Applied Mathematics

Course guarantor:

David Vrba

Lecturer:

David Vrba

Tutor:

Matouš Brunát, David Vrba

Supervisor:

Department of Biomedical Technology

Synopsis:

The course Applied Mathematics combines both theoretical knowledge and practical skills. Theoretical knowledge is necessary to formulate a mathematical model and then to solve decision-making and optimization problems in economic processes. Practical knowledge is trained by solving concrete situations using sample examples, where students are introduced to specific methods and techniques of mathematical data analysis.

Requirements:

1. Active participation in seminars (maximum 2 absences allowed)

2. Min. 50% pass rate in the credit test (max. 20 points)

Examination conditions: the examination consists of two parts: written and oral.

The written part contains 4 examples (max. 60 points).

In the oral part of the exam (max. 20 points) the student defends the grade from the written part. The student must demonstrate that he/she understands the content and the logical connections.

Assessment according to the ECTS scale for the sum of all points (max. 100 points)

Syllabus of lectures:

Numbers and functions: natural numbers, integers, real numbers, intervals, number systems, functions, polynomials, functions of two or more variables, compound and inverse functions. Balancing the production process, plotting the price graph of a function, writing a linear price function.

- Sequences (arithmetic, geometric), series, limits of a sequence, convergence, divergence, proper/non-proper limit, limit at a non-proper point. Interest, appreciation of investments.

- Limits, continuity and derivative: Derivative as rate of change, as directive tangent to a curve, concept of limits, counting with limits, infinities, non-proper limits, continuity of functions.

- Derivative of constant, linear and power functions, rules for calculating derivative of sum, difference, product and quotient functions, derivative of composite function, partial derivative. Marginal price, maximization of earnings, minimization of stacking cost

- Progress of a function of one variable: definitional domain, local and absolute extremes, monotone functions. Plotting and analysis of construction cost.

- The course of a function of one variable: evenness, oddness, convexity, concavity and inflection points

- Fundamentals of integral calculus I: methods of computing integral, properties of integral, indefinite integral, estimation of population of a populated area.

- Fundamentals of integral calculus II: definite integral, implicit integral; integral as a generalized sum, integral as area under a graph, comparing future income, finding the total price from the marginal price, calculating the investment difference, comparing Lorentz curves.

- Mathematical solution of optimization problem I - local extremes of functions of one variable, solution by derivatives. Local extremes of functions of two variables. Profit maximization, optimal placement of stocks.

- Mathematical solution of optimization problem II - functions of three or more variables. Finding the bounded extremum, Lagrange multiplier. Lagrange multiplier for construction optimization, utility maximization, optimal resource allocation, minimization of construction cost.

- Least squares methods and regression analysis.

- Matrix calculus: matrices and vectors, matrix operations, commutative, associative and distributive law, unit and zero matrices, transpose and inverse matrices. Gaussian elimination, determinant and methods of its calculation.

- Introduction to game theory and models of decision games. Selected game theory - Definition of game, Prisoner's dilemma, oligopolies, Nash equilibrium, cake cutting (game).

- Introduction to differential equations: definition of DR, types of DR, intuition, simple equations and models, direction field, general solution, solution with initial condition, finding the yield of a differential equation

Syllabus of tutorials:

- Numbers and functions: natural numbers, integers, real numbers, intervals, number systems, functions, polynomials, functions of two or more variables, compound and inverse functions, goniometric functions

- Sequences (arithmetic, geometric), series, limits of a sequence, convergence, divergence, proper/non-proper limit, limit at a non-proper point

- Limits, continuity and derivative: derivative as rate of change, as directive of a tangent to a curve, concept of limits, counting with limits, infinities, eigenlimits, continuity of functions

- Derivative of constant, linear and power functions, rules for calculating derivative of sum, difference, product and quotient functions, derivative of composite function, partial derivative

- The progression of a function of one variable: definitional domain, local and absolute extremes, monotone functions

- The progression of a function of one variable: evenness, oddness, convexity, concavity and inflection points

- Fundamentals of integral calculus I: methods of calculating the integral, properties of the integral, indefinite integral

- Fundamentals of integral calculus II: definite integral, implicit integral; integral as a generalized sum, integral as the area under a graph

- Mathematical solution of optimization problem I - local extremes of functions of one variable, solution by derivatives. Local extremes of functions of two variables

- Mathematical solution of optimization problem II - functions of three or more variables. Finding the bounded extremum, Lagrange multiplier

- Least squares methods and regression analysis

- Matrix calculus: matrices and vectors, matrix operations, commutative, associative and distributive laws, unit and zero matrices, transpose and inverse matrices. Gaussian elimination, determinant and methods of its calculation

- Introduction to game theory and models of decision games. Selected game theories - Definition of a game, Prisoner's dilemma, oligopolies, Nash equilibrium, cutting the cake (game)

- Introduction to differential equations: definitions of DR, types of DR, intuition, simple equations and models, direction field, general solutions, solutions with initial condition

Study Objective:

Study materials:

Required literature

[1] TKADLEC, Josef. Differential and integral calculus of functions of one variable. Prague: Czech Technical University Publishing House, 2004. ISBN 80-01-03039-3.

[2] OLŠÁK, P. Linear algebra [online], teaching text FEL CTU, 2007. Available from: http://petr.olsak.net/linal.html

[3] The Princeton companion to mathematics. Editor Timothy GOWERS, editor June BARROW - GREEN, editor Imre LEADER. Princeton: Princeton University Press, 2008. ISBN 0691118809.

Recommended literature

[4] BINMORE, K. G. Game theory: --and how it can change your life. Prague: Dokořán, 2014. Aliter (Argo: Dokořán). ISBN 978-80-7363-549-7.

[5] Eichhorn, W., Gleißner, W. Mathematics and methodology for economics: applications, problems and solutions. New York, NY: Springer Berlin Heidelberg, 2016. ISBN 9783319233529.

[6] Hoffmann L., Bradley G., Sobecki D. and Price M., Calculus For Business, Economics, and the Social and Life Sciences 11th Edition, McGraw-Hill 2013, ISBN 978-0073532387.

Study aids

Handouts designed for combined study and presentations posted on the course website.

Note:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomKL:B-537 Vrba D. 10:00–11:50 (lecture parallel1) Kladno FBMI
Tue	roomKL:B-720 Brunát M. 10:00–11:50 (lecture parallel1 parallel nr.2) Kladno FBMI roomKL:B-720 Brunát M. 12:00–13:50 (lecture parallel1 parallel nr.3) Kladno FBMI
Wed
Thu	roomKL:B-420 Vrba D. 14:00–15:50 (lecture parallel1) Kladno FBMI
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Systematic Integration of Prosesses of Healthcare (compulsory course)