Mathematics and statistics 1000-MWC-MAT2
The lecture objective is to familiarize the students with the following issues:
1. Vectors in R2/R3 and analytic geometry:
-scalar and vector products;
-linear independence of vectors;
-line going through 2 points, parallelity and orthogonality of the lines in R3;
-the distance in R2, metrics;
2. Solving linear systems of equations:
-determinant of matrix;
-Cramer's rule;
3. Complex numbers:
-geometric interpretation,
-operations on complex numbers: addition, subtraction, multiplication, division, involution, evolution, solving simple equations,
4. Basic elementary functions:
-exponential and logarithmic function;
-logarithmic scale and its applications;
5. Number sequences and series:
-the limit of a sequence, the convergence of an infinite sequence;
-geometric sequence and series, simple tests of convergence of infinite series;
6. The rudiments of financial mathematics:
-simple and compound interest,
-credits with constant and decreasing instalments
-deposit accounts,
7. Properties of continuous functions;
8. Differential calculus and its applications:
-definition of the derivative, derivative as a line tangent to the graph of a function;
-derivatives of elementary functions, basic operations on derivatives;
-finding the properties of a function from its derivative, finding the maximum and minimum of a function, optimization;
-Taylor expansion, finding the approximate value of a function;
-partial derivative and gradient of the function of several variables;
9. Integral calculus:
-indefinite and definite integral;
-calculating simple integrals;
-applications of integrals: computing the area under the graph of a function, finding the length of a curve, volume of solid of revolution, calculating the path;
-definition of improper integral;
10. The rudiments of the probability theory:
-conditional probability;
-total probability and Bayes' formula;
-the distribution of a random variable, binomial distribution, uniform distribution, normal (Gaussian) distribution;
-expectation and variance of a random variable;
-Central Limit Theorem
-Law of Small Numbers;
All issues will be illustrated, where possible, with the examples coming from natural sciences.
Type of course
Mode
Learning outcomes
Upon the course completion student:
--is able to compute: vector and scalar product in R3 and R2, length of a vector, area of a figure which has edges that are defined by two vectors, volume of a solid which has edges defined by three vectors, determinant of a matrix, solve a linear system of equations by Cramer's rule,
--can discriminate between linearly dependent and independent vectors and how it influences the determinant of a matrix,
--understands the idea of a complex numbers, is able to compute n-th roots of a complex number, perform basic operations and understands the geometrical interpretation,
--possesses general knowledge of basic issues of mathematical analysis (sequence, series, convergence of a sequence/series, derivative, integral),
--is able to calculate simple limits of sequences, derivatives and simple integrals of functions,
--possesses general knowledge of financial mathematics: is able to calculate the instalment of a credit with constant or decreasing instalments, can compare different deposit accounts,
--understands the notion of the derivative and integral of a continuous function,
--is able to apply derivatives to: finding the maximum and minimum of a function, finding a tangent line to a given function, optimization problems,
--possesses skill of applying integral calculus to finding areas;
--displays: the knowledge of basic issues of the probability theory, ability of applying in practice the formulae of conditional and total probability, Bayes formula and Central Limit Theorem and Law of Small Numbers,
--possesses the ability of understanding to some degree the mathematics appearing in science articles, regarding natural sciences.
Assessment criteria
Fall semester:
In fall semester there will be assessed students' work during classes (activeness, doing homework). Moreover student has to pass the written assessment. The points for activeness during classes and for the written assessment will be the basis for fall semester mark.
Spring semester:
In spring semester the basis of a final mark will be: points for activeness during classes and points for a final exam.
Practical placement
-
Bibliography
Marek Bodnar, Zbiór zadań z matematyki dla biologów, Wydawnictwa Uniwersytetu Warszawskiego, Warszawa 2008.
Adam Łomnicki, Wprowadzenie do statystyki dla przyrodników, Wydawnictwo Naukowe PWN, Warszawa 2007.
J. Jakubowski, R. Sztencel, Wstęp do teorii prawdopodobieństwa, SCRIPT, Warszawa, 2001.
K. Kuratowski, Rachunek różniczkowy i całkowy, PWN, Warszawa 1979.