Mathematical analysis in examples and tasks. Part 1

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Mathematical Analysis ◽  
Continuous Functions ◽  
Advanced Mathematics ◽  
Hyperbolic Functions ◽  
Integral Calculus ◽  
Postgraduate Students ◽  
Background Material ◽  
Improper Integrals ◽  
Sets Theory ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 1 of the proposed cycle of workshops on the following topics: theory of sets, theory of limits, theory of continuous functions; differential calculus of functions of one variable, its application to the study of the properties of functions and graph; integral calculus of functions of one variable: indefinite, definite, improper integrals; hyperbolic functions; applications of integral calculus to the analysis and solution of practical problems. For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the obtained knowledge provides exercises and problems with answers and guidance. For teachers, students and postgraduate students studying advanced mathematics.

The mathematical analysis. Volume 1

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Mathematical Analysis ◽  
Differential Calculus ◽  
Real Variable ◽  
Hyperbolic Functions ◽  
Integral Calculus ◽  
Postgraduate Students ◽  
First Semester ◽  
Improper Integrals ◽  
Basic Concepts ◽  
Sets Theory

The aim of the tutorial is to help students to master the basic concepts and methods of the study of calculus. Volume 1 explores the following topics: theory of sets, theory of limits; differential calculus of functions of one variable; investigation of the properties of functions and graphing; integral calculus of functions of one real variable (indefinite, definite and improper integrals), the technique of integration; hyperbolic functions; applications to the analysis and solution of practical problems. These topics are studied in universities, as a rule, in the first semester in the framework of self-discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". Great attention is paid to comparison of these methods, the proper choice of study design tasks, analyze complex situations that arise in the study of these branches of mathematical analysis. For teachers, students and postgraduate students studying mathematical analysis.

Mathematical analysis in examples and tasks. Part 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Graduate Students ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Advanced Mathematics ◽  
Analytic Geometry ◽  
Several Variables ◽  
Background Material ◽  
Basic Concepts ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 2 of the proposed cycle of workshops on the following topics: analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Fourier series; applications to the analysis and solution of applied problems. These topics are studied in universities, usually in the second semester in the discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the acquired knowledge in each section designed exercises and tasks with answers and guidance. It is recommended that teachers, students and graduate students studying advanced mathematics.

The mathematical analysis. Volume 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Quality Control ◽  
Fourier Series ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Analytic Geometry ◽  
Postgraduate Students ◽  
Several Variables ◽  
Basic Concepts ◽  
Control Knowledge

The aim of the tutorial is to help students to master the basic concepts and methods of the study of calculus. In volume 2 we study analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Taylor series and Maclaurin, and Fourier series; applications to the analysis and solution of applied problems. Great attention is paid to comparison of these methods, the proper choice of study design tasks, analyze complex situations that arise in the study of these branches of mathematical analysis. For self-training and quality control knowledge given test questions. For teachers, students and postgraduate students studying mathematical analysis.

Economic and mathematical and econometric modeling: Computer workshop

2020 ◽  
Author(s):  
Vasiliy Kolpakov
Keyword(s):  
Economic Condition ◽  
Federal State ◽  
Economic Factors ◽  
Econometric Modeling ◽  
Mathematical Research ◽  
Mathematical Methods ◽  
Postgraduate Students ◽  
Educational Standard ◽  
State Educational ◽  
Independent Work

The textbook presents mathematical research methods and models of economic objects and processes designed for the analysis and prediction of economic factors and develop control solutions as in the deterministic conditions, and in conditions of some uncertainty, and dynamics. Each Chapter of the book consists of a theoretical framework, discussed in detail several examples and tasks for independent work. As workbench simulation uses standard office the program Excel and Mathcad. Tutorial focused on independent performance of students individual tasks on disciplines "Economic-mathematical methods" and "Econometrics". Meets the requirements of Federal state educational standard of higher education of the last generation. The publication is intended for students and postgraduate students in economic disciplines. It can also be useful as they perform final qualifying works. The book will be useful for practitioners engaged in the analysis of the current financial and economic condition and future development of firms and businesses.

scholarly journals Calculus for the intermediate point associated with a mean value theorem of the integral calculus

2020 ◽  
Vol 28 (1) ◽  
pp. 59-66
Author(s):  
Emilia-Loredana Pop ◽  
Dorel Duca ◽  
Augusta Raţiu
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Integral Calculus ◽  
Intermediate Point

AbstractIf f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).In this paper, we study the approaching of the point c towards a, when b approaches a.

scholarly journals ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY

2013 ◽  
Vol 57 (1) ◽  
pp. 90-101
Author(s):  
Raffaele Pisano ◽  
Paolo Bussotti
Keyword(s):  
Mathematics Education ◽  
Middle Ages ◽  
Mathematical Analysis ◽  
Mathematical Reasoning ◽  
Basic Knowledge ◽  
16Th Century ◽  
Advanced Mathematics ◽  
Scientific Education ◽  
Late Middle Ages ◽  
The Social

Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathematics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of mathematical analysis was profound: there were two complex examinations in which the theory was as important as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and mathematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance humanists, and with respect to mathematics education nowadays is discussed. Particularly the ways through which mathematical knowledge was spread in Italy between late Middle ages and early Modern age is shown. At that time, the term “scientific education” corresponded to “teaching of mathematics, physics”; hence something different from what nowadays is called science education, NoS, etc. Moreover, the relationships between mathematics education and civilization in Italy between the 12th and the 16th century is also popularized within the Abacus schools and Niccolò Tartaglia. These are significant cases because the events connected to them are strictly interrelated. The knowledge of such significant relationships between society, mathematics education, advanced mathematics and scientific knowledge can be useful for the scholars who are nowadays engaged in mathematics education research. Key words: Abacus schools, mathematics education, science & society, scientific education, Tartaglia

A consistent combinatory logic with an inverse to equality

1980 ◽  
Vol 45 (3) ◽  
pp. 529-543 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Mathematical Analysis ◽  
Continuous Functions ◽  
Combinatory Logic ◽  
New Haven ◽  
Yale University ◽  
Free System ◽  
Abstraction Principle ◽  
Unit Class ◽  
Excluded Middle ◽  
Comprehension Axiom

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

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