MATHEMATICS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- FABIO STUMBO
- Credits
- 12
- Didactic period
- Annualità Singola
Training objectives
- The course aims to introduce students to the study of the calculus. Moreover, the course will provide essential implements to apply derivatives to the study of functions, finding largest and smallest values of a function, to obtain qualitative information about the shape of the graph, to calculate the area under a curve. Moreover, the course will provide some of the basic definitions and elementary algebraic operations on matrices and on determinants of matrix. Finally, the aim is to introduce students to the descriptive statistical analysis of data both from the point of view of the numerical distribution and of the geostatistical analysis.
At the end of the course, the student should get the ability to solve maximum and minimum problems, to obtain the equation of the tangent line, to sketch quite accurately a graph of a function, to solve systems of linear equations, to determine the convergence or divergence of series. Moreover, the student will acquire the necessary skills to apply methods of basic statistical analysis and of spatial interpolation for the realization of thematic maps for the comprehension and interpretation of environmental data. Prerequisites
- The student must have notions of analytic geometry and elementary algebra, in particular: Cartesian coordinate system, polynomials, linear and quadratic equations and inequalities, rational and irrational systems of equations and of inequalities, even with the absolute values.
Course programme
- The course is divided in three modules A, B and C, for a total of 96 hours among lectures and exercises. In particular the module A) is 36 hours of lectures dedicated to calculus; the module B) is 36 hours on integration, matrices, linear systems, infinite numerical sequences and infinite numerical series; the module C) is of 24 hours about statistical and geostatistical analysis.
Module A) 36 hours among lessons and exercises on calculus.
Infinite sequences. Convergent and divergent sequences. Limits. Monotone sequences.
Real function and its graph; continuous functions definition and continuity theorems; definition of limits; derivative of a function. Slope of the tangent line. Rules of differentiation; higher derivatives. Larges and smallest values of a function. Shape of the graph. Tests for local maxima and minima. Limits at infinity. Sketching the graphs. Inverse and composite functions. The chain rule for differentiating composite functions.
Module B) 36 hours among lessons and exercises on integration, matrices, infinite sequences and infinite series.
Primitive functions and antiderivatives. Integral of a function. Area under a curve. Basic properties. The fundamental theorem of calculus. Indefinite and definite integrals. The substitution rule, integration by parts. Natural logarithms. Logarithmic function. The number e. Logarithms to different bases. Exponential functions. Trigonometric functions and inverse trigonometric functions.
Definition of a matrix, of the determinant of a square matrix. Matrix rank and how to find the rank of a matrix. Number of solutions of a linear system of equations (Rouché Capelli Theorem). Cramer’s rule for a system of linear equations. Solving a system of linear Equations by Gaussian Elimination.
Infinite series. Convergent and divergent series. Properties of convergent series. Series with positive terms. The ratio test, theorem on alternating series. The comparison test. Harmonic series. Geometric series.
Module C) 24 hours among lessons and exercises on Statistics and Geostatistics
Univariate statistics: variables and statistical inference; frequency distribution; measures of relative standing, central tendency, dispersion; graphical representation; probability; probability distribution of environmental data.
Multivariate statistics: variance, covariance and correlation; regression analysis.
Geostatistics: deterministic method for spatial interpolation, spatial covariance, variograms, probabilistic methods, Kriging. Didactic methods
- The course is organized as follows:
Lectures (in classroom and/or streaming) on the course topics. During the lessons it is expected a continuous interaction with students and a revision on the prerequisites.
Exercises performed by the teacher at the blackboard. Sometimes students have to solve a problem in the classroom in an allotted time, after which the exercises are solved by the teacher at the blackboard. Learning assessment procedures
- In order to check the achieving-level of educational goals described above, the examination consists of three tests, one for each of the modules A, B and C.
An oral exam is optional.
Each test is formed by exercises and theoretical questions about the topics taught in the course, similar to those seen during the course.
A score of 18 points on 30 is needed to pass each test.
Use of calculators is admitted; it is only allowed to consult personally handwritten notes, written on an A4 sheet that can hold all the knowledge deemed useful for passing the test.
Use of notebooks, tablets and smartphones is forbidden.
In the optional oral exam, questions will focused at first on the exercises of the written test.
However, the teacher can ask the student to explain basic concepts developed during the lessons: during the oral section the knowledge of the basic concepts and results of the theory and the ability of linking different subjects related to geometry are evaluated, rather than the ability of “repeating” specific topics tackled in the cours.
The final score is the weighted average of the three scores of the modules A, B and C. Reference texts
- P. Marcellini, C. Sbordone, Esercitazioni di analisi matematica, vol. 1, parti 1 e 2, Napoli, Liguori Editore
P. Marcellini, C. Sbordone, Elementi di analisi matematica 1, Napoli, Liguori Editore
G. Strang, Algebra lineare, Apogeo Education
S.R. Ross, Introduzione alla statistica, Apogeo Education
Course Classroom Code: rnia5w7
