solving differential equations. When the differential calculus of a function of one variable takes place, the solution becomes simpler. Although, it should be noted, it obeys the same basic rules. In order to study a function in practice by differential calculus, it is enough to follow the already existing algorithm, which is given in high school and only slightly complicated when new variables are introduced.

Differential calculus of a function of several variables

Basic definition and concepts.

1. The image of a function of two variables, the domain of definition and change of the function.

2. Partial derivatives, their geometric meaning.

3. Derivatives of higher orders.

4. Differential of a function of two variables, approximate calculations using the differential.

5. Tangent plane and normal to the surface.

Variable zhttps://pandia.ru/text/80/329/images/image001_141.gif" width="13" height="13">G by law (rule) f : (x, y) → z(z = f(x, y) ) a one-to-one correspondence is established.

Lots of G called function scope z = f(x, y) and denoted

Lots of Z called function scope z = f(x, y) and denoted E(z).

A function of two variables can be denoted:

but) explicitly z = f(x, y); z = φ (x, y); z = z(x, y);

b) implicitly F(x, y, z(x, y))=0.

If ( x0,y0)https://pandia.ru/text/80/329/images/image003_67.gif" width="76 height=24" height="24">; E(z) ≥ 0.

schedule functions the spirit of variables is a surface in space .

https://pandia.ru/text/80/329/images/image006_41.gif" width="83" height="29">; depict on a plane howe

the set of points of the domain of definition of these functions.

1) The law (rule) of the correspondence of a function and pairs of independent variables z = f(x, y) is logarithmic, so (x - y)>0, i.e x > y. Domain is the set of points in the plane howe, lying under the line y = x, not including the points belonging to the line, so it is shown as a dotted line.

Change area according to the law of functional dependence z .

2) Law (rule) of conformity z = f(x, y) ,

that's why (y - x2) ≥ 0, i.e y ≥ x2. Domain –

set of plane points howe lying inside

parabolas y ≥ x2, including points belonging to

parabola (boundary of the area). Change area on

the law of functional dependence z ≥ 0.

Definition of partial derivatives of a function of two variables and their geometric meaning.

Partial derivative functions z= f(x, y) are called the limits of the ratio of increments of the function z = z(x, y) to the increment of the corresponding argument along the directions Oh or OU at Δ x → 0 And Δ y → 0 respectively:

Partial derivative with respect to x:

when calculating consider x = const.

Geometrically

https://pandia.ru/text/80/329/images/image014_30.gif" width="108" height="24"> , where α is the angle of the tangent to the surface at the point with the direction of the x-axis;

Where β is the angle of the tangent to the surface at the point with the direction of the y axis.

Differentiation rules And tabular derivatives functions of one variable fully fair for a function of two and several variables.

For a function of two variables z = f(x, y) there are two

first order partial derivatives : https://pandia.ru/text/80/329/images/image017_23.gif" width="89" height="44 src=">, which are also functions of two variables and can be differentiated by variables X And y. Let's find four second order partial derivatives :

Note that mixed derivatives higher orders are equal (Schwarz's theorem): , that is, different derivatives

second order - three: , .

Third derivatives for a function of two variables ( z = f(x, y)) - eight: but four of them are different, since the mixed derivatives when differentiating in any order are equal:

Let's find the first derivatives:

https://pandia.ru/text/80/329/images/image037_12.gif" width="139" height="27">Find the second mixed derivatives:

we see that, that is, we checked the Schwartz theorem and showed that.

Differential and its geometric meaning. Approximate calculations using the differential. Tangent plane and surface normal.

The total differential of a function z= f(x, y) is the linear part of the increment of the function (up to the tangent plane to the surface at the point (x0; y0)):

This formula is used for approximate calculations of a function at a point.

For example, you need to calculate the value of the function in, where

= 1.02 = 1 + 0.02 , but y0 = 2.97 = 3 - 0.03 : accept for X= 1 , and for y = 3;

behind Δ X And Δ at should choose Δ x = 0.02 And Δ y = – 0.03 so that the calculation error is the smallest (does not follow in this example for Δ at choose a value Δ y = 0.97, and for y = 2, presenting a point y0 = 2.97 =2 + 0,97).

Example 2 Calculate the value https://pandia.ru/text/80/329/images/image049_4.gif" width="79" height="33"> and note that it must be calculated at the point x0 = 0.98; y0 = 1.05.

Let's use the opportunity to carry out calculations using the differential. Imagine a point x0 = 0.98 = 1 - 0.02; y0 = 1.05 = 1 + 0.05 and denote x = 1; y = 1; Δх = - 0.02; Δу = 0.05.

Let's calculate the partial derivatives of the function = ; . Then .

For and we calculate

https://pandia.ru/text/80/329/images/image057_3.gif" width="376" height="41 src=">.

Calculating this value on the calculator, we get https://pandia.ru/text/80/329/images/image058_4.gif" width="192 height=48" height="48">0.0003 .

From the definition of a differential, one can also extract it geometric meaning.

If A(x, y)https://pandia.ru/text/80/329/images/image001_141.gif" width="13" height="13">planeZ(x, y) = z(A) + a(x- xA) + b(y- yA), and the surface of the graph of the function merges with the plane in the vicinity of the point A(x, y), then such a plane is called tangent plane to the surface at this point.
Or tangent plane equation a(x-xA)+b(y-yA)+(-1)(z- zA)=0 And normal vector to her who believe normal vector to the surface at the point A(x, y).

Ministry of Education of the Republic of Belarus

Ministry of Education and Science of the Russian Federation

PUBLIC INSTITUTION

HIGHER PROFESSIONAL EDUCATION

BELARUSIAN-RUSSIAN UNIVERSITY

Department of Higher Mathematics

Differential calculus of functions of one and several variables.

Methodical instructions and tasks control work №2

for part-time students

all specialties

committee of the Methodological Council

Belarusian-Russian University

Approved by the department "Higher Mathematics" "_____" ____________ 2004,

protocol no.

Compiled by: Chervyakova T.I., Romskaya O.I., Pleshkova S.F.

Differential calculus of functions of one and several variables. Guidelines and assignments for test No. 2 for part-time students. The paper contains methodological recommendations, control tasks, examples of solving problems in the section "Differential calculus of functions of one and several variables". Tasks are intended for students of all specialties absentee form learning.

Educational edition

Differential calculus of functions of one and several variables

Technical editor A.A. Podoshevko

Computer layout N.P. left-handed

Reviewers L.A. Novik

Responsible for the release of L.V. Pletnev

Signed for printing. Format 60×84 1/16. Offset paper. Screen printing. Conv. oven l. . Uch.-ed. l. . Circulation copies. Order No._________

Publisher and printing design:

State Vocational Education Institution

"Belarusian-Russian University"

License LV No. 243 dated March 11, 2003, License LP No. 165 dated January 8, 2003.

212005, Mogilev, Mira Ave., 43

© GUVPO "Belarusian-Russian

University", 2004

Introduction

These guidelines contain material for studying the section "Differential calculus of a function of one and several variables."

The control work is carried out in a separate notebook, on the cover of which the student should clearly write the number, the name of the discipline, indicate his group, surname, initials and record book number.

The variant number corresponds to the last digit of the record book. If the last digit of the grade book is 0, the option number is 10.

The solution of problems must be carried out in the sequence indicated in the control work. In this case, the condition of each problem is completely rewritten before its solution. Be sure to leave margins in the notebook.

The solution of each problem should be stated in detail, the necessary explanations should be given along the way with reference to the formulas used, the calculations should be carried out in a strict order. Bring the solution of each problem to the answer required by the condition. At the end of the control work, indicate the literature used in the performance of the control work.

Inself-study questions

Derivative of a function: definition, designation, geometric and mechanical meanings. Equation of tangent and normal to a plane curve.

Continuity of a differentiable function.

Rules for differentiating a function of one variable.

Derivatives of complex and inverse functions.

Derivatives of basic elementary functions. Derivative table.

Differentiation of parametrically and implicitly defined functions. Logarithmic differentiation.

Function differential: definition, notation, connection with derivative, properties, form invariance, geometric meaning, application of function values ​​in approximate calculations.

Derivatives and differentials of higher orders.

Theorems of Fermat, Rolle, Lagrange, Cauchy.

The Bernoulli-L'Hopital rule, its application to the calculation of limits.

Monotonicity and extrema of a function of one variable.

Convexity and inflections of the graph of a function of one variable.

Asymptotes of the graph of a function.

Complete exploration and plotting of a function of one variable.

The largest and smallest values ​​of the function on the segment .

The concept of a function of several variables.

Limit and continuity of FNP.

Private derivatives of FNP.

Differentiability and the total differential of FNP.

Differentiation of complex and implicitly given FNPs.

Partial derivatives and total differentials of higher orders FNP.

Extremes (local, conditional, global) FNP.

Directional derivative and gradient.

Tangent plane and surface normal.

Solution of a typical variant

Task 1. Find derivatives of functions:

	b) ;	in) ;
G)		e)

Solution. When solving tasks a)-c), we apply the following differentiation rules:

1)
; 2)
;

3)
; 4)

5)
6)

7)
;

8) if , i.e.
is a complex function
.

Based on the definition of the derivative and the rules of differentiation, a table of derivatives of the main elementary functions has been compiled.

1 ,	8 ,
2 ,	9 ,
3 ,	10 ,
4 ,	11 ,
5 ,	12 ,
6 ,	13 .
7 ,

Using the rules of differentiation and the table of derivatives, we find the derivatives of these functions:

Answer:

Answer:

Answer:

This function is exponential. We apply the method of logarithmic differentiation. Let's log the function:

.

Let's apply the property of logarithms:
. Then
.

Differentiate both sides of the equality with respect to :

;

;

;

.

The function is defined implicitly in the form
. Differentiate both sides of this equation, assuming function from :

We express from the equation :

.

The function is set parametrically
The derivative of such a function is found by the formula:
.

Answer:

Task 2. Find the fourth order differential of a function
.

Solution. Differential
is called a first-order differential.

Differential
is called a second-order differential.

The nth order differential is determined by the formula:
, where n=1,2,…

Let us find successively derivatives.

Task 3. At what point in the graph of the function
tangent to it is parallel to the line
? Make a drawing.

Solution. By condition, the tangents to the graph and the given line are parallel, so the slopes of these lines are equal to each other.

Slope of a straight line
.

Slope of the tangent to the curve at some point we find from the geometric meaning of the derivative:

, where  is the slope of the tangent to the graph of the function
at point .

.

To find the slope coefficients of the desired lines, we compose the equation

.

Solving it, we find the abscissas of the two points of contact:
And
.

From the equation of the curve, we determine the ordinates of the touch points:
And
.

Let's make a drawing.

Answer: (-1;-6) and
.

Comment : equation of the tangent to the curve at a point
looks like:

the equation of the normal to the curve at a point has the form:

.

Task 4. Conduct a complete study of the function and build its graph:

.

Solution. To fully study the function and build its graph, the following exemplary scheme is used:

find the scope of the function;

investigate the function for continuity and determine the nature of the break points;

to investigate the function for even and odd, periodicity;

find the points of intersection of the graph of the function with the coordinate axes;

examine the function for monotonicity and extremum;

find intervals of convexity and concavity, inflection points;

find the asymptotes of the function graph;

to refine the graph, it is sometimes advisable to find additional points;

plot the function according to the data obtained.

We apply the above scheme to study this function.

The function is neither even nor odd. The function is not periodic.

Dot
- point of intersection with the x-axis.

With y-axis:
.

Point (0;-1) - the point of intersection of the graph with the Oy axis.

We find the derivative.

at
and does not exist at
.

Critical points:
And
.

We investigate the sign of the derivative of the function on the intervals .

The function decreases over intervals
; increases - on the interval
.

We find the second derivative.

at
and does not exist for .

Critical points of the second kind: and
.

The function is convex on the interval
, the function is concave on the intervals
.

inflection point,
.

Let us prove this by examining the behavior of the function near the point .

Let's find oblique asymptotes

Then
- horizontal asymptote

Let's find additional points:

Based on the data obtained, we build a graph of the function.

Task 5. Let's formulate the Bernoulli-L'Hopital rule as a theorem.

Theorem: if two functions
And
:

.

Find the limits by applying the Bernoulli-L'Hopital rule:

but)
; b)
; in)
.

Solution. but) ;

in)
.

Let's apply the identity
. Then

Task 6. Given a function
. To find , ,
.

Solution. Let's find partial derivatives.

Total differential of a function
calculated by the formula:

.

Answer:
,
,
.

Task 7 Differentiate:

Solution. but) The derivative of a complex function is found by the formula:

;
;

Answer:

b) If the function is given implicitly by the equation
, then its partial derivatives are found by the formulas:

,
.

,
,
.

;
.

Answer:
,
.

Task 8 Find local, conditional or global extrema of a function:

Solution. but) Let's find the critical points of the function by solving the system of equations:

- critical point.

We apply sufficient conditions for an extremum.

Let's find the second partial derivatives:

;
;
.

We compose the determinant (discriminant):

Because
, then at the point M 0 (4; -2) the function has a maximum.

Answer: Z max \u003d 13.

b)
, provided that
.

To compose the Lagrange function, we apply the formula

- this function

Communication equation. can be shortened. Then. Left and right limits. Theorems... Document

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Mathematics part 4 differential calculus of functions of several variables differential equations series

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Maths. Part 4 differentialcalculusfunctionsseveralvariables. Differential equations. Rows: Educational ... mathematical analysis", " differentialcalculusfunctionsonevariable" and "Integral calculusfunctionsonevariable". GOALS AND...

An extension of the calculus of functions of a variable is multivariate analysis, when differential calculus of functions of several variables- functions that are integrated and differentiated affect not one, but several variables.

The differential calculus of functions of several variables involves the following typical operations:

1. Continuity and limits.

Many pathological and illogical results that are not characteristic of a function of one variable result from the study of continuity and limits in multidimensional spaces. For example, there are two variable scalar functions that have points in the domain of definition, which give a specific limit when approached along a straight line, and when approached along a parabola give a completely different limit. To zero, the function tends to zero when passing along any straight line that passes through the origin. Due to the fact that the limits do not coincide along different trajectories, there is no single limit.

When the variables x tend, the limit function has a certain number. If the limit value of a function at a certain point exists and is equal to the particular value of the function, then such a function is called continuous at a given point. If a function is continuous on the set of points, then it is called continuous on the set of points.

2. Finding the partial derivative.

The partial derivative of several variables means the derivative of one variable, and all other variables are considered constants.

3. Multiple integration.

The multiple integral extends the notion of an integral to functions of several variables. To calculate the volumes and areas of regions in space and plane, double and triple integrals are used. According to the Tonelli-Fubini theorem, the multiple integral can also be calculated as an iterated integral.

All this makes it possible to perform differential calculus of functions of several variables.

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Function of n variables Variable u is called a function of n variables (arguments) x, y, z, …, t, if each system of values ​​x, y, z, …, t, from the range of their changes (the domain of definition), corresponds to a certain value u. The domain of a function is the set of all points at which it has certain real values. For a function of two variables z=f(x, y), the domain of definition represents a certain set of points in the plane, and for a function of three variables u=f(x, y, z) it represents a certain set of points in space.

Function of two variables A function of two variables is a law according to which each pair of values ​​of independent variables x, y (arguments) from the domain of definition corresponds to the value of the dependent variable z (function). This function is denoted as follows: z = z(x, y) or z= f(x, y) , or another standard letter: u=f(x, y) , u = u (x, y)

Partial derivatives of the first order The partial derivative of the function z \u003d f (x, y) with respect to the independent variable x is the finite limit calculated at a constant y The partial derivative with respect to y is the finite limit calculated at a constant x For partial derivatives, the usual rules and differentiation formulas are valid.

The total differential of the function z =f(x, y) is calculated by the formula The total differential of the function of three arguments u =f(x, y, z) is calculated by the formula

Partial derivatives of higher orders Partial derivatives of the second order of the function z =f(x, y) are partial derivatives of its partial derivatives of the first order. Similarly, partial derivatives of the third and higher orders are defined and denoted.

Higher-order differentials The second-order differential of a function z=f(x, y) is the differential of its shallow differential. Higher-order differentials are calculated by the formula There is a symbolic formula

Differentiation of complex functions Let z=f(x, y), where x=φ(t), y=ψ(t) and the functions f(x, y), φ(t), ψ(t) are differentiable. Then the derivative of the complex function z=f[φ(t), ψ(t)] is calculated by the formula

Differentiation of implicit functions Derivatives of an implicit function of two variables z=f(x, y), given by the equation F(x, y, z)=0, can be calculated by the formulas

The extremum of the function The function z=f(x, y) has a maximum (minimum) at the point M 0(x 0; y 0) if the value of the function at this point is greater (less) than its value at any other point M(x; y ) of some neighborhood of the point M 0. If the differentiable function z=f(x, y) reaches an extremum at the point M 0(x 0; y 0), then its first-order partial derivatives at this point are equal to zero, i.e. (necessary extremum conditions).

Let M 0(x 0; y 0) be a stationary point of the function z=f(x, y). Let's designate AND make up the discriminant Δ=AC B 2. Then: If Δ>0, then the function has an extremum at the point M 0, namely, a maximum at A 0 (or C>0); If Δ

Antiderivative function The function F(x) is called antiderivative for the function f(x) on the interval X=(a, b) if at each point of this interval f(x) is the derivative for F(x), i.e. From this definition it follows that the problem of finding the antiderivative is inverse to the problem of differentiation: for a given function f(x), it is required to find a function F(x) whose derivative is equal to f(x).

Indefinite integral The set of all antiderivatives of the function F(x)+C for f(x) is called the indefinite integral of the function f(x) and is denoted by the symbol. Thus, by definition, where C is an arbitrary constant; f(x) integrand; f(x) dx integrand; x variable of integration; indefinite integral sign.

Properties of the indefinite integral 1. The differential of the indefinite integral is equal to the integrand, and the derivative of the indefinite integral is equal to the integrand: 2. The indefinite integral of the differential of some function is equal to the sum of this function and an arbitrary constant:

3. The constant factor can be taken out of the integral sign: 4. The indefinite integral of the algebraic sum of a finite number of continuous functions is equal to the algebraic sum of the integrals of the terms of the functions: 5. If, then and where u=φ(x) is an arbitrary function that has a continuous derivative

Basic Integration Methods Direct Integration Method An integration method in which a given integral is reduced to one or more table integrals by identical transformations of the integrand (or expression) and applying the properties of the indefinite integral is called direct integration.

When reducing this integral to a tabular one, the following transformations of the differential are often used (the operation of “bringing under the sign of the differential”):

Variable change in the indefinite integral (substitution integration) The substitution integration method consists in introducing a new integration variable. In this case, the given integral is reduced to a new integral, which is tabular or reducible to it. Let it be required to calculate the integral. Let's make a substitution x = φ(t), where φ(t) is a function that has a continuous derivative. Then dx=φ "(t)dt and based on the invariance property of the indefinite integral integration formula, we obtain the integration formula by substitution

Integration by parts Formula of integration by parts The formula makes it possible to reduce the calculation of the integral to the calculation of the integral, which may turn out to be much simpler than the original one.

Integration of Rational Fractions A rational fraction is a fraction of the form P(x)/Q(x), where P(x) and Q(x) are polynomials. A rational fraction is called proper if the degree of the polynomial P(x) is lower than the degree of the polynomial Q(x); otherwise, the fraction is called an improper fraction. The simplest (elementary) fractions are regular fractions of the following form: where A, B, p, q, a are real numbers.

The first integral of the simplest type IV fraction on the right side of the equation is easily found by substituting x2+px+q=t, and the second is transformed as follows: Setting x+p/2=t, dx=dt we get and denoting qp 2/4=a 2 ,

Integration of rational fractions using decomposition into simple fractions Before integrating the rational fraction P(x)/Q(x), the following algebraic transformations and calculations must be done: 1) If an improper rational fraction is given, then select an integer part from it, i.e. in the form where M(x) is a polynomial, and P 1(x)/Q(x) is a proper rational fraction; 2) Expand the denominator of the fraction into linear and quadratic factors: where р2/4 q

3) Decompose the correct rational fraction into simple fractions: 4) Calculate the indefinite coefficients A 1, A 2, ..., Am, ..., B 1, B 2, ..., Bm, ..., C 1, C 2, ..., Cm, ... , for which we bring the last equality to a common denominator, equate the coefficients at the same powers of x in the left and right parts of the resulting identity and solve the system of linear equations with respect to the desired coefficients.

Integration of the simplest irrational functions 1. Integrals of the form where R is a rational function; m 1, n 1, m 2, n 2, … integers. Using the substitution ax+b=ts, where s is the least common multiple of the numbers n 1, n 2, ..., the specified integral is converted into an integral of a rational function. 2. An integral of the form Such integrals, by selecting a square from a square trinomial, are reduced to table integrals 15 or 16

3. Integral of the form To find this integral, we select in the numerator the derivative of the square trinomial, which is under the root sign, and expand the integral into the sum of integrals:

4. Integrals of the form By substituting x α=1/t, this integral is reduced to the considered item 2. 5. An integral of the form where Рn(х) is a polynomial of the nth degree. An integral of this kind is found using the identity where Qn 1(x) is a polynomial (n 1) of the th degree with indefinite coefficients, λ is a number. Differentiating the indicated identity and reducing the result to a common denominator, we obtain the equality of two polynomials, from which we can determine the coefficients of the polynomial Qn 1(x) and the number λ.

6. Integrals of differential binomials where m, n, p are rational numbers. As P. L. Chebyshev proved, integrals of differential binomials are expressed in terms of elementary functions only in three cases: 1) p is an integer, then this integral is reduced to the integral of a rational function using the substitution x=ts, where s is the least common multiple denominators of fractions m and n. 2) (m+1)/n is an integer, in this case this integral is rationalized using the substitution a+bxn=ts; 3) (m+1)/n+р is an integer, in this case the substitution ax n+b=ts leads to the same goal, where s is the denominator of the fraction р.

Integration of trigonometric functions Integrals of the form where R is a rational function. Under the integral sign is a rational function of sine and cosine. In this case, the universal trigonometric substitution tg(x/2)=t is applicable, which reduces this integral to the integral of the rational function of the new argument t (table p. 1). There are other substitutions as shown in the following table:

The definite integral of the function f(x) on a segment is the limit of integral sums, provided that the length of the largest partial segment Δхi tends to zero. The numbers a and b are called the lower and upper limits of integration. Cauchy's theorem. If the function f(x) is continuous on the segment , then the definite integral exists

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Rules for calculating definite integrals 1. Newton Leibniz formula: where F(x) is the antiderivative for f(x), i.e. F(x)‘= f(x). 2. Integration by parts: where u=u(x), v=v(x) are continuously differentiable functions on the segment .

3. Change of variable where x=φ(t) is a function that is continuous together with its derivative φ' (t) on the segment α≤t≤β, a= φ(a), b= φ(β), f[φ( t)] – the function is continuous on [α; β] 4. If f(x) is an odd function, i.e. f(x)= f(x), then If f(x) is an even function, i.e. f(x)=f(x) , then.

Improper integrals Improper integrals are: 1) integrals with infinite limits; 2) integrals of unbounded functions. The improper integral of the function f (x) in the range from a to + infinity is defined by the equality If this limit exists and is finite, then the improper integral is called convergent; if the limit does not exist or is equal to infinity, divergent If the function f(x) has an infinite discontinuity at a point from the segment and is continuous for a≤x

In the study of the convergence of improper integrals, one of the signs of comparison is used. 1. If the functions f(x) and φ(x) are defined for all х≥а and are integrable on the segment , where А≥а, and if 0≤f(x)≤φ(x) for all х≥а, then from convergence of the integral implies the convergence of the integral, and 2. 1 If for x→+∞ the function f(x)≤0 is infinitesimal of order p>0 compared to 1/x, then the integral converges for p>1 and diverges for p≤1 2. 2 If the function f(x) ≥ 0 is defined and continuous in the interval a ≤ x

Calculating the area of ​​a flat figure The area of ​​a curvilinear trapezoid bounded by a curve y=f(x), straight lines x=a and x=b and a segment of the OX axis is calculated by the formula Area of ​​a figure bounded by a curve y=f 1(x) and y=f 2( x) and straight lines x=a and x=b is found by the formula and a segment of the OX axis is calculated by the formula where t 1 and t 2 are determined from the equation a = x (t 1), b = x (t 2) two polar radii θ=α, θ=β (α

Calculating the arc length of a plane curve If the curve y=f(x) on the segment is smooth (i.e., the derivative y'=f'(x) is continuous), then the length of the corresponding arc of this curve is found by the formula (t), y=y(t) [x(t) and y(t) are continuously differentiable functions] the length of the arc of the curve, corresponding to the monotonic change in the parameter t from t 1 to t 2, is calculated by the formula If a smooth curve is given in polar coordinates by the equation ρ=ρ(θ), α≤θ≤β, then the length of the arc is equal to.

Calculation of body volume 1. Calculation of body volume from known cross-sectional areas. If the cross-sectional area of ​​the body, a plane perpendicular to the OX axis, can be expressed as a function of x, i.e., in the form S=S(x) (a≤x≤b), the volume of the body part enclosed between the planes perpendicular to the OX axis x= a and x=b, is found by formula 2. Calculation of the volume of a body of revolution. If a curvilinear trapezoid bounded by the curve y=f(x) and straight lines y=0, x=a, x=b rotates around the OX axis, then the volume of the body of revolution is calculated by the formula If the figure bounded by the curves y1=f 1(x) and y2=f 2(x) and straight lines x=a, x=b, rotates around the OX axis, then the volume of the subject of rotation is equal.

Calculation of the surface area of ​​rotation If the arc of a smooth curve y=f(x) (a≤х≤b) rotates around the OX axis, then the area of ​​the surface of rotation is calculated by the formula If the curve is given by the parametric equations x=x(t), y=y(t ) (t 1≤t≤t 2), then.

Basic concepts A differential equation is an equation that relates independent variables, their function and derivatives (or differentials) of this function. If there is one independent variable, then the equation is called ordinary, but if there are two or more independent variables, then the equation is called a partial differential equation.

First order equation The functional equation F(x, y, y) = 0 or y = f(x, y) connecting the independent variable, the desired function y(x) and its derivative y (x), is called the first order differential equation . A solution of a first-order equation is any function y= (x), which, being substituted into the equation together with its derivative y = (x), turns it into an identity with respect to x.

General solution of a first-order differential equation A general solution of a first-order differential equation is a function y = (x, C) that, for any value of the parameter C, is a solution to this differential equation. The equation Ф(x, y, C)=0, which defines the general solution as an implicit function, is called the general integral of the differential equation.

Equation solved with respect to the derivative If the equation of the 1st order is solved with respect to the derivative, then it can be represented as Its general solution is geometrically a family of integral curves, i.e. a set of lines corresponding to different values ​​of the constant C.

Statement of the Cauchy problem The problem of finding a solution to a differential equation that satisfies the initial condition at is called the Cauchy problem for a first order equation. Geometrically, this means: find the integral curve of the differential equation passing through the given point.

Separated Variable Equation A differential equation is called a separated variable equation. A 1st order differential equation is called an equation with separable variables if it has the form: To solve the equation, divide both its parts by the product of functions, and then integrate.

Homogeneous Equations A first-order differential equation is called homogeneous if it can be reduced to the form y = or to the form where and are homogeneous functions of the same order.

First order linear equations A first order differential equation is called linear if it contains y and y‘ to the first degree, i.e., it has the form. Such an equation is solved using the substitution y=uv, where u and v are auxiliary unknown functions that are found by substituting auxiliary functions into the equation and certain conditions are imposed on one of the functions.

Bernoulli's equation Bernoulli's equation is a first-order equation that has the form

2nd order differential equations A 2nd order equation has the form Or A general solution of a second order equation is a function that, for any values ​​of the parameters, is a solution to this equation.

Cauchy problem for the 2nd order equation If the 2nd order equation is solved with respect to the second derivative, then for such an equation the following problem takes place: find a solution of the equation that satisfies the initial conditions: and This problem is called the Cauchy problem for the 2nd order differential equation.

Existence and uniqueness theorem for a solution to a second-order equation If in an equation a function and its partial derivatives with respect to arguments and are continuous in some domain containing a point, then there also exists a unique solution of this equation that satisfies the conditions and.

2nd Order Equations Allowing Reduction of Order The simplest 2nd order equation is solved by double integration. An equation that does not explicitly contain y is solved by substitution, an equation that does not contain x is solved by substitution, .

Linear Homogeneous Equations A second-order linear homogeneous differential equation is an equation. If all the coefficients of this equation are constant, then the equation is called an equation with constant coefficients.

Properties of Solutions to a Linear Homogeneous Equation Theorem 1. If y(x) is a solution to an equation, then Cy(x), where C is a constant, is also a solution to this equation.

Properties of Solutions to a Linear Homogeneous Equation Theorem 2. If and are solutions to an equation, then their sum is also a solution to this equation. Consequence. If and is a solution to an equation, then the function is also a solution to that equation.

Linearly dependent and linearly independent functions Two functions and are called linearly dependent on some interval if it is possible to choose such numbers and not equal to zero at the same time that the linear combination of these functions is identically equal to zero on this interval, i.e.

If such numbers cannot be chosen, then the functions and are called linearly independent on the indicated interval. Functions will be linearly dependent if and only if their ratio is constant, i.e.

Theorem on the structure of the general solution of a linear homogeneous equation of the 2nd order If linearly independent partial solutions of the 2nd order LOE, then their linear combination where and are arbitrary constants, is a general solution of this equation.

Linear homogeneous equation of the 2nd order with constant coefficients The equation is called the characteristic equation of a linear equation. It is obtained from the LOE by replacing the derivative with the power k corresponding to the order.