## MATHEMATICS mutuatoMATEMATICA

A.Y. Credits
2022/2023 12
Lecturer Email Office hours for students
Alessia Elisabetta Kogoj Wednesday and Thursday 13.00-14.00 and on demand
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

### Assigned to the Degree Course

Geology and Land-Use Planning (L-34 / L-21)
Curriculum: COMUNE
Date Time Classroom / Location
Date Time Classroom / Location

### Learning Objectives

The course provides the student with the main notions, results and methodologies of basic mathematics, real function of a single variable, probability and statistics, linear algebra and matrix theory, with particular attention to logical deduction and analysis of the arguments of the course.

### Program

Elementary set theory: union, intersection, difference, complement. Connective and quantifier. Cartesian product of two or more sets.
Functions between sets: domain, range, graph, image and counter-image of an element and a set. Injections, surjections, 1-1 correspondence. Composition of functions. Inverse function.
Numerical sets (N, Z, Q, R) and their main properties. Absolute value. Total ordering of the sets N, Z, Q, R.
Equations and inequalities.
Supremum and infimum, maximum and minimum. Intervals, disks.
Elementary functions. powers, exponential, logarithm, goniometric functions.

Definition of limit at a point. Continuous functions at a point, in a set.
The derivative. Derivative rules: sum, difference, product, ratio and composition of two functions.
Derivatives of elementary functions. Sign of derivative and monotonicity. Maximum and minimum points. Concavity and convexity. De l'Hôpital theorem and application to limits. Hierarchy of infinities.
Sketching the graph of a function.

The inverse problem of differentiation. Riemann integral for functions of a real variable. Integral function. Properties.
Set of primitives of a continuous function. The Fundamental Theorem of Integral Calculus. Integration methods: decomposition, substitution, parts.

Differential equations. Solutions to linear differential equations.

Probablity calculus: Probability space, events. Conditional probability. Independence. Law of total probability. Bayes rule. Examples, problems and applications.

Random variables: Independent random variables. Expected value, variance and their properties. Special discrete random variables: Bernoulli, binomial and Poisson distributions. Special continuous distributions:  Gaussian distributions, Chi-squared, the t-distribution, the F-distribution.

Elements of Statistics: Mean, median and mode. Variance. The least squares method. Interpolation techniques.

Systems of linear equations. Resolution methods. Cramer method and Gauss method. Square and rectangular matrices and row or column vectors. Operations between matrices and vectors. Determinant of a matrix.
Invertible matrices and inverse matrix. Transposed matrix and orthogonal matrices. Eigenvectors and eigenvalues.

### Bridging Courses

There are no mandatory prerequisites. It is recommended to take the exam of Calculus during the first academic year.

### Learning Achievements (Dublin Descriptors)

Knowledge and understanding: The student will achieve a deep understanding of the structure of mathematical reasoning, both in a general context and in the context of the arguments of the course. The student will achieve the mastery of the computational methods of basic mathematics and of the other arguments of the course.

Applying knowledge and understanding: The student will learn how to apply the knowledge: to analyze and understand resuls and methods regarding both the arguments of the course and arguments similar to those of the course; to use a clear and correct mathematical formulation of problems pertaining or similar to those of the course.

Making judgements: The student will be able to construct and develop a logical reasoning pertaining the arguments of the course with a clear understanding of hypotheses, theses and rationale of the reasoning.

Communication skills: The student will achieve: the mastery of the lexicon of basic mathematics and of the arguments of the course; the skill of working on these arguments autonomously or in a work group context; the skill of easily fit in a team working on the arguments of the course; the skill of expose problems, ideas and solutions about the arguments of the course both to a expert and non-expert audience, both in written or oral form.

Learning skills: The student will be able: to autonomously deepen the arguments of the course and of other but similar mathematical and scientific theories; to easily reper literature and other material about the arguments of the course and similar theories and to add knoledges by a correct use of the bibliographic material.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Supporting Activities

The teaching material and specific communications from the lecturer can be found, together with other supporting activities, inside the Moodle platform › blended.uniurb.it

### Didactics, Attendance, Course Books and Assessment

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Abate, Matematica e statistica. Le basi per le scienze della vita, McGraw-Hill

Bramanti - Pagani - Salsa, Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli

Assessment

The knowledge, understanding and ability to communicate are assessed with a written exam with open questions. After having passed the written exam students can attempt an oral examination.

### Additional Information for Non-Attending Students

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Abate, Matematica e statistica. Le basi per le scienze della vita, McGraw-Hill

Bramanti - Pagani - Salsa, Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli

Assessment

The knowledge, understanding and ability to communicate are assessed with a written exam with open questions. After having passed the written exam students can attempt an oral examination.

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