Differential calculus. Differential and integral calculus

A circle arises, the most prominent representatives of which were the brothers Bernoulli (Jacob and Johann) and Lopital. In , using the lectures of I. Bernoulli, L'Hopital wrote the first textbook outlining the new method as applied to the theory of plane curves. He called him Analysis of infinitesimals, thus giving one of the names to the new branch of mathematics. The presentation is based on the concept of variables, between which there is some connection, due to which a change in one entails a change in the other. In Lopital, this connection is given with the help of flat curves: if M (\displaystyle M) is a moving point of a plane curve, then its Cartesian coordinates x (\displaystyle x) and y (\displaystyle y), called the abscissa and ordinate of the curve, are variables, and the change x (\displaystyle x) entails change y (\displaystyle y). The concept of a function is absent: wishing to say that the dependence of the variables is given, Lopital says that "the nature of the curve is known." The concept of differential is introduced as follows:

An infinitesimal part, by which a variable is continuously increasing or decreasing, is called its differential ... To denote the differential of a variable, which itself is expressed by one letter, we will use the sign or symbol d (\displaystyle d). ... An infinitesimal part, by which the differential of a variable value continuously increases or decreases, is called ... the second differential.

These definitions are explained geometrically, with Fig. infinitesimal increments are depicted as finite. Consideration is based on two requirements (axioms). First:

It is required that two quantities, differing from each other only by an infinitesimal amount, can be taken [when simplifying expressions?] indifferently one instead of the other.

Hence it turns out x + d x = x (\displaystyle x+dx=x), Further

D x y = (x + d x) (y + d y) − x y = x d y + y d x + d x d y = (x + d x) d y + y d x = x d y + y d x (\displaystyle dxy=(x+dx)(y+dy)- xy=xdy+ydx+dxdy=(x+dx)dy+ydx=xdy+ydx)

The second requirement is:

It is required that one can consider a curved line as a collection of an infinite set of infinitely small straight lines.

The continuation of each such line is called a tangent to the curve. Exploring a tangent through a point M = (x , y) (\displaystyle M=(x,y)), L'Hopital attaches great importance to the quantity

y d x d y − x (\displaystyle y(\frac (dx)(dy))-x),

reaching extreme values at the inflection points of the curve, while the ratio d y (\displaystyle dy) to d x (\displaystyle dx) no special significance is attached.

Finding extremum points is noteworthy. If with a continuous increase in the abscissa x (\displaystyle x) ordinate y (\displaystyle y) first increases and then decreases, then the differential d y (\displaystyle dy) initially positive compared to d x (\displaystyle dx) and then negative.

But any continuously increasing or decreasing quantity cannot turn from positive to negative without passing through infinity or zero ... It follows that the differential of the largest and smallest magnitude must equal zero or infinity.

This formulation is probably not flawless, if we recall the first requirement: let, say, y = x 2 (\displaystyle y=x^(2)), then by virtue of the first requirement

2 x d x + d x 2 = 2 x d x (\displaystyle 2xdx+dx^(2)=2xdx);

at zero, the right side is zero, but the left side is not. Apparently it should have been said that d y (\displaystyle dy) can be transformed in accordance with the first requirement so that at the maximum point d y = 0 (\displaystyle dy=0). . In the examples, everything is self-evident, and only in the theory of inflection points does Lopital write that d y (\displaystyle dy) equals zero at the maximum point, when divided by d x (\displaystyle dx) .

Further, with the help of differentials alone, conditions for an extremum are formulated and a large number of complex problems are considered, mainly related to differential geometry on the plane. At the end of the book, in ch. 10, what is now called L'Hopital's rule is stated, although in a not quite ordinary form. Let the value of the ordinate y (\displaystyle y) curve is expressed as a fraction whose numerator and denominator vanish at . Then the point of the curve with x = a (\displaystyle x=a) has an ordinate y (\displaystyle y), equal to the ratio of the numerator differential to the denominator differential, taken at x = a (\displaystyle x=a).

According to L'Hopital's idea, what he wrote was the first part of the Analysis, while the second was supposed to contain integral calculus, that is, a way to find the connection of variables by the known connection of their differentials. Its first exposition is given by Johann Bernoulli in his Mathematical lectures on the integral method. Here a way of taking most elementary integrals is given and methods for solving many first-order differential equations are indicated.

Pointing to the practical usefulness and simplicity of the new method, Leibniz wrote:

What a man versed in this calculus can get in just three lines, other most learned men have been compelled to look for, following complex detours.

Euler

Leonard Euler

The changes that took place over the next half century are reflected in Euler's extensive treatise. The presentation of the analysis opens the two-volume "Introduction", which contains research on various representations of elementary functions. The term "function" first appears only in Leibniz, but it was Euler who put it forward to the first roles. The original interpretation of the concept of a function was that a function is an expression for counting (German Rechnungsausdrϋck) or analytic expression.

The function of a variable quantity is an analytic expression made up in some way of this variable quantity and numbers or constant quantities.

Emphasizing that “the main difference between functions lies in the way they are composed of variables and constants,” Euler enumerates the actions “by which quantities can be combined and mixed with each other; these actions are: addition and subtraction, multiplication and division, exponentiation and extraction of roots; the solution of [algebraic] equations should also be included here. In addition to these operations, called algebraic, there are many others, transcendental, such as: exponential, logarithmic and innumerable others, delivered by integral calculus. Such an interpretation made it possible to easily deal with multi-valued functions and did not require an explanation of which field the function is considered over: the expression for the count is defined for the complex values of the variables even when this is not necessary for the problem under consideration.

Operations in expression were allowed only in a finite number, and the transcendent penetrated with the help of an infinitely large number ∞ (\displaystyle \infty ). In expressions, this number is used along with natural numbers. For example, such an expression for the exponent is considered valid

e x = (1 + x ∞) ∞ (\displaystyle e^(x)=\left(1+(\frac (x)(\infty ))\right)^(\infty )),

in which only later authors saw the transition to the limit. Various transformations were made with analytic expressions, which allowed Euler to find representations for elementary functions in the form of series, infinite products, etc. Euler transforms expressions for counting in the same way as they do in algebra, not paying attention to the possibility of calculating the value of a function at a point for each from written formulas.

In contrast to L'Hôpital, Euler considers transcendental functions in detail and, in particular, the two most studied classes of them - exponential and trigonometric. He discovers that all elementary functions can be expressed using arithmetic operations and two operations - taking the logarithm and the exponent.

The very course of the proof perfectly demonstrates the technique of using the infinitely large. Having determined the sine and cosine using the trigonometric circle, Euler deduces the following from the addition formulas:

(cos ⁡ x + − 1 sin ⁡ x) (cos ⁡ y + − 1 sin ⁡ y) = cos ⁡ (x + y) + − 1 sin ⁡ (x + y) , (\displaystyle (\cos x+(\ sqrt (-1))\sin x)(\cos y+(\sqrt (-1))\sin y)=\cos ((x+y))+(\sqrt (-1))\sin ((x +y)))) 2 cos ⁡ n x = (cos ⁡ x + − 1 sin ⁡ x) n + (cos ⁡ x − − 1 sin ⁡ x) n (\displaystyle 2\cos nx=(\cos x+(\sqrt (-1)) \sin x)^(n)+(\cos x-(\sqrt (-1))\sin x)^(n))

Assuming n = ∞ (\displaystyle n=\infty ) and z = n x (\displaystyle z=nx), he gets

2 cos ⁡ z = (1 + − 1 z ∞) ∞ + (1 − − 1 z ∞) ∞ = e − 1 z + e − − 1 z (\displaystyle 2\cos z=\left(1+(\ frac ((\sqrt (-1))z)(\infty ))\right)^(\infty )+\left(1-(\frac ((\sqrt (-1))z)(\infty )) \right)^(\infty )=e^((\sqrt (-1))z)+e^(-(\sqrt (-1))z)),

discarding infinitesimal values of a higher order. Using this and a similar expression, Euler also obtains his famous formula

e − 1 x = cos ⁡ x + − 1 sin ⁡ x (\displaystyle e^((\sqrt (-1))x)=\cos (x)+(\sqrt (-1))\sin (x) ).

Having indicated various expressions for functions that are now called elementary, Euler proceeds to consider curves in the plane, drawn with a free movement of the hand. In his opinion, it is not possible to find a single analytic expression for every such curve (see also the String Controversy). In the 19th century, at the suggestion of Casorati, this statement was considered erroneous: according to the Weierstrass theorem, any continuous curve in the modern sense can be approximately described by polynomials. In fact, Euler was hardly convinced by this, because we still need to rewrite the passage to the limit using the symbol ∞ (\displaystyle \infty ).

Euler's presentation of the differential calculus begins with the theory of finite differences, followed in the third chapter by a philosophical explanation that "an infinitesimal quantity is exactly zero", which most of all did not suit Euler's contemporaries. Then, differentials are formed from finite differences with an infinitesimal increment, and from Newton's interpolation formula, Taylor's formula. This method essentially goes back to the work of Taylor (1715). In this case, Euler has a stable relation d k y d x k (\displaystyle (\frac (d^(k)y)(dx^(k)))), which, however, is considered as the ratio of two infinitesimals. The last chapters are devoted to approximate calculation using series.

In the three-volume integral calculus, Euler introduces the concept of an integral as follows:

The function whose differential = X d x (\displaystyle =Xdx), is called its integral and is denoted by the sign S (\displaystyle S) placed in front.

On the whole, this part of Euler's treatise is devoted to the more general problem of integrating differential equations from a modern point of view. In doing so, Euler finds a number of integrals and differential equations that lead to new functions, for example, Γ (\displaystyle \Gamma )-functions, elliptic functions, etc. A rigorous proof of their non-elementarity was given in the 1830s by Jacobi for elliptic functions and by Liouville (see elementary functions).

Lagrange

The next major work that played a significant role in the development of the concept of analysis was Theory of analytic functions Lagrange and an extensive retelling of Lagrange's work, done by Lacroix in a somewhat eclectic manner.

Wishing to get rid of the infinitesimal altogether, Lagrange reversed the connection between the derivatives and the Taylor series. By an analytic function, Lagrange understood an arbitrary function investigated by methods of analysis. He designated the function itself as , giving a graphical way to write the dependence - earlier, Euler managed with only variables. To apply the methods of analysis, according to Lagrange, it is necessary that the function expands into a series

f (x + h) = f (x) + p h + q h 2 + … (\displaystyle f(x+h)=f(x)+ph+qh^(2)+\dots ),

whose coefficients will be new functions x (\displaystyle x). It remains to name p (\displaystyle p) derivative (differential coefficient) and denote it as f ′ (x) (\displaystyle f"(x)). Thus, the concept of a derivative is introduced on the second page of the treatise and without the aid of infinitesimals. It remains to note that

f ′ (x + h) = p + 2 q h + … (\displaystyle f"(x+h)=p+2qh+\dots ),

so the coefficient q (\displaystyle q) is the double derivative of the derivative f (x) (\displaystyle f(x)), i.e

q = 1 2 ! f ″ (x) (\displaystyle q=(\frac (1)(2}f""(x)} !} etc.

This approach to the interpretation of the concept of derivative is used in modern algebra and served as the basis for the creation of the Weierstrass theory of analytic functions.

Lagrange operated on such series as formal and obtained a number of remarkable theorems. In particular, for the first time and quite rigorously he proved the solvability of the initial problem for ordinary differential equations in formal power series.

The question of estimating the accuracy of approximations supplied by partial sums of the Taylor series was first posed by Lagrange: at the end Theories of analytic functions he derived what is now called Taylor's Lagrange remainder formula. However, in contrast to modern authors, Lagrange did not see the need to use this result to justify the convergence of the Taylor series.

The question of whether the functions used in analysis can really be expanded in a power series subsequently became the subject of discussion. Of course, Lagrange knew that at some points elementary functions may not expand into a power series, but at these points they are in no sense differentiable. Koshy in his Algebraic analysis gave the function as a counterexample

f (x) = e − 1 / x 2 , (\displaystyle f(x)=e^(-1/x^(2)),)

extended by zero at zero. This function is everywhere smooth on the real axis and has zero Maclaurin series at zero, which, therefore, does not converge to the value f (x) (\displaystyle f(x)). Against this example, Poisson objected that Lagrange defined a function as a single analytic expression, while in Cauchy's example the function is given differently at zero, and when x ≠ 0 (\displaystyle x\not =0). It was only at the end of the 19th century that Pringsheim proved that there exists an infinitely differentiable function given by a single expression for which the Maclaurin series diverges. An example of such a function is the expression

Ψ (x) = ∑ k = 0 ∞ cos ⁡ (3 k x) k ! (\displaystyle \Psi (x)=\sum \limits _(k=0)^(\infty )(\frac (\cos ((3^(k)x)))(k}} !}.

Further development

Differential calculus

The differential calculus studies the definition, properties, and applications of derivative functions. The process of finding the derivative is called differentiation. Given a function and a point in its domain, the derivative at that point is a way of encoding the fine-scale behavior of that function near that point. By finding the derivative of a function at each point in the domain, one can define a new function called derivative function or simply derivative from the original function. In mathematical language, a derivative is a linear mapping that has one function as input and another as output. This concept is more abstract than most of the processes studied in elementary algebra, where functions usually have one number as input and another as output. For example, if the doubling function is given an input of three, the output will be six; if the input to a quadratic function is three, the output will be nine. The derivative can also have a quadratic function as input. This means that the derivative takes all the information about the squaring function, that is: when two is input, it gives four as output, it converts three to nine, four to sixteen, and so on, and uses this information to obtain another function. (The derivative of a quadratic function is just the doubling function.)

The most common symbol for denoting a derivative is an apostrophe-like mark called a prime. So the derivative of the function f there is f′, pronounced "f stroke". For example, if f(x) = x 2 is a squaring function, then f′(x) = 2x is its derivative, this is the doubling function.

If the function input is time, then the derivative is the change with respect to time. For example, if f is a function that depends on time, and it gives the output of the position of the ball in time, then the derivative f determines the change in the position of the ball over time, that is, the speed of the ball.

Indefinite integral is an primitive, that is, the operation inverse to the derivative. F is an indefinite integral of f in the case when f is a derivative of F. (This use of uppercase and lowercase letters for a function and its indefinite integral is common in calculus.)

Definite integral input function and output values is a number that is equal to the area of the surface bounded by the function graph, the x-axis, and two straight line segments from the function graph to the x-axis at the points of the output values. In technical terms, the definite integral is the limit of the sum of the areas of rectangles, called the Riemann sum.

An example from physics is the calculation of the distance traveled while walking at any given time.

D i s t a n c e = S p e e d ⋅ T i m e (\displaystyle \mathrm (Distance) =\mathrm (Speed) \cdot \mathrm (Time) )

If the speed is constant, the multiplication operation is sufficient, but if the speed varies, then we must apply a more powerful method of calculating the distance. One of these methods is an approximate calculation by breaking down time into separate short periods. Then multiplying the time in each interval by any one of the speeds in that interval and then summing all the approximate distances (Riemann sum) traveled in each interval, we get the total distance traveled. The basic idea is that if you use very short intervals, then the speed at each of them will remain more or less constant. However, the Riemann sum only gives an approximate distance. To find the exact distance, we must find the limit of all such Riemann sums.

If a f(x) on the diagram on the left represents the change in speed over time, then the distance traveled (between moments a and b) is the area of the shaded area s.

For an approximate estimate of this area, an intuitive method is possible, consisting in dividing the distance between a and b into a certain number of equal segments (segments) of length Δx. For each segment, we can choose one function value f(x). Let's call this value h. Then the area of the rectangle with base Δx and height h gives distance (time Δx multiplied by the speed h) passed in this segment. Each segment is associated with the average value of the function on it f(x)=h. The sum of all such rectangles gives an approximation of the area under the curve, which is an estimate of the total distance travelled. Decrease Δx will give more rectangles and in most cases be a better approximation, but to get an accurate answer we must calculate the limit at Δx tending to zero.

The integration symbol is ∫ (\displaystyle \int ), an extended letter S(S stands for "sum"). The definite integral is written as:

∫ a b f (x) d x . (\displaystyle \int _(a)^(b)f(x)\,dx.)

and reads: "the integral of a before b functions f from x on x". The notation proposed by Leibniz dx is intended to divide the area under the curve into an infinite number of rectangles such that their width Δx is an infinitesimal quantity dx. In the formulation of the calculus based on limits, the notation

∫ a b … d x (\displaystyle \int _(a)^(b)\ldots \,dx)

should be understood as an operator that takes a function as input and outputs a number equal to the area. dx is not a number and cannot be multiplied by f(x).

The indefinite integral, or antiderivative, is written as:

∫ f (x) d x . (\displaystyle \int f(x)\,dx.)

Functions that differ by a constant have the same derivatives, and therefore the antiderivative of a given function is actually a family of functions that differ only by a constant. Since the derivative of the function y = x² + C, where C- any constant, equal to y′ = 2x, then the antiderivative of the latter is determined by the formula:

∫ 2 x d x = x 2 + C . (\displaystyle \int 2x\,dx=x^(2)+C.)

Undefined type constant C in the antiderivative is known as the constant of integration.

Newton-Leibniz theorem

Newton's - Leibniz's theorem, which is also called main theorem of analysis states that differentiation and integration are mutually inverse operations. More precisely, it concerns the value of antiderivatives for certain integrals. Since it is generally easier to calculate the antiderivative than to apply the definite integral formula, the theorem provides a practical way to calculate definite integrals. It can also be interpreted as an exact statement that differentiation is the inverse of integration.

The theorem says: if the function f continuous on the segment [ a, b] and if F there is a function whose derivative is equal to f on the interval ( a, b), then:

∫ a b f (x) d x = F (b) − F (a) . (\displaystyle \int _(a)^(b)f(x)\,dx=F(b)-F(a).)

In addition, for any x from the interval ( a, b)

d d x ∫ a x f (t) d t = f (x) . (\displaystyle (\frac (d)(dx))\int _(a)^(x)f(t)\,dt=f(x).)

This insight, made by both Newton and Leibniz, who based their results on the earlier writings of Isaac Barrow, was key to the rapid dissemination of analytic results after their work became known. The fundamental theorem gives an algebraic method for calculating many definite integrals without limiting processes, by finding the antiderivative formula. In addition, a prototype emerged for solving differential equations. Differential equations connect unknown functions with their derivatives, they are used everywhere in many sciences.

Applications

Mathematical analysis is widely used in physics, computer science, statistics, engineering, economics, business, finance, medicine, demography and other areas in which a mathematical model can be built to solve a problem, and it is necessary to find its optimal solution.

In particular, almost all concepts in classical mechanics and electromagnetism are inextricably linked with each other precisely by means of classical mathematical analysis. For example, given the known density distribution of an object, its mass , moments of inertia , as well as the total energy in a potential field can be found using differential calculus. Another striking example of the application of mathematical analysis in mechanics is Newton's second law: historically, it directly uses the term "rate of change" in the formulation "Force \u003d mass × acceleration", since acceleration is the time derivative of speed or the second derivative of time from trajectory or spatial position.

Mathematical analysis is also used to find approximate solutions to equations. In practice, this is the standard way of solving differential equations and finding roots in most applications. Examples are Newton's method, the simple iteration method, and the linear approximation method. For example, when calculating the trajectory of spacecraft, a variant of the Euler method is used to approximate curvilinear motion courses in the absence of gravity.

Bibliography

encyclopedia articles

// Encyclopedic Lexicon: In 17 vols. - St. Petersburg. : Type. A. Plushard, 1835-1841.
// Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 volumes and 4 additional). - St. Petersburg. , 1890-1907.

Educational literature

Standard textbooks

For many years, the following textbooks have been popular in Russia:

Kurant, R. A course in differential and integral calculus (in two volumes). The main methodological finding of the course: first, the main ideas are simply stated, and then they are given rigorous proofs. Written by Courant when he was a professor at the University of Göttingen in the 1920s under the influence of Klein's ideas, then transferred to American soil in the 1930s. The Russian translation of 1934 and its reprinting gives the text according to the German edition, the translation of the 1960s (the so-called 4th edition) is a compilation from the German and American versions of the textbook and is therefore very verbose.
Fikhtengolts G. M. A course in differential and integral calculus (in three volumes) and a problem book.
Demidovich B.P. Collection of problems and exercises in mathematical analysis.
Lyashko I. I. and others. Reference manual for higher mathematics, vol. 1-5.

Some universities have their own guidelines for analysis:

Moscow State University, MehMat:

Arkhipov G. I., Sadovnichiy V. A., Chubarikov V. N. Lectures on Math. analysis.
Zorich V. A. Mathematical analysis. Part I. M.: Nauka, 1981. 544 p.
Zorich V. A. Mathematical analysis. Part II. M.: Nauka, 1984. 640 p.
Kamynin L.I. Course of mathematical analysis (in two volumes). Moscow: Moscow University Press, 2001.

V. A. Ilyin, V. A. Sadovnichiy, Bl. H. Sendov. Mathematical Analysis / Ed.

The student must:

know:

definition of the limit of a function at a point;

properties of the limit of a function at a point;

Remarkable limits formulas;

determination of the continuity of a function at a point,

properties of continuous functions;

definition of the derivative, its geometric and physical meaning; tabular derivatives, differentiation rules;

a rule for calculating the derivative of a complex function; definition of the differential of a function, its properties; definition of derivatives and differentials of higher orders; determination of function extremum, convex function, inflection points, asymptotes;

definition of an indefinite integral, its properties, tabular integrals;

· formulas for integration by means of a change of variable and by parts for the indefinite integral;

definition of a definite integral, its properties, the basic formula of the integral calculus - the Newton-Leibniz formula;

· formulas for integration by means of a change of variable and by parts for a definite integral;

· the geometric meaning of the definite integral, the application of the definite integral.

be able to:

Calculate limits of sequences and functions; disclose uncertainties;

· calculate derivatives of complex functions, derivatives and differentials of higher orders;

find extremums and inflection points of functions;

· conduct a study of functions with the help of derivatives and build their graphs.

Calculate indefinite and definite integrals by the method of change of variable and by parts;

· integrate rational, irrational and some trigonometric functions, apply universal substitution; apply the definite integral to find the areas of plane figures.

Function limit. Function limit properties. Unilateral limits. The limit of the sum, product and quotient of two functions. Continuous functions, their properties. Continuity of elementary and complex functions. Remarkable limits.

Definition of the derivative of a function. Derivatives of basic elementary functions. Function differentiability. Function differential. Derivative of a complex function. Differentiation rules: derivative of sum, product and quotient. Derivatives and differentials of higher orders. Disclosure of uncertainties. Increasing and decreasing functions, conditions for increasing and decreasing. Extrema of functions, a necessary condition for the existence of an extremum. Finding extrema using the first derivative. Convex functions. Inflection points. Asymptotes. Full function study.

Indefinite integral, its properties. Table of basic integrals. Change of variables method. Integration by parts. Integration of rational functions. Integration of some irrational functions. Universal substitution.

Definite integral, its properties. Basic formula of integral calculus. Integration by change of variable and by parts in a definite integral. Applications of a definite integral.

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 main provisions 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 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, led to the emergence of a number of new mathematical disciplines (theory of series, the theory of differential equations, differential geometry, calculus of variations, functional analysis) and significantly expanded the possibilities of applying mathematics to questions of natural science and technology.

The differential calculus is based on such fundamental concepts as real number, function, limit, continuity. These concepts took on a modern form in the course of the development of differential and integral calculus. The main 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 the determination of the speed of movement of a material point along a straight line and the construction of a tangent to a curve. The concept of a differential is related to 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 be easily transferred to functions of a more general nature, including mappings from one Euclidean space to another, mappings of Banach spaces to 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 denote 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 the movement (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 \u003d f (x), then at an arbitrary point in time x it has the coordinate f (x), and at the point in time x + Δx - the coordinate f (x + Δx), where Δx is the increment of time . The number Δy \u003d f (x + Δx) - f (x), called the increment of the function, is the path traveled by the moving point in the time from x to x + Δx. Attitude

called the difference ratio, is the average speed of the 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 when the time interval Δx tends to zero, i.e. limit (2)

The concept of instantaneous velocity leads to the concept of a derivative. The derivative of an arbitrary function y \u003d f (x) at a given fixed point x is called the limit (2) (provided that this limit exists). The derivative of the function y \u003d 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 transition 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 \u003d f (x), at some point M (x, y) (Fig.) also leads to the limit (2). Having given the increment Δx to the argument x and taking the point M' with coordinates (x + Δx, f(x) + Δx) on the curve), determine the tangent at the point M as the limit 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). Drawing a straight line MR parallel to the Ox axis, it is obtained that the slope of the secant MM' is equal to the ratio

In the limit at Δx → 0, the slope of the secant turns into the slope of the tangent, which turns out to be equal to the limit (2), i.e., the derivative f’(x).

A number of other problems of natural science also lead to the concept of a 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 the amount matter during the 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 \u003d f (x) is defined both at the point x itself and in some of its neighborhood, and has a derivative at the point x, then this function is continuous at the point x. An example of a function y \u003d |x|, defined in any neighborhood of the point x \u003d 0, continuous at this point, but not having a derivative at x \u003d 0, shows that the existence of a function at this point, in general, does not follow from the continuity 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 of this domain.

In the case when the function y \u003d 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 given), the concepts of the right and left derivatives of the function y \u003d f (x) are introduced at point x. The right derivative of the function y \u003d f (x) at the point x is defined as the limit (2) provided that Δx tends to zero, remaining positive, and the left derivative is defined as the limit (2) provided that Δx tends to zero, remaining negative . The function y \u003d f (x) has a derivative at a point x if and only if it has right and left derivatives equal to each other at this point. The above function y = |x| has a right derivative equal to 1 at the point x = 0 and a left derivative equal to -1, and since the right and left derivatives are not equal to each other, this function has no 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 a. In addition, the following differentiation rules hold true:

The derivatives of some elementary functions are:

α - any number, x > 0;

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 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 \u003d f (x), the second derivative is 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 \u003d f (x), whose domain of definition contains some neighborhood of the point x, is called differentiable at the point x if its increment at this point, corresponding to the increment of the argument Δx, i.e., the value Δy \u003d f (x + Δx) - f (x) can be represented in the form and is denoted by the symbol dy or df(x). Geometrically, for a fixed value x and a changing increment Δx, the differential is an increment in the ordinate of the tangent, i.e., the segment PM "(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 when fixed value x magnitude dy is a linear function of Δх, and the difference Δу - dy is infinitely small with respect to Δх as Δх → 0. For the function f(х) = x, by definition, dx = Δх, that is, the differential of the independent variable dx coincides with its increment Δх. This allows the expression for the differential to be rewritten as dy=Adx.

For a function of one variable, the concept of a differential is closely related to the concept of a derivative: in order for a function y \u003d 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, while the equality dy = f'(x)dx. The visual meaning of this statement is that the tangent to the curve y \u003d 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 infinitely small neighborhood of the point x is adjacent to the curve y \u003d f (x ) closer than any other straight line. Thus, always A(x) = 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. By virtue of the equality dy = f'(x)dx, the rules for finding differentials follow directly from the corresponding rules for derivatives. Differentials of the second and higher orders are also considered.

Applications. The differential calculus establishes connections between the properties of the function f(x) and its derivatives (or its differentials), which are the content of the main theorems of the differential calculus. These theorems include the assertion 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 frequently used finite increment formula (Lagrange formula) f(b ) - f(a) = f'(ξ)(b - a), where a<ξ0 entails a strict increase in the function, and the condition f '' (x)\u003e 0 - its strict convexity. In addition, the differential calculus allows one to calculate various kinds of limits of functions, in particular the limits of the ratios of two functions, which are uncertainties of the form 0/0 or of the form ∞/∞ (see Disclosure of uncertainties). The differential calculus is especially convenient for studying elementary functions whose derivatives are written out explicitly.

Differential calculus of functions of several variables. The methods of differential calculus 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 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 in a similar way. The value Δu \u003d f (x + Δx, y + Δy) - f (x, y) is called the total increment of the function and at the point M (x, y). If this value can be represented as

where A and B do not depend on Δх and Δу, and α tends to zero at

then the function u = f(x, y) is called 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 \u003d f’x (x, y), B \u003d f’y (x, y), and the increments Δx and Δy can be taken equal to their differentials dx and dy, the total differential du can be written as

Geometrically, the differentiability of a function of two variables u = f(x, y) at a given point M (x, y) means that its graph exists at this point of the tangent plane, 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 a differential is much more important and natural than the concept of partial derivatives. In contrast to 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 that the finite partial derivatives f'x(x, y) and f' y(x, y). A necessary and sufficient condition for the function u = f(x, y) to be differentiable 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 by first taking the increment of the function f(x, y) corresponding to the increment Δx of its first argument, and then taking 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.

The partial derivatives of higher orders are defined similarly. Partial derivatives ∂ 2 f/∂х 2 and ∂ 2 f/∂у 2 , in which both differentiations are carried out in one variable, are called pure, and partial derivatives ∂ 2 f/∂х∂у and ∂ 2 f/∂у∂х - mixed. At every point where both mixed partial derivatives are continuous, they are equal to each other. These definitions and notation carry over to the case of a larger number of variables.

Historical outline. Separate problems of determining the tangents to curves and finding the maximum and minimum values of variables were solved by the mathematicians of Ancient Greece. For example, ways 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 be applied only 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 the instantaneous speed) can be solved using the same mathematical apparatus, using derivatives and differentials. Around 1666, I. Newton developed the method of fluxes (see flux calculus). Newton considered, in particular, two problems of mechanics: the problem of determining the instantaneous speed of motion from a known dependence of the path on time, and the problem of determining the path traveled in a given time from a known instantaneous speed. Newton called continuous functions of time fluents, and the rates of their change - fluctuations. Thus, Newton's main concepts were the derivative (fluxion) and the indefinite integral(fluent). He tried to substantiate the method of fluxions with the help of the theory of limits, which at that time was underdeveloped.

In the mid-1670s, G. W. Leibniz developed convenient algorithms for differential calculus. The main concepts of Leibniz 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", received a number of rules for differentiation, and proposed convenient symbolism. The further development of differential calculus in the 17th century proceeded mainly along the path outlined by Leibniz; the works of J. and I. Bernoulli, B. Taylor, and others played an important 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 build the differential calculus algebraically, using the expansions of functions into power series; he introduced the term "derivative" and the designations y' and f'(x). At the beginning of the 19th century, the problem of substantiating the differential calculus on the basis of the theory of limits was basically solved, mainly thanks to the work of O. Cauchy, B. Bolzano and C. Gauss. Deep analysis 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 - early 20th century.

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: In the 2nd part of the 4th ed. M., 2002; Kudryavtsev L.D. A course of mathematical analysis: In 3 volumes, 5th ed. M., 2003-2006; Fikhtengol'ts G. M. The 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., Sadovnichiy 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.

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

The formation of differential calculus in the form of an applied, and then a scientific method occurred before the emergence of a 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 by a large number of problems from the natural sciences and mathematics, which led to finding the values of the 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 match 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 a 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 has long been used in many educational institutions as one of the main study 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:

Find the scope of the function.
Find the roots of the given equation.
Calculate extremes. To do this, calculate the derivative and the points where it equals zero.
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:

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.
Solution of the formulated equation. This process is simpler than the first point, since it requires only strict mathematical calculations.
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 occurs when building 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 various biological populations and chemical processes in the human body plays an important role in it.

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 cases, 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 textbooks for higher education. 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.