Dipartimento di Scienze della Vita e dell'Ambiente - Guida degli insegnamenti (Syllabus)

Basic elements of Calculus and Analytic Geometry

There will be lectures and exercises for a total of 36 two-hour lessons each (9CFU).

Knowledge:

The course aims to introduce students to some theoretical, methodological and applicative elements of differential and integral calculus for real functions of one real variable and the basic elements of descriptive statistics (frequency distribution, indicators of centrality, dispersion, covariance, linear regression). It aims to provide students with the elements necessary for the understanding of analytical models in use for describing the scientific phenomena and the correct interpretation of the experimental data.

Ability to apply the knowledge:

The course aims to develop the ability to perform studies of functions, derivation, integration and solve simple differential equations. It also develops the ability to perform graphical representations of data and relative statistical analysis.

Soft Skills:

Classroom and individual resolution of many problems and exercises will improve learning ability and independence of judgment. The study of deductive logical topics and the correct use of logical mathematical language develops communication skills.

Mathematics. Sets, Relations and Functions. Composition, invertibility. Natural, Integer, Rational and Real numbers. The Induction principle. Supremum, infimum, maximum, minimum. Modulus and powers. Exponential, logaritmic and angular functions. Limit of real sequences and its properties. Indeterminate forms. Monotone sequences. The Neper's number and related limits. Asymptotic comparison. Limits of real function of real variable. Properties. Indeterminate forms. Monotone functions. Asymptotic comparison. Continuity; The Weierstrass's and the Intermediate Values Theorems. Derivative and Derivative Formulas. Successive Derivative. The Fermat's, Rolle's, Lagrange's and Cauchy's Theorems. Derivative and monotonicity. Convexity. Primitives. The De L'Hospital's Theorems. Asymptotes and the study of the graphs of functions. Definite Integral and its properties. Fundamental Theorem and Formula of the Integral Calculus. Indefinite Integral and integration methods: sum decomposition, by parts and substitution. General Integral for first order linear ordinary differential equations. The Cauchy Problem. The Bernoulli's equations. The Malthus and Verhulst models for the population dynamics.

Statistics: populations, characters and related typologies; Absolute and relative frequence. Modal class, median, mean, quartiles and percentiles, variance, standard deviation. Frequency distribution and its graphical representations. Multivariate distributions, covariance, correlation coefficient; linear regression and least squares method. Use of a spreadsheet with application to the descriptive analysis of a statistical population of data.

Methods for assessing learning outcomes:

The exam consists of a written and an oral test, the tests will concern the topics covered during the course offered in the same academic year.

The registration to the first written test is mandatory, and has to be done on line on the university web page;

The written test consists of a number of problems and questions (from four to five, according to difficulty) concerning all topics treated during the course; this test will last two hours; the student will not be permitted the use of any kind of electronic device, not even a pocket calculator.

A minimum score of at least 15/30 in the written test is required for the admission to the oral test.

The list of the names of the students admitted to the oral test will be published by the teacher on his web page.

The oral test will contain mainly theoretical questions, some of which may be formulated in written form and contain exercises concerning course topics not covered in the written test or course topics in which the student may have shown weaknesses in the written test.

In case of a successful written test, the student may sit for the oral test either in the same session or the next available session, not later.

All written tests have to be correctly and fluently written, well organized, easily readable and with a negligible presence of corrections which must anyway not mar the esthetics of the text.

Honor code: each student pledges that the written tests are entirely his/her own work and that no input from other students or sources has been used; demeanors which are deemed unfair or not in line with these principles entail the failing of the exam.

Criteria for assessing learning outcomes:

In order to pass the exam the student must demonstrate a good understanding of all topics and concepts covered during the course, and which will be published on line as "Final program" or "Exam program" at the end of the course, and to be able to use them in solving typical calculus and statistical problems. Particular attention will be given to evaluate the student's ability to justify rigorously his assertions and in the proper use of logical mathematical language. In the statistic test the student must show knowledge of the statistical indicators used in his work and ability to interpret the results.

Criteria for measuring learning outcomes:

The final grade is attributed out of thirty. The exam is passed when the rating is greater than or equal to 18. It is possible the award of full marks with honors (30 e lode).

Criteria for conferring final mark:

The final score will be given by the teacher on the basis of the score of the written test and of the level of knowledge and comprehension of the topics covered during the course.

P. Marcellini - C. Sbordone, Elementi di Calcolo, Liguori editore. Marcellini - C. Sbordone, Esercitazioni di matematica vol. 1 (parte I e II), Liguori editore

P. Baldi, Introduzione alla probabilità. Con elementi di statistica, Mc Graw-Hill Editore.

Teacher's lecture notes

- Scienze ambientali e protezione civile

**Università Politecnica delle Marche**

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