Basics of Mathematics

obligatory courses

Classroom

The lecture with exercises is intended to familiarise students with the basic issues of mathematical analysis. Students will learn, among others: the concept of the limit of an infinite sequence, a numerical series, and a logarithmic scale. They will become acquainted with the basics of financial mathematics, the concept of derivative of functions of one and many variables and the least squares method. The subject is intended to prepare students to listen to lectures requiring more advanced mathematical equipment.

The thematic scope of the lecture and exercises overlaps and covers the following topics:

1. Logic.

2. Basic elementary functions:

• exponential, logarithmic, square root and polynomial functions, trigonometric functions;

• logarithmic scale and its applications.

3. Number sequences and series:

• mathematical induction

• limit of an infinite sequence, convergence of an infinite sequence;

• geometric series, simple criteria for convergence of numerical series.

4. Basics of financial mathematics:

• simple interest, compound interest; loans with fixed and decreasing installments;

• cash deposits.

5. The concept of injection,surjection, bijection, function composition, continuous function, properties of continuous functions.

6. Calculus and its applications:

• definition of a derivative, tangent line to the graph of a function;

• derivatives of elementary functions, derivative of a composition of functions, basic operations on derivatives;

• determining the properties of a function based on its derivative, determining the smallest/largest value and extremes of the function.

7. Linear regression:

• the concept of multivariate functions, the partial derivative and gradient of multivariable functions;

• least squares method for two parameters.

Dariusz Wrzosek, Matematyka dla biologów, Wydawnictwa Uniwersytetu Warszawskiego, Warszawa 2008.

Marek Bodnar, Zbiór zadań z matematyki dla biologów, Wydawnictwa Uniwersytetu Warszawskiego, Warszawa 2008.

Włodzimierz Krysicki, Lech Włodarski, Analiza matematyczna w zadaniach część 1, Państwowe Wydawnictwo Naukowe, Warszawa 1977.

Upon completion of the course (lecture/exercises), the student:

- knows the basics of logic;

- has knowledge of the basic concepts of mathematical analysis (sequence, series, convergence of a sequence/series, function, continuity of a function, derivative of a function, composition of functions);

- understands the concept of derivative of a continuous function;

- has the ability to calculate simple limits of sequences, test convergence of series, and calculate derivatives of functions;

- knows basic elementary functions: exponential, logarithmic, root and polynomial functions, trigonometric functions;

- knows the concept of logarithmic scale and its applications;

- has basic knowledge of financial mathematics: can determine the instalment of a loan with fixed or decreasing instalments; can compare in terms of profitability the cash deposits offered by different institutions;

- is able to use derivatives to: find the smallest and largest value of a function, find a straight line tangent to a given function at a point, extremes of a function;

- knows and understands the concepts of: functions of many variables, the partial derivative and gradient of functions of many variables; is able to apply the method of least squares in practice;

- additionally achieves learning outcomes described by symbols: K_W09; K_W10; K_U03; K_K03; K_K05

The condition for admission to the exam is to have at most 2 (two) unexcused absences from the exercises.

The basis of the final grade will be the result of the written exam consisting of a test (60 points) and a task part (40 points):

exam test 60.00% of the final grade

exam task 40,00% of the final grade

Homework and tests that check the student's knowledge and mastery of the material on an ongoing basis are not mandatory.

Not applicable