Course detail

Mathematics 1

FSI-Z1M Acad. year: 2024/2025 Winter semester

The course provides an introduction to linear algebra and analytic geometry. It is also devoted to the differential and integral calculus of functions of one variable, in particular, to properties of functions and their derivatives and basic techniques of integration. The main attention is paid to the use of the mathematical apparatus in solving some basic tasks in mathematical models of real problems. The course is the basis for successful completion of subsequent professional technical courses (machine design, technical mechanics, etc.).

Supervisor

doc. Ing. Jiří Šremr, Ph.D.

Department

Institute of Mathematics

Learning outcomes of the course unit

Prerequisites

Knowledge of mathematics at secondary school level.

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Conditions for awarding the course-unit credit (0-100 points, minimum 50 points):

  • submitting all the assigned homework,
  • written test (at least 50 of possible 100 points); students who fail to score 50 points will be allowed to resit the test during the first week of the examination period.

Conditions for passing the exam (0-100 points, minimum 50 points):

  • written test (maximum 80 points),
  • discussion about the test and the oral part of the exam (maximum 20 points),
  • maximum 100 points, the overall classification is given by ECTS grade scale.

Lecture: Attendance at lectures is obligatory and checked, only one unexpected absence is allowed, absence may be compensated for based on an agreement with the teacher.

Seminar: Attendance in seminars is obligatory and checked, only one unexpected absence is allowed, absence may be compensated for based on an agreement with the teacher.

Language of instruction

Czech

Aims

Students will acquire the skills to apply  of theoretical mathematical apparatus in solving some basic tasks appearing in mathematical models of real problems. 

  • Knowledge of the fundamentals of selected mathematical theories, which are needed in mathematical modelling in physics, mechanics, and other technical disciplines.
  • The ability to think logically and systematically, to move from simpler to more complex and to express and argue accurately when solving problems.
  • The ability to apply a suitable mathematical apparatus in solving some basic tasks appearing in mathematical models of real problems.

Specification of controlled education, way of implementation and compensation for absences

The study programmes with the given course

Programme B-KSI-P: Mechanical Engineering Design, Bachelor's
branch ---: no specialisation, 5 credits, compulsory

Programme C-AKR-P: , Lifelong learning
branch CZS: , 5 credits, elective

Type of course unit

 

Lecture

26 hours, compulsory

Teacher / Lecturer

doc. Ing. Jiří Šremr, Ph.D.

Syllabus


  • Basis of mathematical logic (premise, logical connective, quantifiers).

  • Complex numbers (algebraic and trigonometric forms, operations with complex numbers, Euler’s identity).

  • Vector, Cartesian coordinate system (free and bound vector, operations with vectors, scalar and vector products, magnitude of the vector).

  • Matrices (matrix, operations with matrices, determinant, inverse of a matrix, system of linear algebraic equations).

  • Analytic geometry (problems involving straight lines and planes in 2D and 3D spaces, e.g., intersection, distance, angle, etc.).

  • Functions of one real variable (notion of a function, graph, basic properties, basic elementary functions, vector function).

  • Differential calculus of functions of one variable (limit, L´Hospital rule, continuity, derivative, differential, linear and quadratic approximations, Taylor polynomial).

  • Behaviour of functions of one variable (monotonous functions, convex and concave functions, inflection points, local and global extremes, asymptotes).

  • Integral calculus of functions of one variable (Riemann integral, antiderivative, Newton-Leibnitz formula, indefinite integral, basic techniques of integration).

Exercise

39 hours, compulsory

Teacher / Lecturer

doc. Ing. Jiří Šremr, Ph.D.

Syllabus


  • Operations with vectors, scalar and vector products, examples of a possible use in geometry and solid mechanics.

  • Properties of matrices, operations with matrices, solving of systems of linear algebraic equations, eigenvalues and eigenvectors of a matrix.

  • Problems involving straight lines and planes in 2D and 3D spaces.

  • Basic properties of functions of one real variable, vector function, examples of a possible use in geometry and kinematic.

  • Evaluation of basic limits of functions of one variable, derivatives of functions of one variable, linear and quadratic approximations, examples of a possible use in geometry and kinematic.

  • Behaviour of functions of one variable, local and global extremes, examples of a possible use in problems of strength analysis.

  • Evaluation of indefinite and definite integrals of functions of one variable, geometrical and physical applications, examples of a possible use in evaluation of line integrals.