Mathematics 1000-111MAT

We will start the lecture from the description of basic mathematical notions such as sets, numbers, relations, functions. Next, we will describe basic types of functions: linear, exponential and logarythmic as well as logarithms and its applications. The main part on the lecture will be devoted to mathematical analysis. We will define the notion of limit (in the context of sequences and functions), continuity and differentiability of one variable functions. We will try to understand geometrical and physical interpretations of derivative. Next we will describe an extremum and study the dependence between extrema and convexity (concavity) of the function. The theoretical part of the lecture will end with the description of primitive function and the notion of integral and its geometrical interpretation. In the last part of the lecture we will focus on mathematical modelling of biological phenomena. Therefore, it will be necessary to introduce the notion of ordinary differential equations. On the basis we will study some models of population dynamics (equation of Malthus, logistic equation, competition, predation, symbiosis). We will introduce some basic notions of the probability theory and define Markov chains and its applications for models of evolution.