Reading the graph of the derivative of a function. General lesson on the topic: “Using the derivative and its graph to read the properties of functions” Lesson objectives: Develop specific skills and abilities for work. Explanation of the concept of a graph, reading technique

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Lesson objectives:

Educational: To strengthen students’ skills in working with graphs of functions in preparation for the Unified State Exam.

Developmental: to develop students’ cognitive interest in academic disciplines, the ability to apply their knowledge in practice.

Educational: cultivate attention, accuracy, broaden the horizons of students.

Equipment and materials: computer, screen, projector, presentation “Reading graphs. Unified State Exam"

During the classes

1. Frontal survey.

1) <Презентация. Слайды 3,4>.

What is called the graph of a function, the domain of definition and the range of values of a function? Determine the domain of definition and range of values of functions.\

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Which function is called even, odd, properties of the graphs of these functions?

2. Solution of exercises

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Periodic function. Definition.

Solve the problem: Given a graph of a periodic function, x belongs to the interval [-2;1]. Calculate f(-4)-f(-6)*f(12), T=3.

f(-4)=f(-4+T)=f(-4+3)= f(-1)=-1

f(-6)=f(-6+T)= f(-6+3*2)=f(0)=1

f(12)=f(12-4T)= =f(12-3*4)=f(0)=1

f(-4)-f(-6)*f(12)=-1-1*1=-2

2) <Презентация. Слайды 8,9,10>.

Solving inequalities using function graphs.

a) Solve the inequality f(x) 0 if the figure shows a graph of the function y=f(x) given on the interval [-7;6]. Answer options: 1) (-4;-3) (-1;1) (3;6], 2) [-7;-4) (-3;-1) (1;3), 3) , 4 ) (-6;0) (2;4) +

b) The figure shows a graph of the function y=f(x), specified on the segment [-4;7]. Indicate all values of X for which the inequality f(x) -1 holds.

[-0.5;3], 2) [-0.5;3] U , 3) [-4;0.5] U +, 4) [-4;0,5]

c) The figure shows graphs of the functions y=f(x), and y=g(x), specified on the interval [-3;6]. List all values of X for which the inequality f(x) g(x) holds

[-1;2], 2) [-2;3], 3) [-3;-2] U+, 4) [-3;-1] U

3) <Презентация. Слайд 11>.

Increasing and decreasing functions

One of the figures shows a graph of a function increasing on the segment , and the other - decreasing on the segment [-2;0]. Please indicate these drawings.

4) <Презентация. Слайды 12,13,14>.

Exponential and logarithmic functions

a) Name the condition for increasing and decreasing exponential and logarithmic functions. Through what point do the graphs of exponential and logarithmic functions pass, what properties do the graphs of these functions have?

b) One of the pictures shows a graph of the function y=2 -x. Indicate this picture .

The graph of the exponential function passes through the point (0, 1). Since the base of the degree is less than 1, this function must be decreasing. (No. 3)

c) One of the figures shows a graph of the function y=log 5 (x-4). Indicate the number of this schedule.

The graph of the logarithmic function y=log 5 x passes through the point (1;0) , then, if x -4 = 1, then y = 0, x = 1 + 4, x=5. (5;0) – the point of intersection of the graph with the OX axis. If x -4 = 5 , then y=1, x=5+4, x=9,

5) <Презентация. Слайды 15, 16, 17>.

Finding the number of tangents to the graph of a function from the graph of its derivative

a) The function y=f(x) is defined on the interval (-6;7). The figure shows a graph of the derivative of this function. All tangents parallel to the straight line y=5-2x (or coinciding with it) are drawn to the graph of the function. Indicate the number of points on the graph of the function at which these tangents are drawn.

K = tga = f’(x o). By condition k=-2. Therefore, f’(x o) =-2. We draw a straight line y=-2. It intersects the graph at two points, which means that the tangents to the function are drawn at two points.

b) The function y=f(x) is defined on the interval [-7;3]. The figure shows a graph of its derivative. Find the number of points on the graph of the function y=f(x) at which the tangents to the graph are parallel to the x-axis or coincide with it.

The angular coefficient of straight lines parallel to the abscissa axis or coinciding with it is zero. Therefore, K=tg a = f `(x o)=0. The OX axis intersects this graph at four points.

c) Function y=f(x) defined on the interval (-6;6). The figure shows a graph of its derivative. Find the number of points on the graph of the function y=f(x) at which the tangents to the graph are inclined at an angle of 135° to the positive direction of the x-axis.

6) <Презентация. Слайды 18, 19>.

Finding the slope of a tangent from the graph of the derivative of a function

a) The function y=f(x) is defined on the interval [-2;6]. The figure shows a graph of the derivative of this function. Indicate the abscissa of the point at which the tangent to the graph of the function y=f(x) has the smallest slope.

k=tga=f’(x o). The derivative of the function takes the smallest value y=-3 at the point x=2. Therefore, the tangent to the graph has the smallest slope at point x=2

b) The function y=f(x) is defined on the interval [-7;3]. The figure shows a graph of the derivative of this function. Indicate the abscissa of the point at which the tangent to the graph of the function y=f(x) has the greatest angular coefficient.

7) <Презентация. Слайд 20>.

Finding the value of the derivative from the graph of a function

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

f’(x o) =tga. Since in the figure a is an obtuse angle, then tg a< 0.Из прямоугольного треугольника tg (180 0 -a)=3:2. tg (180 0 -a)= 1,5. Следовательно, tg a= -1,5.Отсюда f `(x o)=-1,5

8) <Презентация. Слайд 21>.

Finding the minimum (maximum) of a function from the graph of its derivative

At the point x=4 the derivative changes sign from minus to plus. This means x=4 is the minimum point of the function y=f(x)

At point x=1 the derivative changes sign from plus to minus . This means x=1 is a point maximum functionsy=f(x))

3. Independent work

<Презентация. Слайд 22>.

1 Option

1) Find the domain of definition of the function.

2) Solve the inequality f(x) 0

3) Determine the intervals of decrease of the function.

4) Find the minimum points of the function.

5) Indicate the abscissa of the point at which the tangent to the graph of the function y=f(x) has the largest slope.

Option 2

1) Find the range of values of the function.

2) Solve the inequality f(x) 0

3) Determine the intervals of increase of the function.

Graph of the derivative of the function y=f(x)

4) Find the maximum points of the function.

5) Indicate the abscissa of the point at which the tangent to the graph of the function y=f(x) has the smallest slope.

4. Summing up the lesson

Topic: General review of the mathematics course. Preparation for exams

Lesson: Reading a graph of functions. Problem solving B2

In our life, graphs are found quite often, take, for example, a weather forecast, which is presented in the form of a graph of changes in some indicators, for example, temperature or wind strength over time. We don't think twice about reading this chart, even though it may be the first time we've read a chart in our lives. You can also give an example of a graph of changes in exchange rates over time and many other examples.

So, the first chart we'll look at.

Rice. 1. Illustration of graph 1

As you can see, the graph has 2 axes. The axis pointing to the right (horizontal) is called the axis . The axis pointing upward (vertical) is called the axis .

First, let's look at the axis. In this graph, the number of revolutions per minute of a certain automobile engine is plotted along this axis. It can be equal, etc. There are also divisions on this axis, some of them are indicated by numbers, some of them are intermediate and are not indicated. It’s easy to guess that the first division from zero is , the third is, etc.

Now let's look at the axis. On this graph, along this axis are plotted the numerical values of Newton per meter (), torque values, which are equal, etc. In this case, the division price is equal to .

Now let's turn to the function itself (to the line that is presented on the graph). As you can see, this line reflects how many Newtons per meter, that is, what torque, will be at a specific engine speed per minute. If we take the value 1000 rpm. and from this point on the graph we go to the left, we will see that the line passes through point 20, i.e. the value of torque at 1000 rpm will be equal (Figure 2.2).

If we take the value of 2000 rpm, then the line will pass already at the point (Figure 2.2).

Rice. 2. Determination of torque by the number of revolutions per minute

Now imagine that our task is to find the largest value from this graph. We are looking for the highest point (), accordingly, the lowest torque value in this graph will be considered 0. To find the highest value of the function on the graph, you need to consider the highest value that the function reaches on the vertical axis. We look at which value is highest and look along the vertical axis at what the highest number achieved will be. If we are talking about the smallest value, then we take, on the contrary, the lowest point and look at its value along the vertical axis.

Rice. 3. The largest and smallest value of a function according to the graph

The largest value in this case is , and the smallest value, respectively, is 0. It is important not to confuse and indicate the maximum value correctly, some indicate the maximum value of 4000 rpm, this is not the maximum value, but the point at which the maximum value is taken (point maximum), the greatest value is exactly .

You should also pay attention to the vertical axis, its units of measurement, that is, for example, if instead of Newtons per meter () hundreds of Newtons per meter () were indicated, the maximum value would need to be multiplied by one hundred, etc.

The largest and smallest values of a function are very closely related to the derivative of the function.

If a function increases on the segment under consideration, then the derivative of the function on this segment is positive or equal to zero at a finite number of points, most often it is simply positive. Similarly, if a function decreases on the segment under consideration, then the derivative of the function on this segment is negative or equal to zero at a finite number of points. The converse is true in both cases.

The following example has some difficulties due to the horizontal axis constraint. It is necessary to find the largest and smallest value on the specified segment.

The graph shows the change in temperature over time. On the horizontal axis we see time and days, and on the vertical axis we see temperature. It is necessary to determine the highest air temperature on January 22, i.e. we need to consider not the entire graph, but the part concerning January 22, i.e. from 00:00 January 22 to 00:00 January 23.

Rice. 4. Temperature change graph

By limiting the graph, it becomes obvious to us that the maximum temperature corresponds to point .

A graph of temperature changes over three days is given. On the ox axis - the time of day and day of the month, on the oy axis - the air temperature in degrees Celsius.

We need to consider not the entire schedule, but the part concerning July 13, that is, from 00:00 July 13 to 00:00 July 14.

Rice. 5. Illustration for additional example

If you do not enter the restrictions described above, you may get an incorrect answer, but at a given interval the maximum value is obvious: , and it is reached at 12:00 on July 13.

Example 3: determine on what date five millimeters of rain fell for the first time:

The graph shows daily precipitation in Kazan from February 3 to February 15, 1909. The days of the month are displayed horizontally, and the amount of precipitation in millimeters is displayed vertically.

Rice. 6. Daily precipitation

Let's start in order. On the 3rd, we see that a little more than 0 fell, but less than 1 mm. precipitation, 4 mm of precipitation fell on the 4th, etc. The number 5 first appears on the 11th day. For convenience, you could virtually draw a straight line opposite the five; for the first time it will cross the chart on February 11, this is the correct answer.

Example 4: determine on what date the price of an ounce of gold was the lowest

The graph shows the price of gold at the close of exchange trading for each day from March 5 to March 28, 1996. The days of the month are displayed horizontally, vertically,

accordingly, the price of an ounce of gold in US dollars.

The lines between the points are drawn for clarity only; the information is carried solely by the points themselves.

Rice. 7. Chart of changes in the price of gold on the stock exchange

Additional example: determine at what point on the segment the function takes the greatest value:

The derivative of a certain function is given on the graph.

Rice. 8. Illustration for additional example

The derivative is defined on the interval

As you can see, the derivative of the function on a given segment is negative and equals zero at the left boundary point. As we know, if the derivative of a function is negative, then the function on the interval under consideration decreases, therefore, our function decreases on the entire interval under consideration, in this case, it takes the greatest value in the leftmost boundary. Answer: period.

So, we looked at the concept of a graph of a function, studied what axes on a graph are, how to find the value of a function from a graph, how to find the largest and smallest value.

Mordkovich A.G. Algebra and beginning of mathematical analysis. - M.: Mnemosyne.
Muravin G.K., Muravin O.V. Algebra and beginning of mathematical analysis. - M.: Bustard.
Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of mathematical analysis. - M.: Enlightenment.

Unified State Exam ().
Festival of Pedagogical Ideas ().
Studying is easy.RF ().

The diagram (Figure 9) shows the average monthly air temperature in Yekaterinburg (Sverdlovsk) for each month of 1973. The horizontal axis indicates the months, and the vertical axis indicates the temperature in degrees Celsius. Determine from the diagram the lowest average monthly temperature during the period from May to December 1973 inclusive. Give your answer in degrees Celsius.

Rice. 9. Temperature chart

Using the same graph (Figure 9), determine the difference between the highest and lowest average monthly temperatures in 1973. Give your answer in degrees Celsius.
The graph (Figure 10) shows the heating process of an internal combustion engine at an ambient temperature of 15 degrees. The abscissa axis shows the time in minutes that has elapsed since the engine was started, and the y-axis shows the engine temperature in degrees Celsius. The load can be connected to the motor when the motor temperature reaches 45 degrees. What is the minimum number of minutes that must be waited before connecting the load to the motor?

Rice. 10. Engine warm-up schedule

Next, in class, it is advisable to consider a key task: using the given graph of the derivative, students must come up with (of course, with the help of the teacher) various questions related to the properties of the function itself. Naturally, these issues are discussed, corrected if necessary, summarized, recorded in a notebook, after which the stage of solving these tasks begins. Here it is necessary to ensure that students not only give the correct answer, but are able to argue (prove) it, using the appropriate definitions, properties, and rules.
Let's give an example of such a task: on the board (for example, using a projector), students are presented with a graph of the derivative; 10 tasks were formulated based on it (not entirely correct or duplicate questions were rejected).
The function y = f(x) is defined and continuous on the interval [–6; 6].
Using the graph of the derivative y = f"(x), determine:

1) the number of intervals of increasing function y = f(x);
2) the length of the interval of decreasing function y = f(x);
3) the number of extremum points of the function y = f(x);
4) maximum point of the function y = f(x);
5) critical (stationary) point of the function y = f(x), which is not an extremum point;
6) the abscissa of the graph point at which the function y = f(x) takes the greatest value on the segment;
7) the abscissa of the graph point at which the function y = f(x) takes on the smallest value on the segment [–2; 2];
8) the number of points in the graph of the function y = f(x), at which the tangent is perpendicular to the Oy axis;
9) the number of points on the graph of the function y = f(x), at which the tangent forms an angle of 60° with the positive direction of the Ox axis;
10) the abscissa of the graph point of the function y = f(x), at which the slope of the tangent takes the smallest value.
Answer: 1) 2; 2) 2; 3) 2; 4) –3; 5) –5; 6) 4; 7) –1; 8) 3; 9) 4; 10) –2.
To reinforce the skills of studying the properties of a function, students can take home a task related to reading the same graph, but in one case it is a graph of a function, and in the other, a graph of its derivative.

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The function y = f(x) is defined and continuous on the interval [–6; 5]. The picture shows:
a) graph of the function y = f(x);
b) graph of the derivative y = f"(x).
Determine from the schedule:
1) minimum points of the function y = f(x);
2) the number of intervals of decreasing function y = f(x);
3) the abscissa of the graph point of the function y = f(x), at which it takes the greatest value on the segment;
4) the number of points on the graph of the function y = f(x) at which the tangent is parallel to the Ox axis (or coincides with it).
Answers:
a) 1) –3; 2; 4; 2) 3; 3) 3; 4) 4;
b) 1) –2; 4.6;2) 2; 3) 2; 4) 5.
To carry out control, you can organize work in pairs: each student prepares a derivative graph on a card for his partner in advance and below offers 4-5 questions to determine the properties of the function. During lessons, they exchange cards, complete the proposed tasks, after which everyone checks and evaluates their partner’s work.

Topic: General review of the mathematics course. Preparation for exams

Lesson: Reading a graph of functions. Problem solving B2

1. Explanation of the concept of a graph, reading technique

So, the first chart we'll look at.

Rice. 1. Illustration of graph 1

As you can see, the graph has 2 axes. The axis pointing to the right (horizontal) is called the axis . The axis pointing upward (vertical) is called the axis .

Now let's look at the axis. On this graph, along this axis are plotted the numerical values of Newton per meter (), torque values, which are equal, etc. In this case, the division price is equal to .

If we take the value of 2000 rpm, then the line will pass already at the point (Figure 2.2).

Rice. 2. Determination of torque by the number of revolutions per minute

2. The concept of maximum and minimum values, the method of finding the maximum and minimum values of a function from a graph

Rice. 3. The largest and smallest value of a function according to the graph

The largest and smallest values of a function are very closely related to the derivative of the function.

3. Additional information about the derivative function

4. Solving examples with a constraint along the OX axis

The following example has some difficulties due to the horizontal axis constraint. It is necessary to find the largest and smallest value on the specified segment.

Rice. 4. Temperature change graph

By limiting the graph, it becomes obvious to us that the maximum temperature corresponds to point .

5. Additional example, task from the Unified State Exam

A graph of temperature changes over three days is given. On the ox axis - the time of day and day of the month, on the oy axis - the air temperature in degrees Celsius.

We need to consider not the entire schedule, but the part concerning July 13, that is, from 00:00 July 13 to 00:00 July 14.

Rice. 5. Illustration for additional example

If you do not enter the restrictions described above, you may get an incorrect answer, but at a given interval the maximum value is obvious: , and it is reached at 12:00 on July 13.

6. Solving other examples on reading the graph of a function

Example 3: determine on what date five millimeters of rain fell for the first time:

Rice. 6. Daily precipitation

Example 4: determine on what date the price of an ounce of gold was the lowest

The graph shows the price of gold at the close of exchange trading for each day from March 5 to March 28, 1996. The days of the month are displayed horizontally, vertically,

accordingly, the price of an ounce of gold in US dollars.

The lines between the points are drawn for clarity only; the information is carried solely by the points themselves.

Rice. 7. Chart of changes in the price of gold on the stock exchange

7. Solution of an additional example

Additional example: determine at what point on the segment the function takes the greatest value:

The derivative of a certain function is given on the graph.

Rice. 8. Illustration for additional example

The derivative is defined on the interval

So, we looked at the concept of a graph of a function, studied what axes on a graph are, how to find the value of a function from a graph, how to find the largest and smallest value.

Mordkovich A. G. Algebra and the beginnings of mathematical analysis. - M.: Mnemosyne. Muravin G.K., Muravin O.V. Algebra and the beginnings of mathematical analysis. - M.: Bustard. Kolmogorov A. N., Abramov A. M., Dudnitsyn Yu. P. et al. Algebra and the beginnings of mathematical analysis. - M.: Enlightenment.

Unified State Exam. Festival of pedagogical ideas. Studying is easy. RF.

The diagram (Figure 9) shows the average monthly air temperature in Yekaterinburg (Sverdlovsk) for each month of 1973. The horizontal axis indicates the months, and the vertical axis indicates the temperature in degrees Celsius. Determine from the diagram the lowest average monthly temperature during the period from May to December 1973 inclusive. Give your answer in degrees Celsius.

Rice. 9. Temperature chart

Using the same graph (Figure 9), determine the difference between the highest and lowest average monthly temperatures in 1973. Give your answer in degrees Celsius. The graph (Figure 10) shows the heating process of an internal combustion engine at an ambient temperature of 15 degrees. The abscissa axis shows the time in minutes that has elapsed since the engine was started, and the y-axis shows the engine temperature in degrees Celsius. The load can be connected to the motor when the motor temperature reaches 45 degrees. What is the minimum number of minutes that must be waited before connecting the load to the motor?

Rice. 10. Engine warm-up schedule

General lesson on the topic: “Using the derivative and its graph to read the properties of functions” Lesson objectives: Develop specific skills in working with the graph of a derivative function for their use when passing the Unified State Exam; Develop the ability to read the properties of a function from the graph of its derivative Prepare for the test

Updating of basic knowledge 3. Relationship between the values of the derivative, the slope of the tangent, the angle between the tangent and the positive direction of the OX axis The derivative of the function at the point of tangency is equal to the slope of the tangent drawn to the graph of the function at this point, that is, the tangent of the angle of inclination of the tangent to the positive direction of the axis abscissa. If the derivative is positive, then the angular coefficient is positive, then the angle of inclination of the tangent to the OX axis is acute. If the derivative is negative, then the angular coefficient is negative, then the angle of inclination of the tangent to the OX axis is obtuse. If the derivative is zero, then the slope is zero, then the tangent is parallel to the OX axis

0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f(x) 7 Updating of basic knowledge Sufficient signs of monotonicity of a function. If f (x) > 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f (x) title="Updating background knowledge Sufficient signs of monotonicity of the function. If f (x) > 0 at each point of the interval (a, b), then the function f (x) increases m on this interval. If f(x)

Updating of reference knowledge Internal points of the domain of definition of a function at which the derivative is equal to zero or does not exist are called critical points of this function. Only at these points can the function have an extremum (minimum or maximum, Fig. 5a, b). At points x 1, x 2 (Fig. 5a) and x 3 (Fig. 5b) the derivative is 0; at points x 1, x 2 (Fig. 5b) the derivative does not exist. But they are all extreme points. 5. Application of the derivative to determine critical points and extremum points

Updating of basic knowledge A necessary condition for an extremum. If x 0 is the extremum point of the function f(x) and the derivative of f exists at this point, then f(x 0)=0. This theorem is a necessary condition for an extremum. If the derivative of a function at a certain point is equal to 0, this does not mean that the function has an extremum at this point. For example, the derivative of the function f (x) = x 3 is 0 at x = 0, but this function does not have an extremum at this point. On the other hand, the function y = | x | has a minimum at x = 0, but the derivative does not exist at this point. Sufficient conditions for an extremum. If the derivative, when passing through the point x 0, changes its sign from plus to minus, then x 0 is the maximum point. If the derivative, when passing through the point x 0, changes its sign from minus to plus, then x 0 is the minimum point. 6. Necessary and sufficient conditions for an extremum

Updating of reference knowledge The minimum and maximum values of the continuous function f(x) can be achieved both at the internal points of the segment [a; c], and at its ends. If these values are reached at the internal points of the segment, then these points are extremum points. Therefore, it is necessary to find the values of the function at the extremum points from the segment [a; c], at the ends of the segment and compare them. 7. Using the derivative to find the largest and smallest value of a function

1. Development of knowledge, skills and abilities on the topic Using the following data given in the table, characterize the behavior of the function. Cheat sheet for practical work x(-3;0)0(0;4)4(4;8)8(8;+) f΄(x) f(x)

Characteristics of the behavior of function 1.ODZ: x belongs to the interval from -3 to +; 2.Increases at intervals (-3;0) and (8;+); 3.Decreases on intervals (0;8); 4.Х=0 – maximum point; 5.Х=4 – inflection point; 6.Х=8 – minimum point; 7.f(0) =-3; f(0) =-5; f(0) = 8;

5. Development of knowledge, skills and abilities on the topic The function y = f(x) is defined and continuous on the interval [–6; 6]. Formulate 10 questions to determine the properties of a function from the graph of the derivative y = f"(x). Your task is not just to give the correct answer, but to skillfully argue (prove) it, using the appropriate definitions, properties, and rules.

List of questions (corrected) 1) number of intervals of increasing function y = f(x); 2) the length of the interval of decreasing function y = f(x); 3) the number of extremum points of the function y = f(x); 4) maximum point of the function y = f(x); 5) critical (stationary) point of the function y = f(x), which is not an extremum point; 6) the abscissa of the graph point at which the function y = f(x) takes the greatest value on the segment; 7) the abscissa of the graph point at which the function y = f(x) takes on the smallest value on the segment [–2; 2]; 8) the number of points in the graph of the function y = f(x), at which the tangent is perpendicular to the OU axis; 9) the number of points on the graph of the function y = f(x), at which the tangent forms an angle of 60° with the positive direction of the OX axis; 10) the abscissa of the graph point of the function y = f(x), in which the slope is Answer: 1) 2; 2) 2; 3) 2; 4) –3; 5) –5; 6) 4; 7) –1; 8) 3; 9) 4; 10) –2.

Testing (B8 from the Unified State Exam) 1. The test tasks are presented on the slides. 2. Enter your answers in the table. 3.After completing the test, exchange answer sheets and check your neighbor’s work using the finished results; evaluate. 4.We consider and discuss problem tasks together.

A tangent is drawn to the graph of the function y =f(x) at its point with the abscissa x 0 =2. Determine the slope of the tangent if the figure shows a graph of the derivative of this function. The function y=f(x) is defined on the interval (-5;5). The figure shows a graph of the derivative of this function. Find the number of points on the graph of the function at which the tangents are parallel to the x-axis. 1

The function is defined on the interval (-5;6). The figure shows a graph of its derivative. Indicate the number of points at which the tangents are inclined at an angle of 135° to the positive direction of the x-axis. The function is defined on the interval (-6;6). The figure shows a graph of its derivative. Indicate the number of points whose tangents are inclined at an angle of 45° to the positive direction of the x-axis.

The function y = f(x) is defined on the interval [-6;6]. The graph of its derivative is shown in the figure. Indicate the number of intervals of increasing function y = f(x) on the segment [-6;6]. The function y = f(x) is defined on the interval [-5;5]. The graph of its derivative is shown in the figure. Indicate the number of maximum points of the function y = f(x) on the segment [-5;5].

The function y = f(x) is defined on the interval. The graph of its derivative is shown in the figure. Indicate the number of minimum points of the function y =f(x) on the segment. The function y = f(x) is defined on the interval [-6;6]. The graph of its derivative is shown in the figure. Indicate the number of intervals of decreasing function y=f(x) on the segment [-6;6]. ab

The function y = f(x) is defined on the interval [-6;6]. The graph of its derivative is shown in the figure. Find the intervals of increase of the function y = f(x) on the segment [-6;6]. In your answer, indicate the shortest of the lengths of these intervals. The function y = f(x) is defined on the interval [-5;5]. The graph of its derivative is shown in the figure. Find the intervals of decrease of the function y = f(x) on the segment [-5;5]. In your answer, indicate the largest of the lengths of these intervals.

The function y = f(x) is defined on the interval [-5;4]. The graph of its derivative is shown in the figure. Determine the smallest of those values of X at which the function has a maximum. The function y = f(x) is defined on the interval [-5;5]. The graph of its derivative is shown in the figure. Determine the smallest of those values of X at which the function has a minimum.

The function y = f(x) is defined on the interval (-6,6). The figure shows the derivative of this function. Find the minimum point of the function. The function y = f(x) is defined on the interval (-6,7). The figure shows the derivative of this function. Find the maximum point of the function.

Solution to task 19 Using the graph of the derivative of the function y = f(x), find the value of the function at point x = 5 if f(6) = 8 For x 3 f (x) =k=3, therefore on this interval the tangent is given by the formula y =3x+b. The value of the function at the point of contact coincides with the value of the tangent. By condition f(6) = 8 8=3·6 + b b = -10 f(5) =3·5 -10 = 5 Answer: 5

Summing up the lesson We examined the relationship between the monotonicity of a function and the sign of its derivative, and sufficient conditions for the existence of an extremum. We examined various tasks for reading the graph of a derivative function, which are found in the texts of the unified state exam. All the tasks we have considered are good because they do not take much time to complete. During the unified state exam, this is very important: quickly and correctly write down the answer.

Homework: a task involving reading the same graph, but in one case it is the graph of a function, and in the other it is the graph of its derivative. The function y = f(x) is defined and continuous on the interval [–6; 5]. The figure shows: a) graph of the function y = f(x); b) graph of the derivative y = f"(x). From the graph, determine: 1) the minimum points of the function y = f(x); 2) the number of intervals of decreasing function y = f(x); 3) the abscissa of the point of the graph of the function y = f (x), in which it takes the greatest value on the segment; 4) the number of points on the graph of the function y = f(x), at which the tangent is parallel to the OX axis (or coincides with it).

Literature 1. Textbook Algebra and beginning of analysis, grade 11. CM. Nikolsky, M.K. Potapov and others. Moscow. "Enlightenment" Unified State Examination Mathematics. Typical test tasks. 3. A guide for intensive preparation for the mathematics exam. Graduation, entrance, Unified State Exam at +5. M. "VAKO" Internet resources.