Direction cosines definition. Direction cosines. B) geometric meaning of the derivative

Let the vector be given. Unit vector in the same direction as (unit vector ) is found by the formula:

.

Let the axis forms angles with the coordinate axes
.Direction cosines of the axis The cosines of these angles are called:. If the direction given by a unit vector , then the direction cosines serve as its coordinates, i.e.:

.

The direction cosines are related to each other by the relation:

If the direction given by an arbitrary vector , then find the unit vector of this vector and, comparing it with the expression for the unit vector , get:

Scalar product

Dot product
two vectors And is a number equal to the product of their lengths and the cosine of the angle between them:
.

The scalar product has the following properties:

Hence,
.

Geometric meaning dot product : scalar product of a vector and a unit vector equal to the projection of the vector to the direction determined , i.e.
.

The following table of multiplication of unit vectors follows from the definition of the scalar product:
:

.

If vectors are given by their coordinates
And
, i.e.
,
, then, multiplying these vectors scalarly and using the multiplication table of unit vectors, we obtain the expression for the scalar product
through vector coordinates:

.

Vector artwork

Cross product of a vectorto vector called a vector , the length and direction of which are determined by the conditions:

The vector product has the following properties:

From the first three properties it follows that the vector multiplication of a sum of vectors by a sum of vectors obeys the usual rules for multiplying polynomials. You just need to make sure that the order of the factors does not change.

The basic vectors are multiplied as follows:

If
And
, then taking into account the properties of the vector product of vectors, we can derive a rule for calculating the coordinates of the vector product from the coordinates of the factor vectors:

If we take into account the above rules for multiplying unit vectors, then:

A more compact form of writing an expression for calculating the coordinates of the vector product of two vectors can be constructed by introducing the concept of a determinant of a matrix.

Let us consider the special case when the vectors And belong to the plane
, i.e. they can be represented as
And
.

If the coordinates of the vectors are written in table form as follows:
, then we can say that a square matrix of the second order is formed from them, i.e. size
, consisting of two rows and two columns. Each square matrix a number is assigned that is calculated from the elements of the matrix according to certain rules and is called a determinant. The determinant of a second-order matrix is ​​equal to the difference between the products of the elements of the main diagonal and the secondary diagonal:

.

In this case:

The absolute value of the determinant is thus equal to the area of ​​the parallelogram constructed on the vectors And , both on the sides.

If we compare this expression with the vector product formula (4.7), then:

This expression is a formula for calculating the determinant of a third-order matrix from the first row.

Thus:

Determinant of a third-order matrix is calculated as follows:

and is the algebraic sum of six terms.

The formula for calculating the determinant of a third-order matrix is ​​easy to remember if you use ruleSarrus, which is formulated as follows:

Each term is the product of three elements located in different columns and different rows of the matrix;

The products of elements forming triangles with a side parallel to the main diagonal have a plus sign;

The products of elements belonging to the secondary diagonal and two have a minus sign. products of elements, forming triangles with a side parallel to the side diagonal.

The sum of the squares of the direction cosines is equal to one.

If the direction cosines of the vector are known, then its coordinates can be found using the formulas: Similar formulas apply in the three-dimensional case - if the direction cosines of the vector are known, then its coordinates can be found using the formulas:

9 Linear dependence And linear independence vectors. Basis on the plane and in space

A set of vectors is called system of vectors.

linearly dependent, if there are numbers that are not all equal to zero at the same time, that

A system of vectors is called linearly independent, if equality is possible only for , i.e. When linear combination on the left side of the equality is trivial.

1. One vector also forms a system: at - linearly dependent, and at - linearly independent.

2. Any part of a system of vectors is called subsystem.

1. If a system of vectors includes a zero vector, then it is linearly dependent

2. If a system of vectors has two equal vectors, then it is linearly dependent.

3. If a system of vectors has two proportional vectors, then it is linearly dependent.

4. A system of vectors is linearly dependent if and only if at least one of the vectors is a linear combination of the others.

5. Any vectors included in a linearly independent system form a linearly independent subsystem.

6. A system of vectors containing a linearly dependent subsystem is linearly dependent.

7. If a system of vectors is linearly independent, and after adding a vector to it it turns out to be linearly dependent, then the vector can be expanded into vectors and, moreover, in a unique way, i.e. the expansion coefficients can be found uniquely.

Basis on a plane and in space is called a maximal system of vectors that is linearly independent on a plane or in space (adding another vector to the system makes it linearly dependent).

Thus, a basis on a plane is any two non-collinear vectors taken in a certain order, and a basis in space is any three non-coplanar vectors taken in a certain order.

Let be a basis in space, then, according to T. 3, any vector of space can be decomposed in a unique way into basis vectors: . The expansion coefficients are called the coordinates of the vector in the basis

Writing linear operations on vectors through coordinates:

a) addition and subtraction: - basis

b) multiplication by the number R:

The formulas follow from the properties of linear operations.

10 Coordinates of the vector relative to the basis. Orty

Basis in free vector space V 3 is any ordered triple of non-coplanar vectors.

Let IN :a 1,a 2,a 3– fixed basis in V 3.

Coordinates vector b relative to the basis IN called an ordered triple of numbers ( x, y, z), incl. b=x· a 1 +ya 2 +z· a 3.

Designation:b={x, y, z} B Note: The coordinates of a fixed vector mean the coordinates of the corresponding free vector.

Theorem1: The correspondence between V 3 and R 3 for a fixed basis is one-to-one, i.e. b V 3 ! {x, y, z) R 3 and ( x, y, z) R 3 ! b V 3, incl. b={x, y, z} B

The correspondence between a vector and its coordinates in a given basis has the following properties:

1. Let b 1 ={x 1 , y 1 , z 1} B , b 2 ={x 2 , y 2 , z 2} B b 1 + b 2 ={x 1 + x 2 , y 1 + y 2 , z 1 + z 2} B

2. Let b={x, y, z} B , λR λ b={ λ· x, λ· y, λ· z} B

3. Let b 1 || b 2 , b 1 = {x 1 , y 1 , z 1} B , b 2 ={x 2 , y 2 , z 2} B
(Here: any number).

Unit vector, directed along the X axis, is denoted i, unit vector, directed along the Y axis, is denoted j, A unit vector, directed along the Z axis, is denoted k. Vectors i, j, k are called orts– they have single modules, that is
i = 1, j = 1, k = 1

11 scalar product of vectors. Angle between vectors. Condition for vector orthogonality

This is a number equal to the product of the lengths of these vectors and the cosine of the angle between them.

Dot product of vectors in terms of their coordinates

Dot product of vectors X, Y, Z and :

where is the angle between the vectors and ; if either, then

From the definition of the scalar product it follows that where, for example, is the magnitude of the projection of the vector onto the direction of the vector.

Scalar squared vector:

Properties of the dot product:

Angle between vectors

Conditions for vector orthogonality.

Two vector a and b orthogonal (perpendicular), if their scalar product is equal to zero a· b= 0

So in the case of a plane vector problem

a= (a x ;a y )and b= (b x ;b y )

orthogonal ifa b= a x b x + a y b y = 0

12 vector product of vectors, its properties. Condition for collinearity of vectors

The cross product of a vector and a vector is a vector denoted by a symbol and defined by the following three conditions:

1). The modulus of the vector is equal to , where is the angle between the vectors and ;

2). The vector is perpendicular to each of the vector and ;

3). The direction of the vector corresponds to the “right hand rule”. This means that if the vectors , and are brought to a common origin, then the vector should be directed in the same way as the middle finger of the right hand, the thumb of which is directed along the first factor (that is, along the vector), and the index finger - along the second (that is, along vector). The vector product depends on the order of the factors, namely: .

Cross Product Module equal to area S of a parallelogram built on the vectors and : .

The vector product itself can be expressed by the formula,

where is the unit vector of the vector product.

The cross product vanishes if and only if the vectors and are collinear. In particular, .

If the coordinate axes system is right and the vectors and are specified in this system by their coordinates:

then the vector product of a vector and a vector is determined by the formula

A vector is collinear to a nonzero vector if and only if the coordinates

vectors are proportional to the corresponding coordinates of the vector, i.e.

Linear operations on vectors specified by their coordinates in space are performed in a similar way.

13 mixed product of vectors. Its properties. Condition for coplanarity of vectors

Mixed product of three vectors, , is a number equal to the scalar product of a vector and a vector:

Properties of a mixed product:

3° Three vectors are coplanar if and only if

4° A triple of vectors is right if and only if . If , then the vectors , and form the left triplet of vectors.

10° Jacobi identity:

If the vectors , and are given by their coordinates, then their mixed product is calculated using the formula

Vectors parallel to one plane or lying on the same plane are called coplanar vectors.

Conditions for coplanarity of vectors

Three vectors are coplanar if their mixed product is zero.

Three vectors are coplanar if they are linearly dependent.

15 different types of line and plane equations

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. This first order equation is called general equation straight. Depending on the values constant A, B and C the following special cases are possible:

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

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

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

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

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

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

Direction cosines of a vector.

Direction cosines of vector a are the cosines of the angles that the vector forms with the positive semi-axes of coordinates.

To find the direction cosines of vector a, it is necessary to divide the corresponding coordinates of the vector by the absolute value of the vector.

Property: The sum of the squares of the direction cosines is equal to one.

So in the case of a plane problem direction cosines of the vector a = (ax; ay) are found by the formulas:

An example of calculating direction cosines of a vector:

Find the direction cosines of the vector a = (3; 4).

Solution: |a| =

So in case of a spatial problem direction cosines of the vector a = (ax; ay; az) are found by the formulas:

An example of calculating direction cosines of a vector

Find the direction cosines of the vector a = (2; 4; 4).

Solution: |a| =

The direction of the vector in space is determined by the angles that the vector forms with the coordinate axes (Fig. 12). The cosines of these angles are called direction cosines of the vector: , , . From the properties of projections:, , . Hence, It is easy to show that

2) the coordinates of any unit vector coincide with its direction cosines: .

Denote by alpha, beta and gamma the angles formed by vector a with the positive direction of the coordinate axes (see Fig. 1). The cosines of these angles are called the direction cosines of the vector a.

Since the coordinates a in the Cartesian rectangular coordinate system are equal to the projections of the vector onto the coordinate axes, then a1 = |a|cos(alpha), a2 = |a|cos(beta), a3 = |a|cos(gamma). Hence: cos (alpha)=a1||a|, cos(beta) =a2||a|, cos(gamma)= a3/|a|. In this case |a|=sqrt(a1^2+ a2^2+ a3^2). So cos (alpha)=a1|sqrt(a1^2+ a2^2+ a3^2), cos(beta) =a2|sqrt(a1^2+ a2^2+ a3^2), cos(gamma)= a3/sqrt(a1^2+ a2^2+ a3^2).

It should be noted the main property of direction cosines. The sum of the squares of the direction cosines of a vector is equal to one. Indeed, cos^2(alpha)+cos^2(beta)+cos^2(gamma)= = a1^2|(a1^2+ a2^2+ a3^2)+ a2^2|(a1^2 + a2^2+ a3^2)+ a3^2/(a1^2+ a2^2+ a3^2) = =(a1^2+ a2^2+ a3^2)|(a1^2+ a2^ 2+ a3^2) = 1.

First way

Example: given: vector a=(1, 3, 5). Find its direction cosines. Solution. In accordance with what we found, we write out: |a|= sqrt(ax^2+ ay^2+ az^2)=sqrt(1+9 +25)=sqrt(35)=5.91. Thus, the answer can be written in the following form: (cos(alpha), cos(beta), cos(gamma))=(1/sqrt(35), 3/sqrt(35), 5/(35))=( 0.16;0.5;0.84).

Second way

When finding the direction cosines of vector a, you can use the technique of determining the cosines of angles using the scalar product. In this case, we mean the angles between a and the directing unit vectors of rectangular Cartesian coordinates i, j and k. Their coordinates are (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively. It should be recalled that the scalar product of vectors is defined as follows.

If the angle between the vectors is φ, then the scalar product of two winds (by definition) is a number equal to the product of the moduli of the vectors and cosφ. (a, b) = |a||b|cos f. Then, if b=i, then (a, i) = |a||i|cos(alpha), or a1 = |a|cos(alpha). Further, all actions are performed similarly to method 1, taking into account coordinates j and k.

these are the cosines of the angles that the vector forms with the positive semi-axes of coordinates. Direction cosines uniquely specify the direction of the vector. If a vector has length 1, then its direction cosines are equal to its coordinates. In general, for a vector with coordinates ( a; b; c) direction cosines are equal:

where a, b, g are the angles made by the vector with the axes x, y, z respectively.

21) Decomposition of a vector in unit vectors. The unit vector of the coordinate axis is denoted by , the axes by , and the axes by (Fig. 1).

For any vector that lies in the plane, the following expansion takes place:

If the vector located in space, then the expansion in unit vectors of the coordinate axes has the form:

22)Dot product two non-zero vectors and the number equal to the product of the lengths of these vectors and the cosine of the angle between them is called:

23)Angle between two vectors

If the angle between two vectors is acute, then their scalar product is positive; if the angle between the vectors is obtuse, then the scalar product of these vectors is negative. The scalar product of two nonzero vectors is equal to zero if and only if these vectors are orthogonal.

24) The condition of parallelism and perpendicularity of two vectors.

Condition for vectors to be perpendicular
Vectors are perpendicular if and only if their scalar product is zero. Given two vectors a(xa;ya) and b(xb;yb). These vectors will be perpendicular if the expression xaxb + yayb = 0.

25) Vector product of two vectors.

The vector product of two non-collinear vectors is a vector c=a×b that satisfies the following conditions: 1) |c|=|a| |b| sin(a^b) 2) c⊥a, c⊥b 3) Vectors a, b, c form a right-hand triplet of vectors.

26) Collinear and coplanar vectors..

Vectors are collinear if the abscissa of the first vector is related to the abscissa of the second in the same way as the ordinate of the first is to the ordinate of the second. Given two vectors a (xa;ya) And b (xb;yb). These vectors are collinear if xa = x b And y a = y b, Where R.

Vectors −→ a,−→b and −→ c are called coplanar, if there is a plane to which they are parallel.

27) Mixed product of three vectors. Mixed product of vectors- scalar product of vector a and the vector product of vectors b and c. Find the mixed product of vectors a = (1; 2; 3), b = (1; 1; 1), c = (1; 2; 1).

Solution:

1·1·1 + 1·1·2 + 1·2·3 - 1·1·3 - 1·1·2 - 1·1·2 = 1 + 2 + 6 - 3 - 2 - 2 = 2

28) The distance between two points on a plane. The distance between two given points is equal to the square root of the sum of the squared differences of the same coordinates of these points.

29) Division of a segment in in this regard. If point M(x; y) lies on a line passing through two given points ( , ) and ( , ), and a relation is given in which point M divides the segment , then the coordinates of point M are determined by the formulas

If point M is the midpoint of the segment, then its coordinates are determined by the formulas

30-31. Slope of a straight line is called the tangent of the angle of inclination of this line. The slope of a straight line is usually denoted by the letter k. Then by definition

Equation of a straight line with slope has the form where k- straight line slope, b– some real number. The equation of a straight line with an angular coefficient can be used to define any straight line, not parallel to the axis Oy(for a straight line parallel to the ordinate axis, the angular coefficient is not defined).

33. General equation of a straight line on a plane. Equation of the form There is general equation of a line Oxy. Depending on the values ​​of constants A, B and C, the following special cases are possible:

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

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

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

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

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

34.Equation of a line in segments on a plane in a rectangular coordinate system Oxy has the form where a And b- some non-zero real numbers. This name is not accidental, since absolute values numbers A And b equal to the lengths of the segments that the straight line cuts off on the coordinate axes Ox And Oy respectively (segments are counted from the origin). Thus, the equation of a line in segments makes it easy to construct this line in a drawing. To do this, you should mark the points with coordinates and in a rectangular coordinate system on the plane, and use a ruler to connect them with a straight line.

35. The normal equation of a line has the form

where is the distance from the straight line to the origin;  – the angle between the normal to the line and the axis.

The normal equation can be obtained from the general equation (1) by multiplying it by the normalizing factor, the sign  is opposite to the sign so that .

The cosines of the angles between the straight line and the coordinate axes are called direction cosines,  – the angle between the straight line and the axis,  – between the straight line and the axis:

Thus, the normal equation can be written in the form

Distance from point to a straight line determined by the formula

36. The distance between a point and a line is calculated using the following formula:

where x 0 and y 0 are the coordinates of the point, and A, B and C are coefficients from the general equation of the line

37. Reducing the general equation of a line to normal. The equation and the plane in this context do not differ from each other in anything other than the number of terms in the equations and the dimension of space. Therefore, first I will say everything about the plane, and at the end I will make a reservation about the straight line.
Let the general equation of the plane be given: Ax + By + Cz + D = 0.
;. we get the system: g;Mc=cosb, MB=cosa Let's reduce it to normal looking. To do this, we multiply both sides of the equation by the normalizing factor M. We get: Max+Mvu+MCz+MD=0. In this case MA=cos;.g;Mc=cosb, MB=cosa we obtain the system:

M2 B2=cos2b
M2 C2=cos2g

Adding up all the equations of the system, we get M*(A2 +B2+C2)=1 Now all that remains is to express M from here in order to know by which normalizing factor the original general equation must be multiplied to bring it to normal form:
M=-+1/ROOT KV A2 +B2 +C2
MD must always be less than zero, therefore the sign of the number M is taken opposite to the sign of the number D.
With the equation of a straight line, everything is the same, only from the formula for M you should simply remove the term C2.

Ax + By + Cz + D = 0,

38.General equation of the plane in space is called an equation of the form

Where A 2 + B 2 + C 2 ≠ 0 .

IN three-dimensional space in the Cartesian coordinate system, any plane is described by an equation of the 1st degree (linear equation). And back, any linear equation defines a plane.

40.Equation of a plane in segments. In a rectangular coordinate system Oxyz in three-dimensional space an equation of the form , Where a, b And c– non-zero real numbers are called equation of the plane in segments. Absolute values ​​of numbers a, b And c equal to the lengths of the segments that the plane cuts off on the coordinate axes Ox, Oy And Oz respectively, counting from the origin. Sign of numbers a, b And c shows in which direction (positive or negative) the segments are plotted on the coordinate axes

41) Normal plane equation.

Normal equation plane is called its equation, written in the form

where , , are the direction cosines of the plane normal, e

p is the distance from the origin to the plane. When calculating the direction cosines of the normal, it should be assumed that it is directed from the origin to the plane (if the plane passes through the origin, then the choice of the positive direction of the normal is indifferent).

42) Distance from a point to a plane.Let the plane be given by the equation and a point is given. Then the distance from the point to the plane is determined by the formula

Proof. The distance from a point to a plane is, by definition, the length of the perpendicular drawn from the point to the plane

Angle between planes

Let the planes and be specified by the equations and , respectively. You need to find the angle between these planes.

The planes intersecting form four dihedral angle: two obtuse and two acute or four straight, both obtuse angles are equal to each other, and both acute ones are also equal to each other. We will always search sharp corner. To determine its value, we take a point on the line of intersection of the planes and at this point in each of

planes, we draw perpendiculars to the intersection line.

Property:

cos 2 α + cos 2 β + cos 2 γ = 1

b) definition of linear operations

the sum of two non-collinear vectors is the vector coming from the common origin of vectors along the diagonal of a parallelogram constructed on these vectors

The vector difference is the sum of a vector and a vector opposite to the vector: . Let's connect the beginnings of the vectors and , then the vector is directed from the end of the vector to the end of the vector.

The work a vector by a number is called a vector with modulus , and at and at . Geometrically, multiplication by a number means “stretching” the vector by a factor, maintaining the direction at and changing to the opposite at .

From the above rules for adding vectors and multiplying them by a number, obvious statements follow:

1. (addition is commutative);

2. (addition is associative);

3. (existence of a zero vector);

4. (existence of an opposite vector);

5. (addition is associative);

6. (multiplication by a number is distributive);

7. (vector addition is distributive);

c) scalar product and its basic properties

Dot product two non-zero vectors is a number equal to the product of the lengths of these vectors and the cosine of the angle between them. If at least one of the two vectors is zero, then the angle between them is not defined, and the scalar product is considered equal to zero. The scalar product of vectors and is denoted

, where and are the lengths of the vectors and , respectively, and is the angle between the vectors and .

The scalar product of a vector with itself is called a scalar square.

Properties of the scalar product.

For any vectors and the following are true: properties of the dot product:

the commutative property of a scalar product;

distributive property or ;

associative property or , where is an arbitrary real number;

the scalar square of a vector is always non-negative if and only if the vector is zero.

D) vector product and its properties

vector product vector a to vector b is called a vector c, the length of which is numerically equal to the area of ​​the parallelogram constructed on vectors a and b, perpendicular to the plane of these vectors and directed so that the smallest rotation from a to b around vector c is counterclockwise when viewed from the end vector c

Formulas for calculating the vector product of vectors

Vector artwork two vectors a = (a x; a y; a z) and b = (b x; b y; b z) in the Cartesian coordinate system is a vector whose value can be calculated using the following formulas:

• The cross product of two non-zero vectors a and b is equal to zero if and only if the vectors are collinear.
• Vector c equal vector product non-zero vectors a and b, is perpendicular to these vectors.
• a × b = -b × a
• (k a) × b = a × (k b) = k (a × b)
• (a + b) × c = a × c + b × c

Equation of a straight line on a plane

A) equation of a straight line with an angle coefficient

Slope of a straight line is called the tangent of the angle of inclination of this line.

The slope of a straight line is usually denoted by the letter k. Then by definition.

If the straight line is parallel to the ordinate axis, then the slope does not exist (in this case it is also said that the slope goes to infinity).

A positive slope of a line indicates an increase in its graph of the function, a negative slope indicates a decrease. The equation of a straight line with an angular coefficient has the form y=kx+b, where k is the angular coefficient of the line, b is some real number. Using the equation of a straight line with an angular coefficient, you can specify any straight line that is not parallel to the Oy axis (for a straight line parallel to the ordinate axis, the angular coefficient is not defined).

B) types of straight line equations

The equation called general equation of the line on surface.

Any first degree equation with two variables x And y kind , Where A, IN And WITH– some real numbers, and A And IN are not equal to zero at the same time, defines a straight line in a rectangular coordinate system Oxy on the plane, and every straight line on the plane is given by an equation of the form .

Line equation of the form , where a And b– some real numbers other than zero are called equation of a straight line in segments. This name is not accidental, since the absolute values ​​of numbers A And b equal to the lengths of the segments that the straight line cuts off on the coordinate axes Ox And Oy respectively (segments are counted from the origin).

Line equation of the form , where x And y- variables, and k And b– some real numbers are called equation of a straight line with slope (k– slope)

Canonical equation of a line on a plane in a rectangular Cartesian coordinate system Oxy looks like , where and are some real numbers, and at the same time they are not equal to zero.

Obviously, the straight line defined by the canonical equation of the line passes through the point. In turn, the numbers and in the denominators of the fractions represent the coordinates of the direction vector of this line. Thus, canonical equation straight in a rectangular coordinate system Oxy on the plane corresponds to a straight line passing through a point and having a direction vector .

Parametric equations of a line on a plane look like , where and are some real numbers, and at the same time are not equal to zero, and is a parameter that takes any real values.

Parametric line equations establish an implicit relationship between the abscissas and ordinates of points on a straight line using a parameter (hence the name of this type of line equation).

A pair of numbers that are calculated from the parametric equations of a line for some real value of the parameter represent the coordinates of a certain point on the line. For example, when we have , that is, the point with coordinates lies on a straight line.

It should be noted that the coefficients and for the parameter in parametric equations line are the coordinates of the direction vector of this line

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 equal to zero, the corresponding numerator should be equal to zero. On the plane, the equation of the line written above is simplified:

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

The fraction = k is called slope straight.

C) calculating the angle between two straight lines

if two lines are given y = k 1 x + b 1, y = k 2 x + b 2, then the acute angle between these lines will be defined as

.

Two lines are parallel if k 1 = k 2. Two lines are perpendicular if k 1 = -1/ k 2.

Theorem. The lines Ax + Bу + C = 0 and A 1 x + B 1 y + C 1 = 0 are parallel when the coefficients A 1 = λA, B 1 = λB are proportional. If also C 1 = λC, then the lines coincide. The coordinates of the point of intersection of two lines are found as a solution to the system of equations of these lines.

D) conditions for parallelism and perpendicularity of two straight lines

Conditions for parallelism of two lines:

a) If the lines are given by equations with an angular coefficient, then the necessary and sufficient condition for their parallelism is the equality of their angular coefficients:

k 1 = k 2 .

b) For the case when the lines are given by equations in general view(6), a necessary and sufficient condition for their parallelism is that the coefficients for the corresponding current coordinates in their equations are proportional, i.e.

Conditions for the perpendicularity of two lines:

a) In the case when the lines are given by equations (4) with an angular coefficient, a necessary and sufficient condition for their perpendicularity is that their angular coefficients are inverse in magnitude and opposite in sign, i.e.

This condition can also be written in the form

k 1 k 2 = -1.

b) If the equations of lines are given in general form (6), then the condition for their perpendicularity (necessary and sufficient) is to satisfy the equality

A 1 A 2 + B 1 B 2 = 0.

Function limit

A) sequence limit

The concept of a limit was used by Newton in the second half of the 17th century and by mathematicians of the 18th century such as Euler and Lagrange, but they understood the limit intuitively. The first rigorous definitions of the sequence limit were given by Bolzano in 1816 and Cauchy in 1821.

The number is called limit number sequence , if the sequence is infinitesimal, i.e. all its elements, starting from a certain one, are less than any predetermined positive number in absolute value.

If a number sequence has a limit in the form of a real number, it is called convergent to this number. Otherwise, the sequence is called divergent . If, moreover, it is unlimited, then its limit is assumed to be equal to infinity.

Moreover, if all elements unlimited sequence, starting from a certain number, have a positive sign, then they say that the limit of such a sequence is equal to plus infinity .

If the elements of an unbounded sequence, starting from a certain number, have a negative sign, then they say that the limit of such a sequence is equal to minus infinity .

B) limit of the function

Function limit (limit value of function) V given point, limiting for the domain of definition of a function, is the value to which the value of the function under consideration tends as its argument tends to a given point.

Function limit is a generalization of the concept of a limit of a sequence: initially, the limit of a function at a point was understood as the limit of a sequence of elements of the domain of values ​​of a function composed of images of points of a sequence of elements of the domain of definition of a function converging to a given point (the limit at which is considered); if such a limit exists, then the function is said to converge to the specified value; if such a limit does not exist, then the function is said to diverge.

Function limit- one of the basic concepts of mathematical analysis. The value is called limit (limit value ) of a function at a point if for any sequence of points converging to but not containing one of its elements (that is, in a punctured neighborhood), the sequence of values ​​of the function converges to .

The value is called limit (limit value) functions at the point if for any positive number taken in advance there is a corresponding one positive number such that for all arguments satisfying the condition the inequality .

C) two remarkable limits

· The first remarkable limit:

Consequences

·

·

·

· The second remarkable limit:

Consequences

1.

2.

3.

4.

5. For ,

6.

D) infinitesimal and infinitely large functions

Function y=f(x) called infinitesimal at x→a or when x→∞, if or , i.e. endlessly small function is a function whose limit at a given point is zero.

if function y=f(x) representable with x→a as a sum of a constant number b and infinitesimal magnitude α(x): f (x)=b+ α(x) That .

Conversely, if , then f (x)=b+α(x), Where a(x)– infinitesimal at x→a.

Corollary 1. If and, then.

Corollary 2. If c= const, then .

If the function f(x) is infinitely large at x→a, then function 1 /f(x) is infinitesimal at x→a.

If the function f(x)- infinitesimal at x→a(or x→∞) and does not vanish, then y= 1/f(x) is infinite great function. the simplest properties of infinitely small and infinitely great functions can be written using the following conditional relations: A≠ 0

D) disclosure of uncertainties. L'Hopital's rule

main types of uncertainties: zero divided by zero ( 0 to 0), infinity divided by infinity, zero multiplied by infinity, infinity minus infinity, one to the power of infinity, zero to the power of zero, infinity to the power of zero.

L'Hopital's rule very widely used for limit calculations when there is an uncertainty of the form zero divided by zero, infinity divided by infinity.

These types of uncertainties include the uncertainties zero times infinity and infinity minus infinity.

If and if functions f(x) And g(x) are differentiable in a neighborhood of the point , then

In the case where uncertainty does not disappear after applying L'Hopital's rule, it can be applied again.

Calculation of derivatives

A) differentiation rule complex function

Let it be complex function , where function is an intermediate argument. We will show how to find the derivative of a complex function, knowing the derivative for the function (we will denote it by) and the derivative for the function.

Theorem 1. If a function has a derivative at a point x, and the function has a derivative at the point (), then the complex function at the point x has a derivative, and = .

Otherwise, the derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument.

B) differentiation of a function specified parametrically

Let the function be given in parametric form, that is, in the form:

where the functions and are defined and continuous over a certain interval of variation of the parameter . Let us find the differentials of the right and left sides of each of the equalities:

To find the second derivative, we perform the following transformations:

B) the concept of logarithmic derivative of a function

The logarithmic derivative of a positive function is called its derivative. Since , then according to the rule of differentiation of a complex function we obtain the following relation for the logarithmic derivative:

.

Using the logarithmic derivative it is convenient to calculate the ordinary derivative in cases where logarithm simplifies the form of the function.

The essence of this differentiation is as follows: first, find the logarithm given function, and only then the derivative from it is calculated. Let some function be given. Let's take logarithms of the left and right sides of this expression:

And then, expressing the desired derivative, the result is:

D) derivative inverse function

If y=f(x) and x=g(y) are a pair of mutually inverse functions, and the function y=f(x) has a derivative f"(x), then the derivative of the inverse function g"(x)=1/f" (x).

Thus, the derivatives of mutually inverse functions are reciprocal quantities. Formula for the derivative of the inverse function:

D) derivative of an implicit function

If a function of one variable is described by the equation y=f(x), where the variable y is on the left side, and the right side depends only on the argument x, then they say that the function is given explicitly. For example, the following functions are specified explicitly:

y=sin x,y=x 2+2x+5,y=lncos x.

In many problems, however, the function can be specified implicitly, i.e. as an equation

F(x,y)=0.

to find the derivative y′( x) an implicitly specified function does not need to be converted into an explicit form. To do this, knowing the equation F(x,y)=0, just do the following:

First you need to differentiate both sides of the equation with respect to the variable x, assuming that y− is a differentiable function x and using the rule for calculating the derivative of a complex function. In this case, the derivative of zero (on the right side) will also be equal to zero.
Comment: If the right side is non-zero, i.e. the implicit equation is

f(x,y)=g(x,y),

then we differentiate the left and right sides of the equation.

Solve the resulting equation for the derivative y′( x).

Concept of derivative

A) definition of derivative

Derivative of a function differentiation integration.

y xx

Definition of derivative

Consider the function f(x x 0. Then the function f(x) is differentiable at the point x 0, and her derivative is determined by the formula

f′( x 0)=limΔ x→0Δ yΔ x=limΔ x→0f(x 0+Δ x)−f(x 0)Δ x.

Derivative of a function is one of the basic concepts of mathematics, and in mathematical analysis the derivative, along with the integral, occupies a central place. The process of finding the derivative is called differentiation. The inverse operation - restoring a function from a known derivative - is called integration.

The derivative of a function at a certain point characterizes the rate of change of the function at that point. An estimate of the rate of change can be obtained by calculating the ratio of the change in function Δ y to a corresponding change in the argument Δ x. In the definition of the derivative, such a relationship is considered in the limit under the condition Δ x→0. Let's move on to a more strict formulation:

Definition of derivative

Consider the function f(x), the domain of which contains some open interval around the point x 0. Then the function f(x) is differentiable at the point x 0, and her derivative is determined by the formula

f′( x 0)=limΔ x→0Δ yΔ x=limΔ x→0f(x 0+Δ x)−f(x 0)Δ x.

B) geometric meaning derivative

The derivative of the function, calculated for a given value, is equal to the tangent of the angle formed by the positive direction of the axis and the positive direction of the tangent drawn to the graph of this function at the point with the abscissa:

If a function has a finite derivative at a point, then in the neighborhood it can be approximated linear function

The function is called tangent to at the point Number.

D) table of derivatives of the simplest elementary functions