# MATHEMATICS

## Learning outcomes of the course unit

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.

## Prerequisites

No mandatory propedeuticities.

## Course contents summary

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

## Course contents

Sets and operations with sets: union, intersection, difference, complement. Diagrams of Euler-Venn. Connective and quantifier. Cartesian product of two or more sets. Applications and functions between sets. Image and counter-image of an element and a set. Injections, surjections, 1-1 correspondence. Composition of applications. Inverse function.

Numerical sets (N, Z, Q, R) and their main properties. Operations and their properties: commutative, associative and distributive property. Neutral elements. Absolute value. Total ordering of the sets N, Z, Q, R. Equations and inequalities. Properties of real numbers: completeness. Supremum and infimum, maximum and minimum. Intervals, disks.

Polynomials. Operations on polynomials, powers. Roots of first and second degree polynomials. Polynomial equations and inequalities with absolute, rational and irrational values.

Real numbers and geometry of the line. Geometry of the Cartesian plane. Distance between two points on the Cartesian plane. Representation of lines, circumferences and conics (canonical form). Parallelism and perpedicularity of two lines. Distance of a point from a line. Intersection and tangency between lines and conics. Special points and lines of conics.

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 and rank of a matrix. Rouche'-Capelli theorem for linear systems.

Graphs of elementary functions. Identity, constant, linear functions, powers, absolute value, sign. Polynomial functions. Exponential and logarithm. Properties of powers. The logarithm function as inverse of the exponential. Goniometric functions. Addition, duplication, bisection formulas. Inverse of circular functions. Graphical interpretation of injectivity and surjectivity, of the composition of functions and of the inverse function. Monotone, even, odd functions. Inverse of a monotonic function. Translations and dilatations of function graphs. Equations and inequalities with elementary functions.

Definition of limit in a point. Continuous functions in a point, in a set. Uniqueness of the limit. Theorems on continuous functions. Indefinite forms.

Difference quotient and derivative in a point. Geometric interpretation of the derivative. Relationship between derivative and continuity. Derivative rules: sum, difference, product, ratio and composition of two functions. Derivatives of elementary functions. Theorems on derivatives. Sign of derivative and monotony. Maximum and minimum points. Concavity and convexity. Second derivative and inflection points. De l'Hôpital theorem and application to limits.

Areas and measures. The inverse problem of differentiation. Cauchy integral for functions of a real variable. Conditions for integrability. Integrability of continuous functions. Integral function. Properties: additivity and monotony. Average of a continuous function. Set of primitives of a continuous function. Relationship between primitives, integral function and areas. The Fundamental Theorem of Integral Calculus. Integration methods: decomposition, substitution, parts.

## Recommended readings

Bigatti A.M., Robbiano L. – Matematica di Base – Casa Editrice Ambrosiana

Languasco A. - Analisi Matematica 1. Teoria ed esercizi - Hoepli Editrice

Bigatti A.M., Tamone G. - Matematica di Base. Esercizi Svolti, Testi d'Esame, Richiami di Teoria - Soc. Ed. Esculapio

PDF files of exercises are supplied every week.

## Teaching methods

Theoretical classroom teaching. Classroom exercise including solutions of the weekly proposed exercises. Clarifying individual meeting if required by the student.

## Assessment methods and criteria

Final written examination divided into a preliminary closed-ended questionnaire to evaluate theoretical and applied knowledge and comprehension skills (50' for 10 questions) and a following open-ended questionnaire to evaluate independent judgement and communication skills (90' for 2 exercises). The student can go through with the second questionnaire only passing the first. Mark is expressed in thirtieths (max 14/30 for the first questionnaire, max 18/30 for the second. A total mark greater than 30 means full mark with honour). During the exam is allowed the use of a non--graph calculator and a small handwritten leaflet with the main formulas. Text, answers and solution are at disposal of the student, after agreement with the teacher, till the date of the next exam.

## Other informations

Theretical classroom teaching and in particular classroom exercise are focused on the formation of the self judgement of the student about his/her theoretical and applied knowledge of the arguments, and his/her learning and communication skills.