Presentation on the topic "geometric meaning of the derivative of a function". Algebra presentation "The derivative of a function
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Text content of presentation slides: V.N. Egorova, mathematics teacher, Secondary School No. 1 (part-time) Definition of a derivative. The derivative of a function is one of the most difficult topics in the school curriculum. Not every graduate will answer the question, what is the derivative ASVtg A-? tg B -? ABCP Oral work Tangent is the ratio of the opposite leg to the adjacent
ASVtg A-?tg B -?47ABCHFind the degree measure< В.3Найдите градусную меру < А.Работа устноВычислите tgα, если α = 150°.
The figure shows graphs of three functions. Which one do you think is growing faster? Oral work Kostya, Grisha and Matvey got a job at the same time. Let's see how their income changed during the year: Kostya's income more than doubled in six months. And Grisha's income also increased, but just a little bit. And Matthew's income decreased to zero. The starting conditions are the same, but the rate of change of the function is different. As for Matvey, his income is generally negative. Work orally
Intuitively, we can easily estimate the rate of change of a function. But how do we do it? What we are really looking at is how steeply the graph of the function goes up (or down). In other words, how fast y changes with x. Obviously, the same function at different points can change faster or slower.
The derivative is the rate of change of the function
Problems leading to the concept of a derivative1. The problem of the rate of change of a function A graph of a certain function is drawn. Take a point on it with an abscissa. Draw a tangent to the graph of the function at this point. To estimate the steepness of the function graph, a convenient value is the tangent of the slope of the tangent. We take the angle between the tangent and the positive direction of the OX axis as the slope angle. Find k=tg α∆AMN: ˂ ANM = 90˚, tgα = 𝐴𝑁𝑀𝑁 geometric sense derivativeConspectus
The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point. The geometric meaning of the derivative The derivative of the function is equal to the tangent of the slope of the tangent - this is the geometric meaning of the derivative
S The travel time is equal to tАBU=S / t Tasks leading to the concept of derivative2. The problem of the speed of movement
TASK. A certain body (material point) moves along a straight line, on which the origin, unit of measurement (meter) and direction are given. The law of motion is given by the formula S=s(t), where t is the time (in seconds), s(t) is the position of the body on the straight line (the coordinate of the moving material point) at time t relative to the origin (in meters). Find the speed of the body at time t (in m/s). SOLUTION. Suppose that at the time t the body was at the point MOM=S(t). Let's increment ∆t to the argument t and consider the situation at time t + ∆t . The coordinate of the material point will become different, the body at this moment will be at the point P: OP= s(t+ ∆t) – s(t). This means that in ∆t seconds the body moved from point M to point P. We have: MP=OP – OM = s(t+ ∆t) – s(t). The resulting difference is called the increment of the function: s(t+ ∆t) – s(t)= ∆s. So, MP= ∆s (m). Then the average speed over time: 𝑣av.=∆𝑆∆𝑡
The derivative of a function y = f(x) at a given point x0 is the limit of the ratio of the increment of the function at this point to the increment of the argument, provided that the increment of the argument tends to zero. Derivative notation: 𝑦′𝑥0 or 𝑓′𝑥0 𝑓′𝑥0=lim∆ 𝑥→0∆𝑦∆𝑥 or 𝑓′𝑥0=lim∆𝑥→0∆𝑓∆𝑥 DefinitionSynopsis
Instantaneous speed is the average speed over the interval provided that ∆t→0, that is: х, where ∆х is the increment of the argument. Let's find the increment of the function ∆f(x) = f(x0 + ∆х) – f(x0) → 0 lim∆𝑥→0Δ𝑓(𝑥)Δ𝑥=𝑓′(𝑥)
Example 2. Find the derivative of the function y = x Solution: f(x) = x.1. Take two values of the argument x and x + Δx.2. .3.∆𝑓∆𝑥=∆𝑥∆𝑥=1.4.𝑓′𝑥=lim∆𝑥→0∆𝑓∆𝑥=lim∆𝑥→01=1. Hence, (𝒙)′ = 1 Derivative example Example 3 .Find the derivative of function y = x2Solution: f(x) = x2.1.Take two values of the argument x and x + Δx.2.∆𝑓=𝑓𝑥+∆𝑥−𝑓𝑥=(𝑥+∆𝑥)2−𝑥2=𝑥2 +2𝑥∆𝑥+(∆𝑥)2−𝑥2=∆𝑥(2𝑥+∆𝑥).3.∆𝑓(𝑥)∆𝑥=∆𝑥(2𝑥+∆𝑥)∆𝑥=2𝑥+∆𝑥.4. 2x Derivative calculation example ∆𝑥+𝑚− 𝑘𝑥−𝑚=𝑘𝑥+𝑘∆𝑥−𝑘𝑥=𝑘∆𝑥.3.∆𝑓(𝑥)∆𝑥=𝑘∆𝑥∆𝑥=𝑘.4.𝑓❈ ∆𝑓∆𝑥=lim∆𝑥→0𝑘=𝑘.So, (𝒌𝒙+𝒎)′ = k An example of calculating the derivative and x + Δx.2.∆𝑓=𝑓𝑥+∆𝑥−𝑓𝑥= 1𝑥+∆𝑥−1𝑥=𝑥−𝑥−∆𝑥𝑥(𝑥+∆𝑥)=−∆𝑥𝑥(𝑥+∆.3.∥) 𝑓(𝑥)∆𝑥=−∆𝑥𝑥(𝑥+∆𝑥):∆𝑥=−∆𝑥𝑥(𝑥+∆𝑥)∆𝑥=−1𝑥(𝑥+∆𝑥) →0∆𝑓∆𝑥=lim∆𝑥→0−1𝑥(𝑥+∆𝑥)=−1lim∆𝑥→01𝑥2 +𝑥∆𝑥 = —Lim∆𝑥 → 01Lim∆𝑥 → 0𝑥2+Lim∆𝑥 → 0𝑥∆𝑥 = −1𝑥2. Know, 𝟏𝒙 ′ = — 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 𝟏𝒙𝟐 ′ ′ = 𝟎 𝒙 𝒙 '= 1𝒙𝟐 ′ = 𝟐𝒙𝒌𝒙+++𝟏𝒙𝟐 𝒙 𝒙 𝟐𝒙𝒌𝒙+++ 𝒎′=𝒌𝟏𝒙= −𝟏𝒙𝟐 Finish the sentence: Our today's lesson was about ... In the lesson, I learned that ... In the lesson, I learned ... The derivative of a function at a point is ... the tangent drawn to the function graph at a given point The rate of change of a function is ... I was hard... GOOD FELLOWS!
ppt_y
Attached files
, The geometric meaning of the derivative
Lesson type: learning new material.
The purpose of the lesson: to find out what the geometric meaning of the derivative is, to derive the equation of the tangent to the graph of the function.
Cognitive task: to form an idea of the geometric meaning of the derivative, the ability to draw up an equation for a tangent to a graph of a function at a given point, to find the slope of the tangent to the graph of a function, the angle between the tangent to the graph and the Ox axis.
Developing task: to continue the formation of skills and abilities to work with a scientific text, the ability to analyze information, the ability to systematize, evaluate, use it; development of logical thinking, conscious perception of educational material.
Educational task: increasing interest in the learning process and active perception of educational material, developing communication skills in working in pairs and groups.
Practical task: the formation of critical thinking skills as a creative, analytical, consistent and structured thinking, the formation of self-education skills.
Form of the lesson: a problematic lesson using the technology for the development of critical thinking (TRCM).
Technology used: technology for developing critical thinking, technology for working in collaboration
Techniques used: “Basket of ideas”, “Thick and thin questions”, true, false statements, INSERT, cluster, “Six Thinking Hats”.
Equipment: PowerPoint presentation, interactive whiteboard, handouts (cards, text material, tables), sheets of paper in a cage,
During the classes
Call stage:
1. Introduction of the teacher.
We are working on mastering the topic “Derivative of a function”. You already have knowledge and skills in the technique of differentiation. But why is it necessary to study the derivative of a function?
"Basket of Ideas".
Can you guess where the knowledge gained can be used?
Pupils offer their ideas, which are recorded on the board. We get a cluster, which by the end of the lesson can branch out significantly.
As you can see, we do not have a clear answer to this question. Today we will try to partially answer it. The topic of our lesson is “The geometric meaning of the derivative”.
Activity motivation.
From the open bank of tasks on the FIPI website, materials for preparing for the exam, I chose several tasks that contain the terms “function” and “derivative”. These are tasks B8. They lie in front of you on the desks.
Examples of tasks B8. Exercise. The figures show graphs of functions y = f(x) and tangents to them at a point with abscissa x 0 . Find the value of the derivative of the function f(x) at the point x 0 .
Can you suggest a way to solve these tasks? (Not)
Today we will learn how to solve such tasks and similar ones.
2. Actualization of basic knowledge and skills.
Work in pairs "Make a couple." Application No. 1
There is a table in front of you. Functions and their derivatives are written in disorder in the cells of the table. For each function, find the derivative and write down the correspondence of cell numbers.
Working hours
- 2 minutes each student works independently.
- 2 minutes - work in pairs. Discussion of the results and recording in the answer card.
- 1 minute - check the work.
- What was easy and what didn't work?
- Finding derivatives of what functions caused difficulties?
3. Work with the vocabulary of the lesson.
Lesson vocabulary: derivative; a function differentiable at a point; linear function, graph of a linear function, slope of a straight line, tangent to a graph, tangent of an angle in a right triangle, values of tangents of angles (acute, obtuse).
Guys, ask each other questions using the words of the dictionary at least 4 questions. Questions should not require “yes” or “no” answers.
Then we listen to one question and answer from each pair, the questions should not be repeated.
There are question cards on the tables. They all begin with the words “Do you believe that…”
The answer to the question can only be “yes” or “no”. If “yes”, then to the right of the question in the first column put a “+” sign, if “no”, then a “-” sign. If in doubt, put a “?” sign.
Work in pairs. Working time 3 minutes. (Appendix No. 2)
After listening to the students' answers, the first column of the pivot table on the board is filled in.
Stage of comprehension of the content (10 min.).
Summing up the work with the questions of the table, the teacher prepares the students for the idea that when answering questions, we do not yet know whether we are right or not.
Assignment to groups. Answers to questions can be found by studying the text of §8 pp. 84-87 (or the proposed sheets with the extraction of paragraph material, on which you can freely make handwritten notes), using the INSERT technique - reception of semantic marking of the text.
V - already knew
- thought otherwise
Didn't understand)
Discussion of the text of paragraph §8.
What did you already know, what was new for you, and what did you not understand?
Discussion, clarification of misunderstood.
Group responses to questions:
What is the sign of f "(x 0)?
Reflection stage. Preliminary summing up.
Let's return to the questions considered at the beginning of the lesson and discuss the results. Let's see, maybe our opinion after work has changed.
Students in groups compare their assumptions with the information obtained in the course of working with the textbook, make changes to the table, share thoughts with the class, and discuss the answers to each question.
Call stage.
What do you think, in what cases, in the performance of which tasks can the considered theoretical material be applied?
Estimated answers of students: finding the value of the derivative of the function f (x) at the point x 0 according to the graph of the tangent to the function; the angle between the tangent to the graph of the function at the point x 0 and the x-axis; obtaining the equation of the tangent to the graph of the function.
I propose to start work on algorithms for finding the value of the derivative of the function f (x) at the point x 0 according to the graph of the tangent to the function; the angle between the tangent to the graph of the function at the point x 0 and the x-axis; obtaining the equation of the tangent to the graph of the function.
Make up algorithms:
- finding the value of the derivative of the function f(x) at the point x 0 according to the graph of the tangent to the function;
- the angle between the tangent to the graph of the function at the point x 0 and the x-axis;
- obtaining the equation of the tangent to the graph of the function.
The stage of comprehension of the content.
1) Work on the compilation of algorithms.
Everyone does work in a notebook. And then, having discussed in the group, they come to a consensus. After the work is completed, a representative of each group will defend their work.
Algorithm for finding the value of the derivative of the function f (x) at the point x 0 according to the graph of the tangent to the function.
Finding algorithm the angle between the tangent to the graph of the function at the point x 0 and the x-axis.
.Algorithm for obtaining the equation of the tangent to the function graph
1) Work on the application of what has been learned in practice. (Appendix No. 4)
2) Consideration of tasks B8.
The figure shows the graph of the function y \u003d f (x) and the tangent to it at the point with the abscissa x 0. Find the value of the derivative of the function f(x) at the point x 0
Problem 2. The figure shows the graph of the function y = f(x) and the tangent to it at the point with the abscissa x 0 . Find the value of the derivative of the function f(x) at the point x 0 .
Task 3. The figure shows the graph of the function y = f(x) and the tangent to it at the point with the abscissa x 0 . Find the value of the derivative of the function f(x) at the point x 0 .
Problem 4. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x 0 . Find the value of the derivative of the function f(x) at the point x 0 .
Answers. Task 1. 2. Task 2. -1 Task 3. 0 Task 4. 0.2 .
Reflection.
Let's summarize.
- Self-esteem
“Self-check sheet, self-assessment”
| Last name, first name | Tasks | ||||||
| Independent work "Make a couple" |
“Lesson Dictionary” (for each correct answer 0.5 points) |
“Do you believe that…” (up to 9 p.) |
Answers to questions to the text (for each correct answer 1 point) |
Drawing up an algorithm (up to 3 points) |
Chart Tasks (up to 3 points) |
Training task (up to 6 p.) |
|
| Evaluation criteria: “3” - 20-26 points; “4” - 27 - 32 points; “5” - 33 or more | |||||||
- Why study the derivative of a function? (To study the functions, the speed of various processes in physics, chemistry ...)
- Using the “Six Thinking Hats” technique, mentally putting on a hat of a certain color, we will analyze the work in the lesson. Changing hats will allow us to see the lesson from different perspectives to get the most complete picture.
White hat: information (concrete judgments without emotional overtones).
Red Hat: Emotional judgments without explanation.
Black hat: criticism - reflects problems and difficulties.
Yellow hat: positive judgments.
Green hat: creative judgments, suggestions.
Blue hat: a generalization of what has been said, a philosophical view.
In fact, we have only touched upon the solution of tasks on the use of the geometric meaning of the derivative. Further, even more interesting, diverse and complex tasks await us.
Homework: § 8 pp. 84-88, no. 89-92, 94-95 (even).
Literature
- Zair.Bek S.I. The development of critical thinking in the classroom: a guide for general education teachers. institutions. - M. Education, 2011. - 223 p.
- Kolyagin Yu.M. Algebra and the beginnings of analysis. Grade 11: textbook. for general education institutions: basic and profile levels. – M.: Enlightenment, 2010.
- Open bank of tasks in mathematics http://mathege.ru/or/ege/Main.html?view=TrainArchive
- Open bank of tasks USE/Mathematics http://www.fipi.ru/os11/xmodules/qprint/afrms.php?proj=
Internet sites related to critical thinking
Critical Thinking http://www.criticalthinking.org/
http://www.ct-net.net/ru/rwct_tcp_ru
To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com
Slides captions:
The geometric meaning of the derivative. Tangent equation. f(x)
Using formulas and differentiation rules, find the derivatives of the following functions:
one . What is the geometric meaning of the derivative? 2. Can a tangent be drawn at any point on the graph? Which function is called differentiable at a point? 3 . The tangent is inclined at an obtuse angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function? 4 . The tangent is inclined at an acute angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function? 5 . The tangent is inclined at right angles to the positive direction of the x-axis. What can be said about the derivative?
for differentiable functions: 0 ° ≤ α ≤ 180 ° , α ≠ 90 ° α - obtuse tg α 0 f ´(x 1) >0 position of the tangent is not defined tg α n.a. f ´(x 3) n.a. α = 0 tg α =0 f ´(x 2) = 0
y \u003d f / (x 0) (x - x 0) + f (x 0) (x 0; f (x 0)) - coordinates of the touch point f ´ (x 0) \u003d tg α \u003d k - slope angle tangent tangent at a given point or slope (x; y) - coordinates of any point of the tangent Tangent equation
No. 1. Find the slope of the tangent to the curve at the point with the abscissa x 0 = - 2. Task B8 FBTZ USE
No. 2. Specify the value of the coefficient k at which the graphs of linear functions y = 8x+12 and y = k x - 3 are parallel. Answer: 8. Task B8 FBTZ USE
0 Y X 1 -1 1 -1 №3. The function y \u003d f (x) is defined on the interval (-7; 7). The figure below shows a graph of its derivative. Find the number of tangents to the graph of the function y \u003d f (x) that are parallel to the x-axis. Answer: 3. Task B8 FBTZ USE
No. 4. The figure shows a straight line that is tangent to the graph of the function y \u003d p (x) at the point (x 0; p (x 0)). Find the value of the derivative at the point x 0. Answer: -0.5. Task B8 FBTZ USE
0 Y X 1 -1 1 -1 №5. All tangents parallel to the straight line y=2x+5 or coinciding with it were drawn to the graph of the function f(x). Specify the number of touch points. Answer: 4. Task B8 FBTZ USE
Write the equations of tangents to the graph of the function at the points of its intersection with the x-axis. Independent work
Surname, name Testing Creative task Lesson +,-, :), :(, : |
1 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. Write an equation for the tangent to the graph of the function f (x) \u003d 0.5 -4, if the tangent forms an angle of 45 degrees with the positive direction of the x-axis.
2 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. Write the equation of the tangent to the graph of the function f (x) \u003d, parallel to the straight line y \u003d 9 x - 7.
3 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. The straight line passing through the origin touches the graph of the function y \u003d f (x) at point A (-7; 14). Find.
4 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. The straight line y \u003d -4x-11 is tangent to the graph of the function. Find the abscissa of the point of contact.
Preview:
Lesson script
in algebra and the beginnings of analysis in the 10th grade.
Topic: “The geometric meaning of the derivative. Tangent Equation»
Objectives: 1) to continue the formation of a system of mathematical knowledge and skills on the topic "Tangential Equation", necessary for application in practice, study related disciplines, continuing education;
2) develop computer and multimedia skills curricula to organize their own cognitive activity;
3) develop logical thinking, algorithmic culture, critical thinking;
4) to cultivate tolerance, communication.
During the classes.
- Organizing time.
- Message topics, setting goals for the lesson.
- Checking homework.
- Basic Level Assignments (Scanned Job)
- The students solved the task of practical content of an increased level of complexity by choice. One of the students presents his solution in the form of a multimedia project: “A parabolic bridge is being built connecting points A and B, the distance between which is 200 m. The entrance to the bridge and the exit from the bridge should be straight sections of the path, these sections are directed to the horizon at an angle 150. The indicated lines must be tangent to the parabola. Equate the bridge profile in the given coordinate system"
- Updating of basic knowledge.
- Differentiate functions:
- ()
- y=4()
- y=7x+4()
- y=tg x+ ()
- y=x 3 sinx()
- y=()
- Answer the questions:
- What is the geometric meaning of the derivative?
- Can a tangent be drawn at any point on the graph? Which function is called differentiable at a point?
- The tangent is inclined at an obtuse angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
- The tangent is inclined at an acute angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
- The tangent is inclined at right angles to the positive direction of the OX axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
- What should the graph of a function differentiable at a point look like?
- What is the tangent equation? Explain that in this equation (x 0; f (x 0 )) , f ’ (x 0 ), (x; y)
- Find the slope of the tangent to the curve y=2x 2 +x at the point with abscissa x 0 =-2 (-7).
- Specify the value of the coefficient k at which the graphs of linear functions y = 8x+12 and y = kx – 3 are parallel. (eight)
- The function y \u003d f (x) is defined on the interval (-7; 7). The figure below shows a graph of its derivative. Find the number of tangents to the graph of the function y \u003d f (x) that are parallel to the x-axis. (3)
- The figure shows a straight line that is tangent to the graph of the function y \u003d p (x) at the point (x 0; p(x 0 )). Find the value of the derivative at the point x 0 . (-0,5)
- All tangents parallel to the straight line y=2x+5 or coinciding with it were drawn to the graph of the function f(x). Specify the number of touch points. (4)
- Independent work with selective checking (one student performs the task at the blackboard). Write the equations of tangents to the graph of a function f(x) \u003d 4 - x 2 at the points of its intersection with the x-axis. (y \u003d - + 4x + 8). Demonstration illustration.
- Work in creative groups for 5-6 people.
- Pass computer testing in turn (Additional testing for lesson 5, options 1 and 2 "Lessons of Cyril and Methodius Algebra"). The results are entered into the diagnostic card.
- Complete tasks in notebooks:
1 group
y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b
No. 4. Write the equation of the tangent to the graph of the function f(x) = 0.5 x 2 -4 if the tangent forms an angle of 45 with the x-axis 0 .
2 group
No. 1. What is the geometric meaning of the derivative?
No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?
No. 3. What is the tangent equation?
№ 4. Write the equation of the tangent to the graph of the function f (x) \u003d x 3 /3 parallel to the line y \u003d 9 x - 7.
3 group
No. 1. What is the geometric meaning of the derivative?
No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?
No. 3. What is the tangent equation?
No. 4. The straight line passing through the origin touches the graph of the function
y \u003d f (x) at point A (-7; 14). Find . (Assignment from KIM to prepare for the exam)
4 group
No. 1. What is the geometric meaning of the derivative?
No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?
No. 3. What is the tangent equation?
No. 4. The line y=-4x-11 is tangent to the graph of the function f(x)=x 3+7x2 +7x-6. Find the abscissa of the point of contact. (Assignment from KIM to prepare for the exam)
A report on the work done is carried out at the blackboard by one of the group. It is chosen by the teacher or the group. The mark of the respondent and the self-assessment of each member of the group are entered in the diagnostic card.
- Summing up the lesson. Reflection.
- Homework consists of exercises B8 FBTZ FIPI.
