11th grade Orlova E.V.
"Antiderivative and indefinite integral"
SLIDE 1
Lesson objectives:
Educational : form and consolidate the concept of an antiderivative, find antiderivative functions of different levels.
Developmental: develop the mental activity of students based on the operations of analysis, comparison, generalization, and systematization.
Educational: to form the ideological views of students, to instill a sense of success from responsibility for the results obtained.
Lesson type: learning new material.
Equipment: computer, multimedia board.
Expected learning outcomes: the student must
derivative definition
the antiderivative is defined ambiguously.
find antiderivative functions in the simplest cases
check whether the function is antiderivative on a given time interval.
During the classes
Organizing time SLIDE 2
Checking homework
Communicating the topic, purpose of the lesson, objectives and motivation for learning activities.
On the board:
Derivative - produces a new function.
Antiderivative - “primary image”.
4. Updating knowledge, systematizing knowledge in comparison.
Differentiation - finding the derivative.
Integration - restoration of a function from a given derivative.
Introducing new symbols:
5.Oral exercises:SLIDE 3
Instead of points, put some function that satisfies equality.
Students perform self-tests.
adjusting students' knowledge.
5. Studying new material.
A) Reciprocal operations in mathematics.
Teacher: in mathematics there are 2 mutually inverse operations in mathematics. Let's look at it in comparison. SLIDE 4
B) Reciprocal operations in physics.
Two mutually inverse problems are considered in the mechanics section.
Finding the velocity using a given equation of motion of a material point (finding the derivative of a function) and finding the equation of the trajectory of motion using a known velocity formula.
C) The definition of an antiderivative and an indefinite integral is introduced
SLIDE 5, 6
Teacher: in order for the task to become more specific, we need to fix the initial situation.
D) Table of antiderivatives SLIDE 7
Tasks to develop the ability to find antiderivatives - work in groups SLIDE 8
Tasks to develop the ability to prove that an antiderivative is for a function on a given interval - pair work.
6. Physical exerciseSLIDE 9
7. Primary comprehension and application of what has been learned.SLIDE 10
8. Setting homeworkSLIDE 11
9. Summing up the lesson.SLIDE 12
During the frontal survey, together with the students, the results of the lesson are summed up, the concept of new material is consciously comprehended, in the form of emoticons.
I understood everything, managed to do everything.
I didn’t understand part of it, I didn’t manage everything.
Lesson topic: “Antiderivative and integral” 11th grade (repetition)
Lesson type: lesson on assessment and correction of knowledge; repetition, generalization, formation of knowledge, skills.
Lesson motto : It’s not a shame not to know, it’s a shame not to learn.
Lesson objectives:
- Educational: repeat theoretical material; develop skills in finding antiderivatives, calculating integrals and areas of curvilinear trapezoids.
- Educational: develop independent thinking skills, intellectual skills (analysis, synthesis, comparison, comparison), attention, memory.
- Educational: nurturing the mathematical culture of students, increasing interest in the material being studied, preparing for the UNT.
Lesson outline plan.
I. Organizing time
II. Updating students' basic knowledge.
1. Oral work with the class to repeat definitions and properties:
1. What is called a curved trapezoid?
2. What is the antiderivative for the function f(x)=x2?
3. What is the sign of constancy of function?
4. What is called the antiderivative F(x) for the function f(x) on xI?
5. What is the antiderivative for the function f(x)=sinx?
6. Is the statement true: “The antiderivative of the sum of functions is equal to the sum of their antiderivatives”?
7. What is the main property of the antiderivative?
8. What is the antiderivative for the function f(x)=.
9. Is the statement true: “The antiderivative of the product of functions is equal to the product of their
Prototypes"?
10. What is called an indefinite integral?
11.What is called a definite integral?
12.Name several examples of the application of the definite integral in geometry and physics.
Answers
1. A figure bounded by the graphs of the functions y=f(x), y=0, x=a, x=b is called a curvilinear trapezoid.
2. F(x)=x3/3+C.
3. If F`(x0)=0 on some interval, then the function F(x) is constant on this interval.
4. The function F(x) is called antiderivative for the function f(x) on a given interval if for all x from this interval F`(x)=f(x).
5. F(x)= - cosx+C.
6. Yes, that's right. This is one of the properties of antiderivatives.
7. Any antiderivative for the function f on a given interval can be written in the form
F(x)+C, where F(x) is one of the antiderivatives for the function f(x) on a given interval, and C is
Arbitrary constant.
9. No, that's not true. There is no such property of primitives.
10. If the function y=f(x) has an antiderivative y=F(x) on a given interval, then the set of all antiderivatives y=F(x)+С is called the indefinite integral of the function y=f(x).
11. Difference between values of the antiderivative function at points b and a for the function y = f (x) on the interval [a; b ] is called the definite integral of the function f(x) on the interval [ a ; b ] .
12..Calculation of the area of a curvilinear trapezoid, volumes of bodies and calculation of the speed of a body in a certain period of time.
Application of the integral. (Additionally write down in notebooks)
Quantities
Derivative calculation
Calculation of the integral
s – movement,
A – acceleration
A(t) =
A - work,
F – strength,
N - power
F(x) = A"(x)
N(t) = A"(t)
m – mass of a thin rod,
Linear density
(x) = m"(x)
q – electric charge,
I – current strength
I(t) = q(t)
Q – amount of heat
C - heat capacity
c(t) = Q"(t)
Rules for calculating antiderivatives
- If F is an antiderivative for f, and G is an antiderivative for g, then F+G is an antiderivative for f+g.
If F is an antiderivative of f and k is a constant, then kF is an antiderivative of kf.
If F(x) is an antiderivative for f(x), ak, b are constants, and k0, that is, there is an antiderivative for f(kx+b).
^4) - Newton-Leibniz formula.
5) The area S of a figure bounded by straight lines x-a,x=b and graphs of functions continuous on the interval and such that for all x is calculated by the formula
6) The volumes of bodies formed by the rotation of a curvilinear trapezoid bounded by the curve y = f(x), the Ox axis and two straight lines x = a and x = b around the Ox and Oy axes are calculated accordingly using the formulas:
Find the indefinite integral:(orally)
1.
2.
3.
4.
5.
6.
7.
Answers:
1.
2.
3.
4.
5.
6.
7.
III Solving problems with the class
1. Calculate the definite integral: (in notebooks, one student on the board)
Drawing problems with solutions:
№ 1. Find the area of a curved trapezoid bounded by the lines y= x3, y=0, x=-3, x=1.
Solution.
-∫ x3 dx + ∫ x3 dx = - (x4/4) | + (x4 /4) | = (-3)4 /4 + 1/4 = 82/4 = 20.5
№3. Calculate the area of the figure bounded by the lines y=x3+1, y=0, x=0
№ 5.Calculate the area of the figure bounded by the lines y = 4 -x2, y = 0,
Solution. First, let's plot a graph to determine the limits of integration. The figure consists of two identical pieces. We calculate the area of the part to the right of the y-axis and double it.
№ 4.Calculate the area of the figure bounded by the lines y=1+2sin x, y=0, x=0, x=n/2
F(x) = x - 2cosx; S = F(n/2) - F(0) = n/2 -2cos n/2 - (0 - 2cos0) = n/2 + 2
Calculate the area of curved trapezoids bounded by the graphs of the lines you know.
3. Calculate the areas of the shaded figures from the drawings (independent work in pairs)
Task: Calculate the area of the shaded figure
Task: Calculate the area of the shaded figure
III Lesson summary.
a) reflection: -What conclusions did you draw for yourself from the lesson?
Does everyone have something to work on on their own?
Was the lesson useful to you?
b) analysis of student work
c) At home: repeat the properties of all formulas of antiderivatives, formulas for finding the area of a curvilinear trapezoid, volumes of bodies of revolution. No. 136 (Shynybekov)
OPEN LESSON ON THE TOPIC
« ANIMID AND INDETERMINATE INTEGRAL.
PROPERTIES OF AN INDETERMINED INTEGRAL".
2 hours.
11th grade with in-depth study of mathematics
Problem presentation.
Problem-based learning technologies.
ANIMID AND INDETERMINATE INTEGRAL.
PROPERTIES OF AN INDETERMINED INTEGRAL.
THE PURPOSE OF THE LESSON:
Activate mental activity;
To promote the assimilation of research methods
- ensure a more durable assimilation of knowledge.
LESSON OBJECTIVES:
introduce the concept of antiderivative;
prove the theorem on the set of antiderivatives for a given function (using the definition of an antiderivative);
introduce the definition of an indefinite integral;
prove the properties of the indefinite integral;
develop skills in using the properties of an indefinite integral.
PRELIMINARY WORK:
repeat the rules and formulas of differentiation
concept of differential.
It is proposed to solve problems. The conditions of the tasks are written on the board.
Students give answers to solve problems 1, 2.
(Updating experience in solving problems using differential
citation).
1. Law of body motion S(t), find its instantaneous
speed at any time.
- V(t) = S(t).
2. Knowing that the amount of electricity flowing
through the conductor is expressed by the formula q (t) = 3t 
- 2 t,
derive a formula for calculating the current strength at any
moment of time t.
- I (t) = 6t - 2.
3. Knowing the speed of a moving body at every moment of time,
me, find the law of its motion.
Knowing that the strength of the current passing through the conductor in any
determining the amount of electricity passing
through the conductor.
Teacher: Is it possible to solve problems No. 3 and 4 using
the means we have?
(Creating a problematic situation).
Students' assumptions:
- To solve this problem it is necessary to introduce an operation,
the inverse of differentiation.
The differentiation operation compares a given
function F (x) its derivative.
F(x) = f(x).
Teacher: What is the task of differentiation?
Students' conclusion:
Based on the given function f (x), find such a function
F (x) whose derivative is f (x), i.e.
f (x) = F(x) .
This operation is called integration, more precisely
indefinite integration.
The branch of mathematics that studies the properties of the operation of integrating functions and its applications to solving problems in physics and geometry is called integral calculus.
Integral calculus is a branch of mathematical analysis, together with differential calculus, it forms the basis of the apparatus of mathematical analysis.
Integral calculus arose from the consideration of a large number of problems in natural science and mathematics. The most important of them are the physical problem of determining the distance traveled in a given time using a known, but perhaps variable, speed of movement, and a much more ancient task - calculating the areas and volumes of geometric figures.
What is the uncertainty of this reverse operation remains to be seen.
Let's introduce a definition. (briefly symbolically written
On the desk).
Definition 1. Function F (x) defined on some interval
ke X is called the antiderivative for the given function
on the same interval if for all x 
X
equality holds
F(x) = f (x) or d F(x) = f (x) dx .
For example. (x) = 2x, from this equality it follows that the function
x is antiderivative on the entire number axis
for the 2x function.
Using the definition of an antiderivative, do the exercise
No. 2 (1,3,6). Check that the function F is an antiderivative
noi for the function f if
1) F (x) =
2 cos 2x, f(x) = x 
- 4 sin 2x . 2) F (x) = tan 
x - cos 5x, f(x) = 
+ 5 sin 5x.
3) F (x) = x 
sin x + 
, f (x) = 4x sinx + x cosx + 
.
Students write down the solutions to the examples on the board and comment on them.
ruining your actions.
Is the function x the only antiderivative
for function 2x?
Students give examples
x + 3; x - 92, etc. ,
The students draw their own conclusions:
any function has infinitely many antiderivatives.
Any function of the form x + C, where C is a certain number,
is the antiderivative of the function x.
The antiderivative theorem is written in a notebook under dictation.
teachers.
Theorem. If a function f has an antiderivative on the interval
numeric F, then for any number C the function F + C is also
is an antiderivative of f. Other prototypes
function f on X does not.
The proof is carried out by students under the guidance of a teacher.
a) Because F is an antiderivative for f on the interval X, then
F (x) = f (x) for all x X.
Then for x X for any C we have:
(F(x) + C) = f(x). This means that F (x) + C is also
antiderivative of f on X.
b) Let us prove that the function f of other antiderivatives on X
does not have.
Let us assume that Φ is also antiderivative for f on X.
Then Ф(x) = f(x) and therefore for all x X we have:
F (x) - F (x) = f (x) - f (x) = 0, therefore
Ф - F is constant on X. Let Ф (x) – F (x) = C, then
Ф (x) = F (x) + C, which means any antiderivative
function f on X has the form F + C.
Teacher: what is the task of finding all the prototypes?
nykh for this function?
The students formulate the conclusion:
The problem of finding all antiderivatives is solved
by finding any one: if such a primitive
different is found, then any other is obtained from it
by adding a constant.
The teacher formulates the definition of an indefinite integral.
Definition 2. The set of all antiderivatives of the function f
called the indefinite integral of this
functions.
Designation. 
; 
- read the integral.
= F (x) + C, where F is one of the antiderivatives
for f, C runs through the set
real numbers.
f - integrand function;
f (x)dx - integrand;
x is the integration variable;
C is the constant of integration.
Students study the properties of the indefinite integral independently from the textbook and write them down in their notebooks.
.
Students write down solutions in notebooks, working at the blackboard
1. We recently covered the topic “Derivatives of some elementary functions.” For example:
Derivative of a function f(x)=x 9, we know that f′(x)=9x 8. Now we will look at an example of finding a function whose derivative is known.
Let's say the derivative is given f′(x)=6x 5 . Using knowledge about the derivative, we can determine that this is the derivative of the function f(x)=x 6 . A function that can be determined by its derivative is called an antiderivative. (Give a definition of an antiderivative. (slide 3))
Definition 1: The function F(x) is called the antiderivative of the function f(x) on the interval, if the equality is satisfied at all points of this segment= f(x)
Example 1 (slide 4): Let's prove that for any xϵ(-∞;+∞) function F(x)=x 5 -5x is an antiderivative of the function f(x)=5x 4 -5.
Proof: Using the definition of an antiderivative, we find the derivative of the function
=( x 5 -5x)′=(x 5 )′-(5x)′=5x 4 -5.
Example 2 (slide 5): Let's prove that for any xϵ(-∞;+∞) function F(x)= is not an antiderivative of the function f(x)= .
Prove with students on the board.
We know that finding the derivative is calleddifferentiation. Finding a function from its derivative will be calledintegration. (Slide 6). The goal of integration is to find all antiderivatives of a given function.
For example: (slide 7)
The main property of the antiderivative:
Theorem: If F(x) is one of the antiderivatives for the function f(x) on the interval X, then the set of all antiderivatives of this function is determined by the formula G(x)=F(x)+C, where C is a real number.
(Slide 8) table of antiderivatives
Three rules for finding antiderivatives
Rule #1: If F is an antiderivative for a function f, and G is an antiderivative for g, then F+G is an antiderivative for f+g.
(F(x) + G(x))’ = F’(x) + G’(x) = f + g
Rule #2: If F is an antiderivative of f and k is a constant, then the function kF is an antiderivative of kf.
(kF)’ = kF’ = kf
Rule #3: If F is an antiderivative of f, and k and b are constants (), then the function
Antiderivative for f(kx+b).
The history of the concept of integral is closely connected with problems of finding quadratures. Mathematicians of Ancient Greece and Rome called problems about the quadrature of a particular plane figure problems that we now classify as problems for calculating areas. Many significant achievements of mathematicians of Ancient Greece in solving such problems are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus. Using this method, Eudoxus proved:
1. The areas of two circles are related as the squares of their diameters.
2. The volume of a cone is equal to 1/3 of the volume of a cylinder having the same height and base.
The Eudoxus method was improved by Archimedes and the following things were proven:
1. Derivation of the formula for the area of a circle.
2. The volume of the ball is equal to 2/3 of the volume of the cylinder.
All achievements were proven by great mathematicians using integrals.
Subject: Antiderivative and indefinite integral.
Target: Students will test and consolidate knowledge and skills on the topic “Antiderivative and indefinite integral.”
Tasks:
Educational : learn to calculate antiderivatives and indefinite integrals using properties and formulas;
Developmental : will develop critical thinking, will be able to observe and analyze mathematical situations;
Educational : Students learn to respect other people's opinions and the ability to work in a group.
Expected Result:
They will deepen and systematize theoretical knowledge, develop cognitive interest, thinking, speech, and creativity.
Type : reinforcement lesson
Form: frontal, individual, pair, group.
Teaching methods : partially search-based, practical.
Methods of cognition : analysis, logical, comparison.
Equipment: textbook, tables.
Student rating: mutual esteem and self-esteem, observation of children in
lesson time.
During the classes.
Call.
Goal setting:
You and I know how to build a graph of a quadratic function, we know how to solve quadratic equations and quadratic inequalities, as well as solve systems of linear inequalities.
What do you think the topic of today's lesson will be?
Creating a good mood in the classroom. (2-3 min)
Drawing the mood:A person’s mood is primarily reflected in the products of his activity: drawings, stories, statements, etc. “My mood”:On a common sheet of Whatman paper, using pencils, each child draws his or her mood in the form of a stripe, a cloud, or a speck (within a minute).
Then the leaves are passed around in a circle. The task of everyone is to determine the mood of the other and complement it, complete it. This continues until the leaves return to their owners.
After this, the resulting drawing is discussed.
III. Frontal survey of students: “Fact or opinion” 17 min
1. Formulate the definition of an antiderivative.
2. Which of the functionsare antiderivatives of the function
3. Prove that the functionis the antiderivative of the functionon the interval (0;∞).
4. Formulate the main property of the antiderivative. How is this property interpreted geometrically?
5. For function
find the antiderivative whose graph passes through the point. (Answer:F(
x) =
tgx
+ 2.)
6. Formulate the rules for finding an antiderivative.
7. State the theorem on the area of a curved trapezoid.
8. Write down the Newton-Leibniz formula.
9. What is the geometric meaning of the integral?
10. Give examples of the application of the integral.
11. Feedback: “Plus-minus-interesting”
IV. Individual-pair work with mutual testing: 10 min
Solve No. 5,6,7
V. Practical work: solve in a notebook. 10 min
Solve No. 8-10
VI. Lesson summary. Giving grades (OdO, OO). 2 minutes
VII. Homework: p. 1 No. 11,12 1 min
VIII. Reflection: 2 min
Lesson:
I was attracted by...
Seemed interesting...
Excited...
Made me think...
Got me thinking...
What impressed you the most?
Will the knowledge acquired in this lesson be useful to you in later life?
What new did you learn in the lesson?
What do you think needs to be remembered?
10. What else needs to be worked on
I taught a lesson in 11th grade on the topic"An antiderivative and an indefinite integral", this is a lesson in reinforcing the topic.
Problems to be solved during the lesson:
will learn to calculate antiderivative and indefinite integrals using properties and formulas; will develop critical thinking, will be able to observe and analyze mathematical situations; Students learn to respect other people's opinions and the ability to work in a group.
After the lesson I expected the following result:
Students will deepen and systematize theoretical knowledge, develop cognitive interest, thinking, speech, and creativity.
Create conditions for the development of practical and creative thinking. Fostering a responsible attitude towards academic work, fostering a sense of respect between students to maximize their abilities through group learning
In my lesson I used frontal, individual, pair, and group work.
I planned this lesson in order to reinforce the concept of antiderivative and indefinite integral with students.
I think it was a good job creating the “Drawing the Mood” poster at the beginning of the lesson.A person’s mood is, first of all, reflected in the products of his activity: drawings, stories, statements, etc. “My mood”: whenOn a common sheet of Whatman paper, using pencils, each child draws his or her mood (within a minute).
Then the Whatman paper is turned in a circle. The task of everyone is to determine the mood of the other and complement it, complete it. This continues until the picture on the Whatman paper returns to its owner.After this, the resulting drawing is discussed. Each child was able to reflect their mood and get to work in the lesson.
At the next stage of the lesson, using the “Fact or Opinion” method, students tried to prove that all the concepts on this topic are fact, but not their personal opinion. When solving examples on this topic, perception, comprehension and memorization are ensured. Integrated systems of leading knowledge on this topic are being formed.
When monitoring and self-testing knowledge, the quality and level of mastery of knowledge, as well as methods of action, are revealed, and their correction is ensured.
I included a partial search task in the structure of the lesson. The guys solved the problems on their own. We checked ourselves in the group. We received individual consultation. I am constantly looking for new techniques and methods of working with children. Ideally, I would like each child to plan their own activities during and after the lesson, to answer the questions: do I want to reach certain heights or not, do I need a high-level education or not. Using this lesson as an example, I tried to show that the child himself can determine both the topic and the course of the lesson.That he himself can adjust his activities and the activities of the teacher so that the lesson and additional classes meet his needs.
When choosing this or that type of task, I took into account the purpose of the lesson, the content and difficulties of the educational material, the type of lesson, methods and methods of teaching, age and psychological characteristics of the students.
In a traditional teaching system, when the teacher presents ready-made knowledge and students passively absorb it, the question of reflection usually does not arise.
I think that the work turned out especially well when compiling the reflection “What did I learn in the lesson...”. This task aroused particular interest and helpedunderstand how best to organize this work in the next lesson.
I think that self-esteem and mutual assessment did not work out; the students overestimated themselves and their friends.
Analyzing the lesson, I realized that the students had a good understanding of the meaning of formulas and their application in solving problems and learned to use different strategies at different stages of the lesson.
I want to conduct my next lesson using the “Six Hats” strategy and conduct a “Butterfly” reflection, which will allow everyoneexpress your opinion, write it down.
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