Elements of mathematical analysis in the Unified State Exam Malinovskaya Galina Mikhailovna [email protected] Reference material Table of derivatives of basic functions.  Rules of differentiation (derivative of a sum, product, quotient of two functions).  Derivative of a complex function.  Geometric meaning of derivative.  Physical meaning of derivative.  Reference material Extremum points (maximum or minimum) of a function specified graphically.  Finding the largest (smallest) value of a function continuous on a given interval.  Antiderivative of function. Newton-Leibniz formula. Finding the area of ​​a curved trapezoid.  Physical applications  1.1 A material point moves rectilinearly according to the law 𝑥 𝑡 = −𝑡 4 +6𝑡 3 +5𝑡 + 23, where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. Find its speed (in meters per second) at time t= 3s.  1.2 A material point moves 1 3 rectilinearly according to the law 𝑥 𝑡 = 𝑡 − 3 3𝑡 2 − 5𝑡 + 3 , where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. At what point in time (in seconds) was its speed equal to 2 m/s? Solution: We are looking for the derivative of x(t) (function of the path with respect to time).  In problem 1.1, substitute its value for t and calculate the speed (Answer: 59).  In problem 1.2, we equate the found derivative to a given number and solve the equation with respect to the variable t. (Answer: 7).  Geometric Applications 2.1 The line 𝑦 = 7𝑥 − 5 is parallel to the tangent to the graph 2 of the function 𝑦 = 𝑥 + 6𝑥 − 8 . Find the abscissa of the tangent point. 2.2 The straight line 𝑦 = 3𝑥 + 1 is tangent to the 2nd graph of the function 𝑎𝑥 + 2𝑥 + 3. Find a. 2.3 The straight line 𝑦 = −5𝑥 + 8 is tangent to the 2nd graph of the function 28𝑥 + 𝑏𝑥 + 15. Find b, given that the abscissa of the point of tangency is greater than 0. 2.4 The line 𝑦 = 3𝑥 + 4 is tangent to graph 2 of the function 3𝑥 − 3𝑥 + 𝑐. Find c. Solution: In problem 2.1, we look for the derivative of the function and equate it to the slope of the straight line (Answer: 0.5).  In problems 2.2-2.4 we compose a system of two equations. In one we equate functions, in the other we equate their derivatives. In a system with two unknowns (variable x and parameter), we look for a parameter. (Answers: 2.2) a=0.125; 2.3) b=-33; 2.4) c=7).   2.5 The figure shows the graph of the function y=f(x) and the tangent to it at the point with abscissa 𝑥0. Find the value of the derivative of the function f(x) at point 𝑥0.  2.6 The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa 𝑥0. Find the value of the derivative of the function f(x) at point 𝑥0.  2.7 The figure shows the graph of the function y=f(x). The straight line passing through the origin touches the graph of this function at the point with abscissa 10. Find the value of the derivative of the function at the point x=10. 𝑥0 = 0 Solution:     The value of the derivative of a function at a point is the tangent of the angle of inclination of the tangent to the graph of the function drawn at this point. We “complete” the right triangle and look for the tangent of the corresponding angle, which we take as positive if the tangent forms an acute angle with the positive direction of the Ox axis (the tangent “increases”) and negative if the angle is obtuse (the tangent decreases). In problem 2.7, you need to draw a tangent through the specified point and the origin. Answers: 2.5) 0.25; 2.6) -0.25; 2.7) -0.6. Reading a graph of a function or a graph of a derivative of a function  3.1 The figure shows a graph of the function y=f(x), defined on the interval (6;8). Determine the number of integer points at which the derivative of the function is positive.  3.2 The figure shows a graph of the function y=f(x), defined on the interval (-5;5). Determine the number of integer points at which the derivative of the function f(x) is negative. Solution: The sign of the derivative is related to the behavior of the function.  If the derivative is positive, then we select that part of the graph of the function where the function increases. If the derivative is negative, then where the function decreases. We select the interval corresponding to this part on the Ox axis.  In accordance with the question of the problem, we either recalculate the number of integers included in a given interval or find their sum.  Answers: 3.1) 4; 3.2) 8.   3.3 The figure shows a graph of the function y=f(x), defined on the interval (-2;12). Find the sum of the extremum points of the function f(x). First of all, we look at what is in the figure: a graph of a function or a graph of a derivative.  If this is a graph of the derivative, then we are only interested in the signs of the derivative and the abscissa of the points of intersection with the Ox axis.  For clarity, you can draw a more familiar picture with the signs of the derivative over the resulting intervals and the behavior of the function.  Answer the question in the problem according to the picture. (Answer: 3.3) 44).   3.4 The figure shows a graph of ′ y=𝑓 (𝑥) - the derivative of the function f(x), defined on the interval (-7;14]. Find the number of maximum points of the function f(x) belonging to the segment [-6;9]  3.5 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-11;11).Find the number of extremum points of the function f(x) belonging to the segment [-10;10] Solution: We look for the intersection points of the derivative graph with the Ox axis, highlighting that part of the axis that is indicated in the problem.  We determine the sign of the derivative on each of the resulting intervals (if the derivative graph is below the axis, then “-”, if above, then “+”).  The maximum points will be those where the sign has changed from “+” to “-”, the minimum points - from “-” to “+”. Both are extremum points.  Answers: 3.4) 1; 3.5) 5.   3.6 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-8;3). At what point of the segment [-3;2] does the function f(x) take on the greatest value.  3.7 The figure shows a graph of ′ y=𝑓 (𝑥) - the derivative of the function f(x), defined on the interval (-8;4). At what point of the segment [-7;-3] does the function f(x) take on the smallest value. Solution:    If the derivative changes sign on the segment under consideration, then the solution is based on the theorem: if a function continuous on a segment has a single extremum point on it and this is a maximum (minimum) point, then the largest (smallest) value of the function on this segment is achieved at at this point. If a function continuous on a segment is monotonic, then it reaches its minimum and maximum values ​​on a given segment at its ends. Answers: 3.6) -3; 3.7) -7.  3.8 The figure shows a graph of the function y=f(x), defined on the interval (-5;5). Find the number of points at which the tangent to the graph of the function is parallel to or coincides with the straight line y=6.  3.9 The figure shows a graph of the function y=f(x) and eight points on the abscissa axis: 𝑥1 ,𝑥2 ,𝑥3 , … , 𝑥12 . At how many of these points is the derivative of f(x) positive?  4.2 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-5;7). Find the intervals of decrease of the function f(x). In your answer, indicate the sum of integer points included in these intervals.  4.5 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-4;8). Find the extremum point of the function f(x), belonging to the segment [-2;6].  4.6 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-10;2). Find the number of points at which the tangent to the graph of the function f(x) is parallel to or coincides with the straight line y=-2x-11. Solution: 4.6 Since the figure shows a graph of the derivative, and the tangent is parallel to this line, the derivative of the function at this point is equal to -2. We look for points on the derivative graph with an ordinate equal to -2 and count their number. We get 5.  Answers: 3.8) 4; 3.9) 5; 4.2) 18; 4.5) 4; 4.6) 5.   4.8 The figure shows a graph of y=𝑓 ′ (𝑥) - the derivative of the function f(x). Find the abscissa of the point at which the tangent to the graph y=f(x) is parallel to or coincides with the abscissa axis. Solution: If a straight line is parallel to the Ox axis, then its slope is zero.  The slope of the tangent is zero, which means the derivative is zero.  We are looking for the abscissa of the point of intersection of the derivative graph with the Ox axis.  We get -3.   4.9 The figure shows a graph of the function y=𝑓 ′ (x) derivative of the function f(x) and eight points on the abscissa axis: 𝑥1 ,𝑥2 ,𝑥3 , … , 𝑥8 . At how many of these points does the derivative of the function f(x) increase? Geometric meaning of the definite integral  5.1 The figure shows a graph of some function y=f(x) (two rays with a common starting point). Using the figure, calculate F(8)-F(2), where F(x) is one of the antiderivatives of the function f(x). Solution:     The area of ​​a curved trapezoid is calculated through a definite integral. The definite integral is calculated using the Newton-Leibniz formula as an increment of the antiderivative. In problem 5.1, we calculate the area of ​​the trapezoid using the well-known geometry course formula (this will be the increment of the antiderivative). In problems 5.2 and 5.3 the antiderivative has already been given. It is necessary to calculate its values ​​at the ends of the segment and calculate the difference.  5.2 The figure shows a graph of some function y=f(x). The function 𝐹 𝑥 = 15 3 2 𝑥 + 30𝑥 + 302𝑥 − is one of the 8 antiderivatives of the function f(x). Find the area of ​​the shaded figure. Solution:     The area of ​​a curved trapezoid is calculated through a definite integral. The definite integral is calculated using the Newton-Leibniz formula as an increment of the antiderivative. In problem 5.1, we calculate the area of ​​the trapezoid using the well-known geometry course formula (this will be the increment of the antiderivative). In Problem 5.2 the antiderivative is already given. It is necessary to calculate its values ​​at the ends of the segment and calculate the difference. Good luck on the Unified State Exam in mathematics 

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 that point. For example, the derivative of the function f (x) = x 3 is equal to 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 on the largest 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.

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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 point 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.

General lesson on the topic:

“Using the derivative and its graph to read the properties of a function”

Lesson type: a general lesson using ICT in the form of a presentation.

Lesson objectives:

Educational:

To promote students' understanding of the use of derivatives in practical tasks;

Teach students to clearly use the properties of functions and derivatives.

Educational:

Develop the ability to analyze a task question and draw conclusions;

Develop the ability to apply existing knowledge in practical tasks.

Educational:

Cultivating interest in the subject;

The need for these theoretical and practical skills to continue studying.

Lesson objectives:

Develop specific skills in working with the graph of a derivative function for their use when passing the Unified State Exam;

Prepare for the test.

Lesson plan.

1. Updating of reference knowledge (BK).

2. Development of knowledge, skills and abilities on the topic.

3. Testing (B8 from the Unified State Examination).

4. Mutual check, giving marks to the “neighbor”.

5. Summing up the lessons of the lesson.

Equipment: computer class, blackboard, marker, tests (2 options).

During the classes.

Org moment.

Teacher . Hello, please sit down.

During the study of the topic “Studying functions using derivatives,” the skills were developed to find the critical points of a function, the derivative, determine the properties of the function with its help, and build its graph. Today we will look at this topic from a different angle: how to determine the properties of the function itself through the graph of the derivative of a function. Our task: to learn to navigate the variety of tasks related to graphs of functions and their derivatives.

In preparation for the Unified State Exam in mathematics, KIMs are given problems on using the derivative graph to study functions. Therefore, in this lesson we must systematize our knowledge on this topic and learn to quickly find answers to the questions of tasks B8.

Slide No. 1.

Subject: “Using the derivative and its graph to read the properties of functions”

Lesson objectives:

Development of knowledge of the application of the derivative, its geometric meaning and the graph of the derivative to determine the properties of functions.

Development of efficiency in performing Unified State Exam tests.

Developing such personality qualities as attentiveness, ability to work with text, ability to work with derivative graphs

2.Updating basic knowledge (BK). Slides No. 4 to No. 10.

Review questions will now appear on the screen. Your task: give a clear and concise answer to each point. The correctness of your answer can be checked on the screen.

( A question first appears on the screen, after the students answer, the correct answer appears for verification.)

List of questions for AOD.

Definition of derivative.

Geometric meaning of derivative.

The 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.

Using the derivative to find intervals of monotonicity of a function.

Application of derivative to determine critical points, extremum points

6 Necessary and sufficient conditions for an extremum

7 . Using the Derivative to Find the Largest and Smallest Values ​​of a Function

(Students answer each item, accompanying their answers with notes and drawings on the board. In case of erroneous and incomplete answers, classmates correct and supplement them. After the students answer, the correct answer appears on the screen. Thus, students can immediately determine the correctness of their answer. )

3. Development of knowledge, skills and abilities on the topic. Slides No. 11 to No. 15.

Students are offered tasks from KIMs of the Unified State Examination in mathematics of previous years, from Internet sites on the use of the derivative and its graph to study the properties of functions. The tasks appear sequentially. Students draw up solutions on the board or by oral reasoning. The correct solution then appears on the slide and is checked against the students' solution. If there is an error in the solution, it is analyzed by the whole class.

Slide No. 16 and No. 17.

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.

Testing (B8 from the Unified State Examination). Slides No. 18 to No. 29. Slide No. 30 – keys to the test.

Teacher : So, we have summarized your knowledge on this topic: we repeated the basic properties of the derivative, solved problems related to the graph of the derivative, analyzed complex and problematic aspects of using the derivative and the graph of the derivative to study the properties of functions.

Now we will test in 2 options. Tasks will appear on the screen in both versions at the same time. You study the question, find the answer, and write it down on your answer sheet. After completing the test, exchange forms and check your neighbor’s work using ready-made answers. Give a rating(up to 10 points – “2”, from 11 to 15 points – “3”, from 16 to 19 points – “4”, more than 19 points – “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.

Hand in your answer forms. The grade for the lesson is already known to you and will be included in the journal.

I think the class has prepared for the test.

Homework will be creative . Slide number 33 .

Slide 12

Symmetry about the straight line y=x

The graphs of these functions increase at > 1 and decrease at 0

Slide 13

One of the figures shows a graph of the function y=2-x. Please indicate this drawing. Graph of an exponential function The graph of an exponential function passes through the point (0, 1). Since the base of the degree is less than 1, this function must be decreasing.

Slide 14

One of the figures shows a graph of the function y=log5 (x-4). Indicate the number of this schedule. The graph of the logarithmic function y=log5x passes through the point (1;0), then if x -4 = 1, then = 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, Graph of the logarithmic function 9 5 1

Slide 15

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’(xo) By condition k = -2. Therefore f’(xo) = -2 We draw a straight line y = -2. It intersects the graph at two points, which means the tangents to the function are drawn at two points. Finding the number of tangents to the graph of a function from the graph of its derivative

Slide 16

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 lines parallel to the abscissa or coinciding with it is zero. Therefore K=tg a = f `(xo)=0 The OX axis intersects this graph at four points. Finding the number of tangents to a function from the graph of its derivative

Slide 17

The function y=f(x) is 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. K = tg 135o= f'(xo) tg 135o=tg(180o-45o)=-tg45o=-1 Therefore f`(xo)=-1 Draw a straight line y=-1. It intersects the graph at three points, which means tangents to the function carried out at three points. Finding the number of tangents to a function from the graph of its derivative

Slide 18

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 angular coefficient k=tg a=f’(xo) The derivative of the function takes the smallest value y=-3 at point x=2. Therefore, the tangent to the graph has the smallest slope at point x=2 Finding the slope of the tangent from the graph of the derivative of the function -3 2

Slide 19

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 at which the tangent to the graph of the function y=f(x) has the largest slope. k=tg a=f’(xo) The derivative of the function takes its greatest value y=3 at point x=-5. Therefore, the tangent to the graph has the largest slope at point x = -5 Finding the slope of the tangent from the graph of the derivative of the function 3 -5

Slide 20

The figure shows a graph of the function y=f(x) and a tangent to it at the point with the abscissa xo. Find the value of the derivative f `(x) at the point xo f ’(xo) =tg a Since in the figure a is an obtuse angle, then tan a

Slide 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) 4 At points x = 1, the derivative changes sign from plus. minusMeanx=1 is the maximum point of the function y=f(x))

Slide 22

Independent work

Fig.11) Find the domain of definition of the function. 2) Solve the inequality f(x) ≥ 0 3) Determine the intervals of decrease of the function. Fig. 2 – graph of the derivative function y=f(x) 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 angle coefficient. Fig.11) Find the range of values ​​of the function. 2) Solve the inequality f(x)≤ 0 3) Determine the intervals of increase of the function. Fig. 2 – graph of the derivative 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. 1 Option 2 Option

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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.\

2) <Презентация. Слайды 5,6>.

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

2. Solution of exercises

1) <Презентация. Слайд 7>.

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.

1. [-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. [-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