This module aims to present the basic notions of mathematical analysis. We introduce real functions of a real variable,
the definitions of limit, continuity and differentiability, together with their main properties and some basic results.
We apply the concepts above to the study of the graph of real functions.
Basic knowledge, generally provided in upper secondary schools.
The course, in order to achieve the expected objectives, mainly takes place through lectures. There are also practical
based lessons, guided exercises with teacher support, and exam simulations with the aim of stimulating a critical
thinking together with an autonomous the approach to problem solving.
The topics and the level of the exercises correspond to the program delivered and to the reference texts indicated.
The exam consists of a written test followed by an oral test. During the written test, students are asked to perform the complete development of some exercises. The time assigned for the written test is two hours. The evaluation of the written test is scored out of thirty. The student pass the written test if the overall evaluation is not less than 15/30. Once passed the written test, the student is entitled to participate to the oral examination only in the same session in which he passed the written examination.
The written test evaluates the critical skills achieved by the students in the arguments treated during the course and the methodological rigor of the resolutions proposed in response to the questions. The oral exam consists of an interview on the topics listed on the course program and it assesses the student's ability to communicate, by an appropriate scientific language, the notions learned, as well as his ability to present the theoretical aspects that underlie the various types of exercises contained in written test. The score of the oral exam will be assigned according to the following evaluation criterions:
30-30 cum laude: Comprehensive, deep and critical knowledge of the course topics, excellent skills in understanding and applying independently the acquired knowledge to solve the proposed questions;
29-27: Comprehensive and deep knowledge of the course topics, very good skills in understanding and applying independently the acquired knowledge to solve the proposed questions;
26-25: Comprehensive knowledge of the course topics, good skills in understanding and applying independently the acquired knowledge to solve the proposed questions;
24-22: Proper knowledge of the course topics, skills in understanding and applying independently the acquired knowledge to solve the proposed questions;
21-18: Basic knowledge of the course topics, basic skills in understanding and applying the acquired knowledge to solve the proposed questions.
Students may use the textbook
Claudio Canuto, Anita Tabacco, Mathematical Analysis I, MYLAB edition with online updates, Pearson, 2022
or one of the editions of the textbook
Claudio Canuto, Anita Tabacco, Mathematical Analysis I, Springer
Sets and operations on them. Numerical sets N, Z, Q, R. Representation of numbers on the real line and the order relation. Absolute value.
Intervals. Bounded sets. Supremum and infimum of a set. Neighborhood of a point: circular, open, and closed neighborhoods.
Functions: domain, graph, image set, preimage, boundedness, injectivity, surjectivity, bijectivity, monotonicity. Some examples. Inverse function. Theorem on the inverse function.
Composition. Translation, scaling. Even, odd, and periodic functions.
Elementary functions: powers, polynomials, and rational functions. Exponential and logarithmic functions. Trigonometric functions and their inverses. Hyperbolic sine and cosine.
Sequences.
Limits, right-hand and left-hand limits. Theorems: uniqueness, sum, product, sign preservation (with proof), first and second comparison theorems (with proof).
Algebra of limits.
Continuity, right and left continuity, discontinuities. Fundamental theorems on continuous functions: results derived from limit theorems, continuity of composite and inverse functions.
Indeterminate forms: sum, product, exponential type.
Local comparison, infinitesimals and infinities.
Asymptotes.
Theorems: zeros, Weierstrass, intermediate value.
Derivatives: definitions, rules, relationship between differentiability and continuity (with proof).
L’Hôpital’s rule. Extrema and critical points of a function. Theorems: Fermat, Rolle, Lagrange (and its consequences), Cauchy. Monotonicity test. Convex functions.
Taylor formula.
Series: basic definitions. Ratio test and Leibniz criterion. Geometric series.
