The aim of studying and bibliography

The main aim of study: The aim of the course is to familiarize students with the basic concepts of mathematical analysis, such as the notion of a real number and the limit of a sequence. Students should understand these concepts and gain practical ability to solve related tasks.

The general information about the module: Content provided during the course include: mathematical induction, the set of real numbers, other sets of numbers and their properties, functions, sequences and their limits.

Bibliography required to complete the module

Bibliography used during lectures

1	A.V. Efimov, V.P. Demidović	Higher mathematics: worked examples and problems with elements of theory: for engineering students Part 1: Linear algebra and fundamentals of mathematical analysis	Moscov: "Mir".	1984
2	R.J. Harshbarger, J.J. Reynolds	Calculus with applications	D. C. Heath & Co, Lexington.	1993
3	K. Kuratowski	Rachunek różniczkowy i całkowy. Funkcje jednej zmiennej	PWN, Warszawa.	1975
4	F. Leja	Rachunek różniczkowy i całkowy	PWN, Warszawa.	1975
5	A.V. Manzhirov, A. Polyanin	Handbook of mathematics for engineers and scientists	Boca Raton: Chapman a.Hall/CRC.	2007
6	W. Rudin	Podstawy analizy matematycznej	PWN, Warszawa.	1982
7	L.A. Trivieri	Basic Mathematics	New York: McGraw-Hill, Inc..	1990
8	L.A. Trivieri	Essential mathematics with applications	New York: Random House, Inc..	1988

Bibliography used during classes/laboratories/others

1	J. Banaś, S. Wędrychowicz	Zbiór zadań z analizy matematycznej	WNT, Warszawa.	2003
2	A.V. Efimov, V.P. Demidović	Higher mathematics: worked examples and problems with elements of theory: for engineering students Part 1: Linear algebra and fundamentals of mathematical analysis	Moscov: "Mir".	1984
3	M. Gewert, Z. Skoczylas	Analiza matematyczna I. Przykłady i zadania	GiS.	dow.
4	R.J. Harshbarger, J.J. Reynolds	Calculus with applications	D. C. Heath & Co, Lexington.	1993
5	L.A. Trivieri	Essential mathematics with applications	New York: Random House, Inc..	1988

Bibliography to self-study

1	M. Gewert, Z. Skoczylas	Analiza matematyczna I. Definicje, twierdzenia, wzory	GiS.	dow.
2	W. Krysicki, L. Włodarski	Analiza matematyczna w zadaniach. Cz. I	PWN, Warszawa.	dow.

Basic requirements in category knowledge/skills/social competences

Formal requirements: The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge: Basic knowledge of mathematics on secondary school level.

Basic requirements in category skills: Ability to use the fundamental mathematical tools in the area of the secondary school level.

Basic requirements in category social competences: The student is prepared to undertake objective and justified actions in order to solve the posed exercise.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
01	knows how to use the principle of mathematical induction and can investigate properties of number sets	lecture, solving classes	test	K_W02+++ K_W04+ K_W05++ K_U01++ K_U02++ K_U03+++ K_U08+++ K_K01+	P6S_KK P6S_UK P6S_UU P6S_UW P6S_WG P6S_WK
02	student knows the basic properties of the set of real numbers, can investigate properties of relations and functions	lecture, solving classes	test	K_W02+ K_W04++ K_W05+++ K_U01++ K_U02++ K_U09+++ K_U11+ K_K01+	P6S_KK P6S_UK P6S_UO P6S_UU P6S_UW P6S_WG P6S_WK
03	knows the basics of the theory of number sequences	lecture, solving classes	written exam	K_W02+++ K_W05++ K_U02++ K_U03++ K_U08++ K_U10+++ K_K01+	P6S_KK P6S_UU P6S_UW P6S_WG P6S_WK
04	student can calculate the limits of functions and investigate continuity of the functions and uniform continuity of functions	lecture, tutorials	test or exam	K_W02++ K_W04+ K_U01+ K_U02+ K_U08+ K_U10+++ K_K01+	P6S_KK P6S_UK P6S_UU P6S_UW P6S_WG P6S_WK

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
1	TK01	Axiomatic real numbers theory. The axiom of supremum. The principle of induction. The set of natural numbers, the set of integers and the set of rational numbers. Archimedes theorem. The greatest lower bound and the upper bound of a set. Irrational numbers.	W01-W12, C01-C12	MEK01 MEK02
1	TK02	The extended set of real numbers.	W13-W16, C13-C16	MEK02
1	TK03	Elements of topology of real axis. Open sets, closed sets, compact sets and connected sets. Accumulation points and isolated points of number sets.	W17-W24, C17-C24	MEK04
1	TK04	Relations and functions. Definition of relation and function. Image and preimage of a set by a function. Invers of a function. Injection, surjection and bijection. Cyclometric functions. Composition of functions.	W25-W32, C25-C32	MEK02
1	TK05	Sequences. Sequence of real numbers. Limit of a sequence. Basic properties of limit of a sequence. Euler number. Accumulation point for sequence.	W33-W48, C33-C48	MEK03
1	TK06	Limit of function. Continuity of function. Uniformly continuous function.	W49-W60, C49-C60	MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 1)		contact hours: 60.00 hours/sem.	complementing/reading through notes: 10.00 hours/sem. Studying the recommended bibliography: 10.00 hours/sem.
Class (sem. 1)	The preparation for a Class: 10.00 hours/sem. The preparation for a test: 20.00 hours/sem.	contact hours: 60.00 hours/sem.	Finishing/Studying tasks: 20.00 hours/sem.
Advice (sem. 1)	The preparation for Advice: 4.00 hours/sem.	The participation in Advice: 4.00 hours/sem.
Exam (sem. 1)	The preparation for an Exam: 20.00 hours/sem.	The written exam: 2.00 hours/sem. The oral exam: 1.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	Written or spoken exam concerning the material covered during lectures.
Class	Two written tests in terms agreed with the students. In order to gain credit for the classes, the student must attend classes and pass both written tests. Activity during tutorials gives a student opportunity to get a higher mark (up to one grade).
The final grade	After the credit of all types of classes the final grade is determined on the basis of the exam grades and the credits of the exercises.

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: yes

1	J. Banaś; J. Madej	Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations	2024
2	J. Banaś; J. Madej	On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations	2024
3	J. Banaś; J. Ochab; T. Zając	On the smoothness of normed spaces	2024
4	A. Ali; J. Banaś; . Mahfoudhi; B. Saadaoui	(P,Q)–ε-Pseudo Condition Spectrum for 2×2 Matrices. Linear Operator and Application	2023
5	J. Banaś; R. Taktak	Measures of noncompactness in the study of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations	2023
6	J. Banaś; V. Erturk; P. Kumar; A. Manickam; S. Tyagi	A generalized Caputo-type fractional-order neuron model under the electromagnetic field	2023
7	J. Banaś; A. Chlebowicz; M. Taoudi	On solutions of infinite systems of integral equations coordinatewise converging at infinity	2022
8	J. Banaś; R. Nalepa	The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space	2022
9	J. Banaś; R. Nalepa; B. Rzepka	The Study of the Solvability of Infinite Systems of Integral Equations via Measures of Noncompactness	2022
10	J. Banaś; W. Woś	Solvability of an infinite system of integral equations on the real half-axis	2021
11	J. Banaś; A. Chlebowicz; W. Woś	On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space	2020
12	J. Banaś; B. Krichen; B. Mefteh	Fixed point theorems in WC-Banach algebras and their applications to infinite systems of integral equations	2020
13	J. Banaś; L. Olszowy	Remarks on the space of functions of bounded Wiener-Young variation	2020
14	J. Banaś; A. Chlebowicz	On solutions of an infinite system of nonlinear integral equations on the real half-axis	2019
15	J. Banaś; B. Rzepka	Ocena efektywności inwestycji	2019
16	J. Banaś; B. Rzepka	Wykłady matematyki finansowej	2019
17	J. Banaś; L. Olszowy	On the equivalence of some concepts in the theory of Banach algebras	2019
18	J. Banaś; M. Krajewska	On solutions of semilinear upper diagonal infinite systems of differential equations	2019
19	J. Banaś; R. Nalepa	A measure of noncompactness in the space of functions with tempered increments on the half-axis and its applications	2019
20	J. Banaś; T. Zając	On a measure of noncompactness in the space of regulated functions and its applications	2019
21	L. Abadias; E. Alvarez; J. Banaś; C. Lizama	Solvability and uniform local attractivity for a Volterra equation of convolution type	2019