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Mathematical Analysis
Major: Computer Sciences (Artificial Intelligence)
Code of subject: 6.122.13.O.007
Credits: 6.00
Department: Computational Mathematics and Programming
Lecturer: Professor Petro Pukach
Semester: 1 семестр
Mode of study: денна
Завдання: The study of an educational discipline involves the formation of competencies in students of education:
integral competence:
IНT is the ability to solve complex specialized tasks and practical problems in the field of computer science or in the learning process, which involves the application of theories and methods of information technologies and is characterized by complexity and uncertainty of conditions.
general competences:
ЗK1 – the ability to abstract thinking, analysis and synthesis.;
ЗK2 – ability to apply knowledge in practical situations;
ЗK7 – the ability to search, process and analyze information from various sources;
professional competences:
ФK1 – the ability to mathematically formulate and investigate continuous and discrete mathematical models, justify the choice of methods and approaches for solving theoretical and applied problems in the field of computer science, analysis and interpretation;
ФK2 – the ability to identify statistical regularities of non-deterministic phenomena, use methods of computational intelligence, in particular statistical, neural network and fuzzy data processing, methods of machine learning and genetic programming, etc.
Learning outcomes: 1. calculate limits of sequences and functions, in particular, reveal basic uncertainties, compare infinitely small and infinitely large functions;
2. calculate standard derivatives, use derivatives to study functions; develop functions according to Taylor's and McLauren's formulas, use development for approximate calculations;
3. find partial derivatives and extremum points of functions of many variables;
4. master the basic techniques and methods of integration, apply integral calculus to solve simple practical problems;
5. To be able to examine numerical and functional series for convergence.
6. To be able to develop functions into power series and Fourier series.
7. Have the skills to calculate multiple and curvilinear integrals.
8. To solve the relevant mathematical problems, bringing the solution to a practically satisfactory result (formulas, numbers, graphs, qualitative conclusions, etc.).
9. Be able to solve differential equations of the first order (with separable variables, homogeneous equations, linear equations, Bernoulli's equation, equations in complete differentials).
10. To be able to solve differential equations of higher orders with constant coefficients.
11. To be able to integrate systems of ordinary differential equations with constant coefficients.1. To study the convergence of numerical and functional series. To evaluate the amount and the remainder of the series.
12. To develop functions in the Taylor, Maclaurin and Fourier series,
to use the development for approximate computations.
13. To calculate double and triple integrals in different coordinate systems.
14. To use multiple integrals to solve practical tasks.
15. To calculate curvilinear and surface integral, to be able to apply curvilinear integrals for solving practical problems.
16. Master the basic facts and methods of mathematical field theory
and use them in practice.
17. Master the basic facts and methods of field theory and ordinary differential equations.
18. Master the basic facts and methods of equations of mathematical physics and use them in practice.
Required prior and related subjects: 1. Linear algebra and analytic geometry
3. Discrete mathematics
4. Probability theory and mathematical statistics
5. Physics
Summary of the subject: The notion of a numerical series. Acceptable series.
Significant numerical series.
Functional series. Power series.
Taylor and Maclaurin series.
Fourier series of periodic functions.
Fourier integral. The transformation of Fourier.
Double integrals.
Triple integrals.
Curvilinear integrals.
Surface Integrals.
Elements of mathematical field theory.
Ordinary differential equations of the first order.
Ordinary differential equations of higher orders.
Linear equations of arbitrary order.
Basic problems for equations of mathematical physics.
Fourier method.
Опис: The educational discipline "Mathematical analysis" consists of sections: "Introduction to analysis. Limits and continuity", "Differential calculus of functions of one variable", "Functions of many variables", "Integral calculus of functions of one variable", "Numerical and functional series", "Fourier series", "Multiple integrals", "Ordinary differential equations ". In the section "Introduction to the analysis. Boundaries and continuity" the following topics are considered: "The concept of the set. Complex numbers", "The limit of a sequence and functions", "Continuity of a function of one variable". The section "Differential calculus of functions of one variable" consists of the topics "Derivatives of functions of one variable", "Application of derivatives". The section "Functions of many variables" consists of the topics "The concept of a function of many variables. Calculation of partial derivatives", "Application of derivatives of functions of many variables". The section "Integral calculus of functions of one variable" consists of the topics "Indefinite integral", "Definite integral", "Application of integral calculus of functions of one variable". In the section "Numerical and functional series" the following topics are considered: "The concept of a numerical series. Positive series", "Significant numerical series", "Functional series. Power series", "Taylor and McLaren series". The section "Fourier series" consists of the topics "Fourier series 2? - periodic functions", "Fourier series 2l - periodic and non-periodic functions". The Multiple Integrals section consists of the topics Double Integrals and Triple Integrals. The section "Ordinary differential equations" consists of the topics "Ordinary differential equations of the first order. Basic concepts" and "Ordinary differential equations of the second order with constant coefficients". "Linear ordinary differential equations of the nth order", "Systems of ordinary differential equations".
Assessment methods and criteria: 1. Settlement and graphic work
2. Control work
3. Work on practical classes
4. Semester Exam
Критерії оцінювання результатів навчання: 1. Settlement and graphic work (10 percents)
2. Control work (25 percents)
3. Work on practical classes (10 percents)
4. Semester Exam (55 percents)
Порядок та критерії виставляння балів та оцінок: 100-88 points - certified with an “excellent” grade - High level: the student demonstrates an in-depth mastery of the conceptual and categorical apparatus of the discipline, systematic knowledge, skills and abilities of their practical application. The mastered knowledge, skills and abilities provide the ability to independently formulate goals and organize learning activities, search and find solutions in non-standard, atypical educational and professional situations. The applicant demonstrates the ability to make generalizations based on critical analysis of factual material, ideas, theories and concepts, to formulate conclusions based on them. His/her activity is based on interest and motivation for self-development, continuous professional development, independent research activities, implemented with the support and guidance of the teacher. 87-71 points - certified with a grade of “good” - Sufficient level: involves mastery of the conceptual and categorical apparatus of the discipline at an advanced level, conscious use of knowledge, skills and abilities to reveal the essence of the issue. Possession of a partially structured set of knowledge provides the ability to apply it in familiar educational and professional situations. Aware of the specifics of tasks and learning situations, the student demonstrates the ability to search for and choose their solution according to the given sample, to argue for the use of a particular method of solving the problem. Their activities are based on interest and motivation for self-development and continuous professional development. 70-50 points - certified with a grade of “satisfactory” - Satisfactory level: outlines the mastery of the conceptual and categorical apparatus of the discipline at the average level, partial awareness of educational and professional tasks, problems and situations, knowledge of ways to solve typical problems and tasks. The applicant demonstrates an average level of skills and abilities to apply knowledge in practice, and solving problems requires assistance, support from a model. The basis of learning activities is situational and heuristic, dominated by motives of duty, unconscious use of opportunities for self-development. 49-00 points - certified with a grade of “unsatisfactory” - Unsatisfactory level: indicates an elementary mastery of the conceptual and categorical apparatus of the discipline, a general understanding of the content of the educational material, partial use of knowledge, skills and abilities. The basis of learning activities is situational and pragmatic interest.
Recommended books: 1. Рудавський Ю.К., Понеділок Г.В. та ін. Математичний аналіз. – Львів.: В-во НУ “ЛП”, 2003.
2. Рудавський Ю.К. та ін. Збірник задач з математичного аналізу. Частина 2. Львів.: В-во НУ “ЛП”, 2007.
3. Костенко І.С., Пукач П.Я., Філь Б.М., Тумашова О. В. Кратні інтеграли. Застосування Maple. Методичні вказівки з курсу “Математичний аналіз” для студентів інженерно – технічних спе-ціальностей. - В-во НУ “Львівська політехніка”.- Львів.- 2010. – 36 с.
4. Вища математика, частина 4 - електронний навчально-методичний комплекс, розміщений у Віртуальному навчальному середовищі Національного університету «Львівська політехніка» http://vns.lp.edu.ua /course/view.php?id=13114 Номер та дата реєстрації: Е41-143-59/2015 від 29.04.2015 р.
5. Рудавський Ю.К. та ін. Теорія рядів. – Львів.: В-во НУ “ЛП”, 2001.
6. Гук В.М. Теорія поля.- Львів: В-во НУ “ЛП”, 2004.
7. Шкіль В.П. Курс математичного аналізу, Київ: Наукова думка, 1995.
8. Вища математика. Збірник задач /За ред. В. П. Дубовика, І.І. Юрика.- Київ: В-во “АСК”, 2004.
9. Білущак Г.І., Дасюк Я.І., Каленюк П.І., Клюйник І.І., Кміть І.Я., Новіков Л.О., Пелех Я.М., Пукач П.Я. Салига Б.О. Кратні, криволінійні та поверхневі інтеграли. Теорія поля. Методичні вказівки та завдання до типових розрахунків для студентів інженерно–технічних спеціальностей. – Львів, 1996.
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