– determination of the derivative of a function f (x) at point x 0 ;
– differential function f (x) at point x 0 .
Derivatives of the simplest elementary functions:
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– rule for differentiating a complex function at a point x 0 , here ;
– rule for differentiating the inverse function at a point;
– Lagrange formula, ;
– Cauchy formula, ;
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– Taylor formula, . 1. Berman G.N. Collection of problems for the course of mathematical analysis. M.: Nauka, 1975.
2. Bermant A.F., Aramonovich I.I. Short course in mathematical analysis. M.: Nauka, 1967.
3. Bolgov V.A., Demidovich B.P., Efimov A.V. and etc. Collection of problems in mathematics for colleges. Part 1, M.: Nauka, 1986.
4. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in examples and problems. M.: Higher School, 1986.
5. Problems and exercises in mathematical analysis for technical colleges. Ed. Demidovich B.P., M.: Nauka, 1968.
6. Zaporozhets G.I. Guide to solving problems in mathematical analysis. M.: Higher School, 1964.
7. Kudryavtsev V.A., Demidovich B.P. Short course of higher mathematics. M.: Nauka, 1985.
8. Mathematics at a technical university. Issue II. Differential calculus of functions of one variable. Ed. Zarubina V.S. and Krischenko A.P., M.: Publishing house of MSTU im. N.E. Bauman, 2001.
9. Minorsky V.P. Collection of problems in higher mathematics. M.: Nauka, 1987.
10. Piskunov N.S. Differential and integral calculus. M.: Nauka, T. 1,2, 1976.
11. Collection of problems in mathematics for technical colleges. Ed. Efimova A.V., M.: Nauka, Part 1-4, 1993-1994.
12. Shchipachev V.S. Higher mathematics. M.: Higher School, 1996.
13. Shchipachev V.S. Problems in higher mathematics. M.: Higher School, 1997.
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Tyumen State Oil and Gas University.
Compiled by: Musakaev N.G., Associate Professor, Ph.D.
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Musakaeva M.F., assistant
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differential calculus
a branch of mathematics that studies derivatives, differentials and their applications to the study of properties of functions. The derivative of the function y = f(x) is the limit of the ratio of the increment?y = y1 - y0 of the function to the increment?x = x1 - x0 of the argument as?x tends to zero (if this limit exists). The derivative is denoted by f?(x) or y?; Thus, the Differential of the function y = f(x) is the expression dy = y?dx, where dx = ?x is the increment of the argument x. It is obvious that y? = dy/dx. The ratio dy/dx is often used as a sign for the derivative. The calculation of derivatives and differentials is called differentiation. If the derivative f?(x) has, in turn, a derivative, then it is called the 2nd derivative of the function f(x) and is denoted f??(x), etc. The basic concepts of differential calculus can be extended to the case of functions of several variables. If z = f(x,y) is a function of two variables x and y, then, having fixed some value for y, we can differentiate z with respect to x; the resulting derivative dz/dx = f?x is called the partial derivative of z with respect to x. The partial derivative dz/dy = f?y, partial derivatives of higher orders, partial and total differentials are defined similarly. For applications of differential calculus to geometry, it is important that the so-called. the slope of the tangent, i.e. tangent of the angle? (see Fig.) between the Ox axis and the tangent to the curve y = f(x) at the point M(x0, y0), is equal to the value of the derivative at x = x0, i.e. f?(x0). In mechanics, the speed of a rectilinearly moving point can be interpreted as the derivative of the path with respect to time. Differential calculus (like integral calculus) has numerous applications.
Differential calculus
a branch of mathematics that studies derivatives and differentials of functions and their applications to the study of functions. Design of D. and. into an independent mathematical discipline is associated with the names of I. Newton and G. Leibniz (second half of the 17th century). They formulated the main provisions of D. and. and clearly indicated the mutually inverse nature of the operations of differentiation and integration. From that time on, D. and. develops in close connection with integral calculus, together with which it forms the main part of mathematical analysis (or infinitesimal analysis). The creation of differential and integral calculus opened a new era in the development of mathematics. It entailed the emergence of a number of mathematical disciplines: the theory of series, the theory of differential equations, differential geometry and the calculus of variations. Methods of mathematical analysis have found application in all branches of mathematics. The field of applications of mathematics to questions of natural science and technology has expanded immeasurably. “Only differential calculus gives natural science the opportunity to depict mathematically not only states, but also processes: motion” (F. Engels, see K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 587). D. and. is based on the following most important concepts of mathematics, the definition and study of which form the subject of introduction to mathematical analysis: real numbers (number line), function, limit, continuity. All these concepts crystallized and received modern content in the course of the development and justification of differential and integral calculus. The main idea of D. and. consists of studying functions in the small. More precisely: D. and. provides an apparatus for studying functions whose behavior in a sufficiently small neighborhood of each point is close to the behavior of a linear function or polynomial. The central concepts of differential theory serve as such an apparatus: derivative and differential. The concept of a derivative arose from a large number of problems in natural science and mathematics that lead to the calculation of limits of the same type. The most important of them are determining the speed of linear motion of a point and constructing a tangent to the curve. The concept of differential is a mathematical expression of the closeness of a function to linear in a small neighborhood of the point under study. Unlike the derivative, it is easily transferred to mappings of one Euclidean space into another and to mappings of arbitrary linear normed spaces and is one of the main concepts of modern nonlinear functional analysis. Derivative. Let it be necessary to determine the speed of a rectilinearly moving material point. If the movement is uniform, then the path traveled by the point is proportional to the time of movement; the speed of such movement can be defined as the path traveled per unit of time, or as the ratio of the path traveled over a certain period of time to the duration of this interval. If the movement is uneven, then the paths traversed by the point in equal time intervals will, generally speaking, be different. An example of uneven motion is given by a body freely falling in a vacuum. The law of motion of such a body is expressed by the formula s = gt2/2, where s ≈ distance traveled from the beginning of the fall (in meters), t ≈ time of fall (in seconds), g ≈ constant value, acceleration of free fall, g » 9.81 m/ sec2. In the first second of the fall the body will travel about 4.9 m, in the second ≈ about 14.7 m, and in the tenth ≈ about 93.2 m, i.e. the fall occurs unevenly. Therefore, the above definition of speed is unacceptable here. In this case, the average speed of movement over a certain period of time after (or before) a fixed moment t is considered; it is defined as the ratio of the length of the path traveled during this period of time to its duration. This average speed depends not only on the moment t, but also on the choice of time period. In our example, the average speed of fall over the period of time from t to t + Dt is equal to This expression, with an unlimited decrease in the time interval Dt, approaches the value gt, which is called the speed of movement at time t. Thus, the speed of movement at any point in time is defined as the limit of the average speed when the period of time decreases indefinitely. In the general case, these calculations must be carried out for any moment of time t, the time interval from t to t + Dt and the law of motion expressed by the formula s = f (t). Then the average speed of movement for the period of time from t to t + Dt is given by the formula Ds/Dt, where Ds = f (t + Dt) ≈ f (t), and the speed of movement at time t is equal to The main advantage of speed at a given time, or instantaneous speed, before average speed is that it, like the law of motion, is a function of time t, and not a function of the interval (t, t + Dt). On the other hand, instantaneous speed is a certain abstraction, since it is the average, not the instantaneous speed that can be directly measured. The problem also leads to an expression of type (*) (see. rice.) constructing a tangent to a plane curve at some point M. Let the curve Г be the graph of the function y = f (x). The position of the tangent will be determined if its angular coefficient is found, i.e., the tangent of the angle a formed by the tangent with the Ox axis. Let us denote by x0 the abscissa of point M, and by x1 = x0 + Dх ≈ abscissa of point M
The angular coefficient of the secant MM1 is equal to
where Dy = M1N = f (x0 + Dx) ≈ f (x0) ≈ increment of the function on the segment. Defining the tangent at point M as the limiting position of the secant MM1, when x1 tends to x0, we obtain
Abstracting from the mechanical or geometric content of the above problems and highlighting the common solution method for them, we come to the concept of derivative. The derivative of the function y = f (x) at point x is the limit (if it exists) of the ratio of the increment of the function to the increment of the argument, when the latter tends to zero, so that
With the help of the derivative, in addition to those already discussed, a number of important concepts of natural science are defined. For example, the current strength is defined as the limit
where Dq ≈ positive electric charge transferred through the circuit cross section during time Dt; the rate of a chemical reaction is defined as the limit
where DQ ≈ change in the amount of substance over time Dt; In general, the time derivative is a measure of the rate of a process, applicable to a wide variety of physical quantities.
The derivative of the function y = f (x) is denoted by f" (x), y", dy/dx, df/dx or Df (x). If the function y = f (x) has a derivative at the point x0, then it is defined both at the point x0 itself and in some neighborhood of this point and is continuous at the point x0. The opposite conclusion would, however, be incorrect. For example, a function continuous at every point
the graph of which is the bisectors of the first and second coordinate angles; at x = 0 it has no derivative, because the ratio Dу/Dх has no limit when Dx ╝ 0: if Dх > 0, this ratio is equal to +1, and if Dx< 0, то оно равно -1. Более того, существуют непрерывные функции, не имеющие производной ни в одной точке (см. Непрерывная функция).
The operation of finding the derivative is called differentiation. On the class of functions that have a derivative, this operation is linear.
Table of formulas and differentiation rules
(C)` = 0; (xn)` = nxn-1;
(ax)` = ax ln a and (ex)` = ex;
(logax)` = 1/x ln a and (ln x)` = 1/x;
(sin x)` = cos x; (cos x)` = √ sin x;
(tg x)` = 1/cos2x; (ctg x)` = √ 1/sin2x;
(arc tan x)` = 1/(1 + x2).
` = f `(x) ╠ g`(x);
` = Cf `(x);
` = f``(x) g (x) + f (x) g `(x);
if y = f (u) and u = j(x), i.e. y = f, then dy/dx = (dy/du)×(du/dx) = f¢ (u)j¢(x) .
Here C, n and a ≈ constants, a > 0. This table, in particular, shows that the derivative of any elementary function is again an elementary function.
If the derivative f" (x), in turn, has a derivative, then it is called the second derivative of the function y = f (x) and is denoted
y", f" (x), d2y/dx2, d2f/dx2 or D2f (x).
For a point moving rectilinearly, the second derivative characterizes its acceleration.
Derivatives of a higher (integer) order are defined similarly. The derivative of order n is denoted
yn, fn (x), dny/dxn, dnf/dxn or Dnf (x).
Differential. A function y = f (x), the domain of which contains a certain neighborhood of the point x0, is called differentiable at the point x0 if its increment
Dy = f (x0 + Dx) - f (x0)
can be written in the form
Dу = АDх + aDх,
where A = A (x0), a = a(x, x0) ╝ 0 for x ╝ x0. In this and only in this case, the expression ADx is called the differential of the function f (x) at the point x0 and is denoted dy or df (x0). Geometrically, the differential (with a fixed value of x0 and a varying increment of Dx) represents the increment of the ordinate of the tangent, i.e., the segment NT (see. rice.). The differential dy is a function of both the point x0 and the increment Dx. They say that the differential is the main linear part of the increment of the function, meaning by this that, for a fixed x0, dy is a linear function of Dx and the difference Dy - dy is infinitesimal with respect to Dx. For the function f (x) º x we have dx = Dх, i.e. the differential of the independent variable coincides with its increment. Therefore, they usually write dy = Adx. There is a close connection between the differential of a function and its derivative. In order for a function of one variable y = f (x) to have a differential at the point x0, it is necessary and sufficient that it has a (finite) derivative f" (x0) at this point, and the equality dy = f" (x0) dx is true . The visual meaning of this proposition is that the tangent to the curve y = f (x) at the point with the abscissa x0, as the limiting position of the secant, is also a line that, in an infinitesimal neighborhood of the point x0, adjoins the curve more closely than any other line. Thus, always A (x0) = f" (x0); the notation dy/dx can be understood not only as a designation for the derivative f" (x0), but also as the ratio of the differentials of the dependent and independent variables. Due to the equality dy = f" (x0) dx, the rules for finding differentials directly follow from the corresponding rules for finding derivatives.
Higher order differentials are also considered. In practice, with the help of differentials, approximate calculations of function values are often made, and calculation errors are also assessed. Let, for example, we need to calculate the value of the function f (x) at point x if f (x0) and f" (x0) are known. By replacing the increment of the function with its differential, we obtain an approximate equality
f (x1) » f (x0) + df (x0) = f (x0) + f" (x0) (x1 - x0).
The error of this equality is approximately equal to half of the second differential of the function, i.e.
1/2 d2f = 1/2 f" (x0)(x1 √ x0)
Applications. In D. and. connections are established between the properties of a function and its derivatives (or differentials), expressed by the basic theorems of dynamic theory. These include Rolle’s theorem, Lagrange’s formula f (a) ≈ f (b) = f" (c)(b ≈ a), where a< с < b (подробнее см. Конечных приращений формула), и Тейлора формула.
These proposals allow the methods of D. and. conduct a detailed study of the behavior of functions that are sufficiently smooth (that is, have derivatives of a sufficiently high order). In this way, it is possible to study the degree of smoothness, convexity and concavity, increase and decrease of functions, their extrema, find their asymptotes, inflection points (see Inflection point), calculate the curvature of a curve, find out the nature of its singular points, etc. For example, the condition f" (x) > 0 entails a (strict) increase in the function y = f (x), and the condition f" (x) > 0 ≈ its (strict) convexity. All extremum points of a differentiable function belonging to the interior of its domain of definition are located among the roots of the equation f" (x) = 0.
The study of functions using derivatives is the main application of dynamic theory. In addition, D. and. allows you to calculate various kinds of limits of functions, in particular limits of the form 0/0 and ¥/¥ (see Indefinite expression, L'Hopital's rule). D. and. It is especially convenient for studying elementary functions, because in this case, their derivatives are written out explicitly.
D. and. functions of many variables. Methods D. and. are used to study functions of several variables. For a function of two independent variables z = f (x, y), the partial derivative with respect to x is the derivative of this function with respect to x for constant y. This partial derivative is denoted z"x, f"x (x, y), ╤z/╤x or ╤f (x, y)/╤x, so
The partial derivative of z with respect to y is defined and denoted similarly. Magnitude
Dz = f (x + Dx, y + Dy) - f (x, y)
is called the complete increment of the function z = f (x, y). If it can be represented in the form
Dz = ADx + ВDу + a,
where a ≈ an infinitesimal of a higher order than the distance between the points (x, y) and (x + Dx, y + Dу), then the function z = f (x, y) is said to be differentiable. The terms ADx + BDу form the complete differential dz of the function z = f (x, y), with A = z"x, B = z"y. Instead of Dx and Dy we usually write dx and dy, so
Geometrically, the differentiability of a function of two variables means that its graph has a tangent plane, and the differential represents the increment of the applicate of the tangent plane when the independent variables receive increments dx and dy. For a function of two variables, the concept of differential is much more important and natural than the concept of partial derivatives. Unlike functions of one variable, for functions of two variables the existence of both first-order partial derivatives does not guarantee the differentiability of the function. However, if the partial derivatives are also continuous, then the function is differentiable.
Partial derivatives of higher orders are defined similarly. The partial derivatives ╤2f/╤x2 and ╤2f/╤у2, in which differentiation is carried out with respect to one variable, are called pure, and the partial derivatives ╤2f/╤x╤y and ╤2f/╤у╤х≈ mixed. If mixed partial derivatives are continuous, then they are equal to each other. All these definitions and notations carry over to the case of a larger number of variables.
Historical reference. Individual problems about determining tangents to curves and finding the maximum and minimum values of variables were solved by mathematicians of Ancient Greece. For example, methods were found to construct tangents to conic sections and some other curves. However, the methods developed by ancient mathematicians were applicable only in very special cases and were far from the ideas of D. and.
The era of creation of D. and. As an independent branch of mathematics, one should consider the time when it was understood that these special problems, together with a number of others (especially the problem of determining instantaneous speed), are solved using the same mathematical apparatus - with the help of derivatives and differentials. This understanding was achieved by I. Newton and G. Leibniz.
Around 1666, I. Newton developed the fluxion method (see Fluxion calculus). Newton formulated the main tasks in terms of mechanics: 1) determining the speed of movement based on the known dependence of the path on time; 2) determination of the distance traveled during a given time using a known speed. Newton called a continuous variable fluent (current), its speed ≈ fluxion. Thus, Newton’s main concepts were the derivative (fluxion) and the indefinite integral as an antiderivative (fluentia). He sought to substantiate the method of fluxions with the help of the theory of limits, although the latter was only outlined by him.
In the mid-70s. 17th century G. Leibniz developed a very convenient algorithm for D. and. Leibniz's main concepts were the differential as an infinitesimal increment of a variable and the definite integral as the sum of an infinitely large number of differentials. Leibniz owns the notation for the differential dx and the integral òydx, a number of differentiation rules, convenient and flexible symbolism, and, finally, the term “differential calculus” itself. Further development of D. and. first followed the path outlined by Leibniz; The works of the brothers J. and I. Bernoulli, B. Taylor and others played a major role at this stage.
The next stage in the development of D. and. there were works by L. Euler and J. Lagrange (18th century). Euler first began to present it as an analytical discipline, independent of geometry and mechanics. He again put forward D. and. as a basic concept. derivative. Lagrange tried to build D. and. algebraically, using the expansion of functions into power series; In particular, he was responsible for the introduction of the term “derivative” and the designation y" or f" (x). At the beginning of the 19th century. the problem of substantiating D. and. was satisfactorily solved. based on the theory of limits. This was accomplished mainly thanks to the work of O. Cauchy, B. Bolzano and C. Gauss. A more in-depth analysis of the initial concepts of D. and. was associated with the development of set theory and the theory of functions of a real variable in the late 19th and early 20th centuries.
Lit.: Story. Willeitner G., History of mathematics from Descartes to the mid-19th century, trans. from German, 2nd ed., M., 1966; Stroik D. Ya., Brief sketch of the history of mathematics, trans. from German, 2nd ed., M., 1969; Cantor M., Vorlesungen über Geschichte der Mathematik, 2 Aufl., Bd 3≈4, Lpz. ≈ V., 1901≈24.
The works of the founders and classics of D. and. Newton I., Mathematical works, trans. from Latin, M. ≈ L., 1937; Leibniz G., Selected excerpts from mathematical works, trans. from Latin, “Advances in Mathematical Sciences”, 1948, vol. 3, century. 1; L'Hopital G. F. de, Analysis of infinitesimals, translated from French, M. ≈ Leningrad, 1935; Euler L., Introduction to the analysis of infinitesimals, translated from Latin, 2nd ed., vol. 1, M., 1961; his, Differential calculus, translated from Latin, M. ≈ Leningrad, 1949; Cauchy O. L., Summary of lessons on differential and integral calculus, translated from French, St. Petersburg, 1831; by him, Algebraic analysis, translated from French, Leipzig, 1864.
Textbooks and teaching aids on D. and. Khinchin A. Ya., Short course in mathematical analysis, 3rd ed., M., 1957; by him, Eight Lectures on Mathematical Analysis, 3rd ed., M. ≈ Leningrad, 1948; Smirnov V.I., Course of Higher Mathematics, 22nd ed., vol. 1, M., 1967; Fikhtengolts G. M., Course of differential and integral calculus, 7th ed., vol. 1, M., 1969; La Vallée-Poussin C. J. de, Course of analysis of infinitesimals, trans. from French, vol. 1, L. ≈ M., 1933; Kurant R., Course of differential and integral calculus, trans. with him. and English, 4th ed., vol. 1, M., 1967; Banach S., Differential and integral calculus, trans. from Polish, 2nd ed., M., 1966; Rudin U., Fundamentals of mathematical analysis, trans. from English, M., 1966.
Edited by S. B. Stechkin.
Wikipedia
Differential calculus
Differential calculus- a branch of mathematical analysis that studies the concepts of derivative and differential and how they are applied to the study of functions.
DIFFERENTIAL CALCULUS, a branch of mathematical analysis that studies derivatives, differentials and their application to the study of functions. Differential calculus developed as an independent discipline in the 2nd half of the 17th century under the influence of the works of I. Newton and G. W. Leibniz, in which they formulated the basic principles of differential calculus and noted the mutually inverse nature of differentiation and integration. Since that time, differential calculus has developed in close connection with integral calculus, constituting together with it the main part of mathematical analysis (or infinitesimal analysis). The creation of differential and integral calculus opened a new era in the development of mathematics, entailed the emergence of a number of new mathematical disciplines (the theory of series, the theory of differential equations, differential geometry, calculus of variations, functional analysis) and significantly expanded the possibilities of applications of mathematics to issues of natural science and technology.
Differential calculus is based on such fundamental concepts as real number, function, limit, continuity. These concepts took on a modern form during the development of differential and integral calculus. The basic ideas and concepts of differential calculus are associated with the study of functions in the small, i.e., in small neighborhoods of individual points, which requires the creation of a mathematical apparatus for studying functions whose behavior in a sufficiently small neighborhood of each point of their domain of definition is close to the behavior of a linear function or polynomial. This apparatus is based on the concepts of derivative and differential. The concept of a derivative arose in connection with a large number of different problems in natural science and mathematics, leading to the calculation of limits of the same type. The most important of these tasks is determining the speed of movement of a material point along a straight line and constructing a tangent to the curve. The concept of differential is associated with the possibility of approximating a function in a small neighborhood of the point under consideration by a linear function. Unlike the concept of a derivative of a function of a real variable, the concept of a differential can easily be transferred to functions of a more general nature, including mappings of one Euclidean space into another, mappings of Banach spaces into other Banach spaces, and serves as one of the basic concepts of functional analysis.
Derivative. Let the material point move along the Oy axis, and x denotes the time counted from some initial moment. The description of this movement is given by the function y = f(x), which assigns to each moment of time x the coordinate y of the moving point. This function in mechanics is called the law of motion. An important characteristic of motion (especially if it is uneven) is the speed of the moving point at each moment of time x (this speed is also called instantaneous speed). If a point moves along the Oy axis according to the law y = f(x), then at an arbitrary moment of time x it has the coordinate f(x), and at the moment of time x + Δx - the coordinate f(x + Δx), where Δx is the time increment . The number Δy = f(x + Δx) - f(x), called the increment of the function, represents the path traveled by the moving point during the time from x to x + Δx. Attitude
called the difference ratio, is the average speed of movement of a point in the time interval from x to x + Δx. The instantaneous speed (or simply speed) of a moving point at time x is the limit to which the average speed (1) tends as the time interval Δx approaches zero, i.e. limit (2)
The concept of instantaneous speed leads to the concept of derivative. The derivative of an arbitrary function y = f(x) at a given fixed point x is called limit (2) (provided that this limit exists). The derivative of the function y = f(x) at a given point x is denoted by one of the symbols f’(x), y’, ý, df/dx, dy/dx, Df(x).
The operation of finding a derivative (or moving from a function to its derivative) is called differentiation.
The problem of constructing a tangent to a plane curve, defined in the Cartesian coordinate system Oxy by the equation y = f(x), at some point M (x, y) (Fig.) also leads to the limit (2). Having given the argument x an increment Δx and taking a point M' on the curve with coordinates (x + Δx, f(x) + Δx)), the tangent at point M is determined as the limiting position of the secant MM' as the point M' tends to M (i.e. .as Δx tends to zero). Since the point M through which the tangent passes is given, the construction of the tangent is reduced to determining its angular coefficient (i.e., the tangent of the angle of its inclination to the Ox axis). By drawing the straight line MR parallel to the Ox axis, we find that the angular coefficient of the secant MM’ is equal to the ratio
In the limit, as Δx → 0, the angular coefficient of the secant turns into the angular coefficient of the tangent, which turns out to be equal to the limit (2), i.e., the derivative f’(x).
A number of other problems in natural science also lead to the concept of derivative. For example, the current strength in a conductor is defined as the limit lim Δt→0 Δq/Δt, where Δq is the positive electric charge transferred through the cross section of the conductor in time Δt, the rate of a chemical reaction is defined as lim Δt→0 ΔQ/Δt, where ΔQ is the change in quantity substances over time Δt and, in general, the derivative of some physical quantity with respect to time is the rate of change of this quantity.
If the function y = f(x) is defined both at the point x itself and in some neighborhood of it, and has a derivative at the point x, then this function is continuous at the point x. An example of a function y = |x|, defined in any neighborhood of the point x = 0, continuous at this point, but not having a derivative at x = 0, shows that the continuity of the function at a given point does not, generally speaking, imply the existence at this point derivative. Moreover, there are functions that are continuous at every point of their domain of definition, but do not have a derivative at any point in this domain of definition.
In the case when the function y = f(x) is defined only to the right or only to the left of the point x (for example, when x is the boundary point of the segment on which this function is defined), the concepts of right and left derivatives of the function y = f(x) are introduced. at point x. The right derivative of the function y = f(x) at point x is defined as limit (2) provided that Δx tends to zero while remaining positive, and the left derivative is defined as limit (2) provided that Δx tends to zero while remaining negative . The function y = f(x) has a derivative at a point x if and only if it has equal right and left derivatives at this point. The above function y =|x| has at the point x = 0 a right derivative equal to 1 and a left derivative equal to -1, and since the right and left derivatives are not equal to each other, this function does not have a derivative at the point x = 0. In the class of functions that have a derivative, the operation differentiation is linear, i.e. (f(x) + g(x))' = f'(x) + g'(x), and (αf(x))' = αf'(x) for any number α. In addition, the following differentiation rules are valid:
The derivatives of some elementary functions are:
α - any number, x > 0;
n = 0, ±1, ±2,
n = 0, ±1, ±2,
The derivative of any elementary function is again an elementary function.
If the derivative f'(x), in turn, has a derivative at a given point x, then the derivative of the function f'(x) is called the second derivative of the function y = f(x) at the point x and is denoted by one of the symbols f''(x ), y'', ÿ, d 2 f/dx 2 , d 2 y/dx 2 , D 2 f(x).
For a material point moving along the Oy axis according to the law y = f(x), the second derivative represents the acceleration of this point at time x. Derivatives of any integer order n are defined similarly, denoted by the symbols f (n) (x), y (n), d (n) f/dx (n), d (n) y/dx (n), D (n) f (x).
Differential. A function y = f(x), the domain of which contains a certain neighborhood of the point x, is called differentiable at the point x if its increment at this point corresponds to the increment of the argument Δx, i.e. the value Δy = f(x + Δx) - f (x) can be represented in the form Δy = AΔх + αΔх, where A = A(x), α = α(x, Δх) → 0 as Δх → 0. In this case, the expression AΔх is called the differential of the function f(x) at point x and is denoted by the symbol dy or df(x). Geometrically, for a fixed value of x and a changing increment Δx, the differential is the increment of the ordinate of the tangent, i.e., the segment RM" (Fig.). The differential dy is a function of both the point x and the increment Δx. The differential is called the main linear part of the increment of the function, since at for a fixed value of x, the value dy is a linear function of Δx, and the difference Δу - dy is infinitesimal relative to Δx for Δx → 0. For the function f(x) = x, by definition dx = Δx, that is, the differential of the independent variable dx coincides with its increment Δx This allows you to rewrite the expression for the differential in the form dy=Adx.
For a function of one variable, the concept of differential is closely related to the concept of derivative: in order for the function y = f(x) to have a differential at a point x, it is necessary and sufficient that it has a finite derivative f'(x) at this point, and the equality dy = f'(x)dx. The visual meaning of this statement is that the tangent to the curve y = f(x) at the point with the abscissa x is not only the limiting position of the secant, but also the straight line, which in an infinitesimal neighborhood of the point x adjoins the curve y = f(x ) tighter than any other straight line. Thus, A(x) always = f’(x) and the notation dy/dx can be understood not only as a designation for the derivative f’(x), but also as the ratio of the differentials of the function and the argument. Due to the equality dy = f’(x)dx, the rules for finding differentials directly follow from the corresponding rules for derivatives. Differentials of the second and higher orders are also considered.
Applications. Differential calculus establishes connections between the properties of the function f(x) and its derivatives (or its differentials), which constitute the content of the main theorems of differential calculus. Among these theorems are the statement that all extremum points of a differentiable function f(x) lying inside its domain of definition are among the roots of the equation f'(x) = 0, and the often used finite increment formula (Lagrange formula) f(b ) - f(a) = f'(ξ)(b - a), where a<ξ
Differential calculus of functions of several variables. Differential calculus methods are used to study functions of several variables. For a function of two variables u = f(x, y), its partial derivative with respect to x at the point M (x, y) is the derivative of this function with respect to x for a fixed y, defined as
and denoted by one of the symbols f’(x)(x,y), u’(x), ∂u/∂x or ∂f(x,y)’/∂x. The partial derivative of the function u = f(x,y) with respect to y is defined and denoted similarly. The quantity Δu = f(x + Δx, y + Δy) - f(x,y) is called the total increment of the function at the point M (x, y). If this quantity can be represented in the form
where A and B do not depend on Δх and Δу, and α tends to zero as
then the function u = f(x, y) is said to be differentiable at the point M(x, y). The sum AΔx + BΔy is called the total differential of the function u = f(x, y) at the point M(x, y) and is denoted by the symbol du. Since A = f'x(x, y), B = f'y(x, y), and the increments Δx and Δy can be taken equal to their differentials dx and dy, then the total differential du can be written in the form
Geometrically, the differentiability of a function of two variables u = f(x, y) at a given point M (x, y) means that its graph has a tangent plane at this point, and the differential of this function is the increment of the applicate of the point of the tangent plane corresponding to the increments dx and dy independent variables. For a function of two variables, the concept of differential is much more important and natural than the concept of partial derivatives. Unlike a function of one variable, for a function of two variables u = f(x, y) to be differentiable at a given point M(x, y), it is not sufficient for the existence at this point of finite partial derivatives f'x(x, y), and f' y(x, y). A necessary and sufficient condition for the differentiability of the function u = f(x, y) at the point M (x, y) is the existence of finite partial derivatives f'x(x, y) and f'y(x, y) and tending to zero at
quantities
The numerator of this quantity is obtained if we first take the increment of the function f(x, y), corresponding to the increment Δx of its first argument, and then take the increment of the resulting difference f(x + Δx, y) - f(x, y), corresponding to the increment Δy of its second arguments. A simple sufficient condition for the differentiability of the function u = f(x, y) at the point M(x, y) is the existence of continuous partial derivatives f'x(x, y) and f'y(x, y) at this point.
Partial derivatives of higher orders are defined similarly. The partial derivatives ∂ 2 f/∂х 2 and ∂ 2 f/∂у 2, for which both differentiations are carried out with respect to one variable, are called pure, and the partial derivatives ∂ 2 f/∂х∂у and ∂ 2 f/∂у∂х - mixed. At every point at which both mixed partial derivatives are continuous, they are equal to each other. These definitions and notations carry over to the case of a larger number of variables.
Historical sketch. Separate problems of determining tangents to curves and finding the maximum and minimum values of variables were solved by mathematicians of Ancient Greece. For example, methods were found to construct tangents to conic sections and some other curves. However, the methods developed by ancient mathematicians were far from the ideas of differential calculus and could only be used in very special cases. By the middle of the 17th century, it became clear that many of the problems mentioned, along with others (for example, the problem of determining instantaneous speed) could be solved using the same mathematical apparatus, using derivatives and differentials. Around 1666, I. Newton developed the fluxion method (see Fluxion calculus). Newton considered, in particular, two problems of mechanics: the problem of determining the instantaneous speed of movement from a known dependence of the path on time and the problem of determining the distance traveled during a given time from a known instantaneous speed. Newton called continuous functions of time fluents, and the rates of their change - fluxions. Thus, Newton's main concepts were derivative (fluxion) and indefinite integral (fluentia). He tried to substantiate the fluxion method using the theory of limits, which at that time was insufficiently developed.
In the mid-1670s, G. W. Leibniz developed convenient algorithms for differential calculus. Leibniz's main concepts were the differential as an infinitesimal increment of a function and the definite integral as the sum of an infinitely large number of differentials. He introduced the notation of differential and integral, the term “differential calculus”, obtained a number of rules of differentiation, and proposed convenient symbolism. The further development of differential calculus in the 17th century followed mainly the path outlined by Leibniz; The works of J. and I. Bernoulli, B. Taylor and others played a major role at this stage.
The next stage in the development of differential calculus is associated with the works of L. Euler and J. Lagrange (18th century). Euler first began to present differential calculus as an analytical discipline, independent of geometry and mechanics. He again used the derivative as the basic concept of differential calculus. Lagrange tried to construct differential calculus algebraically, using expansions of functions in power series; he introduced the term “derivative” and the notations y’ and f’(x). At the beginning of the 19th century, the problem of substantiating differential calculus on the basis of the theory of limits was largely solved, mainly thanks to the work of O. Cauchy, B. Bolzano and C. Gauss. A deep analysis of the initial concepts of differential calculus was associated with the development of set theory and the theory of functions of real variables in the late 19th and early 20th centuries.
Lit.: History of mathematics: In 3 vols. M., 1970-1972; Rybnikov K. A. History of mathematics. 2nd ed. M., 1974; Nikolsky S. M. Course of mathematical analysis. 6th ed. M., 2001: Zorich V. A. Mathematical analysis: Part 2, 4th ed. M., 2002; Kudryavtsev L.D. Course of mathematical analysis: In 3 volumes, 5th ed. M., 2003-2006; Fikhtengolts G. M. Course of differential and integral calculus: In 3 volumes. 8th ed. M., 2003-2006; Ilyin V. A., Poznyak E. G. Fundamentals of mathematical analysis. 7th ed. M., 2004. Part 1. 5th ed. M., 2004. Part 2; Ilyin V. A., Sadovnichy V. A., Sendov Bl. X. Mathematical analysis. 3rd ed. M., 2004. Part 1. 2nd ed. M., 2004. Part 2; Ilyin V. A., Kurkina L. V. Higher mathematics. 2nd ed. M., 2005.
into an independent mathematical discipline is associated with the names of I. Newton and G. Leibniz (second half of the 17th century). They formulated the main provisions Differential calculus and clearly indicated the mutually inverse nature of the operations of differentiation and integration. From now on Differential calculus develops in close connection with integral calculus , together with which it forms the main part of mathematical analysis (or infinitesimal analysis). The creation of differential and integral calculus opened a new era in the development of mathematics. It entailed the emergence of a number of mathematical disciplines: the theory of series, the theory of differential equations, differential geometry and the calculus of variations. Methods of mathematical analysis have found application in all branches of mathematics. The field of applications of mathematics to questions of natural science and technology has expanded immeasurably. “Only differential calculus gives natural science the opportunity to depict mathematically not only states, but also processes: motion” (F. Engels, see K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 587).Differential calculus is based on the following most important concepts of mathematics, the definition and study of which form the subject of introduction to mathematical analysis: real numbers (number line), function , limit , continuity . All these concepts crystallized and received modern content in the course of the development and justification of differential and integral calculus. main idea Differential calculus consists of studying functions in the small. More precisely: Differential calculus provides an apparatus for studying functions whose behavior in a sufficiently small neighborhood of each point is close to the behavior of a linear function or polynomial. This apparatus is provided by the central concepts Differential calculus: derivative and differential. The concept of a derivative arose from a large number of problems in natural science and mathematics that lead to the calculation of limits of the same type. The most important of them are determining the speed of linear motion of a point and constructing a tangent to the curve. The concept of differential is a mathematical expression of the closeness of a function to linear in a small neighborhood of the point under study. Unlike the derivative, it is easily transferred to mappings of one Euclidean space to another and to mappings of arbitrary linear normed spaces and is one of the basic concepts of modern nonlinear functional analysis .
Derivative. Let it be necessary to determine the speed of a rectilinearly moving material point. If the movement is uniform, then the path traveled by the point is proportional to the time of movement; the speed of such movement can be defined as the path traveled per unit of time, or as the ratio of the path traveled over a certain period of time to the duration of this interval. If the movement is uneven, then the paths traversed by the point in equal time intervals will, generally speaking, be different. An example of uneven motion is given by a body freely falling in a vacuum. The law of motion of such a body is expressed by the formula s = gt 2/2, where s- distance traveled since the beginning of the fall (in meters), t- fall time (in seconds), g- constant value, free fall acceleration, g» 9.81 m/sec 2. During the first second of the fall the body will travel about 4.9 m, for the second - about 14.7 m, and for a tenth - about 93.2 m, i.e. the fall occurs unevenly. Therefore, the above definition of speed is unacceptable here. In this case, the average speed of movement is considered over a certain period of time after (or before) a fixed moment t; it is defined as the ratio of the length of the path traveled during this period of time to its duration. This average speed depends not only on the moment t, but also on the choice of time period. In our example, the average rate of fall over a period of time from t before t+D t equal to
This expression for an unlimited decrease in the time interval D t approaches the value GT, which is called the speed of movement at the moment of time t. Thus, the speed of movement at any point in time is defined as the limit of the average speed when the period of time decreases indefinitely.
In general, these calculations must be carried out for any moment in time. t, time period from t before t+D t and the law of motion expressed by the formula s = f(t). Then the average speed of movement over a period of time from t before t+D t is given by the formula /D t, where D s = f(t+D t) - f(t), and the speed of movement at the moment of time t equal to
The main advantage of speed at a given time, or instantaneous speed, over average speed is that it, like the law of motion, is a function of time t, not an interval function ( t, t+D t). On the other hand, instantaneous speed is a certain abstraction, since it is the average, not the instantaneous speed that can be directly measured.
The problem also leads to an expression of type (*) (see. rice. ) construction tangent to a plane curve at some point M. Let the curve Г be the graph of the function at = f(x). The position of the tangent will be determined if its angular coefficient is found, i.e. the tangent of the angle a formed by the tangent with the axis Ox. Let us denote by x 0 abscissa point M, and through x 1 = x 0+D X- abscissa of the point M 1. Angular coefficient of the secant MM 1 equals
The operation of finding the derivative is called differentiation. On the class of functions that have a derivative, this operation is linear.
Table of formulas and differentiation rules
These proposals allow methods Differential calculus conduct a detailed study of the behavior of functions that are sufficiently smooth (that is, have derivatives of a sufficiently high order). In this way it is possible to investigate the degree of smoothness, convexity and concavity , increasing and decreasing functions , their extremes , find them asymptotes , inflection points (see Inflection point), calculate curvature curve, find out its nature singular points etc. For example, the condition f"(x) > 0 entails a (strict) increase in the function at = f(x), and the condition f"(x) > 0 - its (strict) convexity. All extremum points of a differentiable function belonging to the interior of its domain of definition are located among the roots of the equation f"(x) = 0.
The study of functions using derivatives is the main application Differential calculus Besides, Differential calculus allows you to calculate various kinds of limits of functions, in particular limits of the form 0/0 and ¥/¥ (see. Undefined expression , L'Hopital's rule ). Differential calculus It is especially convenient for studying elementary functions, because in this case, their derivatives are written out explicitly.
Differential calculus functions of many variables. Methods Differential calculus are used to study functions of several variables. For a function of two independent variables z = f (X, at) partial derivative with respect to X the derivative of this function with respect to X at constant at. This partial derivative is denoted z"x, f" x(x, y), ¶ z/¶ X or ¶ f(x, y)/¶ x, So
The partial derivative is defined and denoted similarly z By at. Magnitude
D z = f(x+D x, y+D y) - f(x, y)
is called the complete increment of the function z = f(x, y). If it can be represented in the form
D z = A D x + IN D at+ a,
where a is an infinitesimal of a higher order than the distance between points ( X, at) And ( X+D X, at+D at), then they say that the function z = f(x, y) is differentiable. Components A D X + IN D at form a total differential dz functions z = f(x, y), and A = z"x, = z" y. Instead of D x and D y usually write dx And dy, So
Geometrically, the differentiability of a function of two variables means that its graph has a tangent plane, and the differential represents the increment of the applicate of the tangent plane when the independent variables receive increments dx And dy. For a function of two variables, the concept of differential is much more important and natural than the concept of partial derivatives. Unlike functions of one variable, for functions of two variables the existence of both first-order partial derivatives does not guarantee the differentiability of the function. However, if the partial derivatives are also continuous, then the function is differentiable.
Partial derivatives of higher orders are defined similarly. Partial derivatives ¶ 2 f/¶ x 2 And ¶ 2 f/¶ at 2, in which differentiation is carried out with respect to one variable, are called pure, and partial derivatives ¶ 2 f/¶ x¶ y And ¶ 2 f/¶ at¶ X- mixed. If mixed partial derivatives are continuous, then they are equal to each other. All these definitions and notations carry over to the case of a larger number of variables.
Historical reference. Individual problems about determining tangents to curves and finding the maximum and minimum values of variables were solved by mathematicians of Ancient Greece. For example, methods were found to construct tangents to conic sections and some other curves. However, the methods developed by ancient mathematicians were applicable only in very special cases and are far from ideas Differential calculus
The era of creation Differential calculus As an independent branch of mathematics, one should consider the time when it was understood that these special problems, together with a number of others (especially the problem of determining instantaneous speed), are solved using the same mathematical apparatus - using derivatives and differentials. This understanding was achieved by I. Newton and G. Leibniz.
Around 1666 I. Newton developed the fluxion method (see. Fluxion calculus ). Newton formulated the main tasks in terms of mechanics: 1) determining the speed of movement based on the known dependence of the path on time; 2) determination of the distance traveled during a given time using a known speed. Newton called a continuous variable fluent (current), its speed - fluxion. Thus, Newton’s main concepts were the derivative (fluxion) and the indefinite integral as an antiderivative (fluentia). He sought to substantiate the method of fluxions with the help of the theory of limits, although the latter was only outlined by him.
In the mid-70s. 17th century G. Leibniz developed a very convenient algorithm Differential calculus Leibniz's main concepts were the differential as an infinitesimal increment of a variable and the definite integral as the sum of an infinitely large number of differentials. Leibniz's notation for the differential dx and integral ò ydx, a number of differentiation rules, convenient and flexible symbolism and, finally, the term “differential calculus” itself. Further development Differential calculus first followed the path outlined by Leibniz; The works of the brothers Ya. and I. played a major role at this stage. Bernoulli , B. Taylor and etc.
The next stage in development Differential calculus there were works by L. Euler and J. Lagrange (18th century). Euler first began to present it as an analytical discipline, independent of geometry and mechanics. He again put forward as a basic concept Differential calculus derivative. Lagrange tried to build Differential calculus algebraically, using the expansion of functions into power series; In particular, he was responsible for the introduction of the term “derivative” and the designation y" or f"(x). At the beginning of the 19th century. the problem of justification was satisfactorily solved Differential calculus based on the theory of limits. This was accomplished mainly thanks to the work of O. Cauchy , B. Bolzano and K. Gauss . A deeper analysis of the original concepts Differential calculus was associated with the development of set theory and the theory of functions of a real variable in the late 19th and early 20th centuries.
Lit.: Story. Willeitner G., History of mathematics from Descartes to the mid-19th century, trans. from German, 2nd ed., M., 1966; Stroik D. Ya., Brief sketch of the history of mathematics, trans. from German, 2nd ed., M., 1969; Cantor M., Vorlesungen über Geschichte der Mathematik, 2 Aufl., Bd 3-4, Lpz. - V., 1901-24.
Works of the founders and classics Differential calculus Newton I., Mathematical works, trans. from Latin, M. - L., 1937; Leibniz G., Selected excerpts from mathematical works, trans. from Latin, “Advances in Mathematical Sciences”, 1948, vol. 3, century. 1; L'Hopital G. F. de, Analysis of infinitesimals, translated from French, M. - Leningrad, 1935; Euler L., Introduction to the analysis of infinitesimals, translated from Latin, 2nd ed., vol. 1, M., 1961; his own, Differential calculus, translated from Latin, M. - L., 1949; Cauchy O. L., Summary of lessons on differential and integral calculus, translated from French, St. Petersburg, 1831; by him, Algebraic analysis, translated from French, Leipzig, 1864.
Textbooks and tutorials on Differential calculus Khinchin A. Ya., Short course in mathematical analysis, 3rd ed., M., 1957; by him, Eight lectures on mathematical analysis, 3rd ed., M. - L., 1948; Smirnov V.I., Course of Higher Mathematics, 22nd ed., vol. 1, M., 1967; Fikhtengolts G. M., Course of differential and integral calculus, 7th ed., vol. 1, M., 1969; La Vallée-Poussin C. J. de, Course of analysis of infinitesimals, trans. from French, vol. 1, L. - M., 1933; Kurant R., Course of differential and integral calculus, trans. with him. and English, 4th ed., vol. 1, M., 1967; Banach S., Differential and integral calculus, trans. from Polish, 2nd ed., M., 1966; Rudin U., Fundamentals of mathematical analysis, trans. from English, M., 1966.
Edited by S. B. Stechkin.
Article about the word " Differential calculus" in the Great Soviet Encyclopedia has been read 24,920 times
Ministry of Science and Education
Department "I&VT"
EXPLANATORY NOTE
For course work
Subject: Higher mathematics
On the topic: Differential calculus
Taldykorgan 2008
Introduction
1. The subject of mathematics and the main periods of its development. Mathematics is one of the most important fundamental sciences. The word "mathematics" comes from the Greek word "mathema", which means knowledge. Mathematics arose at the very first stages of human development in connection with the practical activities of people. Since ancient times, people, performing various works, have been faced with the need to isolate and form certain sets of objects, plots of land, housing needs of objects, housing premises.
Firstly, in all these cases it was necessary to establish quantitative estimates of the sets under consideration, measure their areas and volumes, compare, calculate, and transform. According to the definition given by F. Engels:
MATHEMATICS is a science that studies quantitative relationships and spatial forms of the real world.
2. Basic mathematical concepts such as number, geometric figure, function, derivative, integral, random event and its probability, etc. Over its history, mathematics, which developed in close connection with the development of people's production activities and social culture, turned into a harmonious deductive science, presented as a powerful apparatus for studying the world around us.
Academician A.N. Kalinov identified four main developments in the history of mathematics.
The first is the period of the birth of mathematics, the beginning of which lies and is lost in the depths of thousands of years of human history and continues until the 6th – 5th centuries BC. During this period, arithmetic was created, as well as the rudiments of geometry. Mathematical information of this period consists mainly of a set of rules for solving various practical problems.
The second period is of elementary mathematics, i.e. mathematics, constant quantities (VI – V centuries BC – XVII centuries AD). Already at the beginning of this period (about 300 BC), Euclid creates the theory of three books ("Euclid's Element" - the first of the large theoretical studies in mathematics that have come down to us), in which, in particular, the system of axioms is studied deductively all elementary geometry. Al-Khwarizmi’s work “Kibat al-Jarap al-Mukabana”, published in the 9th century, contains general methods for solving problems that reduce to control of the first and second degrees. In the 15th century, instead of loud expressions, they began to use the signs + and -, signs of degrees, roots, and brackets. In the 16th century, F. Viet used letters to denote data and unknown quantities. By the middle of the 17th century, modern algebraic symbolism had basically developed, and this created the foundations of a formal mathematical language.
The third period is the period of creation of mathematics of variable quantities (XVII century - mid-XIX century). Starting from the 17th century, in connection with the study of quantitative relations in the process of their change, the concepts of a variable quantity and function were brought to the fore. In this period, in the works of R. Descartes, analytical geometry was created on the basis of the world research of the method of system coordinates. In the works of I. Newton and G. W. Leibniz, he completed the creation of differential integral calculus.
The fourth period is modern mathematicians. Its beginning should be attributed to the twenties of the 19th century - this period begins with the works of E. Gauss, which contained the ideas of the theory of algebraic structures, V.I. Lobachevsky, who discovered the first non-Euclidean geometry - Lobachevsky geometry.
Subsequently, the axiomatic method became widespread, and work on the substantiation of mathematics, mathematical logic, and mathematical modeling entered a new phase. The creation of computers in the middle of the last century led not only to a deeper and wider application of mathematics in other fields of knowledge, in technical sciences, in matters of organization and management of production, but also to the emergence of the development of new areas of theoretical and applied mathematical functions. The penetration of methods of modern mathematics and computers into other sciences and practice is so universal and profound that one of the abilities of the current stage of development of human culture is considered to be the process of mathematization of knowledge and computerization of all spheres of work and life of people.
3. The concept of mathematical modeling. When studying the quantitative characteristics of complex objects and phenomena processes, the method of mathematical modeling is used, which consists in the fact that the patterns under consideration are formed in mathematical language and studied using appropriate mathematical means. The mathematical module of the object being studied is written using mathematical symbols and consists of a set of equations, inequalities, formulas, program algorithms (for computers), which include variable and constant quantities, various operations, functions, perhaps their derivatives, and other mathematical concepts. The method of composing the simplest mathematical models is the well-known method from a high school mathematics course, the method of solving problems using equations and systems of equations - the resulting equation or system of equations is a mathematical model of a given problem. These were examples of problems with a single solution - deterministic problems. However, there are often problems that have many solutions. In such cases, in practice the question arises of finding a solution that is most suitable for a particular point of view. Such solutions are called optimal solutions.
An optimal solution is defined as a solution for which a certain function, called the objective function, takes the largest and smallest values under given constraints. The objective function is made up of the problem conditions, and it expresses the value that needs to be optimized (i.e., maximize or minimize) - for example, the profit received, expenses, resources, etc.
It turns out that a wide class, in particular control problems, consists of problems in mathematical models in which conditions on the variables create inequality or equality. The theory and methods for solving such problems constitute a branch of mathematics known as “Mathematical Programming.”
If the constraints and the objective function are numerous of the first degree (linear), then such problems constitute a branch of mathematical programming.
Mathematical models of large derivative systems, as a rule, have a complex structure. In particular, in them the number of variables and inequalities or equations can number several tens or even hundreds of powers and have a rather complex form. Such problems are solved in computer centers using large computers.
Following A.N. Tikhonov, in the process of solving real problems using the mathematical modeling method, we calculate the following five stages:
1. Construction of a qualitative model, i.e. consideration of phenomena, identification of main factors and establishment of patterns that take place in the following phenomenon.
2. Construction of a mathematical model, i.e. translation into the language of mathematical states, established qualitative patterns of phenomena. At the same stage of the state, the target function, i.e. such a numerical characteristic of variables, the largest or smallest value of which corresponds to the best situation from the point of view of the previous decision.
3. Solution of the resulting problem. Due to the fact that mathematical models are often quite huge, calculations are carried out using computers in computer centers.
4. Comparison of the calculation results are unsatisfactory, then proceed to the second cycle of the modeling process, i.e. repeat steps 1, 2, 3 with due clarifications of the information until a satisfactory agreement is reached with the available data about the modulated object.
Mathematical methods must be used in solving major problems, such as: financial relations, national economic planning, the use of atomic energy for broad purposes, the creation of large air and spacecraft for various purposes, ensuring the long-term operation of scientific expeditions in space, etc.
However, it would be a mistake to think that mathematical methods are needed only for solving large problems. When studying science in high school, we encounter the applications of mathematical methods and calculations in solving specific various problems. Similar tasks are encountered in the daily work of technical specialists, economists, and technologists. Therefore, workers in the national economy, no matter what field they work in, must master basic research methods and calculation techniques, oral, written, and machine calculations. Specialists must have a complete understanding of the capabilities of a modern computer.
In high school we became familiar with the basic theories of equations, their systems, vectors, differential and integral calculus and their applications in solving practical problems.
The purpose of studying mathematics in secondary specialized institutions is to deepen knowledge in the studied sections and become familiar with some new sections of mathematics (analytical geometry, probability theory, etc.), which enrich the general culture, develop logical thinking, and are widely used in mathematical modeling of problems that a modern specialist encounters in his daily activities.
Model curriculum
Model curriculum is a document intended to implement state requirements for the minimum content and level of training of graduate educational institutions of secondary specialized education. It defines a general list of disciplines, and the required amount of time for their implementation, the types and minimum duration of practice, an approximate list of classrooms, laboratories and workshops. The curriculum also provides for course design in no more than three disciplines during the entire period of study. The types of practical training and their duration are determined in accordance with the standard educational practice for a given specialty. The schedule of the educational process is of a recommendatory nature and can be adjusted by the educational institution, subject to mandatory observance of the duration of theoretical training, examination sessions, as well as the timing of the winter and summer holidays that end the academic year (see Table 1).