semmelweis university 2 0 1 2 / 2 0 1 3
semmelweis university 2 0 1 2 / 2 0 1 3 semmelweis university 2 0 1 2 / 2 0 1 3
MATHEMATICS University Pharmacy, Department of Pharmacy Administration Tutor: Dr. Andrea Meskó In the first year of the curriculum two hours of lectures are given to pharmacist students under the title above. The lectures are accompanied by practicals to help a better understanding and to get experienced in solving problems and exercises. The title covers two, more or less independent, subjects. The majority of the lectures (over 60 per cent) is devoted to (classical) mathematics, the smaller part, however, in which biostatistics are given, is not of less importance. The aim of learning classical mathematics is to understand biological, chemical, and physical processes dealt with in the subjects mentioned. The most appropriate mathematical model for the processes in nature are functions of one or more variables. To obtain the proper function for a particular process a differential equation is to be solved. The notion and the way of solution of differential equations is the central point of the course. The others namely limits, differential and integral calculus, discussion of functions, series etc. are, however, necessary preparatory steps for getting acquainted with differential equations. Biostatistics, the other subject under this title, is a more recent branch of sciences. Its importance is permanently increasing in each field where data are present, i.e. quite everywhere in scientific work. Pharmacological investigations, clinical trials, epidemiological studies (etc, etc.) cannot be carried on without the statistical analysis of the data obtained. The results of the above mentioned studies are always derived by statistical inference. Statistics is an indispensable part of any research from planning the experiment to interpretation of the results. Statistical methods are essential even for students in their laboratory work. MATHEMATICS University Pharmacy, Department of Pharmacy Administration Tutor: Dr. Andrea Meskó First Semester Lectures: 2 hours per week Practicals: 2 hours per week Differential and differential coefficient. Rules for derivations of functions. The derivative of the power function. Derivation of composite and inverse functions. Differentiability of the elementary functions. Higher order derivatives. Application of differentiation for calculation of limits of fractions. An iterative method to solve equations (Newton-method). Expansion of differentiable functions to power series. The Taylor series of exp x, sin x, cos x, ln x and other functions. Qualitative examination of functions. Roots, extremes and inflexion points. The multiplicity of a root. The complete discussion of elementary functions. Integration as the inverse operation of derivation. The indefinite integral. Integration of power functions. Integration of simple elementary functions. SEMMELWEIS UNIVERSITY / FACULTY OF PHARMACY Faculty of Pharmacy 397
SEMMELWEIS UNIVERSITY / FACULTY OF PHARMACY 398 Integration of products (the rule of “partial integration”). Integration of composite functions. Integration of rational fractions. Area under a curve: the definite integral. Improprious integrals. The concept of a differential equation. Differential equations arising in physics, chemistry, biology, botanics and other fields. The homogeneous linear differential equation with constant coefficients: solution and proof of unicity. Separation of variables as the method of solution. General and particular solutions. Introduction of new variables. Nonlinear differential equations of the first order. Differential equations of the chemical reactions of 0th, 1st and 2nd order. Functions of several variables. Partial derivatives of first and second order. Differentiability and exact differential. Application of exact differential in error calculations. Maxima and minima of two-variable functions. Different kinds of integration of functions of several variables. Integration along a line. Point functions and independence of the integral of the path. Calculation of the integral along different curves. MATHEMATICS University Pharmacy, Department of Pharmacy Administration Tutor: Dr. Andrea Meskó Second Semester Lectures: 2 hours per week Practicals: 1 hour per week Introduction and information. The most common calculations in laboratory. Some hints for numerical calculations. Biometrics and/or biostatistics. Statistical inference. Frequency distributions. Theoretical distribution and probability. The normal distribution. Measures of central tendency (mode, median, mean etc.) Applications of the weighted mean. Measures of dispersion. Standard deviation and variance. The coefficient of variation. Error bounds. The standard error of the mean. The concept of “regression line”. The linear regression: coefficients, interpretation, application. The correlation coefficient: formula and interpretation. Uses and misuses of correlation coefficient. Lack of correlation vs. independence. Spurious correlations. Coefficient of determination. Sampling distributions. Important distributions derived from the normal one: t F, and chis-quared distributions. The use of statistical tables. Theoretical background of statistical inference. Qualitative and quantitative conclusions. Estimation; confidence interval for the expected value. Testing hypotheses. The concept of “significance”. Errors of the first and of the second kind. The t-tests. Analysis of variance. The Ftest. Discrete and dichotomous distributions; variables on a nominal scale. The Poisson distribution. Analysis of qualitative data. Counting tables. Measures of association and statistical tests in fourfold tables. Sets (finite and infinite). Natural, integral, rational, real and complex numbers.
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SEMMELWEIS UNIVERSITY / FACULTY OF PHARMACY<br />
398<br />
Integration of products (the rule of “partial integration”). Integration of<br />
composite functions. Integration of rational fractions.<br />
Area under a curve: the definite integral. Improprious integrals.<br />
The concept of a differential equation. Differential equations arising in physics,<br />
chemistry, biology, botanics and other fields. The homogeneous linear<br />
differential equation with constant coefficients: solution and proof of unicity.<br />
Separation of variables as the method of solution. General and particular<br />
solutions. Introduction of new variables.<br />
Nonlinear differential equations of the first order.<br />
Differential equations of the chemical reactions of 0th, 1st and 2nd order.<br />
Functions of several variables. Partial derivatives of first and second order.<br />
Differentiability and exact differential. Application of exact differential in error<br />
calculations. Maxima and minima of two-variable functions.<br />
Different kinds of integration of functions of several variables. Integration along<br />
a line. Point functions and independence of the integral of the path.<br />
Calculation of the integral along different curves.<br />
MATHEMATICS<br />
University Pharmacy, Department of Pharmacy Administration<br />
Tutor: Dr. Andrea Meskó<br />
Second Semester<br />
Lectures: 2 hours per week<br />
Practicals: 1 hour per week<br />
Introduction and information. The most common calculations in laboratory. Some hints for<br />
numerical calculations.<br />
Biometrics and/or biostatistics. Statistical inference. Frequency distributions. Theoretical<br />
distribution and probability. The normal distribution.<br />
Measures of central tendency (mode, median, mean etc.) Applications of the weighted mean.<br />
Measures of dispersion.<br />
Standard deviation and variance. The coefficient of variation. Error bounds. The standard error of<br />
the mean.<br />
The concept of “regression line”. The linear regression: coefficients, interpretation, application.<br />
The correlation coefficient: formula and interpretation. Uses and misuses of correlation coefficient.<br />
Lack of correlation vs. independence. Spurious correlations. Coefficient of determination.<br />
Sampling distributions. Important distributions derived from the normal one: t F, and chis-quared<br />
distributions. The use of statistical tables.<br />
Theoretical background of statistical inference. Qualitative and quantitative conclusions.<br />
Estimation; confidence interval for the expected value.<br />
Testing hypotheses. The concept of “significance”. Errors of the first and of the second kind. The<br />
t-tests.<br />
Analysis of variance. The Ftest. Discrete and dichotomous distributions; variables on a nominal<br />
scale. The Poisson distribution.<br />
Analysis of qualitative data. Counting tables. Measures of association and statistical tests in<br />
fourfold tables.<br />
Sets (finite and infinite). Natural, integral, rational, real and complex numbers.
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