Lesson topic: “Anti-derivative and integral. Open lesson in algebra
Algebra lesson in 12th grade.
Theme of the lesson: “Antiprimitive. Integral"
Goals:
educational
Generalize and consolidate the material on this topic: the definition and property of the antiderivative, the table of antiderivatives, the rules for finding antiderivatives, the concept of the integral, the Newton-Leibniz formula, calculating the areas of figures. To diagnose the assimilation of the system of knowledge and skills and its application to perform practical tasks of a standard level with the transition to a higher level, to promote the development of the ability to analyze, compare, draw conclusions.
Educational
perform tasks of increased complexity, develop general learning skills and teach to think and perform control and self-control
educators
To educate, a positive attitude to learning, to mathematics
Lesson type: Generalization and systematization of knowledge
Forms of work: group, individual, differentiated
Equipment: cards for independent work, for differentiated work, self-control sheet, projector.
During the classes
Organizing time
Goals and objectives of the lesson: To summarize and consolidate the material on the topic “Antiprimitive. Integral - definition and property of the antiderivative, table of antiderivatives, rules for finding antiderivatives, the concept of the integral, the Newton-Leibniz formula, calculating the area of \u200b\u200bfigures. To diagnose the assimilation of the system of knowledge and skills and its application to perform practical tasks of a standard level with the transition to a higher level, to promote the development of the ability to analyze, compare, draw conclusions.
The lesson will be in the form of a game.
Rules:
The lesson consists of 6 stages. Each stage is worth a certain number of points. In the evaluation sheet, you set points for your work at all stages.
Stage 1. Theoretical. Mathematical dictation "Tic-tac-toe".
Stage 2. Practical. Independent work. Find the set of all antiderivatives.
Stage 3. "Um is good, but 2 is better." Work in notebooks and 2 students on the lapels of the board. Find the antiderivative of the function whose graph passes through point A).
4.stage. "Correct mistakes".
5. stage. "Make a word" Calculation of integrals.
6. stage. "Hurry to see." Calculation of the areas of figures bounded by lines.
2. Evaluation sheet.
Mathematicaldictation
Independent work
Oral response
Correct mistakes
Make up a word
hurry to see
9 points
5+1points
1 point
5 points
5 points
20 points
3 min.
5 minutes.
5 minutes.
6 min
2. Updating knowledge:
stage. Theoretical. Mathematical dictation "Tic-tac-toe"
If the statement is true - X, if false - 0
Function F(x) is called antiderivative on a given interval if for all х from this interval the equality
The antiderivative of a power function is always a power function
An antiderivative of a complex function
This is the Newton-Leibniz formula
Area of a curvilinear trapezoid
Antiderivative of the sum of functions = sum of antiderivatives considered on a given interval
Graphs of antiderivative functions are obtained by parallel translation along the X axis by a constant C.
The product of a number times a function is equal to the product of that number times the antiderivative of the given function.
The set of all antiderivatives has the form
Total 9 points
3. Consolidation and generalization
2 stage . Independent work.
"Examples teach better than theory."
Isaac Newton
Find the set of all antiderivatives:
1 option
The set of all primitives The set of all primitivesoption
Self-test.
For correctly completed tasks
Option 1 - 5 points,
for option 2 +1 point
1 point for addition.
stage . "Mind is good, a - 2 is better."
Work on the lapels of the board of two students and all the rest in notebooks.
The task
1 option. Find the antiderivative of the function, the graph of which passes through the point A (3; 2)
Option 2. Find the antiderivative of a function whose graph passes through the origin.
Mutual verification.
For the correct solution -5 points.
stage . If you want, believe - if you want, check.
Task: correct mistakes, if any.
Find exercises with an error:
Stage . Compose a word.
Calculate Integrals
1 option.
option.
Answer: BRAVO
Self-test. For a correctly completed task - 5 points.
stage. "Hurry to see."
calculation areas of figures bounded by lines.
Task: draw a figure and calculate its area.
2 points
2 points
4 points
6 points
6 points
Checked individually with the teacher.
For correctly completed all tasks - 20 points
Summarizing:
The lesson covered the main questions
1. We recently went through the topic "Derivatives of some elementary functions." For example:
Function derivative f(x)=x 9 , we know that f′(x)=9x 8 . Now we will consider an example of finding a function whose derivative is known.
Suppose we are given a derivative f (x)=6x 5 . Using knowledge of the derivative, we can determine what is the derivative of the function f(x)=x 6 . A function that can be determined by its derivative is called antiderivative. (Give a definition of antiderivative. (slide 3))
Definition 1: The function F(x) is called the antiderivative for the function f(x) on the interval, if the equality holds at all points of this segment= f(x)
Example 1 (slide 4): Let's prove that for anyхϵ(-∞;+∞) function F(x)=х 5 -5х is the antiderivative for the function f (x) \u003d 5x 4 -5.
Proof: Using the definition of antiderivative, we find the derivative of the function
\u003d ( x 5 -5x) \u003d (x 5 ) \u003d (5x) \u003d 5x 4 -5.
Example 2 (slide 5): Let's prove that for anyхϵ(-∞;+∞) function F(x)= is not antiderivative for the function f(x)= .
Prove with students on the blackboard.
We know that finding the derivative is calleddifferentiation. Finding a function by 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 the antiderivative of f and G is the antiderivative of g, then F+G is the antiderivative of f+g.
(F(x) + G(x))' = F'(x) + G'(x) = f + g
Rule #2: If F is an antiderivative for f and k is a constant, then the function kF is an antiderivative for kf.
(kF)' = kF' = kf
Rule #3: If F is the antiderivative of f and k and b are constants (), then the function
Antiderivative for f(kx+b).
The history of the concept of an integral is closely connected with the problems of finding quadratures. Mathematicians of Ancient Greece and Rome called the problems of squaring one or another flat figure as problems that we now refer to as problems for calculating areas. Many significant achievements of the mathematicians of Ancient Greece in solving such problems are associated with the use of the exhaustion method proposed by Eudoxus of Knidos. With 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 method of Eudoxus was perfected by Archimedes and the following things were proven:
1. Derivation of the formula for the area of a circle.
2. The volume of the sphere is 2/3 of the volume of the cylinder.
All achievements have been proven by great mathematicians using integrals.
11th grade Orlova E.V.
"The antiderivative and the indefinite integral"
SLIDE 1
Lesson Objectives:
Educational : to form and consolidate the concept of antiderivative, to find antiderivative functions of different levels.
Developing: to develop the mental activity of students, based on the operations of analysis, comparisons, generalization, systematization.
Educational: to form the worldview views of students, to educate from responsibility for the result, a sense of success.
Lesson type: learning new material.
Equipment: computer, multimedia board.
Expected learning outcomes: student must
definition of derivative
antiderivative is defined ambiguously.
find antiderivative functions in the simplest cases
check whether the antiderivative for a function on a given time interval.
During the classes
Organizing time SLIDE 2
Checking homework
The message of the topic, the purpose of the lesson, the tasks and motivation of educational activities.
On the writing board:
Derivative -produces "a new function".
antiderivative - Primary Image.
4. Actualization of knowledge, systematization of knowledge in comparison.
Differentiation-finding the derivative.
Integration is the restoration of a function by a given derivative.
Introduction to new characters:
5. Oral exercises:SLIDE 3
instead of points, put some function that satisfies the equality.
student self-test.
updating students' knowledge.
5. Learning new material.
A) Reciprocal operations in mathematics.
Teacher: in mathematics there are 2 mutually inverse operations in mathematics. Let's take a look at the comparison. SLIDE 4
B) Reciprocal operations in physics.
Two mutually inverse problems are considered in the mechanics section.
Finding the speed according to the given equation of motion of a material point (finding the derivative of the function) and finding the equation for the trajectory of motion using the known formula for speed.
C) The definition of an antiderivative, 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 for the formation of the ability to find the primitive - work in groups SLIDE 8
Tasks for the formation of the ability to prove that the antiderivative is for a function on a given interval - pair work.
6.FizminutkaSLIDE 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, a conscious understanding of the concept of new material can be in the form of emoticons.
Understood everything, managed everything.
partially did not understand (a), did not manage to do everything.
Methodical development of an algebra lesson on the topic: "Anti-derivative and integral"
Topic: "Anti-derivative and integral".
Group: 82 (14-TTOII-118)
Speciality: Technology of catering products.
Type: lesson of generalization and systematization of knowledge .
The form: AND gra.
Goals:
d idactic:
the formation of educational, cognitive and informational competencies, through generalization, systematization of knowledge on the topic “Antiprimitive. Integral”, the formation of skills in finding the area of a curvilinear trapezoid in several ways.
developing:
the formation of informational, general cultural competencies through the development of cognitive activity, interest in the subject, the creative abilities of students, broadening their horizons, developing mathematical speech.
educational:
the formation of communicative competence and the competence of personal self-improvement, through work on communication skills, the ability to work in cooperation, on the development of such personal qualities as organization, discipline.
Means of education:
Technical: PC, projector, screen.
During the classes
Preparatory stage: The group is first divided into two teams.
I. Organizational moment
Hello guys! I am glad to welcome you to the lesson. C the purpose of our lesson is to generalize, systematize knowledge on the topic “Antiprimitive and integral”, prepare for the upcoming test.
The motto of our work: "Explore everything, let your mind come first" - these words belong to the ancient Greek scientist Pythagoras.
We will make an unusual ascent to the top of the "Peak of Knowledge".
The championship will be contested by two groups. Each group has its own instructor, who evaluates the participation rate of each "tourist" in our ascent.
The first group to reach the top of the Knowledge Peak will be the winner.
II. Checking homework: "Check the backpacks."
Before a long journey, you need to check how well you prepared for the ascent. Let's check the homework that was given in the previous lesson:
Find the area of a figure bounded by lines:
, 
Two people take turns coming to the board and briefly explaining the solution they have prepared on the slides. The rest are checking.
I II. Warm up.
It is accepted that a person, preparing for a competition, usually begins his day with exercises, that is, with a warm-up.
We'll do some warm-ups too.
There are 9 test tasks. Each team in turn chooses a question, for the correct answers they receive tokens (slide)
The operation of finding an indefinite integral of some function is called ...
integration;
differentiation;
logarithm;
exponentiation;
root extraction.
Finish the definition:
Indefinite integral of a function y = f (x) is called:
function derivative F (x );
the set of all antiderivatives of a function y = f (x );
set of all derivatives of a function y = f (x );
kind sign.
Newton-Leibniz formula:
Finish the definition:
“A differentiable function F(x) is called an antiderivative for a function f(x) on an interval X if at each point of this interval…”
IV . Mathematical relay.
Now on the road! The ascent to the "Peak of Knowledge" will not be easy, there may be blockages, collapses, and drifts. But there are also halts, where not only tasks are waiting for you. To move forward, you need to show knowledge.
Teamwork. On the last desk of each row there is a sheet with 8 tasks (two questions for each desk). The first pair of students, having completed any two tasks, passes the sheet in front of those sitting. The work is considered completed when the teacher receives a sheet with 8 tasks correctly completed. The same tasks are presented on the slide. You can solve not only your own tasks, but also check the correctness of the decisions of your team members.
The team that solves all the tasks first wins. Checking the work is carried out using a slide. Points earned are cumulative.
And now a halt.
V. Halt.
“A happy accident falls only to prepared minds” (Louis Pasteur) (slide).
Information from the history of integral calculus is read out (slide).
The integral symbol was introduced by Leibniz (1675). This sign is a change of the Latin letter S (the first letter of the word sum). The very word integral was coined by J. Bernoulli (1690). It probably comes from the Latin integero, which translates as to bring back to its previous state, restore. (Indeed, the operation of integration “restores” the function by differentiation of which the integrand was obtained.) The origin of the word integral may be different: the word integer means whole.
During the correspondence, I. Bernoulli and G. Leibniz agreed with the proposal of J. Bernoulli. Then, in 1696, the name of a new branch of mathematics appeared - integral calculus (calculus integralis), which was introduced by I. Bernoulli.
The emergence of problems of integral calculus is associated with finding areas and volumes. A number of problems of this kind were solved by ancient mathematicians.
Greece. Ancient mathematics anticipated the ideas of integral calculus to a much greater extent than differential calculus. A large role in solving such problems was played by an exhaustive method created.
Eudoxus of Cnidus (c. 408 - c. 355 BC) and widely used
Archimedes (c. 287 - 212 BC).
In the 17th century, many discoveries related to integral calculus were made. So, P. Fermat already in 1629 solved the problem of squaring any curve. However, despite the significance of the results obtained by mathematicians.
XVII century, there was no calculus yet. It was necessary to identify the general ideas underlying the solution of many particular problems, as well as to establish a connection between the operations of differentiation and integration, which gives a fairly accurate algorithm. This was done by Newton and Leibniz, who independently discovered a fact known to you under the name of the Newton-Leibniz formula.
The Russian mathematicians M. V. Ostrogradsky (1801-1862) and V. Ya. Bunyakovsky took part in the development of the integral calculus.
The solution of this problem is associated with the names of O. Cauchy, one of the greatest mathematicians, the German scientist B. Riemann (1826 - 1866), the French mathematician G. Darboux (1842 - 1917).
Answers to many questions related to the existence of areas and volumes of figures were obtained with the creation of measure theory by K. Jordan (1826 - 1922).
Various generalizations of the concept of the integral were already at the beginning of our century proposed by the French mathematicians A. Lebesgue (1875 - 1941) and
A. Danjoy (1884 - 1974) by the Soviet mathematician A. Ya. Khichin (1894 -1959).
VI. The most difficult climb.
The next task is supposed to be done in writing, so students work in notebooks.
A task. In how many ways can you find the area of a figure bounded by lines (slide).
, , ,
Who has suggestions? (the figure consists of two curvilinear trapezoids and a rectangle) (choose how to solve the slide).
After discussing this issue, a note appears on the slide:
1 way: S \u003d S 1 + S 2 + S 3
2 way: S \u003d S 1 + S ABCD -S OCD
Two students decide at the blackboard followed by an explanation of the solution, the rest of the students work in notebooks, choosing one of the solution methods (one person from the team).
Output(students do): we found two ways to solve this problem, getting the same result. Discuss which way is easier.
V II. Last climb. Crossword (slide)
Everyone is very tired, but the closer to the goal, the tasks become easier and easier.
Last climb. On the slide is a crossword puzzle. Your task is to solve it. In turn, each team guesses the word they like, writes down the answer.
VSH. Summary of the lesson (slide).
Lesson topic: "Anti-derivative and integral" Grade 11 (review)
Lesson type: lesson of 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:
- Tutorials: repeat theoretical material; to work out the skills of finding antiderivatives, calculating integrals and areas of curvilinear trapezoids.
- Developing: develop independent thinking skills, intellectual skills (analysis, synthesis, comparison, comparison), attention, memory.
- Educational: education of the mathematical culture of students, increasing interest in the material being studied, preparing for the UNT.
Lesson outline plan.
I. Organizing time
II. Updating the basic knowledge of students.
1.Oral work with the class to repeat definitions and properties:
1. What is called a curvilinear trapezoid?
2. What is the antiderivative for the function f(x)=x2.
3. What is the sign of function constancy?
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
Primitives?
10. What is called an indefinite integral?
11. What is called a definite integral?
12. Name a few examples of the use of a definite integral in geometry and physics.
Answers
1. A figure bounded by the graphs of functions y=f(x), y=0, x=a, x=b is called a curvilinear trapezoid.
2. F(x)=x3/3+С.
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 primitives.
7. Any antiderivative for a function f on a given interval can be written as
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, not true. There is no such property of primitives.
10. If the function y \u003d f (x) has an antiderivative y \u003d F (x) on a given interval, then the set of all antiderivatives y \u003d F (x) + C is called the indefinite integral of the function y \u003d f (x).
11. The difference between the values of the antiderivative function at points b and a for the function y \u003d 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 in notebooks)
Quantities
Derivative calculation
Integral calculation
s - displacement,
A - acceleration
A(t) =
A - work,
F - strength,
N - power
F(x) = A"(x)
N(t) = A"(t)
m is the mass of a thin rod,
Line Density
(x) = m"(x)
q - electric charge,
I - current strength
I(t) = q(t)
Q is the amount of heat
C - heat capacity
c(t) = Q"(t)
Rules for computing 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 the antiderivative of f and k is a constant, then kF is the 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 the figure bounded by the straight lines x-a, x=b and the graphs of continuous functions 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 a curve y = f (x), the axis Ox and two straight lines x = a and x = b around the axes Ox and Oy, are calculated respectively by the formulas:
Find the indefinite integral:(orally)
1.
2.
3.
4.
5.
6.
7.
Answers:
1.
2.
3.
4.
5.
6.
7.
III Solving tasks with a class
1. Calculate the definite integral: (in notebooks, one student on the board)
Tasks for drawings with solutions:
№ 1. Find the area of a curvilinear trapezoid bounded by 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 \u200b\u200bthe figure bounded by the lines y \u003d 4 -x2, y \u003d 0,
Solution. First, let's plot a graph to determine the limits of integration. The figure consists of two identical pieces. 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(p/2) - F(0) = p/2 -2cos p/2 - (0 - 2cos0) = p/2 + 2
Calculate the area of curvilinear trapezoids bounded by graphs of lines known to you.
3. Calculate the areas of the shaded figures from the figures (independent work in pairs)
Task: Calculate the area of the shaded figure
Task: Calculate the area of the shaded figure
III The results of the lesson.
a) reflection: -What conclusions did you draw from the lesson for yourself?
Is there something for everyone to work on on their own?
Was the lesson helpful for you?
b) analysis of student work
c) At home: repeat the properties of all the formulas of antiderivatives, the formulas for finding the area of a curvilinear trapezoid, the volumes of bodies of revolution. No. 136 (Shynybekov)