General lesson on the topic:
"Using the derivative and its graph to read the properties of a function"
Lesson type: a generalizing lesson using ICT in the form of a presentation.
Lesson Objectives:
Educational:
To promote the assimilation by students of the use of the derivative in practical tasks;
To teach students to clearly use the properties of a function and a derivative.
Developing:
Develop the ability to analyze the question of the task and draw conclusions;
Develop skills to apply existing knowledge in practical tasks.
Educational:
Raising interest in the subject;
The need for these theoretical and practical skills to continue your studies.
Lesson objectives:
Develop specific skills and abilities for working with a graph of the derivative of a function for their use when passing the exam;
Prepare for the test.
Lesson plan.
1. Actualization of basic knowledge (AKB).
2. Development of knowledge, skills and abilities on the topic.
3. Testing (B8 from the exam).
4. Mutual verification, grading the "neighbor".
5. Summing up the lessons of the lesson.
Equipment: computer class, whiteboard, marker, tests (2 options).
During the classes.
Organizational moment.
Teacher . Hello, have a seat.
In the course of studying the topic “Investigation of functions using the derivative”, the skills were formed to find critical points of a function, a derivative, to determine the properties of a 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 in a variety of tasks related to graphs of functions and their derivatives.
In preparation for the exam in mathematics in KIMs, tasks were given for using the derivative graph to study functions. Therefore, in this lesson, we must systematize our knowledge on this topic and learn how to quickly find answers to the questions of tasks B8.
Slide number 1.
Topic: "Application of the derivative and its graph to read the properties of functions"
Lesson objectives:
Development of the ZUN of the use of the derivative, its geometric meaning and the graph of the derivative to determine the properties of functions.
Development of the efficiency of performing USE tests.
Education of such personality traits as attentiveness, the ability to work with text, the ability to work with a graph of the derivative
2. Actualization of basic knowledge (AKB). Slides #4 to #10.
Questions will now appear on the screen for repetition. Your task: to give a clear and concise answer to each item. The correctness of your answer can be checked on the screen.
( The question first appears on the screen, after the students' answers, the correct answer appears for verification.)
List of questions for AOP.
Definition of a derivative.
The geometric meaning of the 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.
Application of the derivative to find intervals of monotonicity of a function.
Application of the derivative to determine critical points, extremum points
6 .Necessary and sufficient conditions for an extremum
7 . Applying a derivative to find the largest and smallest value of a function
(Students answer each item, accompanying their answers with notes and drawings on the board. With 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 #11 to #15.
Students are offered assignments from the KIMs of the Unified State Examination in mathematics of past years, from sites on the Internet on the use of the derivative and its graph to study the properties of functions. Tasks appear sequentially. Students write their solutions on the board or verbally. Then the correct solution appears on the slide and is checked against the students' solution. If a mistake is made in the decision, then it is analyzed by the whole class.
Slide #16 and #17.
Further in the class, it is advisable to consider the key task: according to the graph of the derivative, the 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, if necessary, corrected, 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, rules.
Testing (B8 from the exam). Slides from number 18 to number 29. Slide number 30 - the keys to the test.
Teacher : So, we summarized your knowledge on this topic: we repeated the basic properties of the derivative, solved problems related to the derivative graph, analyzed the complex and problematic aspects of using the derivative and the derivative graph to study the properties of functions.
Now we will test in 2 options. Tasks will appear on the screen both options, simultaneously. You study the question, find the answer, enter it in the answer sheet. After completing the test, exchange forms and check the work of a neighbor according to ready-made answers. 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 have considered the relationship between the monotonicity of a function and the sign of its derivative, and sufficient conditions for the existence of an extremum. We considered various tasks for reading the graph of the derivative of a function that are found in the texts of the unified state exam. All the tasks we have considered are good in that they do not take much time to complete.
During the unified state exam, it is very important to write down the answer quickly and correctly.
Submit answer sheets. The grade for the lesson is already known to you and will be put in the journal.
I think the class is ready for the test.
Homework will be creative . slide number 33 .
Further in the class, it is advisable to consider the key task: according to the graph of the derivative, the 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, if necessary, corrected, 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, rules.
Let's give an example of such a task: on the board (for example, using a projector), students are offered a graph of the derivative, 10 tasks were formulated on it (not quite correct or duplicate questions were rejected).
The function y = f(x) is defined and continuous on the interval [–6; 6].
From the graph of the derivative y \u003d 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) the maximum point of the function y = f(x);
5) the 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 largest value on the segment ;
7) the abscissa of the graph point at which the function y = f(x) takes the smallest value on the segment [–2; 2];
8) the number of points of the graph of the function y = f(x), in which the tangent is perpendicular to the axis Oy;
9) the number of points in the graph of the function y = f(x), in which the tangent forms an angle of 60° with the positive direction of the Ox axis;
10) the abscissa of the point of the graph of the function y = f (x), in 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 consolidate the skills of studying the properties of a function at home, students can be offered a task related to reading the same graph, but in one case it is a graph of a function, and in the other it is a graph of its derivative.
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The function y = f(x) is defined and continuous on the interval [–6; five]. The figure shows:
a) graph of the function y = f(x);
b) graph of the derivative y \u003d 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 point of the graph of the function y = f(x), in which it takes the largest value on the segment;
4) the number of points in the graph of the function y = f(x) in 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.
For control, work can be organized in pairs: each student prepares a graph of the derivative on a card for his partner in advance and below offers 4-5 questions to determine the properties of the function. At the lessons they exchange cards, perform the proposed tasks, after which each checks and evaluates the work of the partner.
Back forward
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Lesson Objectives:
Educational: To consolidate the skills of students working with function graphs in preparation for the exam.
Developing: to develop students' cognitive interest in academic disciplines, the ability to apply their knowledge in practice.
Educational: to cultivate attention, accuracy, to expand the horizons of students.
Equipment and materials: computer, screen, projector, presentation “Reading graphs. USE"
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 a function? Determine the domain of definition and range of functions.\
2) <Презентация. Слайды 5,6>.
What 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 task: 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 the 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), given on the interval [-4;7]. Indicate all the values of X for which the inequality f(x) -1 is satisfied.
c) The figure shows the graphs of the functions y=f(x),and y=g(x), given on the interval [-3;6]. Indicate all values of X for which the inequality f(x) g(x) is satisfied
3) <Презентация. Слайд 11>.
Increasing and decreasing functions
One of the figures shows a graph of a function that increases on the segment , the other shows a function that decreases on the segment [-2; 0]. List these pictures.
4) <Презентация. Слайды 12,13,14>.
Exponential and logarithmic functions
a) What is the condition for the increase and decrease of the exponential and logarithmic functions. Through which point do the graphs of the exponential and logarithmic functions pass, what property do the graphs of these functions have?
b) One of the figures shows a graph of the function y \u003d 2 -x. Indicate this figure .
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). Specify the number of this chart.
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) – 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. Specify the number of points in the graph of the function where these tangents are drawn.
K = tga = f'(x o). By condition, k \u003d -2. Therefore, f '(x o) \u003d -2. We draw a straight line y \u003d -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 in the graph of the function y=f(x) where the tangents to the graph are parallel to the x-axis or coincide with it.
The angular coefficient of straight lines parallel to the x-axis or coinciding with it is equal to 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), in which the tangents to the graph are inclined at an angle of 135 o to the positive direction of the x-axis.
6) <Презентация. Слайды 18, 19>.
Finding the slope of the 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. Specify the abscissa of the point where 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 \u003d -3 at the point x \u003d 2. Therefore, the tangent to the graph has the smallest slope at the 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. Specify 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 \u003d f (x) and a tangent to it at a point with the abscissa x o. Find the value of the derivative f `(x) at point x o
f'(xo)=tga. Since in 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 x=4, the derivative changes sign from minus to plus. So x=4 is the minimum point of the function y=f(x)
At the point x \u003d 1, the derivative changes sign with plus and minus . So x=1 is a point maximum functions y=f(x))
3. Independent work
<Презентация. Слайд 22>.
1 option
1) Find the scope of the function.
2) Solve the inequality f(x) 0
3) Determine the intervals of decreasing 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 the function.
2) Solve the inequality f(x) 0
3) Determine the intervals of increasing function.
Graph of the derivative of the function y=f(x)
4) Find the maximum points of the function.
5) Specify 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
slide 12
The graphs of these functions increase when a > 1 and decrease at 0
slide 13
One of the figures shows a graph of the function y=2-x. Specify this picture. Graph of an exponential function 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). Specify the number of this chart. The graph of the logarithmic function y=log5x passes through the point (1;0), then, if x -4 =1, toy=0, x=1+4, x=5. (5;0) – 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 are drawn to the graph of the function, parallel to the straight line y=5-2x (or coinciding with it). Specify the number of points in the graph of the function where these tangents are drawn. K = tga = f’(xo) According to the condition k=-2. Therefore, f’(xo)=-2 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 in the graph of the function y=f(x) where the tangents to the graph are parallel to the x-axis or coincide with it. The slope of the lines parallel to the x-axis or coinciding with it is equal to zero. Therefore K=tg a = f `(xo)=0 The OX axis intersects this chart 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 135ok to the positive direction of the x-axis. K = tg 135o= f'(xo) tg 135o=tg(180o-45o)=-tg45o=-1 Therefore f`(xo)=-1 carried out in triplets. 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. Specify 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 the point x=2. Therefore, the tangent to the graph has the smallest slope at the point x \u003d 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. Specify the abscissa of the day, in which the tangent to the graph of the function y=f(x) has the largest slope. k \u003d tg a \u003d f '(xo) Therefore, the tangent to the graph has the largest slope at the point x \u003d -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 \u003d f (x) and a tangent to it at a point with the xo abscissa. Find the value of the derivative f `(x) at the point xo f ’(xo) \u003d tg a Since in the figure a is an obtuse angle, then tg a
slide 21
At the point x=4, the derivative changes sign from minus to plus. Meanx=4 is the minimum point of the function y=f(x) 4 At points x=1, the derivative changes sign from the plus sign. minus Valuex=1 is the maximum point of the function y=f(x))
slide 22
Fig.11) Find the scope of the function. 2) Solve the inequality f(x) ≥ 0 3) Determine the intervals of decreasing function. Fig.2-graph of the derivative of the function y=f(x) 4) Find the minimum points of the function. 5) Specify the abscissa of the point where the tangent to the graph of the function y=f(x) has the largest slope. Fig.11) Find the range of the function. 2) Solve the inequality f(x)≤ 0 3) Determine the intervals of increasing function. Fig.2-graph of the derivative of the function y=f(x) 4) Find the maximum points of the function. 5) Specify the abscissa of the point where the tangent to the graph of the function y=f(x) has the smallest slope. 1 Option 2 Option
Elements of mathematical analysis in the Unified State Examination Malinovskaya Galina Mikhailovna [email protected] Reference material Table of derivatives of the main functions. Differentiation rules (derivative of sum, product, quotient of two functions). Derivative of a complex function. The geometric meaning of the derivative. The physical meaning of the derivative. Reference material Extremum points (maximum or minimum) of a function given graphically. Finding the largest (smallest) value of a function that is continuous on a given interval. An antiderivative of a function. Newton-Leibniz formula. Finding the area of a curvilinear trapezoid. Physical applications 1.1 A material point moves in a straight line 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 The 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 her speed equal to 2 m/s? Solution: We are looking for the derivative x(t) (of the path function in time). In task 1.1 we substitute its value instead of t and calculate the speed (Answer: 59). In problem 1.2, we equate the found derivative to a given number and solve the equation for 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 point of contact. 2.2 The line 𝑦 = 3𝑥 + 1 is tangent to the 2nd graph of the function 𝑎𝑥 + 2𝑥 + 3 . Find a. 2.3 The line 𝑦 = −5𝑥 + 8 is tangent to the 2nd graph of the function 28𝑥 + 𝑏𝑥 + 15 . Find b, given that the abscissa of the tangent point 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 are looking 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 make up 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 a parameter), we are looking 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 the abscissa 𝑥0 . Find the value of the derivative of the function f(x) at the 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 the 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 the 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 slope of the tangent to the graph of the function drawn at the given point. We “finish” a right-angled 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 “grows”) and negative if the angle is obtuse (the tangent decreases). In Problem 2.7, it is necessary 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 where 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 where 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 select the 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 task, we either recalculate the number of integers included in the 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 a derivative, then we are only interested in the signs of the derivative and the abscissas of the points of intersection with the Ox axis. For clarity, you can draw a more familiar figure with the signs of the derivative with respect to the obtained intervals and the behavior of the function. According to the picture, answer the question of the task. (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 interval [-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 interval [-10;10] Solution: We are looking for the points of intersection of the graph of the derivative with the Ox axis, highlighting that part of the axis that is indicated in the problem. Determine the sign of the derivative on each of the obtained intervals (if the graph of the derivative is below the axis, then “-”, if above, then “+”). The maximum points will be those where the sign has changed from "+" to "-", the minimum - from "-" to "+". The extremum points are both. Answers: 3.4) 1; 3.5) 5. 3.6 The figure shows the graph y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-8;3). At what point of the segment [-3; 2] the function f(x) takes the maximum 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] the function f(x) takes 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 the segment has a single extremum point on it and this is the maximum (minimum) point, then the largest (smallest) value of the function on this segment is reached in given point. If a function continuous on a segment is monotonic, then it reaches its minimum and maximum values on the 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 where the tangent to the graph of the function is parallel to the line y=6 or coincides with it. 3.9 The figure shows a graph of the function y=f(x) and eight points on the x-axis: 𝑥1 ,𝑥2 ,𝑥3 , … , 𝑥12 . At how many of these points is the derivative of the function 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 decreasing function f(x). In your answer, indicate the sum of integer points included in these intervals. 4.5 The figure shows the graph 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 the graph y=𝑓 ′ (𝑥) - the derivative of the function f(x), defined on the interval (-10;2). Find the number of points where the tangent to the graph of the function f(x) is parallel to or coincides with the line y=-2x-11. Solution: 4.6 Since the graph of the derivative is shown in the figure, and the tangent is parallel to this line, the derivative of the function at this point is -2. We are looking for points on the graph of the derivative 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 the graph y=𝑓 ′ (𝑥) - the derivative of the function f(x). Find the abscissa of the point where the tangent to the graph y=f(x) is parallel to the x-axis or coincides with it. Solution: If the line is parallel to the Ox axis, then its slope is zero. The slope of the tangent is zero, so the derivative is zero. We are looking for the abscissa of the point of intersection of the graph of the derivative with the axis Ox. We get -3. 4.9 The figure shows the graph of the function y=𝑓 ′ (x) the derivative of the function f(x) and eight points on the x-axis: 𝑥1 ,𝑥2 ,𝑥3 , … , 𝑥8 . At how many of these points does the derivative of the function f(x) increase? Geometric meaning of a 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, compute F(8)-F(2), where F(x) is one of the antiderivatives of f(x). Solution: The area of a curvilinear trapezoid is calculated through a definite integral. The definite integral is calculated by the Newton-Leibniz formula as an increment of the antiderivative. In problem 5.1, we calculate the area of the trapezoid according to the well-known geometry course formula (this will be the increment of the antiderivative). In Problems 5.2 and 5.3, an antiderivative is already 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 curvilinear trapezoid is calculated through a definite integral. The definite integral is calculated by the Newton-Leibniz formula as an increment of the antiderivative. In problem 5.1, we calculate the area of the trapezoid according to the well-known geometry course formula (this will be the increment of the antiderivative). In Problem 5.2, an 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 exam in mathematics
