The slope of the straight line in the figure how to find. The equation of a straight line with a slope: theory, examples, problem solving. equation of a line passing through two given points

Numerically equal to the tangent of the angle (constituting the smallest rotation from the Ox axis to the Oy axis) between the positive direction of the x-axis and the given straight line.

The tangent of an angle can be calculated as the ratio of the opposite leg to the adjacent one. k is always equal to , that is, the derivative of the straight line equation with respect to x.

With positive values ​​of the angular coefficient k and zero value of the shift coefficient b line will lie in the first and third quadrants (in which x and y both positive and negative). At the same time, large values ​​of the angular coefficient k a steeper straight line will correspond, and a smaller one - a flatter one.

Lines and are perpendicular if , and parallel when .

Notes

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See what the "Line Slope" is in other dictionaries:

slope (straight)- — Topics oil and gas industry EN slope … Technical Translator's Handbook

- (mathematical) number k in the equation of a straight line on the plane y \u003d kx + b (see Analytical geometry), characterizing the slope of the straight line relative to the abscissa axis. In a rectangular coordinate system U. to. k \u003d tg φ, where φ is the angle between ... ... Great Soviet Encyclopedia

A branch of geometry that studies the simplest geometric objects by means of elementary algebra based on the method of coordinates. The creation of analytical geometry is usually attributed to R. Descartes, who outlined its foundations in the last chapter of his ... ... Collier Encyclopedia

The measurement of reaction time (RT) is probably the most revered subject in empirical psychology. It originated in the field of astronomy, in 1823, with the measurement of individual differences in the speed at which a star was perceived to cross the telescope's line of sight. These … Psychological Encyclopedia

A branch of mathematics that gives methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and ... Collier Encyclopedia

This term has other meanings, see Direct (meanings). A straight line is one of the basic concepts of geometry, that is, it does not have an exact universal definition. In a systematic presentation of geometry, a straight line is usually taken as one ... ... Wikipedia

Representation of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Representation of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Not to be confused with the term "Ellipsis". Ellipse and its foci Ellipse (other Greek ἔλλειψις disadvantage, in the sense of lack of eccentricity up to 1) the locus of points M of the Euclidean plane, for which the sum of the distances from two given points F1 ... ... Wikipedia

Tasks for finding the derivative of the tangent are included in the exam in mathematics and are met there annually. At the same time, the statistics of recent years show that such tasks cause certain difficulties for graduates. Therefore, if a student expects to get decent scores based on the results of passing the exam, then he should definitely learn how to cope with the tasks from the section “Angle factor of a tangent as the value of the derivative at the point of contact”, prepared by the specialists of the Shkolkovo educational portal. Having dealt with the algorithm for solving them, the student will be able to successfully overcome the certification test.

Basic moments

Starting to solve USE problems on this topic, it is necessary to recall the basic definition: the derivative of a function at a point is equal to the slope of the tangent to the graph of the function at this point. This is the geometric meaning of the derivative.

Another important definition needs to be refreshed. It sounds like this: the slope equals the tangent of the angle of inclination of the tangent to the x-axis.

What other important points should be noted in this topic? When solving problems for finding the derivative in the USE, it must be remembered that the angle that the tangent forms can be less, more than 90 degrees, or equal to zero.

How to prepare for the exam?

In order for the tasks in the USE on the topic “The slope of the tangent as the value of the derivative at the point of contact” to be given to you quite easily, use the information on this section on the Shkolkovo educational portal when preparing for the final test. Here you will find the necessary theoretical material, collected and clearly presented by our experts, and you will also be able to practice the exercises.

For each task, for example, tasks on the topic "The angular coefficient of the tangent as the tangent of the angle of inclination", we wrote down the correct answer and the solution algorithm. At the same time, students can perform exercises of various levels of complexity online. If necessary, the task can be saved in the "Favorites" section in order to discuss its solution with the teacher later.

Numerically equal to the tangent of the angle (constituting the smallest rotation from the Ox axis to the Oy axis) between the positive direction of the x-axis and the given straight line.

The tangent of an angle can be calculated as the ratio of the opposite leg to the adjacent one. k is always equal to , that is, the derivative of the straight line equation with respect to x.

With positive values ​​of the angular coefficient k and zero value of the shift coefficient b line will lie in the first and third quadrants (in which x and y both positive and negative). At the same time, large values ​​of the angular coefficient k a steeper straight line will correspond, and a smaller one - a flatter one.

Lines and are perpendicular if , and parallel when .

Notes

Wikimedia Foundation. 2010 .

• Ifit (king of Elis)
• List of Decrees of the President of the Russian Federation "On awarding state awards" for 2001

See what the "Line Slope" is in other dictionaries:

slope (straight)- — Topics oil and gas industry EN slope … Technical Translator's Handbook

Slope- (mathematical) number k in the equation of a straight line on the plane y \u003d kx + b (see Analytical geometry), characterizing the slope of the straight line relative to the abscissa axis. In a rectangular coordinate system U. to. k \u003d tg φ, where φ is the angle between ... ... Great Soviet Encyclopedia

Line equations

ANALYTIC GEOMETRY- a branch of geometry that explores the simplest geometric objects by means of elementary algebra based on the method of coordinates. The creation of analytical geometry is usually attributed to R. Descartes, who outlined its foundations in the last chapter of his ... ... Collier Encyclopedia

Reaction time- The measurement of reaction time (RT) is probably the most venerable subject in empirical psychology. It originated in the field of astronomy, in 1823, with the measurement of individual differences in the speed at which a star was perceived to cross the telescope's line of sight. These … Psychological Encyclopedia

MATHEMATICAL ANALYSIS- a section of mathematics that provides methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and ... Collier Encyclopedia

Straight- This term has other meanings, see Direct (meanings). A straight line is one of the basic concepts of geometry, that is, it does not have an exact universal definition. In a systematic presentation of geometry, a straight line is usually taken as one ... ... Wikipedia

Straight line- Image of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Direct- Image of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Minor axis- Not to be confused with the term "Ellipsis". Ellipse and its foci Ellipse (other Greek ἔλλειψις disadvantage, in the sense of lack of eccentricity up to 1) the locus of points M of the Euclidean plane, for which the sum of the distances from two given points F1 ... ... Wikipedia

In Cartesian coordinates, every straight line is defined by a first degree equation and, conversely, every first degree equation defines a straight line.

Type equation

is called the general equation of a straight line.

The angle defined as shown in Fig. is called the angle of inclination of the straight line to the x-axis. The tangent of the angle of inclination of the straight line to the x-axis is called the slope of the straight line; it is usually denoted by the letter k:

The equation is called the equation of a straight line with a slope; k is the slope, b is the value of the segment that the straight line cuts off on the Oy axis, counting from the origin.

If the straight line is given by the general equation

,

then its slope is determined by the formula

The equation is the equation of a straight line that passes through the point (, ) and has a slope k.

If the line passes through the points (, ), (, ), then its slope is determined by the formula

The equation

is the equation of a straight line passing through two points (, ) and (, ).

If the slope coefficients of two straight lines are known, then one of the angles between these straight lines is determined by the formula

.

A sign of parallelism of two lines is the equality of their angular coefficients:.

A sign of perpendicularity of two lines is the ratio , or .

In other words, the slopes of perpendicular lines are reciprocal in absolute value and opposite in sign.

4. General equation of a straight line

The equation

Ah+Wu+C=0

(where A, B, C can have any values, as long as the coefficients A, B were not zero both at once) represents straight line. Any straight line can be represented by an equation of this type. Therefore it is called the general equation of a straight line.

If a BUTX, then it represents a line, parallel to the x-axis.

If a AT=0, that is, the equation does not contain at, then it represents a line, parallel to the OY axis.

Kogla AT is not equal to zero, then the general equation of a straight line can be resolve relative to ordinateat , then it is converted to the form

(where a=-A/B; b=-C/B).

Similarly, when BUT different from zero, the general equation of a straight line can be solved with respect to X.

If a FROM=0, that is, the general equation of a straight line does not contain a free term, then it represents a straight line passing through the origin

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

Equation of a line passing through a given point A(x 1 , y 1) in a given direction, determined by the slope k,

y - y 1 = k(x - x 1). (1)

This equation defines a pencil of lines passing through a point A(x 1 , y 1), which is called the center of the beam.

6. equation of a straight line passing through two given points.

. Equation of a straight line passing through two points: A(x 1 , y 1) and B(x 2 , y 2) is written like this:

The slope of a straight line passing through two given points is determined by the formula

7. Equation of a straight line in segments

If in the general equation of the line , then dividing (1) by , we obtain the equation of the line in the segments

where , . The line intersects the axis at the point , the axis at the point .

8. Formula: Angle between lines on a plane

At Goal α between two straight lines given by the equations: y=k 1 x+b 1 (first line) and y=k 2 x+b 2 (second line), can be calculated by the formula (the angle is measured from the 1st line to the 2nd counterclock-wise ):

9. Mutual arrangement of two straight lines on a plane.

Let both now equations straight lines are written in general form.

Theorem. Let

- general equations two straight lines coordinate Oxy plane. Then

1) if , then straight and match;

2) if , then the lines and

parallel;

3) if , then straight intersect.

Proof. The condition is equivalent to the collinearity of normal vectors direct data:

Therefore, if , then straight intersect.

If , then , , and the equation straight takes the form:

Or , i.e. straight match. Note that the coefficient of proportionality , otherwise all the coefficients of the total equations would be zero, which is impossible.

If straight do not coincide and do not intersect, then the case remains, i.e. straight are parallel.

The theorem has been proven.

Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight line or a curved line. That is, the derivative characterizes the rate of change of the function at a particular point in time. Remember the general rules by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described there.

Learn to distinguish between problems in which the slope needs to be calculated in terms of the derivative of a function. In tasks, it is not always suggested to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x, y). You may also be asked to find the slope of the tangent at point A(x, y). In both cases, it is necessary to take the derivative of the function.

Take the derivative of the given function. You don't need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x). Take the derivative according to the methods outlined in the article mentioned above:

Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of the function is equal to the slope at a certain point. In other words, f "(x) is the slope of the function at any point (x, f (x)). In our example: