Which line on the plane is determined by the equation. Book: The equation of a line on a plane. Angle between lines on a plane
Equation of a line on a plane
Main questions of the lecture: equations of a line on a plane; various forms of the equation of a straight line on a plane; angle between straight lines; conditions of parallelism and perpendicularity of lines; distance from a point to a line; second-order curves: circle, ellipse, hyperbola, parabola, their equations and geometric properties; equations of a plane and a straight line in space.
An equation of the form is called the equation of a straight line in general view.
If expressed in this equation, then after the replacement and we get an equation called the equation of a straight line with slope factor, and , where is the angle between the straight line and the positive direction of the x-axis. If in general equation straight line to transfer the free coefficient to the right side and divide by it, then we get the equation in segments
Where and are the points of intersection of the straight line with the abscissa and ordinate axes, respectively.
Two lines in a plane are called parallel if they do not intersect.
Lines are called perpendicular if they intersect at a right angle.
Let two straight lines and be given.
To find the point of intersection of the lines (if they intersect) it is necessary to solve the system with these equations. The solution of this system will be the point of intersection of the lines. Let's find the conditions relative position two straight lines.
Because 
, then the angle between these lines is found by the formula
From this it can be obtained that for , the lines will be parallel, and for , they will be perpendicular. If the lines are given in a general form, then the lines are parallel under the condition and perpendicular under the condition
The distance from a point to a line can be found using the formula
Normal equation of a circle:
An ellipse is the locus of points on a plane, the sum of the distances from which to two given points, called foci, is a constant value.
The canonical equation of an ellipse is:
. The vertices of the ellipse are the points , , ,. The eccentricity of an ellipse is the ratio
A hyperbola is the locus of points on a plane, the modulus of the difference in distances from which to two given points, called foci, is a constant value.
The canonical equation of a hyperbola has the form:
where is the major semiaxis, is the minor semiaxis, and . Foci are in points . The vertices of the hyperbola are the points , . The eccentricity of a hyperbola is the ratio
The straight lines are called the asymptotes of the hyperbola. If , then the hyperbola is called isosceles.
From the equation we obtain a pair of intersecting lines and .
A parabola is the locus of points on a plane, from each of which the distance to a given point, called the focus, is equal to the distance to a given line, called the directrix, is a constant value.
Canonical parabola equation
The straight line is called the directrix, and the point is called the focus.
The concept of functional dependency
The main questions of the lecture: sets; basic operations on sets; definition of a function, its area of existence, methods of setting; basic elementary functions, their properties and graphs; numerical sequences and their limits; limit of a function at a point and at infinity; infinitesimal and infinitely large quantities and their properties; basic theorems about limits; wonderful limits; continuity of a function at a point and on an interval; properties of continuous functions.
If each element of the set is associated with a well-defined element of the set, then they say that a function is given on the set. In this case, it is called an independent variable or argument, and a dependent variable, and the letter denotes the law of correspondence.
The set is called the domain of definition or existence of the function, and the set is called the domain of the function.
There are the following ways to define a function
1. Analytical method, if the function is given by a formula of the form
2. The tabular method is that the function is given by a table containing the values of the argument and the corresponding values of the function
3. The graphical method consists in displaying the function graph - a set of points in the plane, the abscissas of which are the values of the argument, and the ordinates are the corresponding function values
10.1. Basic concepts
A line on a plane is considered (given) as a set of points that have some geometric property inherent only to them. For example, a circle of radius R is the set of all points in the plane that are at a distance - R from some fixed point O (the center of the circle).
The introduction of a coordinate system on the plane allows you to determine the position of a point on the plane by setting two numbers - its coordinates, and determine the position of the line on the plane using an equation (i.e., an equality relating the coordinates of the points of the line).
Line equation(or curve) on the Oxy plane is such an equation F(x;y) = 0 with two variables, which is satisfied by the coordinates x and y of each point of the line and not satisfied by the coordinates of any point not lying on this line.
The x and y variables in the line equation are called the current coordinates of the line points.
The line equation allows the study of the geometric properties of the line to be replaced by the study of its equation.
So, in order to establish whether the point A (x 0; y 0) lies on a given line, it is enough to check (without resorting to geometric constructions) whether the coordinates of the point A satisfy the equation of this line in the chosen coordinate system.
The problem of finding the intersection points of two lines given by the equations F 1 (x 1; y 1) = 0 and F 2 (x 2; y) = 0 is reduced to finding points whose coordinates satisfy the equations of both lines, i.e., is reduced to solving a system of two equations with two unknowns:
If this system has no real solutions, then the lines do not intersect.
The concept of the equation of a line in a polar coordinate system is introduced in a similar way.
The equation F(r; φ)=O is called the equation of a given line in the polar coordinate system if the coordinates of any point lying on this line, and only they, satisfy this equation.
A line on a plane can be defined using two equations:
where x and y are the coordinates of an arbitrary point M(x; y) lying on a given line, and t is a variable called a parameter; the parameter t determines the position of the point (x; y) on the plane.
For example, if x \u003d t + 1, y \u003d t 2, then the point (3; 4) corresponds to the value of the parameter t \u003d 1 on the plane, because x \u003d 1 + 1 \u003d 3, y \u003d 22 - 4.
If the parameter t changes, then the point on the plane moves, describing the given line. This way of defining a line is called parametric, and equations (10.1) - parametric equations lines.
To pass from the parametric equations of the line to an equation of the form F(x;y) = 0, the parameter t must be eliminated from the two equations in some way.
For example, from equations by substituting t = x
into the second equation, it is easy to obtain the equation y \u003d x 2; or y-x 2 = 0, i.e., of the form F(x; y) = 0. However, we note that such a transition does not always possible.
A line on a plane can be specified by the vector equation r=r(t), where t is a scalar variable parameter. Each value t 0 corresponds to a certain vector r=r(t) planes. When the parameter t changes, the end of the vector r=r(t) describes some line (see Fig. 31).
Line vector equation r=r(t) in the Oxy coordinate system, two scalar equations (10.1) correspond, i.e., the equations of projections onto the coordinate axes of the vector equation of the line are its parametric equations. I The vector equation and parametric equations of the I line have a mechanical meaning. If a point moves on a plane, then these equations are called equations of motion, and the line is called the trajectory of the point, while the parameter t is time. So, any line on the plane corresponds to some equation of the form F(x; y) = 0.
To any equation of the form F (x; y) \u003d 0, generally speaking, there corresponds a certain line, the properties of which are determined by this equation (the expression “generally speaking” means that what has been said admits exceptions. So, the equation (x-2) 2 + (y-3) 2 \u003d 0 corresponds not to a line, but to a point (2; 3); to the equation x 2 + y 2 + 5 \u003d 0 on the plane does not correspond to any geometric image).
IN analytical geometry two main problems arise on the plane. First: knowing the geometric properties of the curve, find its equation) second: knowing the equation of the curve, study its shape and properties.
Figures 32-40 show examples of some curves and their equations.
10.2. Equations of a straight line on a plane
The simplest of the lines is the straight line. different ways assignments of a straight line correspond in a rectangular coordinate system different types its equations.
Line Equation with Slope
Let an arbitrary straight line not parallel to the Oy axis be given on the Oxy plane. Its position is completely determined by the ordinate b of the point N(0; b) of intersection with the Oy axis and the angle a between the Ox axis and the straight line (see Fig. 41).
At an angle a (0 The definition of the tangent of an angle implies the equality We introduce the notation tg a=k , we obtain the equation which is satisfied by the coordinates of any point M(x; y) of the line. It can be seen that the coordinates of any point P (x; y) lying outside the given line do not satisfy equation (10.2). The number k = tga is called the slope of the line, and equation (10.2) is the equation of the line with the slope. If the line passes through the origin, then b = 0 and, therefore, the equation of this line will look like y=kx . If the line is parallel to the Ox axis, then a \u003d 0, therefore, k \u003d tga \u003d 0 and equation (10.2) takes the form y \u003d b. If the straight line is parallel to the Oy axis, then equation (10.2) loses its meaning, because for it the slope In this case, the equation of a straight line will look like Where a- abscissa of the point of intersection of the line with the axis Ox. Note that equations (10.2) and (10.3) are equations of the first degree. General equation of a straight line. Consider a first degree equation for x and y in general form where A, B, C are arbitrary numbers, and A and B are not equal to zero at the same time. Let us show that equation (10.4) is the equation of a straight line. Two cases are possible. If B = 0, then equation (10.4) has the form Ax + C = O, and A ¹ 0 i.e. . This is the equation of a straight line parallel to the Oy axis and passing through the point If B ¹ 0, then from equation (10.4) we obtain So, equation (10.4) is the equation of a straight line, it is called the general equation of a straight line. Some special cases of the general equation of a straight line: 1) if A = 0, then the equation is reduced to the form. This is the equation of a straight line parallel to the x-axis; 2) if B \u003d 0, then the straight line is parallel to the Oy axis; 3) if С = 0, then we get . The equation is satisfied by the coordinates of the point O(0;0), the straight line passes through the origin. Equation of a line passing through a given point in a given direction Let a straight line pass through a point and its direction is determined by the slope k. The equation of this straight line can be written as , where b is an unknown quantity. Since the line passes through the point, then the coordinates of the point satisfy the equation of the line:. From here. Substituting the value of b into the equation, we obtain the desired equation of the straight line: , i.e. Equation (10.5) with different values of k is also called the equations of a pencil of straight lines centered at a point From this pencil, it is impossible to determine only a straight line parallel to the Oy axis. Equation of a line passing through two points Let the line pass through the points and . The equation of a straight line passing through the point M 1 has the form where k is a yet unknown coefficient. Since the line passes through the point , then the coordinates of this point must satisfy equation (10.6): . Here we find . Substituting the found value of k into equation (10.6), we obtain the equation of a straight line passing through the points M 1 and M 2 . It is assumed that in this equation If x 2 \u003d x 1 is a straight line passing through the points and parallel to the y-axis. Its equation is . If y 2 = y 1 then the equation of a straight line can be written as , straight line M1 M2 parallel to the x-axis. Equation of a straight line in segments This equation is called equation of a straight line in segments, since the numbers α and b indicate which segments the line cuts off on the coordinate axes. Equation of a straight line passing through a given point perpendicular to a given vector Let's find the equation of a straight line passing through a given point perpendicular to a given non-zero vector . Equation (10.8) is called the equation of a straight line passing through a given point perpendicular to a given vector. A vector perpendicular to a line is called the normal vector of that line. Equation (10.8) can be rewritten as where A and B are the coordinates of the normal vector, is the free term. Equation (10.9) is the general equation of a straight line (see (10.4)). Polar equation of a straight line For any point on this line we have: On the other side, Hence, The resulting equation (10.10) is the equation of a straight line in polar coordinates. Normal equation of a straight line Let the line be determined by setting p and α (see Fig. 45). Consider a rectangular coordinate system. We introduce the polar system, taking the pole and the polar axis. The equation of a straight line can be written as But, due to the formulas connecting rectangular and polar coordinates, we have: , . Consequently, equation (10.10) of a straight line in a rectangular coordinate system takes the form Equation (10.11) is called normal equation of a straight line. We multiply all terms of equation (10.4) by some factor . We get . This equation should turn into equation (10.11). Therefore, the equalities must be satisfied: , , . From the first two equalities, we find e. Equation of a line on a plane.
As is known, any point on the plane is determined by two coordinates in some coordinate system. Coordinate systems can be different depending on the choice of basis and origin. Definition. Line equation is called the ratio y=f(x ) between the coordinates of the points that make up this line. Note that the line equation can be expressed in a parametric way, that is, each coordinate of each point is expressed through some independent parametert.
A typical example is the trajectory of a moving point. In this case, time plays the role of a parameter. Equation of a straight line on a plane.
Definition.
Any line in the plane can be given by a first order equation Ah + Wu + C = 0, moreover, the constants A, B are not equal to zero at the same time, i.e. A 2 + B 2¹
0. This first order equation is called the general equation of a straight line.
Depending on the values of the constants A, B and C, the following special cases are possible: C = 0, A ¹ 0, B ¹ 0 - the line passes through the origin A = 0, B ¹ 0, C ¹ 0 ( By + C \u003d 0) - a straight line is parallel to the Ox axis B = 0, A ¹ 0, C ¹ 0 ( Ax + C = 0) - a straight line parallel to the Oy axis B \u003d C \u003d 0, A ¹ 0 - the line coincides with the Oy axis A = C = 0, B ¹ 0 - the line coincides with the Ox axis The equation of a straight line can be presented in various forms depending on any given initial conditions. The distance from a point to a line.
Theorem.
If a point M(x 0, y 0) is given, then the distance to the line Ax + Vy + C \u003d 0 is defined as
Proof.
Let the point M 1 (x 1, y 1) be the base of the perpendicular dropped from the point M to the given line. Then the distance between points M and M 1: (1)
Coordinates x 1 and y 1 can be found as a solution to the system of equations: The second equation of the system is the equation of a straight line passing through a given point M 0 perpendicular to a given straight line. If we transform the first equation of the system to the form: A(x - x 0) + B(y - y 0) + Ax 0 + By 0 + C = 0, then, solving, we get:
Substituting these expressions into equation (1), we find: The theorem has been proven. Example. Determine the angle between the lines: y=-3x+7; y = 2 x + 1. K 1 \u003d -3; k 2 = 2tg j = ; j = p /4. Example. Show that the lines 3x - 5y + 7 = 0 and 10x + 6y - 3 = 0 are perpendicular. Find: k 1 = 3/5, k 2 = -5/3, k 1 k 2 = -1, hence the lines are perpendicular. Example. Given the vertices of the triangle A(0; 1), B(6;5), C (12; -1). Find the equation for the height drawn from vertex C. This article is a continuation of the line on the plane section. Here we turn to the algebraic description of a straight line using the equation of a straight line. The material of this article is the answer to the questions: “What equation is called the equation of a straight line and what form does the equation of a straight line have in a plane”? Page navigation. Let Oxy be fixed on the plane and a straight line be given in it. A straight line, like any other geometric figure, consists of points. In a fixed rectangular coordinate system, each point of the line has its own coordinates - the abscissa and the ordinate. So the relationship between the abscissa and the ordinate of each point of a straight line in a fixed coordinate system can be given by an equation, which is called the equation of a straight line on a plane. In other words, equation of a straight line in a plane in the rectangular coordinate system Oxy there is some equation with two variables x and y that turns into an identity when the coordinates of any point of this line are substituted into it. It remains to deal with the question of what form the equation of a straight line on a plane has. The answer to it is contained in the next paragraph of the article. Looking ahead, we note that there are various forms of writing the equation of a straight line, which is explained by the specifics of the tasks being solved and the method of setting a straight line on a plane. So, let's start a review of the main types of the equation of a straight line on a plane. The form of the equation of a straight line in the rectangular coordinate system Oxy on the plane is given by the following theorem. Theorem. Any equation of the first degree with two variables x and y of the form , where A , B and C are some real numbers, and A and B are not equal to zero at the same time, defines a straight line in the rectangular coordinate system Oxy on the plane, and any straight line on the plane is given by an equation of the form The equation Let us explain the meaning of the theorem. Given an equation of the form Look at the drawing. On the one hand, we can say that this line is determined by the general equation of a straight line of the form The general equation of a straight line is called complete, if all numbers A, B and C are non-zero, otherwise the general equation of a straight line is called incomplete. An incomplete equation of a straight line form defines a straight line passing through the origin. When A=0, the equation Thus, any straight line on a plane in a given rectangular coordinate system Oxy can be described using the general equation of a straight line for a certain set of values of the numbers A, B and C. Normal vector of a straight line given by a general equation of a straight line of the form All equations of lines, which are given in the following paragraphs of this article, can be obtained from the general equation of a line, and can also be reduced back to the general equation of a line. We recommend further study of the article. There, the theorem formulated at the beginning of this paragraph of the article is proved, graphic illustrations are given, solutions of examples for compiling the general equation of a straight line are analyzed in detail, the transition from the general equation of a straight line to equations of another type and vice versa is shown, and other characteristic problems are also considered. A straight line equation, where a and b are some non-zero real numbers, is called equation of a straight line in segments. This name is not accidental, since the absolute values of the numbers a and b are equal to the lengths of the segments that the straight line cuts off on the coordinate axes Ox and Oy, respectively (the segments are measured from the origin). Thus, the equation of a straight line in segments makes it easy to build this straight line in a drawing. To do this, mark points with coordinates and in a rectangular coordinate system on the plane, and use a ruler to connect them with a straight line. For example, let's build a straight line given by an equation in segments of the form . Marking the dots You can get detailed information about this type of equation of a straight line in a plane in the article. A straight line equation, where x and y are variables and k and b are some real numbers, is called equation of a straight line with a slope(k is the slope factor). The equations of a straight line with a slope are well known to us from a high school algebra course. This kind of equation of a straight line is very convenient for research, since the variable y is an explicit function of the argument x. The definition of the slope of the straight line is given through the definition of the angle of inclination of the straight line to the positive direction of the axis Ox . Definition.
The angle of inclination of the straight line to the positive direction of the x-axis in a given rectangular Cartesian coordinate system, Oxy is called the angle measured from the positive direction of the Ox axis to the given straight line counterclockwise. If the straight line is parallel to the abscissa axis or coincides with it, then the angle of its inclination is considered equal to zero. Definition.
Slope of a straight line is the tangent of the slope of this straight line, that is, . If the line is parallel to the y-axis, then the slope goes to infinity (in this case, it is also said that the slope does not exist). In other words, we cannot write the equation of a line with a slope for a line parallel to or coinciding with the Oy axis. Note that the straight line defined by the equation passes through a point on the y-axis. Thus, the equation of a straight line with a slope determines a straight line on a plane that passes through a point and forms an angle with the positive direction of the abscissa axis, and . As an example, let's draw a straight line defined by an equation of the form . This line passes through the point and has a slope Note that it is very convenient to search in the form of an equation of a straight line with a slope. Canonical equation of a straight line in a plane in a rectangular Cartesian coordinate system Oxy has the form It is obvious that the straight line, defined by the canonical equation of the straight line, passes through the point. In turn, the numbers and , standing in the denominators of the fractions, are the coordinates of the directing vector of this line. Thus, the canonical equation of a straight line in the rectangular coordinate system Oxy on the plane corresponds to a straight line passing through a point and having a direction vector . For example, let's draw a straight line on the plane corresponding to the canonical straight line equation of the form The canonical straight line equation is used even when one of the numbers or is equal to zero. In this case, the entry is considered conditional (since the denominator contains zero) and should be understood as Detailed information about the equation of a straight line in the canonical form, as well as detailed solutions to typical examples and problems are collected in the article. Parametric equations of a straight line on a plane look like Parametric equations of a straight line establish an implicit relationship between the abscissas and ordinates of the points of a straight line using a parameter (hence the name of this type of straight line equations). A pair of numbers , which are calculated by the parametric equations of the straight line for some real value of the parameter , is the coordinates of some point on the straight line. For example, when we have It should be noted that the coefficients and at the parameter in the parametric equations of the straight line are the coordinates of the directing vector of this straight line. Equation of a line on a plane Main questions of the lecture: equations of a line on a plane; various forms of the equation of a straight line on a plane; angle between straight lines; conditions of parallelism and perpendicularity of lines; distance from a point to a line; second-order curves: circle, ellipse, hyperbola, parabola, their equations and geometric properties; equations of a plane and a straight line in space. An equation of the form is called the equation of a straight line in general form. If we express in this equation , then after replacing and we get the equation , called the equation of a straight line with a slope, and , where is the angle between the straight line and the positive direction of the x-axis. If, in the general equation of a straight line, we transfer the free coefficient to the right side and divide by it, then we get the equation in segments Where and are the points of intersection of the straight line with the abscissa and ordinate axes, respectively. Two lines in a plane are called parallel if they do not intersect. Lines are called perpendicular if they intersect at a right angle. Let two straight lines and be given. To find the point of intersection of the lines (if they intersect) it is necessary to solve the system with these equations. The solution of this system will be the point of intersection of the lines. Let us find the conditions for the mutual arrangement of two lines. Because From this it can be obtained that for , the lines will be parallel, and for , they will be perpendicular. If the lines are given in a general form, then the lines are parallel under the condition and perpendicular under the condition The distance from a point to a line can be found using the formula Normal equation of a circle: An ellipse is the locus of points on a plane, the sum of the distances from which to two given points, called foci, is a constant value. The canonical equation of an ellipse is: A hyperbola is the locus of points on a plane, the modulus of the difference in distances from which to two given points, called foci, is a constant value. The canonical equation of a hyperbola has the form: where is the major semiaxis, is the minor semiaxis, and . Foci are in points The straight lines are called the asymptotes of the hyperbola. If , then the hyperbola is called isosceles. From the equation we obtain a pair of intersecting lines and . A parabola is the locus of points on a plane, from each of which the distance to a given point, called the focus, is equal to the distance to a given line, called the directrix, is a constant value. Canonical parabola equation The straight line is called the directrix, and the point is called the focus. The concept of functional dependency The main questions of the lecture: sets; basic operations on sets; definition of a function, its area of existence, methods of setting; basic elementary functions, their properties and graphs; numerical sequences and their limits; limit of a function at a point and at infinity; infinitesimal and infinitely large quantities and their properties; basic theorems about limits; wonderful limits; continuity of a function at a point and on an interval; properties of continuous functions. If each element of the set is associated with a well-defined element of the set, then they say that a function is given on the set. In this case, it is called an independent variable or argument, and a dependent variable, and the letter denotes the law of correspondence. The set is called the domain of definition or existence of the function, and the set is called the domain of the function. There are the following ways to define a function 1. Analytical method, if the function is given by a formula of the form 2. The tabular method is that the function is given by a table containing the values of the argument and the corresponding values of the function 3. The graphical method consists in displaying the function graph - a set of points in the plane, the abscissas of which are the values of the argument, and the ordinates are the corresponding function values 4. Verbal method, if the function is described by the rule of its compilation. Main properties of the function 1. Even and odd. A function is called even if for all values from the domain of definition and odd if 2. Monotony. A function is called increasing (decreasing) on the interval if the larger value of the argument from this interval corresponds to the larger (smaller) value of the function. 3. Limited. A function is called bounded on an interval if there exists a positive number such that for any . Otherwise, the function is called unbounded. 4. Periodicity. A function is called periodic with a period if for any of the domain of the function Classification of functions. 1. Inverse function. Let there be a function of an independent variable defined on a set with a range of values . Let us assign to each a unique value for which . Then the resulting function defined on the set with range is called inverse. 2. Complex function. Let a function be a function of a variable defined on a set with a range of values , and the variable in turn be a function. The following functions are most commonly used in economics. 1. The utility function and the preference function - in the broad sense of the dependence of utility, that is, the result, the effect of some action on the level of intensity of this action. 2. Production function - the dependence of the result of production activity on the factors that caused it. 3. The output function (a particular type of production function) is the dependence of the volume of production on the beginning or consumption of resources. 4. Cost function (a particular type of production function) - the dependence of production costs on the volume of production. 5. Functions of demand, consumption and supply - the dependence of the volume of demand, consumption or supply for individual goods or services on various factors. If, according to some law, each natural number is assigned a well-defined number, then they say that a numerical sequence is given. The numbers are called the members of the sequence, and the number is the common member of the sequence. A number is called the limit of a numerical sequenceif for any small number there is such a number (depending on) that equality is true for all members of the sequence with numbers. The limit of a numerical sequence is denoted. A sequence that has a limit is called convergent, otherwise it is divergent. A number is called the limit of the function for if for any small number there is such a positive number that for all such that the inequality is true. Limit of a function at a point. Let the function be given in some neighborhood of the point , except, perhaps, the point itself. The number is called the limit of the function at , if for any, even arbitrarily small, there is such a positive number (depending on ) that for all and satisfying the condition the inequality is true. This limit is denoted by . A function is called an infinitesimal value at if its limit is zero. Properties of infinitesimals 1. The algebraic sum of a finite number of infinitesimal quantities is an infinitesimal quantity. 2. The product of an infinitely small value by a bounded function is an infinitesimal quantity 3. The quotient of dividing an infinitesimal quantity by a function whose limit is different from zero is an infinitesimal quantity. The concept of the derivative and differential of a function The main questions of the lecture: problems leading to the concept of a derivative; definition of derivative; geometric and physical meaning of the derivative; the concept of a differentiable function; basic rules of differentiation; derivatives of basic elementary functions; derivative of a complex and inverse function; derivatives of higher orders, basic theorems of differential calculus; L'Hopital's theorem; disclosure of uncertainties; increasing and decreasing function; function extremum; convexity and concavity of the function graph; analytical signs of convexity and concavity; inflection points; vertical and oblique asymptotes of the graph of the function; the general scheme of the study of the function and the construction of its graph, the definition of a function of several variables; limit and continuity; partial derivatives and differential functions; directional derivative, gradient; extremum of a function of several variables; the largest and smallest values of the function; conditional extremum, Lagrange method. The derivative of a function is the limit of the ratio of the increment of the function to the increment of the independent variable when the latter tends to zero (if this limit exists) If a function at a point has a finite derivative, then the function is said to be differentiable at that point. A function that is differentiable at each point of the interval is called differentiable on this interval. The geometric meaning of the derivative: the derivative is the slope (tangent of the slope angle) of the tangent reduced to the curve at the point. Then the equation of the tangent to the curve at the point takes the form The mechanical meaning of the derivative: the derivative of the path with respect to time is the speed of a point at a moment of time: The economic meaning of the derivative: the derivative of the volume of output with respect to time is the productivity of labor at the moment Theorem. If a function is differentiable at a point, then it is continuous at that point. The derivative of a function can be found in the following way 1. Let's increment the argument and find the incremented value of the function 2. Find the increment of the function. 3. We make the ratio. 4. We find the limit of this relation at, that is (if this limit exists). Differentiation rules 1. The derivative of a constant is zero, that is. 2. The derivative of the argument is 1, that is. 3. The derivative of the algebraic sum of a finite number of differentiable functions is equal to the same sum of the derivatives of these functions, that is. 4. The derivative of the product of two differentiable functions is equal to the product of the derivative of the first factor by the second plus the product of the first factor by the derivative of the second, that is 5. The derivative of the quotient of two differentiable functions can be found by the formula: Theorem. If and are differentiable functions of their variables, then the derivative of the complex function exists and is equal to the derivative of the given function with respect to the intermediate argument and multiplied by the derivative of the intermediate argument itself with respect to the independent variable, that is Theorem. For a differentiable function with a derivative that is not equal to zero, the derivative of the inverse function is equal to the reciprocal of the derivative of this function, that is, . The elasticity of a function is the limit of the ratio of the relative increment of the function to the relative increment of the variable at: The elasticity of a function shows approximately how many percent the function will change when the independent variable changes by one percent. Geometrically, this means that the elasticity of the function (in absolute value) is equal to the ratio of the tangential distances from a given point of the graph of the function to the points of its intersection with the axes and . The main properties of the elasticity function: 1. The elasticity of a function is equal to the product of the independent variable and the rate of change of the function 2. The elasticity of the product (quotient) of two functions is equal to the sum (difference) of the elasticities of these functions: , 3. Elasticity of mutually inverse functions - mutually inverse quantities: The elasticity of a function is used in the analysis of demand and consumption. Fermat's theorem. If a function differentiable on an interval reaches its maximum or minimum value at an interior point of this interval, then the derivative of the function at this point is equal to zero, that is, . Rolle's theorem. Let the function satisfy the following conditions: 1) is continuous on the segment ; 2) differentiable on the interval ; 3) at the ends of the segment takes equal values, that is, . Then inside the segment there is at least one such point at which the derivative of the function is equal to zero: . Lagrange's theorem. Let the function satisfy the following conditions 1. Continuous on the segment . 2. Differentiable on the interval ; Then inside the segment there is at least one such point at which the derivative is equal to the increment of the function divided by the increment of the argument on this segment, that is Theorem. The limit of the ratio of two infinitely small or infinitely large functions is equal to the limit of the ratio of their derivatives (finite or infinite), if the latter exists in the indicated sense. So, if there is an uncertainty of the form or , then Theorem (sufficient condition for the function to increase) If the derivative of a differentiable function is positive inside some interval X, then it increases on this interval. Theorem (sufficient condition for a function to decrease), If the derivative of a differentiable function is negative inside some interval, then it decreases on this interval. A point is called a maximum point of a function if the inequality is true in some neighborhood of the point. A point is called a minimum point of a function if the inequality is true in some neighborhood of the point. The values of the function at the points and are called the maximum and minimum of the function, respectively. The maximum and minimum of a function are combined by the common name of the extremum of the function. For a function to have an extremum at a point, its derivative at that point must be equal to zero or not exist. The first sufficient condition for an extremum. Theorem. If, when passing through a point, the derivative of a differentiable function changes its sign from plus to minus, then the point is the maximum point of the function, and if from minus to plus, then the minimum point. Scheme of studying a function for an extremum. 1. Find the derivative. 2. Find the critical points of the function at which the derivative or does not exist. 3. Examine the sign of the derivative to the left and right of each critical point and draw a conclusion about the presence of extrema of the function. 4. Find extrema (extreme values) of the function. The second sufficient condition for an extremum. Theorem. If the first derivative of a twice differentiable function is equal to zero at some point , and the second derivative at this point is positive, that is, the minimum point of the function , if negative, then the maximum point. To find the largest and smallest values on the segment, we use the following scheme. 1. Find the derivative. 2. Find the critical points of the function at which or does not exist. 3. Find the values of the function at critical points and at the ends of the segment and choose the largest and smallest of them. A function is called upward convex on the interval X if the segment connecting any two points of the graph lies under the graph of the function. A function is called downward convex on the interval X if the segment connecting any two points of the graph lies above the graph of the function. Theorem. A function is convex down (up) on the interval X if and only if its first derivative on this interval is monotonically increasing (decreasing). Theorem. If the second derivative of a twice differentiable function is positive (negative) inside some interval X, then the function is convex down (up) on this interval. The inflection point of the graph of a continuous function is the point that separates the intervals in which the function is convex downwards and upwards. Theorem (necessary inflection condition). The second derivative of a twice differentiable function at the inflection point is equal to zero, that is, . Theorem (sufficient condition for inflection). If the second derivative of a twice differentiable function changes sign when passing through a certain point, then there is an inflection point of its graph. Scheme of studying the function for convexity and inflection points: 1. Find the second derivative of the function. 2. Find points at which the second derivative or does not exist. 3. Examine the sign of the second derivative to the left and right of the found points and draw a conclusion about the convexity intervals and the presence of inflection points. 4. Find the function values at the inflection points. When examining a function for plotting their graphs, it is recommended to use the following scheme: 1. Find the domain of the function. 2. Investigate the function for evenness - oddness. 3. Find vertical asymptotes 4. Investigate the behavior of the function at infinity, find horizontal or oblique asymptotes. 5. Find extrema and intervals of monotonicity of the function. 6. Find the convexity intervals of the function and the inflection points. 7. Find points of intersection with the coordinate axes and, possibly, some additional points that refine the graph. The differential of a function is the main, linear with respect to part of the increment of the function, equal to the product of the derivative and the increment of the independent variable. Let there be variables, and each set of their values from some set X corresponds to one well-defined value of the variable . Then we say that a function of several variables is given Variables are called independent variables or arguments, - dependent variable. The set X is called the domain of the function. The multidimensional analogue of the utility function is the function Also, for the case of variables, the concept of a production function is generalized, expressing the result of production activity from the factors that caused it. less than by definition and are continuous at the point itself. Then the partial derivatives., and find the critical points of the function. 3. Find partial derivatives of the second order, calculate their values at each critical point and, using a sufficient condition, draw a conclusion about the presence of extrema. Find extrema (extreme values) of the function. Literature 1. Higher mathematics for economists: Textbook for universities / Ed. N.Sh. Kremer. – M.: UNITI, 2003. 2.E.S. Kochetkov, S.O. Smerchinskaya Theory of Probability in Problems and Exercises / M. INFRA-M 2005. 3. Higher mathematics for economists: Workshop / Ed. N.Sh. Kremer. - M.: UNITI, 2004. Part 1, 2 4. Gmurman V.E. Guide to solving problems in probability theory and mathematical statistics. M., Higher School, 1977 5. Gmurman V.E. Theory of Probability and Mathematical Statistics. M., Higher School, 1977 6. M.S. Crass Mathematics for economic specialties: Textbook / M. INFRA-M 1998. 7. Vygodsky M.Ya. Handbook of higher mathematics. - M., 2000. 8. Berman G.N. Collection of problems on the course of mathematical analysis. – M.: Nauka, 1971. 9.A.K. Kazashev Collection of problems in higher mathematics for economists - Almaty - 2002 10. Piskunov N.S. Differential and integral calculus. - M .: Nauka, 1985, T. 1.2. 11.P.E. Danko, A.G. Popov, T.Ya. Kozhevnikov Higher Mathematics in Exercises and Problems / M. ONIKS-2005. 12.I.A. Zaitsev Higher Mathematics / M. Higher School-1991 13. Golovina L.I. Linear algebra and some of its applications. – M.: Nauka, 1985. 14. Zamkov O.O., Tolstopyatenko A.V., Cheremnykh Yu.N. Mathematical methods of economic analysis. – M.: DIS, 1997. 15. Karasev A.I., Aksyutina Z.M., Savelyeva T.I. Course of higher mathematics for economic universities. - M .: Higher School, 1982 - Ch 1, 2. 16. Kolesnikov A.N. A short course in mathematics for economists. – M.: Infra-M, 1997. 17.V.S. Shipatsev Task book on higher mathematics-M. High School, 2005
(10.2)
does not exist.
(10.4)
. This is the equation of a straight line with a slope 
|.
(10.5)
(10.6)
(10.7)
Let the straight line intersect the Ox axis at a point, and the Oy axis at a point (see Fig. 42). In this case, equation (10.7) takes the form
Let us take an arbitrary point M(x; y) on the line and consider a vector (see Fig. 43). Since the vectors and are perpendicular, their scalar product is equal to zero: that is
(10.9)
Let's find the equation of a straight line in polar coordinates. Its position can be determined by indicating the distance ρ from the pole O to the given straight line and the angle α between the polar axis Op and the axis l passing through the pole O perpendicular to the given line (see Fig. 44).
(10.10)
(10.11)
Let us show how to bring equation (10.4) straight to the form (10.11).
. The factor λ is called normalizing factor. According to the third equality, the sign of the normalizing factor is opposite to the sign of the free term C of the general equation of the straight line.
.
.
Equation of a straight line on a plane - definition.
General equation of a straight line.
.
called the general equation of a straight line on surface.corresponds to a straight line on a plane in a given coordinate system, and a straight line on a plane in a given coordinate system corresponds to an equation of a straight line of the form .
, since the coordinates of any point of the depicted line satisfy this equation. On the other hand, the set of points in the plane defined by the equation , give us a straight line shown in the drawing.sets a straight line parallel to the abscissa axis Ox , and when B=0 - parallel to the ordinate axis Oy ., has coordinates .Equation of a straight line in segments.
and connect them.
Equation of a straight line with a slope.
radians (60 degrees) to the positive direction of the Ox axis. Its slope is .
Canonical equation of a straight line on a plane.
, where and are some real numbers, and and are not equal to zero at the same time.
. It is obvious that the point belongs to the line, and the vector is the directing vector of this line.
. If , then the canonical equation takes the form 
and defines a line parallel to the y-axis (or coinciding with it). If , then the canonical equation of the line takes the form 
and defines a straight line parallel to the x-axis (or coinciding with it).Parametric equations of a straight line on a plane.
, where and are some real numbers, and and are not equal to zero at the same time, and is a parameter that takes any real values.
, that is, the point with coordinates lies on a straight line.
, then the angle between these lines is found by the formula
. The vertices of the ellipse are the points , , ,. The eccentricity of an ellipse is the ratio. The vertices of the hyperbola are the points , . The eccentricity of a hyperbola is the ratio
. Otherwise, the function is called a generic function.
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:
.
.
.
, that is .
.
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., which expresses the dependence on the purchased goods.