470-8721/02 – Mathematical Analysis I (MAINT)
| Gurantor department | Department of Applied Mathematics | Credits | 6 |
| Subject guarantor | prof. RNDr. Jiří Bouchala, Ph.D. | Subject version guarantor | prof. RNDr. Jiří Bouchala, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Compulsory |
| Year | 1 | Semester | winter |
| | Study language | English |
| Year of introduction | 2019/2020 | Year of cancellation | |
| Intended for the faculties | USP, FMT, FS, HGF | Intended for study types | Bachelor |
Subject aims expressed by acquired skills and competences
Students will get basic practical skills for work with fundamental concepts, methods and applications of differential and integral calculus of one-variable real functions.
Teaching methods
Lectures
Tutorials
Summary
In the first part of this subject, there are fundamental properties of the set of real numbers mentioned. Further, basic properties of elementary functions are recalled. Then limit of sequence, limit of function, and continuity of function are defined and their basic properties are studied. Differential and integral calculus of one-variable real functions is essence of this course.
Compulsory literature:
L. Gillman, R. H. McDowell: Calculus, New York, W.W. Norton & Comp. Inc. 1973.
M. Demlová, J. Hamhalter: Calculus I, skripta ČVUT Praha 1996.
J. Bouchala, M. Sadowská: Mathematical Analysis I (www.am.vsb.cz/bouchala)
Recommended literature:
J. Stewart: Calculus, Belmont, California, Brooks/Cole Pub. Comp. 1987.
Way of continuous check of knowledge in the course of semester
Written and oral exam.
E-learning
Other requirements
Additional requirements for students are not.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
Lectures:
Real numbers and numerical sets. Supremum and infimum. Real functions of one real variable.
Elementary functions. Sequences of real numbers. Limit of a sequence. Limit of a function. Continuity of a function. Differential and derivative of a function.
Fundamental theorems of differential calculus. Taylor polynomial. Investigation of the behavior of functions.
Primitive functions and indefinite integral.
Methods of integration (integration by parts, substitution, partial fraction decomposition).
Integration of special classes of functions.
Riemann integral. Integral with a variable upper limit.
Computation of definite integrals.
Applications. Improper integrals.
Seminars:
Logical connectives and basic terms of propositional logic. Applications of the mathematical induction principle. Identification of supremum and infimum in various types of sets.
Definition of a function. Increasing, decreasing, periodic functions.
Injective functions, finding inverse functions. Graph representation of a function.
Applications of properties of elementary functions in solving equations and inequalities, and other problems.
Calculation of limits of sequences, discussion of the concept of limit of a function.
Techniques for computing limits of functions.
Computation of derivatives and differentials of functions.
Construction of Taylor polynomial and estimation of the remainder in function approximation. Applications of derivatives, differentials, and Taylor polynomial in physics, geometry, and numerical mathematics.
Solving problems on function behavior.
Solving problems from integral calculus using integration by parts and substitution methods.
Solving problems related to the decomposition of a rational fractional function into partial fractions.
Practice of special substitutions in the integration of certain classes of functions.
Computation of definite integrals. Applications.
Calculation of improper integrals. Use of convergence criteria for improper integrals.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction