Mathematics I (FSI-1M)

COURSE-UNIT CREDIT REQUIREMENTS: The course includes seminars and exercises in the computer lab. There are two written tests within the seminars. Students may achieve max 12 points in each of these two tests, i.e. 24 points altogether. The course-unit credit is conditional on obtaining at least 6 points in each written test. If the minimum number of points is not achieved, students may repeat the test during the first two weeks of the examination period.

FORM OF EXAMINATIONS:
The exam has a written part (at most 75 points) and an oral part (at most 25 points)

WRITTEN PART OF EXAMINATION (at most 75 points)
In a 120-minute written test, students have to solve the following four problems:
Problem 1: Functions and their properties: domains, graphs (at most 10 points)
Problem 2: In linear algebra (at most 20 points)
Problem 3: In differential calculus (at most 20 points)
Problem 4: In integral calculus (at most 25 points)
The above problems can also contain a theoretical question. During the written exam, students are allowed to use a self-prepared handwritten cheat sheet not exceeding two pages A4.

ORAL PART OF EXAMINATION (max 25 points)
• Discussion based on the written test: students have to explain how they solved each problem. Should the student fail to explain it sufficiently, the test results will not be accepted and will be classified by 0 points.
• Possible theoretic question.
• Possible simple problem to be solved straight away.
• The results achieved in the written tests in seminars may be taken into account within the oral examination.



FINAL CLASSIFICATION:
0-49 points: F
50-59 points: E
60-69 points: D
70-79 points: C
80-89 points: B
90-100 points: A



Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is fully at the discretion of the teacher.

Week 1: Basics of mathematical logic and set operations, matrices and determinants (transposing, adding, and multiplying matrices, common matrix types).
Week 2: Matrices and determinants (determinants and their properties, regular and singular matrices, inverse to a matrix, calculating the inverse to a matrix using determinants), systems of linear algebraic equations (Cramer's rule, Gauss elimination method).
Week 3: More about systems of linear algebraic equations (Frobenius theorem, calculating the inverse to a matrix using the elimination method), vector calculus (operations with vectors, linear combinations and linear independence of vectors, scalar (dot) product, vector (cross) product, scalar triple (box) product).
Week 4: Eigenvalues of a matrix, the notion of a function (domain and range, bounded functions, even and odd functions, periodic functions, monotonous functions, composite functions, one-to-one functions, inverse functions).
Week 5: Basic elementary functions (exponential, logarithm, general power, trigonometric functions and cyclometric (inverse to trigonometric functions), polynomials (root of a polynomial, the fundamental theorem of algebra, multiplicity of a root, product breakdown of a polynomial), introducing the notion of a rational function.
Week 6: Sequences and their limits, limit of a function, continuous functions.
Week 7: Derivative of a function (basic problem of differential calculus, notion of derivative, calculating derivatives, geometric applications of derivatives), calculating the limit of a function using L' Hospital rule.
Week 8: Monotonous functions, maxima and minima of functions, points of inflection, convex and concave functions, asymptotes, sketching the graph of a function.
Week 9: Differential of a function, Taylor polynomial, parametric and polar definitions of curves and functions (parametric definition of a derivative, transforming parametric definitions into polar ones and vice versa).
Week 10: Primitive function (antiderivative) (definition, properties and basic formulas), integrating by parts, method of substitution.
Week 11: Calculating a primitive function by the method of substitution in some of the elementary functions. Riemann integral (basic problem of integral calculus, definition and properties of the Riemann integral).
Week 12: Calculating the Riemann integral (Leibniz-Newton' s formula). Applications of the definite integral (mainly surface area of a plane figure, length of a curve, volume and lateral surface area of a rotational body).
Week 13: Improper integral.