Mathematics 2
Aperçu des sections
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1- Information about the scale
Djilali bounaama – khemis meliana-
university
Faculty of Economic, Commercial and Management Sciences.
college
Common torso
section
first year.
level
Mathematics 2
The scheduled scale
methodology.
Education Unit
Lecture + oriented works
Lesson type
Second
hexagon
02.
coefficient
04
credit
3 hours
Courier size per week
14 weeks in the first hexagon about 42 hours
Courier Size
an hour and a half
Lecture
an hour and a half
oriented works
All your concerns about the lecture or business addressed via professional email will be answered within 48 hours.
e-mail
Communication Window
Mathematics Lesson 2 is one of the systematic teaching units of the composition offerings in the LMD system, taught in the second semester of the first year of the Bachelor of Common Core, Mathematics is a set of abstract knowledge resulting from logical conclusions applied to groups. quantitative topics, helping us to understand, compare and understand the world around us and develop the ability to think properly and reasonably, Mathematics is involved in many other fields such as technology and science, helping researchers and scholars find solutions to problems and make the right decisions.
Mathematics helps provide logical solutions and address everyday issues such as project planning, project management, budgets and even discussion management.
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contact info
Professor Benfreiha Najat
Department of Commercial Sciences, Faculty of Economic and Commercial Sciences and Management Sciences, Djilali bou naama University, Khamis Milaina
Email:
All your concerns about the lecture or business addressed via professional email are answered within 48 hours.
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At the end of this standard, the student must be able to identify, explain, discuss, and conclude. The desired goals of this standard are as follows:
• Knowing what mathematics is 02.
• Enabling the student to be empowered in the areas of research, interpretation, and the ability to make sound decisions.
• Identify quantitative methods used in data processing, linear functions, and dealing with matrices.
• The student will gain a clear vision of the most important aspects of mathematics, which will enable him to work in different fields of life, such as economics and trade.
• Enabling the student to deal with methods of thinking and form and develop sound mathematical trends.
• Accurate and correct understanding of the most important topics of mathematics 02.
• Providing the student with a range of experiences in the field of mathematics to help him present the results of qualitative economic research in a specific, clear, concise and accurate quantitative manner.
Through this class we will try to present math 2, in a simplified and clear manner, where: upon completion of this axis the student will be familiar with the objectives of the axis based on Bloom's cognitive levels:
1. Level of knowledge and memory: Students at this level recover information from memory (Tribal gains), where students save definitions associated with math theme 1, acquire the necessary skills that enable them to use optimal methods of counting and analysis that help them to view data. Students are given multiple choice questions, asked to answer them and can be given empty filling questions, with the aim of conjuring their tribal gains related to mathematics.
2. Level of assimilation and understanding: The student clarifies the basic characteristics that allow him to review the various variables and concepts in the axis. Here, the student gives some varied questions based on what has been used and understood for the study.
3. Application Level: Students learn about various concepts related to Mathematics 2, use this on quantitative measurements of different economic phenomena and interpret them, and ask students to clarify the concepts by providing relevant examples of reality and working life.
4. Analysis level: Students distinguish between methods of statistical work in field studies, have the student study the most important mathematical rules, and analyze them.
5. Level of Installation and Creation: Students search for the role of mathematics in various economic activities through careful examination, and here students brainstorm to find the causes of the problem and how to solve it based on the benefits of the lectures provided.
6. Evaluation level: The student studies differential equations, functions with two variables, and matrices with all their properties based on the resources and tools available to him, as well as solving equation sentences using matrices and other mathematical methods. We develop a final exercise in which he identifies the various operations around what was discussed in the lessons
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This subject requires the student to have prior knowledge of the concept of functions and derivatives, as well as integrals and basic methods for numbers, in addition to a good knowledge of the mathematical aspect. Also, a good understanding of the quantitative, arithmetic and mathematical methods prior to this subject. In order for the student to be able to comprehend this standard, he must be familiar with: Functions Derivatives, integrals, and matrices. This makes it easier for the student to deduce or estimate to solve problems and help in decision making. The student may also have to expand his cognitive interests to reveal the added value of what he learned about mathematics lessons.
All who study mathematics should acquire and possess many experiences, skills and abilities, which allow you to have good control over the scale, among which are:
- Desire for self-development and diligence in specialization.
- Mathematical and computational skills.
- Acquaintance with various mathematical concepts and terminology related to mathematics.
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Dear student, you can view a detailed mathematics lesson plan through the following file:
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Introduction:
Mathematics is a study mainly related to representation and reasoning about abstract objects such as numbers and spaces. It is mainly used in many fields such as natural sciences, engineering, medicine, and several other fields. It also works to develop new specializations in the field of mathematics, such as statistics, and thus there is no dividing line between theoretical and applied mathematics.
Mathematics Standard 02 is directed to first-year bachelor’s students. It has been divided into five axes, including 16 lessons divided over 14 weeks. The first axis includes differential equations of the first order and second order, and it is a continuation of the fifth axis: Integrals, which is the last axis in Mathematics 01 for the hexagram. the first.
The second axis includes functions with two variables by presenting general concepts and then addressing partial derivation of the first and second order.
As for the third axis, it included matrices and working on them, where we will provide a definition of the matrix, its types, and the most important operations on it.
In the fourth axis, we will discuss the determinants and inverses of the matrix, of the second and third order, and we will present the properties of determinants, and finding the inverses of the matrix in several ways.
Through the fifth axis, we will learn about solving a set of linear equations, by writing the sentence in matrix form and solving it in several ways.
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The following map is a detailed explanation of the mathematics lesson for first-year students.
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Differential equations first appeared with the invention of calculus
Before Newton and Leibniz. In the year 1761, Isaac Newton wrote in the second chapter of his book Method
flows, by including three types of differential equations, and in 1695 Jacob Bernoulli proposed the Bernoulli differential equation
Which Leibniz tried to solve by simplifying it the following year.
The first studies date back to what are currently called homogeneous first-order differential equations
Until the year 1614, Alexis Clairaut developed Lagrange and Euler
The method used by Clairau
Through this chapter we will try to present differential equations, in a simplified and clear manner, where: upon completion of this axis the student will be familiar with the objectives of the axis based on Bloom's knowledge levels:
1. Level of knowledge and remembrance: Students at this level recover information from memory (tribal gains), where students save definitions associated with the subject of mathematics, acquire the necessary skills that enable them to use optimal methods of counting and the student is given multiple choice questions, and is asked to answer them.
2. Level of assimilation and understanding: The student clarifies the basic characteristics that allow him to review the various theories and concepts in the axis. Here, the student gives some varied questions based on what has been used and understood for the study.
3. Students learn about various concepts related to differential equations, and use this to quantify and interpret different economic phenomena, and ask students to clarify the concepts by providing relevant examples of reality and working life.
4. Level of Analysis: Students distinguish between field working methods, having the student study the most important methods of dissemination.
5. Students investigate the role of differential equations in various economic activities through careful examination, and here students brainstorm to find the causes of the problem and how to solve it from the benefits of the lectures provided.
6. The student studies the concepts of first- and second-order differential equations, based on the resources and tools available to them and measure them. We develop a final exercise in which the various processes around the axis are determined.
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Let y be a function of the variable x that can be differentiated n times. We call each differential equation of order n
An equation of the form:
Where: y
is the nth derivative of y
The general form of y(x) that satisfies equation (1) is called the general solution of the differential equation. We call a special solution to the differential equation. Every solution satisfies some special conditions, and these conditions are called the initial conditions.
Order and degree of the differential equation:
The order of the differential equation: It is the order of the highest differential coefficient in the equation
Degree of the differential equation: It is the degree (strength) of the highest differential coefficient in the equation, provided that all differential coefficients are free of fractional powers.
Example:
• y’’’2+2y’3-5y=0 is a third-order and second-order differential equation.
• y’+xy=x2 is a differential equation of first order and first order.
• y’’’+y’3+y=cos x is a third-order and first-order differential equation.
Types of differential equations:
Differential equations can be divided into two parts, ordinary differential equations and partial differential equations.
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There are many quantities whose value depends on another quantity or quantities. Such quantities are called dependent variables, while the quantities that depend on them are called independent variables. A function is called one variable if the number of independent variables in it is one. A function is called two-variable if the number of independent variables in it is two. A function is called three-variable if the number of independent variables in it is three. ........ And so on........ A function is called n-variable if the number of variables is Independent in which n.
Whereas in the previous chapter we studied differential equations and discussed their types and ways of solving them, the goal of this chapter is partial derivation of the first and second order.
Upon completion of this axis, the student will be familiar with the objectives of the axis based on Bloom’s levels of knowledge:
1. Knowledge and memory level: Students at this level retrieve and organize information from memory (a priori acquisitions), where students become familiar with the concept of functions with two variables, and provide them with the necessary skills that enable them to use optimal methods to calculate partial derivatives of the first and second order, and the student is given Multiple choice questions, and he is asked to answer them. They can also be given fill-in-the-blank questions, the goal of which is to restore his tribal gains related to the axis.
2. The level of comprehension and understanding: The student identifies and identifies the various variables and concepts related to the second axis. Here we give the student some various questions based on what was learned and evaluated for the lesson.
3. Application level: The student is able to perform operations to identify functions with two variables and apply them
4. Level of analysis: The student distinguishes various calculation methods related to countries with two variables.
5. Structure level: Students organize information in different ways through brainstorming.
6. Evaluation level: The student evaluates and evaluates all the information acquired so that decisions are made based on the specified criteria. We develop a final exercise in which he determines the various operations of functions with two variables.
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We call a function of two variables each defined function from the set ℛ such that it encloses
Each binary (x.y) has a single numerical value, let it be z. We symbolize functions with two variables as follows:
Example:
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Dear student if you cannot solve exercises you have to re-read the lesson and apply examples.
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Although actual matrix arithmetic did not appear until the beginning of the nineteenth century, However, matrices, as rows of numbers, have a long history of applications in solving Linear equations. In the Chinese text "The Nine Chapters on the Art of Mathematics", written in the second century BC, which is the first known example of the use of tables in solving sentences of equations, the concept of the determinant was also introduced.
Once the array is defined, many operations can be performed on it. At the simplest level, two matrices with the same dimensions can be combined using two of the most common mathematical operations: addition and subtraction. There is also the process of multiplying a matrix by a constant number, which somewhat mimics the traditional understanding of multiplication.
Upon completion of this axis, the student will be familiar with the objectives of the axis based on Bloom’s levels of knowledge:
1. Knowledge and memory level: Students at this level retrieve and organize information from memory (prior acquisitions), where students learn about the concept of matrices and their types, and provide them with the necessary skills that enable them to perform basic operations on them. The student is given multiple choice questions, and is asked to answer them. They can also give them fill-in-the-blank questions, aimed at restoring their tribal gains related to the axis.
2. The level of comprehension and understanding: The student identifies and identifies the various variables and concepts related to the third axis. Here we give the student some various questions based on what was learned and evaluated for the lesson.
3. Application level: The student can perform operations to learn about applications about matrices and operations on them.
4. Analysis level: The student distinguishes the various types of matrices as well as their basic operations.
5. Structure level: Students organize information in different ways through brainstorming.
6. Evaluation level: The student evaluates and evaluates all the acquired information in order to make decisions based on the specified criteria. We develop a final exercise in which he identifies the various matrix operations.
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A matrix of rank m×n is a rectangular system consisting of m×n arranged elements, distributed in lines and columns, and surrounded by parentheses. We symbolize this set of elements with a capital letter:
Example:
We call the elements of the matrix for which the line number is equal to the column number as diagonal elements.
If the matrix is square, then the diagonal elements form what is called the diagonal of the matrix, which is called...
Sometimes the main diameter of this matrix
We say that matrix B is diagonal if it is a square and all of its elements are zero except for the diagonal elements.
Example
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Dear student if you cannot solve exercises you have to re-read the lesson and apply examples.
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Matrices have been used for a long time in solving linear equations, and are the oldest form to use
Matrices in Solving Equations is a Chinese text called “The Nine Chapters on the Art of Mathematics,” which was written around the third century B.C. It also includes a study of the principle of determinants.
The inverse matrix of a matrix A is denoted by A−1. The inverse of matrix A is the process of searching for matrix B referred to above. A matrix is invertible if and only if its determinant differs from zero. Thus, it is irreversible if and only if its determinant equals zero.
Upon completion of this axis, the student will be familiar with the objectives of the axis based on Bloom’s levels of knowledge:
1. Knowledge and memory level: Students at this level retrieve and organize information from memory (prior acquisitions), where students learn about the determinants and inverses of a matrix, and provide them with the necessary skills that enable them to find the inverses of a matrix. The student is given multiple choice questions, and is asked to answer them. They can also be given fill-in-the-blank questions, the goal of which is to restore his tribal gains related to the axis.
2. The level of comprehension and comprehension: The student identifies and identifies the various variables and concepts related to the fourth axis. Here we give the student some various questions based on what was learned and evaluated for the lesson.
3. Application level: The student is able to perform operations on the determinants and inverses of matrices and learn about various methods for doing so.
4. Level of analysis: The student distinguishes the various operations related to the axis.
5. Structure level: Students organize information in different ways through brainstorming.
6. Evaluation level: The student evaluates and evaluates all the information acquired so that decisions are made based on the specified criteria. We develop a final exercise in which he identifies the various operations of determinants and inverses of the matrix.
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Define the determinant of an array
Let A be a square matrix. We define the determinant of a matrix to be a real number, which we denote by det (A).
Which is determined according to the rank of the matrix.
The determinant of a matrix of rank 1.1 is the number itself.
For matrix rank 2, its determinant is:
Example:
Find the value of the determinant
the solution:
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Dear student if you cannot solve exercises you have to re-read the lesson and apply examples.
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The origin of linear algebra dates back to the second half of the seventeenth century, during the study of... Sentences of equations; Among the first mathematicians who contributed in this field are Leibintz and who
After him, the scientist Maclaurin gave the relationships that allow solving sets of linear equations With two unknowns and three unknowns.
As for the study of the general case of sets of linear equations, credit goes to the scientist Cramer in the second half of the eighteenth century. As a result of these studies, the concept of the determinant of rank arosen We mention that Laplace and Vandermonde who worked in this field and later came up with the concept
The matrix was created by Gauss. The development of matrix theory allowed the emergence in the mid-nineteenth century of the concept of n-dimensional radial space. He was the first to talk about this radiological concept Cayley The final definition of radial spaces was developed by the scientist Peano in 1888
Upon completion of this axis, the student will be familiar with the objectives of the axis based on Bloom’s levels of knowledge:
1. Knowledge and memory level: Students at this level retrieve and organize information from memory (prior acquisitions), where students learn basic concepts about writing a set of equations in matrix form, and provide them with the necessary skills that enable them to solve a set of linear equations in several ways, and are given The student is given multiple choice questions, and is asked to answer them. They can also be given fill-in-the-blank questions, the goal of which is to restore his prior gains related to the axis.
2. The level of comprehension and comprehension: The student identifies and identifies the various variables and concepts related to the fifth axis. Here we give the student some various questions based on what was learned and evaluated for the lesson.
3. Application level: The student is able to perform operations on sets of linear equations and solve them with methods such as Gauss, Cramer, or determinants....
4. Analysis level: The student distinguishes various operations on solutions to sentences of linear equations.
5. Structure level: Students organize information in different ways through brainstorming.
6. Evaluation level: The student evaluates and evaluates all the information acquired in order to make decisions based on the specified criteria. We develop a final exercise in which he identifies the various processes for solving a set of linear equations.
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We call the sentence m a linear equation for n unknowns, each sentence of the following form:
The set of linear equations can be written in terms of matrices in the following matrix form:
AX=B
Where A represents the matrix of coefficients (proverbs):
x represents the matrix of the column of unknowns (in the set of linear equations):
B represents the column matrix of numbers on the second side of the set of linear equations:
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Dear student upload the above file and read it well
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Dear student if you cannot solve exercises you have to re-read the lesson and apply examples.
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Complete the final evaluation on time
Important Note: The time limit for completion of the final evaluation is 60 minutes and two attemptsDear student in case you hide in resolving your exam by revising the lesson and applying examples
Dear student, you can then check your answer using the model answer for the exam.
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he student must see all the references at his disposal in order to ensure the good way to acquire all the targeted competencies and thus the proven success. Among the references we place in the students' hands are the following:
Mas' adan Hadi and Hanashi Farah, General Mathematics of Economists, edition 1, Noran Publishing and Distribution, Tibsa, Algeria, 2021.
· Hussein Yassin Tamah, Mathematics for Economics, Administrative and Financial Sciences, T1, Dar Al Safa for Publishing and Distribution, Amman, Jordan, 2010
· .Jacques Dixmier (2001), Cours de mathématiques du premier cycle 1re année– Avec exercices corrigès, Deuxième édition, Dunod, Paris, France.
4. Jacques Vélu (2020), Mathématiques Générales - Cours et exercices corrigès, Dunod, Paris, France.
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My wishes to dear students for success, thank you.
