Analytic geometry. Lines on a plane and their equations What is the equation of a line on a plane

We will consider a line on a plane as the locus of points M(x, y) satisfying a certain condition.

If we write down in a Cartesian coordinate system a property that all points on a line have, connecting the coordinates and some constants, we can obtain an equation of the form: F(x, y) = 0 or .

Example. Write the equation of a circle with center at point C(x 0 , y 0) and radius R.

A circle is the geometric locus of points equidistant from point C. Let’s take point M with current coordinates. Then |CM| = R or or .

If the center of the circle is at the origin, then x 2 + y 2 = R 2 .

Not every equation of the form F(x, y) = 0 defines a line in the indicated sense: x 2 + y 2 = 0 is a point.

Straight on a plane.

Lines on a given plane are a special case of lines in space. Therefore, their equations can be obtained from the corresponding equations of lines in space.

General equation of a straight line on a plane. Equation of a straight line with an angular coefficient.

Any straight line in the XOY plane can be defined as the line of intersection of the Ax + By + Cz + D = 0 plane with the XOY plane: z = 0.

- straight line in the XOY plane: Ax + By + D = 0.

The resulting equation is called the general equation of the line. In the future we will write it in the form:

Ax + By + C = 0 (1)

1) Let , then or y = kx + b (2) – equation of a straight line with an angular coefficient. Let's find out the geometric meaning of k and b.

Let's put x = 0. Then y = b is the initial ordinate of the line.

Let's put y = 0. Then ; - slope coefficient of a straight line.

Special cases: a) b = 0, y=kx – the line passes through the origin; b) k = 0, y = b – straight line parallel to the OX axis; b) if B = 0, then Ax + C = 0, ,

This is the locus of points with constant abscissas equal to a, i.e. the straight line is perpendicular to the OX axis.

Equation of a straight line in segments.

Let the general equation of the line be given: Ax + By + C = 0, and . Let's divide both sides by –C:

or (3),

Where ; . This is the equation of a line in segments. Numbers a and b are the values of the segments cut off on the coordinate axes.

Equation of a line passing through a given point with a given slope.

Let a point M 0 (x 0 , y 0) lying on a straight line L and an angular coefficient k be given. Let's write the equation:

Here b is unknown. Let's find it, taking into account that M 0 L:

y 0 = kx 0 + b (**).

Subtract term by term from (1) (2):

y – y 0 = k(x – x 0) (4).

The equation of a line passing through a given point in a given direction.

Equation of a line passing through two given points.

Let two points M 1 (x 1 , y 1) and M 2 (x 2 , y 2) L be given. Let us write equation (4) in the form: y – y 1 = k(x – x 1). Because M 2 L, then y 2 – y 1 = k(x 2 – x 1). Let's divide it term by term:

(5),

This equation makes sense if , . If x 1 = x 2, then M 1 (x 1, y 1) and M 2 (x 1, y 2). If y 2 = y 1, then M 1 (x 1, y 1); M 2 (x 2, y 1).

Thus, if one of the denominators in (5) becomes zero, the corresponding numerator must be set equal to zero.

Example. M 1 (3, 1) and M 2 (-1, 4). Write the equation of the line passing through these points. Find k.

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 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 line in space.

An equation of the form is called an equation of a straight line in general form.

If we express in this equation, then after replacement we obtain an equation called the equation of a straight line with an angular coefficient, and where is the angle between the straight line and the positive direction of the abscissa axis. If in the general equation of a straight line we transfer the free coefficient to the right side and divide by it, we obtain an equation in segments

Where and are the points of intersection of the 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 right angles.

Let two 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 to this system will be the point of intersection of the lines. Let us find the conditions for the relative position of two lines.

Because , then the angle between these lines is found by the formula

From this we can conclude that when the lines will be parallel, and when they will be perpendicular. If the lines are given in general form, then the lines are parallel under the condition and perpendicular under the condition

The distance from a point to a straight line can be found using the formula

Normal equation of a circle:

An ellipse is the geometric 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 has the form:

. 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 semimajor axis, is the semiminor axis and . Focuses are at points . The vertices of a hyperbola are the points , . The eccentricity of a hyperbola is the ratio

The straight lines are called asymptotes of the hyperbola. If , then the hyperbola is called equilateral.

From the equation we obtain a pair of intersecting lines and .

A parabola is the geometric 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 straight line, called the directrix, and 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 dependence

Main questions of the lecture: sets; basic operations on sets; definition of a function, its domain of existence, methods of assignment; basic elementary functions, their properties and graphs; number sequences and their limits; limit of a function at a point and at infinity; infinitely small 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 a set is associated with a completely specific element of the set, then they say that a function is defined on the set. In this case, it is called the independent variable or argument, and the dependent variable, and the letter denotes the law of correspondence.

A set is called the domain of definition or existence of a function, and a set is called the domain of values of a function.

There are the following ways to specify 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 specified by a table containing the values of the argument and the corresponding values of the function

3. The graphical method consists of depicting a graph of a function - a set of points on the plane, the abscissas of which are the values of the argument, and the ordinates are the corresponding values of the function

4. Verbal method, if the function is described by the rule for its composition.

Basic properties of a function

1. Even and odd. A function is called even if for all values from the domain of definition and odd if . Otherwise, the function is called a general function.

2. Monotony. A function is said to be increasing (decreasing) on the interval if a larger value of the argument from this interval corresponds to a larger (smaller) value of the function.

3. Limited. A function is said to be bounded on an interval if there is a positive number such that for any . Otherwise the function is called unbounded.

4. Frequency. A function is called periodic with period if for any of the domain of definition 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 associate each with a single value at which . Then the resulting function defined on a set with a range of values 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 is a function.

The following functions are most often used in economics.

1. Utility function and preference function - in a broad sense, the dependence of utility, that is, the result, 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 determined it.

3. Output function (a particular type of production function) – the dependence of production volume on the beginning or consumption of resources.

4. Cost function (a particular type of production function) – the dependence of production costs on production volume.

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 associated with a very specific number, then they say that a number sequence is given.

Numbers are called members of a sequence, and a number is a common member of the sequence.

A number is called the limit of a number sequence if for any small number there is a number (depending on) such that the equality is true for all members of the sequence with numbers. The limit of a number sequence is denoted by .

A sequence having a limit is called convergent, otherwise it is called divergent.

A number is called the limit of a function at if for any small number there is a positive number such that for all such numbers 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. A number is called the limit of a function at , if for any, even arbitrarily small, there is a positive number (depending on ) such that for all and satisfying the condition the inequality . This limit is designated .

A function is called infinitesimal if its limit is zero.

Properties of infinitesimal quantities

1. The algebraic sum of a finite number of infinitesimal quantities is an infinitesimal quantity.

2. The product of an infinitesimal quantity and a bounded function is an infinitesimal quantity

3. The quotient of dividing an infinitesimal quantity by a function whose limit is non-zero is an infinitesimal quantity.

The concept of derivative and differential of a function

The main questions of the lecture: problems leading to the concept of derivative; definition of derivative; geometric and physical meaning of derivative; concept of 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 functions; extremum of a function; convexity and concavity of the graph of a function; analytical signs of convexity and concavity; inflection points; vertical and oblique asymptotes of the graph of a function; general scheme for studying a function and constructing its graph, defining 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 a 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 as 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.

Geometric meaning of the derivative: the derivative is the slope (tangent of the angle of inclination) 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

Mechanical meaning of the derivative: the derivative of a path with respect to time is the speed of a point at a moment in time:

The economic meaning of the derivative: the derivative of the volume of production with respect to time is labor productivity 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 using the following scheme

1. Give the argument an increment and find the incremented value of the function .

2. Find the increment of the function.

3. We create a relationship.

4. Find the limit of this ratio at, that is (if this limit exists).

Rules of differentiation

1. The derivative of a constant is zero, that is.

2. The derivative of the argument is equal to 1, that is.

3. The derivative of an 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 using the formula:

Theorem. If and are differentiable functions of their variables, then the derivative of a complex function exists and is equal to the derivative of this 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 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 a function to the relative increment of a 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 a function (in absolute value) is equal to the ratio of the tangent distances from a given point on the graph of the function to the points of its intersection with the and axes.

Basic 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 , that is .

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 reciprocal functions – reciprocal quantities:

The elasticity function is used in the analysis of demand and consumption.

Fermat's theorem. If a function differentiable on an interval reaches its greatest or minimum value at an internal 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) 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 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 quotient of dividing the increment of the function by the increment of the argument on this segment, that is .

Theorem. The limit of the ratio of two infinitesimal 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 uncertainty of the form or , then

Theorem (sufficient condition for the function to increase)

If the derivative of a differentiable function is positive inside a certain interval X, then it increases over this interval.

Theorem (sufficient condition for a function to decrease), If the derivative of a differentiable function is negative inside a certain interval, then it decreases on this interval.

A point is called a maximum point of a function if the inequality holds in some neighborhood of the point.

A point is called a minimum point of a function if the inequality holds 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 united by the common name of the extremum of the function.

In order for a function to have an extremum at a point, its derivative at this 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 the 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 for studying a function at an extremum.

1. Find the derivative.

2. Find the critical points of the function at which the derivative or does not exist.

3. Investigate 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 the 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 it is negative, then it is the maximum point.

To find the largest and smallest values on a 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 select the largest and smallest from them.

A function is said to be convex upward on the interval X if the segment connecting any two points on the graph lies under the graph of the function.

A function is called convex downward on the interval X if the segment connecting any two points on the graph lies above the graph of the function.

Theorem. A function is convex downward (upward) on the interval X if and only if its first derivative monotonically increases (decreases) on this interval.

Theorem. If the second derivative of a twice differentiable function is positive (negative) inside some interval X, then the function is convex downward (upward) on this interval.

The inflection point of the graph of a continuous function is the point separating the intervals in which the function is convex downward and upward.

Theorem (necessary condition for inflection). 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 its sign when passing through a certain point, then there is an inflection point in its graph.

Scheme for studying a function for convexity and inflection points:

1. Find the second derivative of the function.

2. Find the points at which the second derivative or does not exist.

3. Investigate 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 values of the function at the inflection points.

When studying functions to construct their graphs, it is recommended to use the following scheme:

1. Find the domain of definition of the function.

2. Investigate the function for evenness - oddness.

3. Find vertical asymptotes

4. Investigate the behavior of a function at infinity, find horizontal or oblique asymptotes.

5. Find extrema and intervals of monotonicity of the function.

6. Find the intervals of convexity of the function and inflection points.

7. Find the points of intersection with the coordinate axes and, possibly, some additional points that clarify the graph.

The differential of a function is the principal, relatively linear part of the increment of a function, equal to the product of the derivative by the increment of the independent variable.

Let there be variable quantities, and each set of their values from a certain 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 definition of the function.

A multidimensional analogue of the utility function is the function , expressing dependence on purchased goods.

Also, in the case of variables, the concept of a production function is generalized, expressing the result of production activity from the factors that determined it. less than by definition and continuous at the point itself. Then partial derivatives, and find the critical points of the function.

3. Find second-order partial derivatives, 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

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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.: UNITY, 2004. Parts 1, 2

4. Gmurman V.E. A 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 for 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. ONICS-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 – Part 1, 2.

16. Kolesnikov A.N. A short course in mathematics for economists. – M.: Infra-M, 1997.

17.V.S. Shipatsev Problem book in higher mathematics-M. Higher school, 2005

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 relation y = f(x) between the coordinates of the points that make up this line.

Note that the equation of a line can be expressed parametrically, that is, each coordinate of each point is expressed through some independent parameter t.

A typical example is the trajectory of a moving point. In this case, the role of the parameter is played by time.

Equation of a straight line on a plane.

Definition. Any straight line on the plane can be specified by a first-order equation

Ax + Wu + C = 0,

Moreover, the constants A and B are not equal to zero at the same time, i.e. A 2 + B 2 ¹ 0. This first order equation is called general equation of a straight line.

Depending on the values of constants A, B and C, the following special cases are possible:

C = 0, A ¹ 0, B ¹ 0 – the straight line passes through the origin

A = 0, B ¹ 0, C ¹ 0 (By + C = 0) - straight line parallel to the Ox axis

B = 0, A ¹ 0, C ¹ 0 (Ax + C = 0) – straight line parallel to the Oy axis

B = C = 0, A ¹ 0 – the straight line coincides with the Oy axis

A = C = 0, B ¹ 0 – the straight line coincides with the Ox axis

The equation of a straight line can be presented in different forms depending on any given initial conditions.

Equation of a straight line from a point and a normal vector.

Definition. In the Cartesian rectangular coordinate system, a vector with components (A, B) is perpendicular to the straight line given by the equation Ax + By + C = 0.

Example. Find the equation of the line passing through the point A(1, 2) perpendicular to the vector (3, -1).

With A = 3 and B = -1, let’s compose the equation of the straight line: 3x – y + C = 0. To find the coefficient C, we substitute the coordinates of the given point A into the resulting expression.

We get: 3 – 2 + C = 0, therefore C = -1.

Total: the required equation: 3x – y – 1 = 0.

Equation of a line passing through two points.

Let two points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2) be given in space, then the equation of the line passing through these points is:

If any of the denominators is zero, the corresponding numerator should be set equal to zero.

On the plane, the equation of the straight line written above is simplified:

if x 1 ¹ x 2 and x = x 1, if x 1 = x 2.

The fraction = k is called slope straight.

Example. Find the equation of the line passing through points A(1, 2) and B(3, 4).

Applying the formula written above, we get:

Equation of a straight line using a point and slope.

If the general equation of the straight line Ax + By + C = 0 is reduced to the form:

and denote , then the resulting equation is called equation of a straight line with slope k.

Equation of a straight line from a point and a direction vector.

By analogy with the point considering the equation of a straight line through a normal vector, you can enter the definition of a straight line through a point and the directing vector of the straight line.

Definition. Each non-zero vector (a 1 , a 2), the components of which satisfy the condition Aa 1 + Ba 2 = 0 is called a directing vector of the line

Ax + Wu + C = 0.

Example. Find the equation of a straight line with a direction vector (1, -1) and passing through the point A(1, 2).

We will look for the equation of the desired line in the form: Ax + By + C = 0. In accordance with the definition, the coefficients must satisfy the conditions.

Target: Consider the concept of a line on a plane, give examples. Based on the definition of a line, introduce the concept of an equation of a line on a plane. Consider the types of straight lines, give examples and methods of defining a straight line. Strengthen the ability to translate the equation of a straight line from a general form into an equation of a straight line “in segments”, with an angular coefficient.

Equation of a line on a plane.
Equation of a straight line on a plane. Types of equations.
Methods for specifying a straight line.

1. Let x and y be two arbitrary variables.

Definition: A relation of the form F(x,y)=0 is called equation , if it is not true for any pairs of numbers x and y.

Example: 2x + 7y – 1 = 0, x 2 + y 2 – 25 = 0.

If the equality F(x,y)=0 holds for any x, y, then, therefore, F(x,y) = 0 is an identity.

Example: (x + y) 2 - x 2 - 2xy - y 2 = 0

They say that the numbers x are 0 and y are 0 satisfy the equation , if when substituting them into this equation it turns into a true equality.

The most important concept of analytical geometry is the concept of the equation of a line.

Definition: The equation of a given line is the equation F(x,y)=0, which is satisfied by the coordinates of all points lying on this line, and not satisfied by the coordinates of any of the points not lying on this line.

The line defined by the equation y = f(x) is called the graph of f(x). The variables x and y are called current coordinates, because they are the coordinates of a variable point.

Some examples line definitions.

1) x – y = 0 => x = y. This equation defines a straight line:

2) x 2 - y 2 = 0 => (x-y)(x+y) = 0 => points must satisfy either the equation x - y = 0, or the equation x + y = 0, which corresponds on the plane to a pair of intersecting straight lines that are bisectors of coordinate angles:

3) x 2 + y 2 = 0. This equation is satisfied by only one point O(0,0).

2. Definition: Any straight line on the plane can be specified by a first-order equation