Definition of derivative, its geometric meaning. What is a derivative? Definition and meaning of a derivative function

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

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

Figure 1. Function graph

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

Point N lying on a curve can tend to M from any side. So, if in Figure 1 the tangent is given the opposite direction, the angle $\alpha $ will change by the amount $\pi $, which will significantly affect the tangent of the angle and, accordingly, the angular coefficient.

Conclusion

It follows that the existence of a derivative is associated with the existence of a tangent to the curve y = f(x), and the angular coefficient - tg $\alpha $ = f`(x) is finite. Therefore, the tangent should 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 similar 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 final limit, it is denoted as:

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

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

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

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

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

Figure 3. No derivatives

Example 1

Figure 4 shows a graph of the function and the tangent to the graph at the abscissa point $x_0$. 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 us select two points on the tangent with integer coordinates. Let, for example, these be points F (-3.2) and C (-2.4).

Job type: 7

Condition

The straight 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 tangent point less than zero.

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 point x_0 is equal to the slope of the tangent, that is, y"(x_0)=-24x_0+b=3. On the other hand, the point of tangency belongs simultaneously to both the graph of the function and the tangent, that is, -12x_0^2+bx_0-10= 3x_0 + 2. We obtain 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 abscissa condition, the tangent points are less than zero, so x_0=-1, then b=3+24x_0=-21.

Answer

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The straight 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 tangent point.

Solution

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

We get: x_0 = 4.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level" Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

Solution

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

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

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The straight 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 tangent point is greater than zero.

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 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 point of tangency belongs simultaneously to both the graph of the function and the tangent, that is, 16x_0^2+bx_0+12=- 2x_0-4 We obtain 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 abscissa condition, the tangent points are greater than zero, so x_0=1, then b=-2-32x_0=-34.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

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

Solution

The straight line y=6 is parallel to the Ox axis. Therefore, we find points at which the tangent to the graph of the function 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 Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

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 tangent point.

Solution

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

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

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 point x_0.

Solution

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

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

Let us consider some function f(x), which will be continuous over the entire interval of consideration. On the interval under consideration, we select the point x 0, as well as the value of the function at this point.

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

It is also worth understanding that each x corresponds to eigenvalue functions 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: ∆у = f(x 0 + ∆x) - f(x 0).

Let's pay attention to the additional information that is available on the graph - this 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 α. It can be easily determined that the degree measure of the angle LKN is also equal to α.

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

That is, the tangent of the secant angle 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 a function to the increment of the argument on infinitesimal intervals.

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

Geometric meaning of derivative

If you find the derivative of any function at a certain point, 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 tangential slope angle is denoted by the letter φ and is determined by the coefficient k in the equation of the straight line: y = kx + b.

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

What is a derivative?
Definition and meaning of a derivative function

Many will be surprised by the unexpected placement of this article in my author’s course on the derivative of a function of one variable and its applications. After all, as it has been since school: the standard textbook first of all gives the definition of a derivative, its geometric, mechanical meaning. Next, students find derivatives of functions by definition, and, in fact, only then they perfect the technique of differentiation using derivative tables.

But from my point of view, the following approach is more pragmatic: first of all, it is advisable to UNDERSTAND WELL limit of a function, and, in particular, infinitesimal quantities. The fact is that the definition of derivative is based on the concept of limit, which is poorly considered in school course. That is why a significant part of young consumers of the granite of knowledge do not understand the very essence of the derivative. Thus, if you have little knowledge of differential calculus or a wise brain for long years successfully got rid of this baggage, please start with function limits. At the same time, master/remember their solution.

The same practical sense dictates that it is advantageous first learn to find derivatives, including derivatives of complex functions. Theory is theory, but, as they say, you always want to differentiate. In this regard, it is better to work through the listed basic lessons, and maybe master of differentiation without even realizing the essence of their actions.

I recommend starting with the materials on this page after reading the article. The simplest problems with derivatives, where, in particular, the problem of the tangent to the graph of a function is considered. But you can wait. The fact is that many applications of the derivative do not require understanding it, and it is not surprising that the theoretical lesson appeared quite late - when I needed to explain finding increasing/decreasing intervals and extrema functions. Moreover, he was on the topic for quite a long time. Functions and graphs”, until I finally decided to put it earlier.

Therefore, dear teapots, do not rush to absorb the essence of the derivative like hungry animals, because the saturation will be tasteless and incomplete.

The concept of increasing, decreasing, maximum, minimum of a function

Many teaching aids lead to the concept of derivative using some practical problems, and I also came up with interesting example. Imagine that we are about to travel to a city that can be reached in different ways. Let’s immediately discard the curved winding paths and consider only straight highways. However, straight-line directions are also different: you can get to the city along a smooth highway. Or along a hilly highway - up and down, up and down. Another road goes only uphill, and another one goes downhill all the time. Extreme enthusiasts will choose a route through a gorge with a steep cliff and a steep climb.

But whatever your preferences, it is advisable to know the area or at least locate it topographic map. What if such information is missing? After all, you can choose, for example, a smooth path, but as a result stumble upon a ski slope with cheerful Finns. It’s not a fact that the navigator and even satellite image will provide reliable data. Therefore, it would be nice to formalize the relief of the path using mathematics.

Let's look at some road (side view):

Just in case, I remind you of an elementary fact: travel happens from left to right. For simplicity, we assume that the function continuous in the area under consideration.

What are the features of this graph?

At intervals function increases, that is, each next value of it more previous one. Roughly speaking, the schedule is on down up(we climb the hill). And on the interval the function decreases– each next value less previous, and our schedule is on top down(we go down the slope).

Let us also pay attention to singular points. At the point we reach maximum, that is exists such a section of the path where the value will be the largest (highest). At the same point it is achieved minimum, And exists its neighborhood in which the value is the smallest (lowest).

We will look at more strict terminology and definitions in class. about the extrema of the function, but for now let's study one more important feature: at intervals the function increases, but it increases With at different speeds . And the first thing that catches your eye is that the graph soars up during the interval much more cool, than on the interval . Is it possible to measure the steepness of a road using mathematical tools?

Rate of change of function

The idea is this: let's take some value (read "delta x"), which we'll call argument increment, and let’s start “trying it on” to various points on our path:

1) Let's look at the leftmost point: passing the distance, we climb the slope to a height (green line). The quantity is called function increment, and in this case this increment is positive (the difference in values ​​along the axis is greater than zero). Let's create a ratio that will be a measure of the steepness of our road. It is obvious that this is quite specific number, and, since both increments are positive, then .

Attention! Designations are ONE symbol, that is, you cannot “tear off” the “delta” from the “X” and consider these letters separately. Of course, the comment also concerns the function increment symbol.

Let's explore the nature of the resulting fraction more meaningfully. Let us initially be at a height of 20 meters (at the left black point). Having covered the distance of meters (left red line), we will find ourselves at an altitude of 60 meters. Then the increment of the function will be meters (green line) and: . Thus, on every meter this section of the road height increases average by 4 meters...forgot your climbing equipment? =) In other words, the constructed relationship characterizes the AVERAGE RATE OF CHANGE (in this case, growth) of the function.

Note : numeric values The example under consideration corresponds to the proportions of the drawing only approximately.

2) Now let's go the same distance from the rightmost black point. Here the rise is more gradual, so the increment (crimson line) is relatively small, and the ratio compared to the previous case will be very modest. Relatively speaking, meters and function growth rate is . That is, here for every meter of the path there are average half a meter of rise.

3) A little adventure on the mountainside. Let's look at the top black dot located on the ordinate axis. Let's assume that this is the 50 meter mark. We overcome the distance again, as a result of which we find ourselves lower - at the level of 30 meters. Since the movement is carried out top down(in the “counter” direction of the axis), then the final the increment of the function (height) will be negative: meters (brown segment in the drawing). And in this case we are already talking about rate of decrease Features: , that is, for every meter of path of this section, the height decreases average by 2 meters. Take care of your clothes at the fifth point.

Now let's ask ourselves the question: what value of the “measuring standard” is best to use? It’s completely understandable, 10 meters is very rough. A good dozen hummocks can easily fit on them. No matter the bumps, there may be a deep gorge below, and after a few meters there is its other side with a further steep rise. Thus, with a ten-meter we will not get an intelligible description of such sections of the path through the ratio .

From the above discussion the following conclusion follows: how less value , the more accurately we describe the road topography. Moreover, the following facts are true:

– For anyone lifting points you can select a value (even if very small) that fits within the boundaries of a particular rise. This means that the corresponding height increment will be guaranteed to be positive, and the inequality will correctly indicate the growth of the function at each point of these intervals.

- Likewise, for any slope point there is a value that will fit completely on this slope. Consequently, the corresponding increase in height is clearly negative, and the inequality will correctly show the decrease in the function at each point of the given interval.

– A particularly interesting case is when the rate of change of the function is zero: . Firstly, zero height increment () is a sign of a smooth path. And secondly, there are other interesting situations, examples of which you see in the figure. Imagine that fate has brought us to the very top of a hill with soaring eagles or the bottom of a ravine with croaking frogs. If you take a small step in any direction, the change in height will be negligible, and we can say that the rate of change of the function is actually zero. This is exactly the picture observed at the points.

Thus, we have come to an amazing opportunity to perfectly accurately characterize the rate of change of a function. After all, mathematical analysis makes it possible to direct the increment of the argument to zero: , that is, to make it infinitesimal.

As a result, another one arises logical question: is it possible to find for the road and its schedule another function, which would let us know about all the flat sections, ascents, descents, peaks, valleys, as well as the rate of growth/decrease at each point along the way?

What is a derivative? Definition of derivative.
Geometric meaning of derivative and differential

Please read carefully and not too quickly - the material is simple and accessible to everyone! It’s okay if in some places something doesn’t seem very clear, you can always return to the article later. I will say more, it is useful to study the theory several times in order to thoroughly understand all the points (the advice is especially relevant for “technical” students, for whom higher mathematics plays a significant role in the educational process).

Naturally, in the very definition of the derivative at a point we replace it with:

What have we come to? And we came to the conclusion that for the function according to the law is put in accordance other function, which is called derivative function(or simply derivative).

The derivative characterizes rate of change functions How? The idea runs like a red thread from the very beginning of the article. Let's consider some point domain of definition functions Let the function be differentiable at a given point. Then:

1) If , then the function increases at the point . And obviously there is interval(even a very small one), containing a point at which the function grows, and its graph goes “from bottom to top”.

2) If , then the function decreases at the point . And there is an interval containing a point at which the function decreases (the graph goes “top to bottom”).

3) If , then infinitely close near a point the function maintains its speed constant. This happens, as noted, with a constant function and at critical points of the function, in particular at minimum and maximum points.

A bit of semantics. What does the verb “differentiate” mean in a broad sense? To differentiate means to highlight a feature. By differentiating a function, we “isolate” the rate of its change in the form of a derivative of the function. What, by the way, is meant by the word “derivative”? Function happened from function.

The terms are very successfully interpreted by the mechanical meaning of the derivative :
Let us consider the law of change in the coordinates of a body, depending on time, and the function of the speed of movement of a given body. The function characterizes the rate of change of body coordinates, therefore it is the first derivative of the function with respect to time: . If the concept of “body movement” did not exist in nature, then there would be no derivative concept of "body speed".

The acceleration of a body is the rate of change of speed, therefore: . If the initial concepts of “body motion” and “body speed” did not exist in nature, then there would not exist derivative concept of “body acceleration”.

Lesson objectives:

Students should know:

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

Lesson objectives:

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

Educational: to form a scientific worldview in students.

Developmental: to develop students’ cognitive interest, creativity, will, memory, speech, attention, imagination, perception.

Methods of organizing educational and cognitive activities:

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

Lesson Plan

1. Oral exercises (find the derivative)
2. Student message on the topic “Causes of mathematical analysis”.
3. Learning new material
4. Phys. Just a minute.
5. Solving tasks.
6. Laboratory work.
7. Summing up the lesson.
8. Commenting on homework.

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

“A person only achieves something where he believes in his own strength”

L. Feuerbach

Organization of the class throughout the lesson, students' readiness 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.

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 management system would have been impossible without the sufficient development of arithmetic and algebra, because collecting taxes, organizing army supplies, building palaces and pyramids, and creating irrigation systems required complex calculations. During the Renaissance, connections between different parts of the medieval world expanded, trade and crafts developed. A rapid rise in the technical level of production begins, and new sources of energy that are not associated with the muscular efforts of humans or animals are being used industrially. In the XI-XII centuries, fulling and weaving machines appeared, and in the middle of the XV - a printing press. Due to the need for the rapid development of social production during this period, the essence of the natural sciences, which had been descriptive since ancient times, changed. The goal of natural science is an in-depth study of natural processes, not objects. Mathematics, which operated with constant quantities, corresponded to the descriptive natural science of antiquity. 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 of creating mathematical analysis. What is “mathematical analysis”? How can one characterize and predict the characteristics of any process? Use these features? To penetrate deeper into the essence of a particular phenomenon?

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

Let us introduce several concepts 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 the 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=f(x). Let's draw a secant through any two points, for example, secant AM. (Fig.2)

Angular coefficient 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 necessarily have to be time.

So, the tangent of the angle of inclination of the secant tg = .

We are interested in the dependence of changes in quantities over a shorter period of time. Let us direct 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 straight line AM is approaching some straight line AB, which is tangent to the graph of the function y = 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.

Mechanical meaning of derivative.

The tangent of the tangent angle is a value showing the instantaneous rate of change of the function at a given point, that is, a new characteristic of the process being studied. Leibniz called this quantity derivative, and Newton said that the derivative itself is called the instantaneous speed.

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

The angular coefficient of the tangent to the curve f(x) = x 3 at point x 0 – 1 is the value of the derivative of this function at x = 1. f’(1) = 3x 2 ; f’(1) = 3.

No. 91 (3.5) – dictation.

No. 92(1) – on the board if desired.

No. 92 (3) – independently with oral testing.

No. 92 (5) – at the board.

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

Goal: to develop the concept of “mechanical meaning of a derivative.”

Applications of derivatives to mechanics.

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

1. Average speed of movement over a specified period of time;
2. Velocity and acceleration at time t 04
3. Moments of stopping; whether the point after the moment of stopping continues to move in the same direction or begins to move in the opposite direction;
4. Highest speed movements over a specified period of time.

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

Before starting work, a conversation on the following questions:

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

A sample of student work.

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

Figure 4

In the opposite direction.

Let's draw a schematic diagram of the speed. The highest speed is achieved at the point

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

Figure 5

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

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

Used Books

• Textbook Algebra and beginnings of analysis.
  Authors: Yu.M. Kolyagin, M.V. Tkacheva, N.E. Fedorova, M.I. Shabunina.
  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: