Geometric meaning of the derivative presentation of the Ulama. Presentation on the topic "geometric meaning of the derivative function." Practical research work Geometric meaning of derivative
, Geometric meaning of 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 the graph of a function at a given point, to find the angular coefficient of the tangent to the graph of a function, the angle between the tangent to the graph and the Ox axis.
Developmental task: to continue the formation of skills and abilities to work with scientific text, the ability to analyze information, the ability to systematize, evaluate, and 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 for working in pairs and groups.
Practical task: developing critical thinking skills as creative, analytical, consistent and structured thinking, developing self-education skills.
Lesson form: problem-based lesson using technology for the development of critical thinking (TRKM).
Technology used: technology for developing critical thinking, technology for working in collaboration
Techniques used: “Basket of ideas”, “Thick and thin questions”, true and false statements, INSERT, cluster, “Six thinking hats”.
Equipment: PowerPoint presentation, interactive whiteboard, handouts (cards, text material, tables), squared sheets of paper,
During the classes
Call stage:
1. Teacher's introduction.
We are working on mastering the topic “Derivative of a function”. You already have knowledge and skills in differentiation techniques. But why is it necessary to study the derivative of a function?
“Basket of Ideas.”
Suggest where the knowledge gained can be used?
Students offer their ideas, which are recorded on the board. We get a cluster that can branch out significantly by the end of the lesson.
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 “Geometric meaning of derivatives”.
Motivation for activity.
From open bank assignments on the FIPI website, preparation materials for the Unified State Exam, I chose several assignments 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 the functions y = f(x) and tangents to them at the point with the 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? (No)
Today we will learn how to solve such tasks and similar ones.
2. Updating basic knowledge and skills.
Work in pairs “Make a pair.” Appendix No. 1
There is a table in front of you. Functions and their derivatives are written in disarray in the cells of the table. For each function, find the derivative and write down the correspondence of cell numbers.
Working hours
- Each student works independently for 2 minutes.
- 2 minutes - work in pairs. Discuss the results and write down the answers on the card.
- 1 minute – check the work.
- What was easy and what didn't work out?
- Finding the derivatives of which functions caused difficulties?
3. Working with the lesson dictionary.
Lesson vocabulary: derivative; function differentiable at a point; linear function, graph of a linear function, slope of a line, tangent to the graph, tangent of an angle in a right triangle, values of tangents of angles (acute, obtuse).
Guys, ask each other questions using vocabulary words at least 4 questions. Questions should not require “yes” or “no” answers.
Then one by one to the question asked and we listen to the answer from each pair; questions should not be repeated.
You have cards with questions on your 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. Operating time 3 minutes. (Appendix No. 2)
After hearing the students' answers, the first column of the summary table on the board is filled in.
Stage of understanding the content (10 min.).
By summing up the work with the questions in the table, the teacher prepares students for the idea that when answering questions, we do not yet know whether we are right or wrong.
Group assignment. 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 - method of semantic marking of text.
V - already knew
– - thought differently
Didn't understand)
Discussion of the text of paragraph §8.
What did you already know, what is new to you, and what did you not understand?
Discussion, clarification of what is not understood.
Group answers to questions:
What sign does f "(x 0) have?
Reflection stage. Preliminary summary.
Let's return to the questions discussed at the beginning of the lesson and discuss the results obtained. Let's see, maybe our opinion has changed after work.
Students in groups compare their assumptions with the information obtained from working with the textbook, make changes to the table, share their thoughts with the class, and discuss the answers to each question.
Call stage.
In what cases and when performing what tasks do you think the discussed theoretical material can be applied?
Expected student answers: finding the value of the derivative of the function f(x) at point x 0 from the graph of the tangent to the function; the angle between the tangent to the graph of the function at point x 0 and the Ox axis; obtaining the equation of the tangent to the graph of a function.
I propose to start working on algorithms for finding the value of the derivative of the function f(x) at point x 0 using the graph of the tangent to the function; the angle between the tangent to the graph of the function at point x 0 and the Ox axis; obtaining the equation of the tangent to the graph of the function.
Create algorithms:
- finding the value of the derivative of the function f(x) at 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 point x 0 and the Ox axis;
- obtaining the equation of the tangent to the graph of the function.
Stage of understanding the content.
1) Work on compiling algorithms.
Everyone does the work in a notebook. And then, after discussing in the group, they come to a consensus. After completing the work, a representative from each group defends their work.
An algorithm for finding the value of the derivative of the function f(x) at point x 0 using the graph of the tangent to the function.
Finding algorithm the angle between the tangent to the graph of the function at point x0 and the Ox axis.
.Algorithm for obtaining the equation of the tangent to the graph of a function
1) Work on applying what has been learned in practice. (Appendix No. 4)
2) Review of tasks B8.
The figure shows a graph of the function y = f(x) and a 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 a graph of the function y = f(x) and a 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 3. The figure shows a graph of the function y = f(x) and a 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. Problem 1. 2. Problem 2. -1 Problem 3. 0 Problem 4. 0.2 .
Reflection.
Let's summarize.
- Self-esteem
“Self-test, self-assessment sheet”
| Last name, first name | Tasks | ||||||
| Independent work “Make a pair” |
“Lesson vocabulary” (for each correct answer 0.5 points.) |
“Do you believe that...” (up to 9 points) |
Answers to questions about the text (for each correct answer 1 point.) |
Drawing up an algorithm (up to 3 points) |
Tasks according to schedules (up to 3 points) |
Training task (up to 6 points) |
|
| Evaluation criteria: “3” - 20-26 points; “4” - 27 – 32 points; “5” - 33 or more | |||||||
- Why is it necessary to study the derivative of a function?
- (To study functions, 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 (specific judgments without emotional connotation).
Red hat: emotional judgments without explanation.
Black hat: criticism – reflects problems and difficulties.
Yellow hat: positive judgments.
Green hat: creative judgments, suggestions.
Blue hat: generalization of what has been said, philosophical view.
In fact, we have only touched upon solving problems using the geometric meaning of the derivative. Further, even more interesting, varied and complex tasks await us. Homework:
§ 8 pp.84-88, No. 89-92, 94-95 (even).
- Literature
- Zaire.Bek S.I. Development of critical thinking in the classroom: a manual for general education teachers.
- institutions. – M. Education, 2011. – 223 p. Kolyagin Yu.M. Algebra and the beginnings of analysis. 11th grade: educational. for general education institutions: basic and profile levels. – M.: Education, 2010.
- Open bank
math assignments http://mathege.ru/or/ege/Main.html?view=TrainArchive
Open bank of Unified State Examination/Mathematics tasks http://www.fipi.ru/os11/xmodules/qprint/afrms.php?proj=
Websites related to the topic of critical thinking
Municipal budgetary educational institution
Glukhovskaya secondary school
Abstract open lesson in algebra
on the topic of:
“The derivative and its geometric meaning. Derivative in the Unified State Exam"
mathematics and computer science teacher
Dikalov Dmitry Gennadievich
2015
Lesson summary on the topic: Derivative and its geometric meaning
Lesson objectives:
Educational:
- Review the basic concepts of the section “Derivative”
- Teach students how to quickly solve problems on the topic “Derivative” from the Unified State Exam options
Educational:
- Development of cognitive interest, logical thinking, development of memory, attentiveness.
- cultivate interest in the structure of computer networks.
Educational:
- cultivate a conscientious attitude to work and initiative;
- instilling discipline and organization
Lesson type:
- lesson of repetition and consolidation of knowledge
Lesson structure:
- Organizing time;
- updating of background knowledge
- problem solving
- homework
Equipment : program Microsoft presentations Office PowerPoint, presentation, computer, multimedia projector, interactive whiteboard.
Lesson plan:
- Organizational moment (1 min)
- Updating knowledge (5 min)
- Problem solving (34 min)
- Lesson summary (4 min)
- Homework (1 min)
During the classes:
I. Organizational moment
The teacher greets, introduces the topic, goals and progress of the lesson.
II. Updating knowledge
- 1. What is the geometric meaning of the derivative?
- How are the intervals of increasing (decreasing) functions found?
- What is the algorithm for finding extremum points?
- How do stationary points differ from extremum points?
III. Problem solving.
Solving problems on finding the derivative at a point, finding intervals of increasing and decreasing, finding points at which the derivative = 0, finding the largest and smallest values of a function.
Students solve these problems using interactive whiteboard, each task is depicted on a separate slide.
Students discuss the nuances of solving problems as they move through the slides.
The following problems are offered to students to solve independently.
IV. Summing up the lesson.
To summarize the lesson, 1-2 students are called to the board to solve problems from textbook No. 956 (1,2): find the intervals of increasing and decreasing function y = 2x 3 +3x 2 -2
Student solution:
To find the intervals of increase and decrease of a function, we find its derivative:
y`=6x 2 +6x
To find stationary points, we equate the derivative to 0 and solve this equation, we get points x=0 and x=-1. Let us find the extremum points among these points. To do this, we determine the sign of the derivative on each of the three intervals. On the interval x0, the derivative is positive, which means that the function increases on these intervals. On the interval
1
The student writes down the answer.
V. Homework
No. 957, No. 956 (to complete)
Giving grades to students who were active in the lesson.
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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 a function is equal to the tangent of the tangent angle - this is the geometric meaning of the derivative
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Instantaneous speed is the average speed over the interval provided that ∆t→0, i.e.: 𝒍𝒊𝒎∆𝒕→𝟎𝒗av.=𝒍𝒊𝒎∆𝒕→𝟎∆𝑺∆𝒕 Instantaneous speed we look at two values of the argument x0 and ∆ x, where ∆x is the increment of the argument. Let's find the increment of the function ∆f(x) = f(x0 + ∆x) – f(x0) Let's find the ratio of the increment of the function to the increment of the argument ∆𝐟(x)∆x Let's calculate the limit of this ratio at ∆x → 0 lim∆𝑥→0Δ𝑓(𝑥)Δ𝑥=𝑓′(𝑥) Algorithm for finding the derivative (by definition) Example of calculating the derivative Solution Notes
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.∆𝑓=𝑓𝑥+∆𝑥−𝑓𝑥=𝑥+∆𝑥−𝑥=∆𝑥 .3.∆𝑓∆𝑥=∆𝑥∆𝑥=1.4.𝑓′𝑥=lim∆𝑥→0∆𝑓∆𝑥=lim∆𝑥→01=1. So, (𝒙)′ = 1 Example of calculating the derivative Example 3 .Find the derivative of the 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. 𝑓′𝑥=lim∆𝑥→0∆𝑓∆𝑥=lim∆𝑥→0(2𝑥+∆𝑥)=lim∆𝑥→02𝑥+lim∆𝑥→0∆𝑥=2𝑥. So, (𝒙 𝟐)′ = 2x Example of calculating the derivative Example 4. Find the derivative of the function y =𝒌𝒙+𝒎Solution: f(x) = 𝑘𝑥+𝑚.1.Take two values of the argument x and x + Δx.2.∆𝑓=𝑓𝑥+∆𝑥−𝑓𝑥=𝑘𝑥+ ∆𝑥+𝑚− 𝑘𝑥−𝑚=𝑘𝑥+𝑘∆𝑥−𝑘𝑥=𝑘∆𝑥.3.∆𝑓(𝑥)∆𝑥=𝑘∆𝑥∆𝑥=𝑘.4. 𝑓′𝑥=lim∆𝑥→0 ∆𝑓∆𝑥=lim∆𝑥→0𝑘=𝑘. So, (𝒌𝒙+𝒎)′ = k Example of calculating the derivative Example 5. Find the derivative of the function y = 𝟏𝒙Solution: f(x) = 1𝑥.1. Take two values of the argument x and x + Δx.2.∆𝑓=𝑓𝑥+∆𝑥−𝑓𝑥= 1𝑥+∆𝑥−1𝑥=𝑥−𝑥−∆𝑥𝑥(𝑥+∆𝑥)=−∆𝑥𝑥(𝑥+∆ 𝑥).3.∆ 𝑓(𝑥)∆𝑥=−∆𝑥𝑥(𝑥+∆𝑥):∆𝑥=−∆𝑥𝑥(𝑥+∆𝑥)∆𝑥=−1𝑥(𝑥+∆𝑥) .4.𝑓′𝑥= lim∆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 Finish the phrase: Our lesson today was dedicated to... During the lesson I learned that... In the lesson I learned... The derivative of a function at a point is equal to... the tangent drawn to the graph of the function at a given point. The rate of change of a function is... It was difficult for me... WELL DONE!
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