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Course info
KMA / MA1-A
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Course description
Department/Unit / Abbreviation
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KMA
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MA1-A
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Mathematical Analysis 1
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
7
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
4
[Hours/Week]
Tutorial
2
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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English
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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0 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Winter + Summer
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Semester taught
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Winter + Summer
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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English
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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N/A
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The aim of this course is a basic introduction and active understanding of the fundamental concepts of mathematical analysis such as: - infinite sequences and series of real numbers; - functions of one real variable; - differential and integral calculus. This course is lectured in English, its subject is equivalent to KMA/MA1.
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Requirements on student
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Demonstrate knowledge of the definitions, fundamental theorems and their proofs concerning sequences, series, and functions of one real variable. Use rigorous arguments in calculus and ability to apply them in solving problems on the topics in the syllabus. Detailed information may be found on the server http://analyza.kma.zcu.cz.
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Content
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Week 1: Mathematical reasoning - open statements and quantifiers, negating quantified statements; sets and elementary operations; subsets of real numbers; absolute value; maximum, minimum, least upper bound, and greatest lower bound of a subset of real numbers;
Week 2: Sequences of real numbers; subsequences; bounded and monotone sequences; recursively defined sequences; Bolzano-Weierstrass theorem; convergent and divergent sequences; Cauchy sequences;
Week 3: Algebra of limits and fundamental theorems concerning the properties of a limit;
Week 4: Conditions ensuring the convergence of infinite sequences and series;
Week 5: Functions of one real variable; graphical representation; inverse functions; composition of functions; polynomial, trigonometric, exponential, and hyperbolic functions;
Week 6: Local and global behaviour of a function; limits; one-sided limits; algebra of limits;
Week 7: Continuity of a function at a point; points of discontinuity; continuity in a closed interval;
Week 8: Derivative and differential of a function - definition and both the geometrical and the physical meaning; differentiability and continuity of a function;
Week 9: Differentiation from first principles, product rule and chain rule, Rolle's theorem, Langrange's and Cauchy's mean value theorems; stationary points of a function; l'Hospital's rule;
Week 10: Indefinite integral; fundamental theorem of calculus; integration by parts and integration by substitution;
Week 11: Definite integral and its applications; improper integrals; inequalities for integrals;
Week 12: Higher order derivatives and differentials; Taylor's theorem;
Week 13: Applications of differential and integral calculus in solving optimization and physical problems.
Further information and the lecture notes can be found on the web page http://analyza.kma.zcu.cz.
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Activities
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Fields of study
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Studentům je k dispozici kurz v Moodle se všemi podstatnými informacemi a materiály.
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Guarantors and lecturers
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Literature
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Recommended:
http://trial.kma.zcu.cz
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Recommended:
Matematická analýza
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Recommended:
Drábek, Pavel; Míka, Stanislav. Matematická analýza I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-558-8.
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Recommended:
Pultr, Aleš. Matematická analýza I. Praha : Matfyzpress, 1995. ISBN 80-8586-3-09-X.
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Recommended:
Polák, Josef. Přehled středoškolské matematiky. Praha : Prometheus, 1995. ISBN 80-85849-78-X.
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Recommended:
Míková, Marta; Kubr, Milan; Čížek, Jiří. Sbírka příkladů z matematické analýzy I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-568-5.
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Recommended:
Děmidovič, Boris Pavlovič. Sbírka úloh a cvičení z matematické analýzy. Havlíčkův Brod : Fragment, 2003. ISBN 80-7200-587-1.
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On-line library catalogues
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Preparation for comprehensive test (10-40)
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36
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Presentation preparation (report in a foreign language) (10-15)
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10
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Contact hours
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78
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Preparation for an examination (30-60)
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60
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Total
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184
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
There is no prerequisite for this course. Students should be familiar with a high school algebra and trigonometry. |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
By the end of the course, a successful student should be able to:
1. Read mathematical text in English and to use logical constructions; 2. Use rigorous arguments in calculus and ability to apply them in solving problems on the topics in the syllabus; 3. Demonstrate knowledge of the definitions and fundamental theorems concerning sequences, series and continuous and differentiable functions of one real variable; 4. Use both the definition of derivative as a limit and the rules of differentiation to differentiate functions; 5. Sketch the graph of a function using asymptotes, critical points, and the derivative test for increasing/decreasing and concavity properties; 6. Set up max/min problems and use differentiation to solve them; 7. Use l'Hospital's rule; 8. Evaluate integrals using techniques of integration, such as substitution, inverse substitution, partial fractions and integration by parts; 9. Apply integration to compute areas and volumes by slicing; 10. Find the Taylor series expansion of a function near a point, with emphasis on the several starting terms; 11. Use developed theory in solving problems on physical systems.
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills demonstration during practicum |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture supplemented with a discussion |
Interactive lecture |
Task-based study method |
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