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: .
"How to find direction cosines of a vector"
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.
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 direction 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.
DEFINITION
Vector is called an ordered pair of points and (that is, it is known exactly which of the points in this pair is the first).
The first point is called beginning of the vector, and the second is his the end.
The distance between the beginning and end of a vector is called length or vector module.
A vector whose beginning and end coincide is called zero and is denoted by ; its length is considered to be zero. Otherwise, if the length of the vector is positive, then it is called non-zero.
Comment. If the length of a vector is equal to one, then it is called ortom or unit vector and is designated .
EXAMPLE
| Exercise | Check if a vector is  single. |
| Solution | Let's calculate the length of a given vector, it is equal to the square root of the sum of the squares of the coordinates: Since the length of the vector is equal to one, it means that the vector is an orth. |
| Answer | Unit vector. |
A nonzero vector can also be defined as a directed segment.
Comment. The direction of the zero vector is not defined.
Direction cosines of a vector
DEFINITION
Direction cosines of a certain vector are called the cosines of the angles that the vector forms with the positive directions of the coordinate axes.
Comment. The direction of a vector is uniquely determined by its direction cosines.
To find the direction cosines of a vector, it is necessary to normalize the vector (that is, divide the vector by its length):
Comment. The coordinates of a unit vector are equal to its direction cosines.
THEOREM
(Property of direction cosines). The sum of the squares of the direction cosines is equal to one:
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 this relation. 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. Using the equation of a straight line with an angle coefficient, you can specify any straight line that is 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 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
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 are equal to the lengths of the segments that the straight line cuts off at 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. An equation and a 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 bring it to normal form. 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.
The normal equation of a plane is 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 angles: two obtuse and two acute or four right angles, and both obtuse angles are equal to each other, and both acute angles 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 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 to 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 equations of a straight line
The equation 
called general equation straight on surface.
Any first degree equation with two variables x And y type , 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 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 straight 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.
In addition, if all elements of an unbounded sequence, starting from a certain number, have a positive sign, then the limit of such a sequence is said to be 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 positive number such that for all arguments satisfying the condition the inequality is satisfied.
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. an infinitesimal 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 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
