To find out the geometric value of the derivative, consider the graph of the function y = f(x). Take an arbitrary point M with coordinates (x, y) and a point N close to it (x + $\Delta $x, y + $\Delta $y). Let us draw the ordinates $\overline(M_(1) M)$ and $\overline(N_(1) N)$, and draw a line parallel to the OX axis from the point M.

The ratio $\frac(\Delta y)(\Delta x) $ is the tangent of the angle $\alpha $1 formed by the secant MN with the positive direction of the OX axis. As $\Delta $x tends to zero, the point N will approach M, and the tangent MT to the curve at the point M will become the limiting position of the secant MN. Thus, the derivative f`(x) is equal to the tangent of the angle $\alpha $ formed by the tangent to curve at the point M (x, y) with a positive direction to the OX axis - the slope of the tangent (Fig. 1).

Figure 1. Graph of a function

When calculating the values ​​using formulas (1), it is important not to make a mistake in the signs, because increment can be negative.

The point N lying on the curve can approach M from any side. So, if in Figure 1, the tangent is given the opposite direction, the angle $\alpha $ will change by $\pi $, which will significantly affect the tangent of the angle and, accordingly, the slope.

Conclusion

It follows that the existence of the derivative is connected with the existence of a tangent to the curve y = f(x), and the slope -- tg $\alpha $ = f`(x) is finite. Therefore, the tangent must not be parallel to the OY axis, otherwise $\alpha $ = $\pi $/2, and the tangent of the angle will be infinite.

At some points, a continuous curve may not have a tangent or have a tangent parallel to the OY axis (Fig. 2). Then the function cannot have a derivative in these values. There can be any number of such points on the function curve.

Figure 2. Exceptional points of the curve

Consider Figure 2. Let $\Delta $x tend to zero from negative or positive values:

\[\Delta x\to -0\begin(array)(cc) () & (\Delta x\to +0) \end(array)\]

If in this case relations (1) have a finite aisle, it is denoted as:

In the first case, the derivative on the left, in the second, the derivative on the right.

The existence of a limit speaks of the equivalence and equality of the left and right derivatives:

If the left and right derivatives are not equal, then at this point there are tangents that are not parallel to OY (point M1, Fig. 2). At points M2, M3, relations (1) tend to infinity.

For N points to the left of M2, $\Delta $x $

To the right of $M_2$, $\Delta $x $>$ 0, but the expression is also f(x + $\Delta $x) -- f(x) $

For point $M_3$ on the left $\Delta $x $$ 0 and f(x + $\Delta $x) -- f(x) $>$ 0, i.e. expressions (1) are both positive on the left and right and tend to +$\infty $ both when $\Delta $x approaches -0 and +0.

The case of the absence of a derivative at specific points of the line (x = c) is shown in Figure 3.

Figure 3. Absence of derivatives

Example 1

Figure 4 shows the graph of the function and the tangent to the graph at the point with the abscissa $x_0$. Find the value of the derivative of the function in the abscissa.

Solution. The derivative at a point is equal to the ratio of the increment of the function to the increment of the argument. Let's choose two points with integer coordinates on the tangent. Let, for example, these be points F (-3.2) and C (-2.4).

Before reading the information on the current page, we advise you to watch a video about the derivative and its geometric meaning

See also an example of calculating the derivative at a point

The tangent to the line l at the point M0 is the straight line M0T - the limit position of the secant M0M, when the point M tends to M0 along this line (i.e., the angle tends to zero) in an arbitrary way.

The derivative of the function y \u003d f (x) at the point x0 called the limit of the ratio of the increment of this function to the increment of the argument when the latter tends to zero. The derivative of the function y \u003d f (x) at the point x0 and textbooks is denoted by the symbol f "(x0). Therefore, by definition

The term "derivative"(and also "second derivative") introduced J. Lagrange(1797), besides, he gave the designations y’, f’(x), f”(x) (1770,1779). The designation dy/dx is first found in Leibniz (1675).

The derivative of the function y \u003d f (x) at x \u003d xo is equal to the slope of the tangent to the graph of this function at the point Mo (ho, f (xo)), i.e.

where a - tangent angle to the x-axis of a rectangular Cartesian coordinate system.

Tangent equation to the line y = f(x) at the point Mo(xo, yo) takes the form

The normal to the curve at some point is the perpendicular to the tangent at the same point. If f(x0) is not equal to 0, then line normal equation y \u003d f (x) at the point Mo (xo, yo) will be written as follows:

The physical meaning of the derivative

If x = f(t) is the law of rectilinear motion of a point, then x’ = f’(t) is the speed of this motion at time t. Flow rate physical, chemical and other processes is expressed using the derivative.

If the ratio dy/dx at x-> x0 has a limit on the right (or on the left), then it is called the derivative on the right (respectively, the derivative on the left). Such limits are called one-sided derivatives..

Obviously, the function f(x) defined in some neighborhood of the point x0 has a derivative f'(x) if and only if the one-sided derivatives exist and are equal to each other.

Geometric interpretation of the derivative as the slope of the tangent to the graph also applies to this case: the tangent in this case is parallel to the Oy axis.

A function that has a derivative at a given point is called differentiable at that point. A function that has a derivative at every point of a given interval is called differentiable in this interval. If the interval is closed, then there are one-sided derivatives at its ends.

The operation of finding the derivative is called.

Lesson Objectives:

Students should know:

• what is called the slope of a straight line;
• the angle between the line and the x-axis;
• what is the geometric meaning of the derivative;
• the equation of the tangent to the graph of the function;
• a method for constructing a tangent to a parabola;
• be able to apply theoretical knowledge in practice.

Lesson objectives:

Educational: to create conditions for students to master the system of knowledge, skills and abilities with the concepts of the mechanical and geometric meaning of the derivative.

Educational: to form a scientific worldview in students.

Developing: to develop students' cognitive interest, creativity, will, memory, speech, attention, imagination, perception.

Methods of organizing educational and cognitive activities:

• visual;
• practical;
• on mental activity: inductive;
• according to the assimilation of the material: partially exploratory, reproductive;
• by degree of independence: laboratory work;
• stimulating: encouragement;
• control: oral frontal survey.

Lesson Plan

1. Oral exercises (find the derivative)
2. Student's report on the topic “The reasons for the appearance of mathematical analysis”.
3. Learning new material
4. Phys. Minute.
5. Problem solving.
6. Laboratory work.
7. Summing up the lesson.
8. Commenting on homework.

Equipment: multimedia projector (presentation), cards ( laboratory work).

During the classes

“A person achieves something only where he believes in himself”

L. Feuerbach

I. Organizational moment.

The organization of the class throughout the lesson, the readiness of students for the lesson, order and discipline.

Setting learning goals for students, both for the entire lesson and for its individual stages.

Determine the significance of the material being studied both in this topic and in the entire course.

Verbal counting

1. Find derivatives:

" , ()" , (4sin x)", (cos2x)", (tg x)", "

2. Logic test.

a) Insert the missing expression.

5x 3 -6x	15x 2 -6	30x
2sinx	2cosx…
cos2x	… …

II. Student's report on the topic “The reasons for the appearance of mathematical analysis”.

The general direction of the development of science is ultimately determined by the requirements of the practice of human activity. The existence of ancient states with a complex hierarchical system of government would have been impossible without a sufficient development of arithmetic and algebra, because the collection of taxes, the organization of army supplies, the construction of palaces and pyramids, the creation of irrigation systems required complex calculations. During the Renaissance, ties between various parts of the medieval world expanded, trade and crafts developed. A rapid rise in the technical level of production begins, new sources of energy are being used industrially, not connected with the muscular efforts of humans or animals. In the XI-XII centuries, fullers and looms appeared, and in the middle of the XV - a printing press. In connection with the need for the rapid development of social production during this period, the essence of the natural sciences, which have been descriptive since antiquity, changes. The goal of natural science is in-depth study natural processes, not objects. The descriptive natural science of antiquity corresponded to mathematics, which operated with constant values. It was necessary to create a mathematical apparatus that would describe not the result of the process, but the nature of its flow and its inherent patterns. As a result, by the end of the 12th century, Newton in England and Leibniz in Germany completed the first stage in the creation of mathematical analysis. What is "mathematical analysis"? How can one characterize and predict the features of any process? Use these features? To penetrate deeper into the essence of this or that phenomenon?

III. Learning new material.

Let's go along the path of Newton and Leibniz and see how we can analyze the process, considering it as a function of time.

Let us introduce some notions that will help us further.

The graph of the linear function y=kx+ b is a straight line, the number k is called the slope of the straight line. k=tg, where is the angle of a straight line, that is, the angle between this straight line and the positive direction of the Ox axis.

Picture 1

Consider the graph of the function y \u003d f (x). Draw a secant through any two points, for example, the secant AM. (Fig.2)

The slope of the secant k=tg. In a right triangle AMC<МАС = (объясните почему?). Тогда tg = = , что с точки зрения физики есть величина средней скорости протекания любого процесса на данном промежутке времени, например, скорости изменения расстояния в механике.

Figure 2

Figure 3

The term “speed” itself characterizes the dependence of a change in one quantity on a change in another, and the latter does not have to be time.

So, the tangent of the slope of the secant tg = .

We are interested in the dependence of the change in values ​​in a shorter period of time. Let us tend the increment of the argument to zero. Then the right side of the formula is the derivative of the function at point A (explain why). If x -> 0, then point M moves along the graph to point A, which means that line AM approaches some line AB, which is tangent to the graph of the function y \u003d f (x) at point A. (Fig.3)

The angle of inclination of the secant tends to the angle of inclination of the tangent.

The geometric meaning of the derivative is that the value of the derivative at a point is equal to the slope of the tangent to the graph of the function at the point.

The mechanical meaning of the derivative.

The tangent of the slope of the tangent is a value that shows the instantaneous rate of change of the function at a given point, that is, a new characteristic of the process under study. Leibniz called this quantity derivative, and Newton said that the instantaneous speed.

IV. Fizkultminutka.

V. Problem solving.

No. 91(1) page 91 - show on the board.

The slope of the tangent to the curve f (x) \u003d x 3 at the point x 0 - 1 is the value of the derivative of this function at x \u003d 1. f '(1) \u003d 3x 2; f'(1) = 3.

No. 91 (3.5) - under dictation.

No. 92 (1) - on the board at will.

No. 92 (3) - independently with oral verification.

No. 92 (5) - at the board.

Answers: 45 0, 135 0, 1.5 e 2.

VI. Laboratory work.

Purpose: development of the concept of “mechanical meaning of the derivative”.

Applications of the derivative to mechanics.

The law of rectilinear motion of a point x = x(t), t is given.

1. The average speed of movement in the specified period of time;
2. Velocity and acceleration at time t 04
3. stopping points; whether the point continues to move in the same direction after the moment of stopping or starts moving in the opposite direction;
4. The highest speed of movement for a specified period of time.

The work is performed according to 12 options, the tasks are differentiated by the level of complexity (the first option is the lowest level of complexity).

Before starting work, a conversation on the following questions:

1. What physical meaning displacement derivative? (Speed).
2. Can you find the derivative of speed? Is this quantity used in physics? What is it called? (Acceleration).
3. Instant Speed equals zero. What can be said about the movement of the body at this moment? (This is the stopping point).
4. What is the physical meaning of the following statements: the derivative of motion is equal to zero at the point t 0; does the derivative change sign when passing through the point t 0? (The body stops; the direction of movement changes to the opposite).

Sample work for students.

x (t) \u003d t 3 -2 t 2 +1, t 0 \u003d 2.

Figure 4

In the opposite direction.

Let's draw a schematic speed graph. The highest speed is reached at the point

t=10, v (10) =3 10 2 -4 10 =300-40=260

Figure 5

VII. Summing up the lesson

1) What is the geometric meaning of the derivative?
2) What is the mechanical meaning of the derivative?
3) Make a conclusion about your work.

VIII. Commenting on homework.

Page 90. No. 91 (2,4,6), No. 92 (2,4,6,), p. 92 No. 112.

Used Books

• Textbook Algebra and the beginning of analysis.
  Authors: Yu.M. Kolyagin, M.V. Tkacheva, N.E. Fedorova, M.I. Shabunin.
  Edited by A. B. Zhizhchenko.
• Algebra 11th grade. Lesson plans according to the textbook by Sh. A. Alimov, Yu. M. Kolyagin, Yu. V. Sidorov. Part 1.
• Internet resources: http://orags.narod.ru/manuals/html/gre/12.jpg

Lecture: The concept of the derivative of a function, the geometric meaning of the derivative

The concept of the derivative of a function

Consider some function f(x), which will be continuous throughout the entire interval of consideration. On the interval under consideration, we choose the point x 0, as well as the value of the function at this point.

So, let's look at a graph on which we mark our point x 0, as well as the point (x 0 + ∆x). Recall that ∆x is the distance (difference) between two selected points.

It is also worth understanding that each x corresponds to its own value of the function y.

The difference between the values ​​of the function at the point x 0 and (x 0 + ∆x) is called the increment of this function: ∆y \u003d f (x 0 + ∆x) - f (x 0).

Let's pay attention to Additional information, which is on the chart, is the secant, which is called KL, as well as the triangle that it forms with intervals KN and LN.

The angle at which the secant is located is called its angle of inclination and is denoted by α. It can be easily determined that the degree measure of the angle LKN is also equal to α.

Now let's look at the ratios in right triangle tgα = LN / KN = ∆у / ∆х.

That is, the tangent of the slope of the secant is equal to the ratio of the increment of the function to the increment of the argument.

At one time, the derivative is the limit of the ratio of the increment of the function to the increment of the argument on infinitesimal intervals.

The derivative determines the rate at which the function changes over a certain area.

The geometric meaning of the derivative

If you find the derivative of any function at some point, then you can determine the angle at which the tangent to the graph in a given current will be located, relative to the OX axis. Pay attention to the graph - the angle of inclination of the tangent is denoted by the letter φ and is determined by the coefficient k in the straight line equation: y \u003d kx + b.

That is, we can conclude that the geometric meaning of the derivative is the tangent of the slope of the tangent at some point of the function.

Job type: 7

Condition

The line y=3x+2 is tangent to the graph of the function y=-12x^2+bx-10. Find b , given that the abscissa of the touch point is less than zero.

Show Solution

Solution

Let x_0 be the abscissa of the point on the graph of the function y=-12x^2+bx-10 through which the tangent to this graph passes.

The value of the derivative at the point x_0 is equal to the slope of the tangent, i.e. y"(x_0)=-24x_0+b=3. On the other hand, the tangent point belongs to both the graph of the function and the tangent, i.e. -12x_0^2+bx_0-10= 3x_0 + 2. We get a system of equations \begin(cases) -24x_0+b=3,\\-12x_0^2+bx_0-10=3x_0+2. \end(cases)

Solving this system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the condition of the abscissa, the touch points are less than zero, therefore x_0=-1, then b=3+24x_0=-21.

Answer

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=-3x+4 is parallel to the tangent to the graph of the function y=-x^2+5x-7. Find the abscissa of the point of contact.

Show Solution

Solution

The slope of the line to the graph of the function y=-x^2+5x-7 at an arbitrary point x_0 is y"(x_0). But y"=-2x+5, so y"(x_0)=-2x_0+5. Angular the coefficient of the line y=-3x+4 specified in the condition is -3.Parallel lines have the same slopes.Therefore, we find such a value x_0 that =-2x_0 +5=-3.

We get: x_0 = 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

Show Solution

Solution

From the figure, we determine that the tangent passes through the points A(-6; 2) and B(-1; 1). Denote by C(-6; 1) the point of intersection of the lines x=-6 and y=1, and by \alpha the angle ABC (it can be seen in the figure that it is sharp). Then the line AB forms an obtuse angle \pi -\alpha with the positive direction of the Ox axis.

As you know, tg(\pi -\alpha) will be the value of the derivative of the function f(x) at the point x_0. notice, that tg \alpha =\frac(AC)(CB)=\frac(2-1)(-1-(-6))=\frac15. From here, by the reduction formulas, we obtain: tg(\pi -\alpha) =-tg \alpha =-\frac15=-0.2.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=-2x-4 is tangent to the graph of the function y=16x^2+bx+12. Find b , given that the abscissa of the touch point is greater than zero.

Show Solution

Solution

Let x_0 be the abscissa of the point on the graph of the function y=16x^2+bx+12 through which

is tangent to this graph.

The value of the derivative at the point x_0 is equal to the slope of the tangent, that is, y "(x_0)=32x_0+b=-2. On the other hand, the tangent point belongs to both the graph of the function and the tangent, that is, 16x_0^2+bx_0+12=- 2x_0-4 We get a system of equations \begin(cases) 32x_0+b=-2,\\16x_0^2+bx_0+12=-2x_0-4. \end(cases)

Solving the system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the condition of the abscissa, the touch points are greater than zero, therefore x_0=1, then b=-2-32x_0=-34.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The figure shows a graph of the function y=f(x) defined on the interval (-2; 8). Determine the number of points where the tangent to the graph of the function is parallel to the straight line y=6.

Show Solution

Solution

The line y=6 is parallel to the Ox axis. Therefore, we find such points at which the tangent to the function graph is parallel to the Ox axis. On this chart, such points are extremum points (maximum or minimum points). As you can see, there are 4 extremum points.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The line y=4x-6 is parallel to the tangent to the graph of the function y=x^2-4x+9. Find the abscissa of the point of contact.

Show Solution

Solution

The slope of the tangent to the graph of the function y \u003d x ^ 2-4x + 9 at an arbitrary point x_0 is y "(x_0). But y" \u003d 2x-4, which means y "(x_0) \u003d 2x_0-4. The slope of the tangent y \u003d 4x-7 specified in the condition is equal to 4. Parallel lines have the same slopes. Therefore, we find such a value x_0 that 2x_0-4 \u003d 4. We get: x_0 \u003d 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: The geometric meaning of the derivative. Tangent to function graph

Condition

The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x_0. Find the value of the derivative of the function f(x) at the point x_0.

Show Solution

Solution

From the figure, we determine that the tangent passes through the points A(1; 1) and B(5; 4). Denote by C(5; 1) the point of intersection of the lines x=5 and y=1, and by \alpha the angle BAC (it can be seen in the figure that it is sharp). Then the line AB forms an angle \alpha with the positive direction of the Ox axis.