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
Differential calculus is based on the fundamental concepts of mathematics. They are: real number, continuity, function and limit. Over time, they took on a modern look, thanks to integral and differential calculus.
Creation process
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.
Ancient mathematicians used the unit as 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 a derivative and a differential.
The appearance of the concept of a derivative was caused by a large number of problems from the natural sciences and mathematics,which led to finding the values of limits of the same type.
One of the main problems 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 to a function of a general nature, in particular, to the image of one Euclidean space on another.
Derivative
Let the point move in the direction of the Oy axis, take the time 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 motion, especially uneven, is the instantaneous speed. 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.
Due to the emergence of thisvelocity at time, the derivative is introduced. In an arbitrary function, the derivative at a fixed point is called the limit (assuming 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 calculus method is used when examining 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.
Can be represented by the following characters:
f’(x)(x, y), u’(x), ∂u/∂x or ∂f(x, y)’/∂x.
Required Skills
Skills in integration and differentiation are required to successfully study and be able to solve diffuses. To make it easier to understand differential equations, you should have a good understanding of the topic of the derivative and the indefinite integral. It also does not hurt to learn how to find the derivative of an implicitly given function. This is due to the fact that in the process of studying integrals and differentiation will often have to be used.
Types of differential equations
In almost all test papers related to first-order differential equations, 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.
Decision basics
To begin with, you should 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, it was only necessary to substitute this value in place of the unknown.
Differential equation is similar to this. In general, such a first-order equation includes:
- Independent variable.
- The derivative of the first function.
- A 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 matching the given expression. Such a set of functions is often called the general solution of DE.
Integral calculus
Integral calculus is one of the sections of mathematical analysis that studies the concept of an integral, properties and methods of 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 arbitrarysmall 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 sparing 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 increasingly smaller partitions, then the area above and below should converge at a certain value.
It should go back 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 dividingsegment. 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 preimages 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 using the methods of differential calculus, you must follow the already given algorithm:
- Find the scope of a 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
first-order control (otherwise, differentialsingle variable calculus) and their types:
- Separable 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 first-order DE: y'+P(x)y=Q(x).
- Bernoulli differential equation: y'+P(x)y=Q(x)ya.
- Equation with total differentials: P(x, y)dx+Q(x, y)dy=0.
Second order differential equations and their types:
- Linear second order homogeneous differential equation with constant coefficient values: y +py'+qy=0 p, q belongs to R.
- Linear inhomogeneous second-order differential equation with constant coefficients: y +py'+qy=f(x).
- Linear homogeneous differential equation: y +p(x)y'+q(x)y=0, and inhomogeneous second order equation: y+p(x)y'+q(x)y=f(x).
Higher order differential equations and their types:
- Differential equation that can be reduced in order: F(x, y(k), y(k+1),.., y(n)=0.
- Linear higher order homogeneous equation: y(n)+f(n-1)y(n- 1)+…+f1y'+f0y=0, and inhomogeneous: y(n)+f(n-1)y(n-1)+…+f1 y'+f0y=f(x).
Steps in 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 frombiology, economics, sociology, etc. Despite the wide variety of topics, one should adhere to a single logical sequence when solving such problems:
- Compilation of remote control. 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 step, since it requires only strict mathematical calculations.
- Analysis and evaluation of the results. The derived solution should be evaluated to establish the practical and theoretical value of the result.
An example of using 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, we can consider the spread of infection in an isolated society. The 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 type includessusceptible individuals y(t) capable of becoming infected through contact with infected individuals.
- The third species includes immune individuals z(t) that 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. There will be two hypotheses at the core.
The percentage of incidence at a certain time point is x(t)y(t) (based on the theory 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 connection with this, the number of cases increases, and the number of susceptible decreases at a rate that is calculated by the formula ax(t)y(t) (a > 0).
The number of immune individuals that have become immune or died is increasing at a rate that is proportional to the number of cases, bx(t) (b > 0).
As a result, you can make a system of equations taking into account all three indicators and draw conclusions based on it.
Economics example
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 resolve such issues, it is necessarybuild 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 maximum point, 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 problems of finding the extremum of a function with several variables, because economic indicators are made up of many factors. Questions like this are good.studied in the theory of functions of several variables, applying 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.
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 ones are introduced.variables.
