HIGHER MATHEMATICS 2

Abstract of the academic discipline

The purpose of studying the discipline. To provide theoretical and practical training of applicants for the formation of the development of logical and algorithmic thinking, studying the basics of mathematical apparatus, which is necessary for solving theoretical and practical problems of technology, raising the general level of mathematical culture. To teach students to independently work on mathematical literature and its application, to acquire skills in mathematical research of applied problems and to be able to build mathematical models of technical tasks.

Practical significance and use of acquired knowledge. Gaining knowledge of the theoretical foundations of higher mathematics and practical methods of solving relevant problems.
Obtaining knowledge and skills about methods of reducing a real problem to a mathematical model and apply them for the purpose of research and analysis. To uUse the mathematical apparatus for questions related to the specialty and literature on applied mathematics, reference books, tables.
To understand cause-and-effect relationships, to have basic mathematical apparatus, knowledge of modern information technologies and fundamental sciences to the extent necessary for mastering general professional disciplines. Be able to write and speak in your native language and computer skills, etc. Acquisition of skills to solve typical mathematical problems, to use in practice the algorithm for solving typical problems, to be able to systematize typical problems, to find criteria for reducing problems to typical problems, to be able to recognize a typical problem or to reduce it to a typical problem; be able to use various information sources to find procedures for solving typical problems (textbook, reference book, Internet resources).
To possess the deductive method of proving and refuting statements and use in practice the conceptual apparatus of deductive theories, reproduce deductive proofs of theorems and prove the correctness of procedures for solving typical problems. Be able to conduct deductive justifications of the correctness of solving problems and look for logical errors in incorrect deductive reasoning; use mathematical and logical symbols in practice. Be able to see and apply mathematics in real life, understand the content and method of mathematical modeling, be able to build a mathematical model, investigate it using mathematical methods, interpret the results obtained, estimate the calculation error. To understand the need to be persistent in achieving the goal and quality performance of work in the professional field.

Main learning outcomes
PRN#1. Be able to knowledge of the basic forms and laws of abstract and logical thinking, the basics of the methodology of scientific knowledge, the forms and methods of extracting, analyzing, processing and synthesizing information in the subject area of computer science.
PRN#2. Be able to use the modern mathematical apparatus of continuous and discrete analysis, linear algebra, analytical geometry, in professional activities to solve problems of a theoretical and applied nature in the process of designing and implementing informatization objects.

Subjects and types of educational classes
1 week.
Lecture #16 
"Derivative: definition, geometric, physical and mechanical meaning. The equation of the tangent and the normal to the curve. The relationship between differentiability of a function and continuity. Differentiation of sum, product, quotient. Differentiation of a composite function. Table of derivatives".
Practical lesson #16. 
"Differentiation of functions. Table. Derivative of sum, product, quotient, composite function".
Obtaining a task for calculation and graphic work
2 week
Lecture #17. 
"Differentiation of implicit functions and parametrically specified functions. Logarithmic differentiation. Derivatives of higher orders. The mechanical meaning of the second derivative".
Practical lesson #17. 
"Derivatives of higher orders. Differentiation of implicitly specified and parametrically specified functions".
Performing calculation and graphic work. Part 1. Continuity and differentiation of a function.
3 week
Lecture #18. 
"Differential of a function. The concept and geometric content of the differential. Basic theorems and table of differentials. Application of differentials in approximate calculations. Differentials of higher orders".
Practical lesson #18. 
"Differential of the first and higher orders of the function. Differentiation rules. Differentiation of a complex function".
Performing calculation and graphic work. Part 1. Continuity and differentiation of a function.
4 week
Lecture #19. "Investigation of a function using derivatives. Lopital's rule. Fermat's and Rolle's theorems. Theorems of Cauchy and Lagrange. Monotonicity of the function. Local extremum. Sufficient conditions of extremum. Convexity of the graph of the function. Inflection points. Asymptotes of the function graph. Taylor's formula".
Practical lesson #19. 
"The Hospital's rule. Increasing and decreasing functions. Maximum and minimum. The largest and smallest value on the segment. Scheme of the study of the function and construction of the graph. Formula McLaren".
Performing calculation and graphic work. Part 1. Study of functions using derivatives
5 week
Lecture #20.
"Primitive. Indefinite integral. Properties. Table of integrals. Integration of simpler integrals. Changing the variable in the indefinite integral. Integration by parts. Finding integrals containing a quadratic trinomial".
Practical lesson #20. 
"Table integrals. Substituting a variable. Integration in parts".
Performing calculation and graphic work. Part 2. Technique of indefinite integration
6 week
Lecture #21. 
"Integration of rational fractions. Integration of irrationalities. Integration of trigonometric functions. Universal trigonometric substitution. Integration of irrationalities using trigonometric substitutions. Integrals not expressed in terms of elementary functions".
Practical lesson #21. 
"Integration of fractional-rational and some irrational functions. Integration of trigonometric functions. Trigonometric substitutions for some irrationalities"
Performing calculation and graphic work. Part 2. Mixed typical problems for indefinite integration
7 week
Lecture #22. 
"The definite integral. Definition and properties. Integrals with variable upper bound. Theorem on the differentiation of integrals with a variable upper limit. The Newton-Leibnitz formula".
Practical lesson #22. 
"The Newton-Leibnitz formula. Replacement of a variable and integration by parts in the definite integral"
Performing calculation and graphic work. Part 2. Basic rules of definite integration
8 week
Lecture #23. 
"Application of the definite integral. Calculation of area, arc length, volume of a body based on cross-sectional area and body of rotation" Improper integrals
Practical lesson #23. 
Application of the definite integral to problems of geometry and solving problems of physics and mechanics
Modular control work #1.

9 week
Lecture #24.
"Numerical, functional and power series."
Practical lesson #24. 
"Convergence of series. Decomposition of functions into power series. Some applications of power series".
Performing calculation and graphic work. Part 3. Signs of series convergence Decomposition of functions into Taylor series
10 week
Lecture #25. 
Fourier series. Harmonic analysis. Character of convergence of Fourier series. Fourier integral. Fourier transform. Spectral function.
Practical lesson #25. 
"Expansion of even and odd functions into Fourier series."
Performing calculation and graphic work. Part 3. Signs of convergence of Fourier series Decomposition of even and odd functions into Fourier series.
11 week
Lecture #26. 
"Functions of many variables. Partial derivatives and differentials of higher orders. Theorem on equality of mixed derivatives. Application of the differential to approximate calculations. The equation of the tangent plane and the normal to the surface".
Practical lesson #26. 
"Partial derivatives of higher orders. Application of the differential function of many variables".
Performing calculation and graphic work. Part 3. Methods of determining partial derivatives of higher orders and a complex function
12 week
Lecture #27. 
"Extremum of a function of many variables. Stationary points. Extremum conditions are necessary. A sufficient sign of an unconditional extremum at a stationary point".
Practical lesson #27. 
"Finding critical points and determining the extremum of a function of many variables."
Performing calculation and graphic work. Part 3. Directional derivative and gradient of a function of many variables
13 week
Lecture #28. 
"The concept of conditional extremum. Substitution method of finding conditional extremum. Lagrange's method. Finding the largest and smallest values of a function in the domain »
Practical lesson #28. 
"Extremum of a function of 2 variables. Conditional extremum. The largest and smallest value of a function in a limited domain".
Performing calculation and graphic work. Part 3. The method of least squares
14 week
Lecture #29. 
"Ordinary differential equations. Types of equations of the first order and methods of their solution".
Practical lesson #29. 
"Solutions of differential equations of the first order. The Cauchy problem for the first-order LND.
Performing calculation and graphic work. Part 4. Ordinary differential equations of the first order.
15 week
Lecture #30.
"Ordinary differential equations of higher orders. The structure and methods of their solutions. Application of differential equations in applied research. Systems of differential equations and methods of their solution".
Practical lesson #30. 
"Methods of solving ordinary differential equations of higher orders and systems of first-order differential equations."
Performing calculation and graphic work. Part 4. Ordinary differential equations of higher orders. Their application in modeling dynamic linear systems
Modular control work 2.

Individual work of the applicant takes place during the semester and consists of preparation for classroom classes, control measures, individual tasks.
Consultations: are carried out by the teacher during the semester according to the schedule

Assessment of learning outcomes
The evaluation of the results of studies in the discipline is carried out according to the cumulative system, which allows the student to receive a maximum of 100 points during the semester.
Module #1
Completion of the first part of the Calculation-Graphic Work –  10 points
Completion of the second part of the Calculation-Graphic Work – 10 points
Modular test # 1 – perfect execution of 30 points (in each task of the modular test, the maximum number of points for each task is given).
Module #2
Completion of the third part of the Calculation-Graphic Work – 10 points
Completion of the fourth part of the Calculation-Graphic Work – 10 points
Modular test # 2 – perfect execution of 30 points (in each task of the modular test, the maximum number of points for each task is given).

Links to recommended sources of information
Basic literature
1. Усов А.В. Диференціальне числення функції однієї змінної/ А.В. Усов, Ю.О. Кліх, Л.І. Плотнікова, О.М. Дубров. Навчальний посібник. −Одеса: Астропринт, 2017.− 224с. (300 прим.)
2. Усов А.В. Інтегральне числення функції однієї змінної. Теорія рядів. Навч. Пос. – Одеса: Астропринт, 2018. – 224с. (100)
3. Усов А.В. Звичайні диференціальні рівняння. Крайові задачі та теорія стійкості. Конспект лекій. – Одеса: Астропринт, 2020. – 144 с. (120 прим.)
4. Усов А.В Математичні методи моделювання. / А.В. Усов, О.С. Савельєва, І.І. Становська, А.О. Перпері// Підручник / За ред. Становського О.Л. – Одеса Пальміра, 2011. – 500 с.
5. Моделювання та оптимізація систем. – Підручник /В.М. Дубовой, Р.Н. Квєтний, О.І. Михадьов, А.В. Усов/ − Вінниця: ПП «Едельвейс», 2017. – 804 с.
Additional literature
1. Вища математика: Збірник задач: У 2ч. Ч.1: Лінійна і векторна алгебра. Аналітична геометрія. Вступ до математичного аналізу. Диференціальне та інтегральне числення.: Навчальний посібник для студентів вищих технічних навчальних закладів.(Гаврильченко Х.І., Полушкін С.П., Кропив'янський П.С. та ін.; За заг. ред. д-ра техн. наук, проф. Овчинникова П.П.). К.: Техніка, 2004. 279 с. (Вміщено задачі і вправи з вищої математики для самостійної роботи студентів, наведено приклади розв'язання типових задач).
2. Вища математика: Збірник задач: Підручник: У 2 ч. Ч. 2: Звичайні диференціальні рівняння. Операційне числення. Ряди. Рівняння математичної фізики. Стійкість за Ляпуновим. Елементи теорії ймовірностей і математичної статистики. Методи оптимізації і задачі керування. Варіаційне числення. Числові методи: Навч. посіб. для студентів вищ. тех. навч. закл. /(За заг. ред. П.П. Овчинникова.). К. : Техніка, 2004. 376 с. (Вміщено задачі і вправи з вищої математики для самостійної роботи студентів, наведено приклади розв'язання типових задач).
3. «Індивідуальні домашні завдання з вищої математики і методичні вказівки до їхнього виконання для студентів 1 курсу усіх спеціальностей денної форми навчання 2-го семестру»/ Укл.: Усов А.В., Кузьміна В.М,   Одеса: ОНПУ, 2020. 47 с. Рег. номер в журналі обліку МВО 3328 №00011-РС   2020.
4. Грібова, В.В., Перстньова, В.В. Методичні вказівки до практичних занять з вищої математики на тему «Інтегральне перетворення Лапласа та його застосування» для студентів 2 курсу спеціальності 122 денної форми навчання  / В.В. Перстньова ,В.В.Грібова. –  Одеса: ОНПУ, 2017, 25 с., рег.№ МВ07904 від 20.02.2017 
5. Грібова, В.В.,Перстньова, В.В. Методичні вказівки до практичних занять з вищої математики за розділом «Інтегральне перетворення Фур’є» для студентів 2 курсу спеціальності 123 / В.В.Перстньова, В.В.Грібова. –  Одеса: ОНПУ, 2018. –  11 с., рег.№ МВ04564 від 12.06.2012
6. Комлева Т.О.,Плотнікова Л.І., Скрипнік Н.В., Оборський Г.О.,Усов А.В. «Ряди Фур’є в прикладах і вправах»-Одеса.: «Астропринт»,2007
7. Довідник з математики. Частина І.Навчальний посібник з дисципліни «Вища математика». Для студентів технічних спеціальностей за напрямом підготовки 122, 144, 141, 104./Укл. Колмакова Л.М., Комарницький О.Л.-Одеса: ОНПУ, 2017.-80с. Лаб. інф. технологій, рег. № НП08668 від 11.07.2017
8. Індивідуальні домашні завдання з дисципліни «Вища математика», розділ «Лінійна алгебра» та методичні вказівкми до їх виконання. Для здобувачів вищої освіти усіх форм навчання за спеціальністю 122-Комп’ютерні науки та інформаційні технології./ Укл.: Л.М.Колмакова, Ю.Є.Сікіраш.-Одеса: НУОП, 2021.-40 с. Рег. Номер в журналі обліку 8316-РС-2022.