Analysis I 390-FS1-1AM1

Study profile: general academic
Form of study: full-time
Subject type: compulsory
Field and discipline of study: Field of exact and natural sciences, Discipline of physical sciences.
Level of education: first-cycle studies
Year of study/semester: 1st year/2nd year Semester
ECTS Credits: 6
Prerequisites:

Student Workload Balance:
- Participation in lectures (45 hours),
- Participation in tutorials (60 hours),
- Participation in consultations (15 hours),
- Student's own work at home (30 hours),

Quantitative Indicators:
- Student workload related to classes requiring direct teacher involvement - 4.8 ECTS;
- Student workload related to independent work - 1.2 ECTS.

Principles of Artificial Intelligence (AI) Use:
During classes, the use of AI systems is permitted for the following purposes:
1. Machine translation of source texts from foreign languages.
2. Searching for and organizing scientific sources. 3. Create simulations and modeling of physical phenomena discussed in the lecture.

The use of AI systems is prohibited during the exam.
If any violations of the above rules are detected, the student may be held accountable under separate disciplinary regulations.

Scope of topics:
1. Basic information about mathematical proofs, mathematical logic and set theory.
2. Sequences and numerical series. Geometric series. Convergence criteria: d'Alembert, Cauchy. Harmonic series divergence. Euler number e.
3 Functions of one variable. Limit of a function, continuity, differentiability. Properties of the derivative. Chain rule.
4. Local and global extremes. Convexity, asymptotes. investigation of functions of one real variable.
5. Inverse function theorem. Derivative of inverse function. Lagrange's mean value theorem. Taylor's theorem. L'Hospital's rule.
6. Power series. Overview of elementary functions. The exponential function. Logarithm. Trigonometric, hyperbolic and cyclometric functions.
7. Sequences and functional series, uniform convergence.
8. Definite integral (Riemann integral). Approximate methods of calculating integrals. Newton-Leibnitz theorem. Improper integrals.
9. Basic information on the generalization of the notion of the integral (Stieltjes integral, Lebesque integral), sets of zero measure. Integral criterion of series convergence.
10. Basic information about Fouriere series

Term 2023:

1. Basic information about mathematical proofs, mathematical logic and set theory.
2. Sequences and numerical series. Geometric series. Convergence criteria: d'Alembert, Cauchy. Harmonic series divergence. Euler number e.
3 Functions of one variable. Limit of a function, continuity, differentiability. Properties of the derivative. Chain rule.
4. Local and global extremes. Convexity, asymptotes. investigation of functions of one real variable.
5. Inverse function theorem. Derivative of inverse function. Lagrange's mean value theorem. Taylor's theorem. L'Hospital's rule.
6. Power series. Overview of elementary functions. The exponential function. Logarithm. Trigonometric, hyperbolic and cyclometric functions.
7. Sequences and functional series, uniform convergence.
8. Definite integral (Riemann integral). Approximate methods of calculating integrals. Newton-Leibnitz theorem. Improper integrals.
9. Basic information on the generalization of the notion of the integral (Stieltjes integral, Lebesque integral), sets of zero measure. Integral criterion of series convergence.
10. Basic information about Fouriere series

Term 2024:

Type of course

obligatory courses

Prerequisites (description)

A course in differential and integral calculus of functions of one variable. The exercise classes focus on developing computational skills.

Course coordinators

Term 2024:

Jan Cieśliński

Term 2025:

AAntek AAntkowski
Jan Cieśliński

Term 2023:

Jan Cieśliński

Mode

(in Polish) w sali

Learning outcomes

Knowledge: The graduate knows and understands:
KP6_WG2 advanced knowledge of elements of higher mathematics and mathematical methods used in physics;

Skills: The graduate is able to:
KP6_UW6 learn independently, finding necessary information in professional literature, databases, and other sources, and critically evaluate information from unverified sources;
KP6_U01 organize their own work and that of their team;
KP6_UU1 engage in lifelong learning and inspire and organize the learning process of others.

Social Competencies: The graduate is ready to:
KP6_KK1 critically evaluate their knowledge and the content they receive;
KP6_KK2 recognize the importance of knowledge in solving cognitive and practical problems;
KP6_KK3 collaborate with experts when faced with difficulties in independently solving problems.
KP6_KO1 fulfill social obligations and reject disinformation regarding acquired knowledge;

Assessment criteria

During classes, students solve computational problems and are given homework. The emphasis is on acquiring several skills, described as the main learning outcomes. The effects are checked by written tests (two during the semester). Activity in classes and creativity in the approach to solved problems are also assessed. After completing the education in the Mathematical Analysis, there is an exam to verify the acquired knowledge.

Bibliography

W. Rudin: Principles of mathematical analysis
W.Krysicki, L.Włodarski: Analiza matematyczna w zadaniach,
M.Gewert, Z.Skoczylas, Analiza matematyczna I

Term 2023:

W. Rudin: Principles of mathematical analysis
W.Krysicki, L.Włodarski: Analiza matematyczna w zadaniach,
M.Gewert, Z.Skoczylas, Analiza matematyczna I

Term 2024:

W. Rudin: Principles of mathematical analysis
W.Krysicki, L.Włodarski: Analiza matematyczna w zadaniach,
M.Gewert, Z.Skoczylas, Analiza matematyczna I

Additional information

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 390-FS1-1AM1 in USOSweb