
Course unit
MATHEMATICAL ANALYSIS 1 (Iniziali cognome AL)
SC05100190, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/05 
Mathematical Analysis 
8.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
5.0 
40 
85.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
9 Analisi Matematica 1 
01/10/2018 
30/11/2019 
TREU
GIULIA
(Presidente)
VITTONE
DAVIDE
(Membro Effettivo)
MONTEFALCONE
FRANCESCOPAOLO
(Supplente)
MONTI
ROBERTO
(Supplente)

8 Analisi Matematica 1 
01/10/2018 
30/11/2019 
VITTONE
DAVIDE
(Presidente)
TREU
GIULIA
(Membro Effettivo)
MARASTONI
CORRADO
(Supplente)
MONTEFALCONE
FRANCESCOPAOLO
(Supplente)
MONTI
ROBERTO
(Supplente)

Prerequisites:

Elementary functions of one variable: exponential, absolute value, logarithms, trigonometric functions. Cartesian geometry of the plane: lines, conic sections, geometric loci.
Whoever feels he has gaps in his Mathematical formation can consult the Precalculus Course on the EduOpen platform: https://learn.eduopen.org/eduopen/course_details.php?courseid=109 
Target skills and knowledge:

Ability of developing basic arguments about the topological properties of the real line and the completeness axiom.
Computations with complex numbers: trigonometric forms, nth roots.
Limits and continuity: computation of limits and study of the continuity of a function. Knowledge of the fundamental results' proofs (Bolzano's and intermediate value theorems).
Differential calculus: study of the derivative of a function and mastery of the basic results of differential calculus (monotonicity vs. sign of the derivative, convexity study). Ability to perform a function study. Applications to the calculus of limits (Taylor formula, de l'Hôpital rule).
Integration: ability to integrate basic functions and to utilize the formulas for integration by substitution and by parts. ability of integrating rational functions. Knowledge of the meaning of the integral of a function (Riemann sums, areas). Mastery of the Fundamental Theorem of Calculus.
Differential equations. Mastery of the tecniques for solving differential equations with separation of variables, 1st order linear equations, 2nd order linear equations with constant coefficients. Knowledge of the meaning of a Cauchy problem. 
Examination methods:

Written exam mainly made by exercises; an oral exam, mainly dealing with the theoretical part of the course, is optional. The written exam can be replaced by two intermediate partial examinations. 
Assessment criteria:

Mastery of the acquired knowledge and ability in utilizing it for the solution of simple problems. Completeness and clarity of the solutions to the exercises (also of theoretical type) proposed in the written examination. In case of oral examination, mastery of the proofs exposed in the course. 
Course unit contents:

NUMBER SETS
Elements of number theory. Induction. Natural, integer, and rational numbers. The real line, completeness, max, min, inf, sup. Complex numbers and complex roots. Topology of the real line.
REAL FUNCTIONS OF ONE VARIABLE AND LIMITS
Real functions, notion of limit and properties of limits.
SEQUENCES OF REAL NUMBERS
Sequences and countable sets. Limit of a sequence. Topology vs. sequences. Monotone and recursive sequences.
CONTINUITY
Definition of continuity for real functions. Bolzano and Weierstrass theorems. Uniform continuity.
DIFFERENTIATION
Differentiation. Monotonicity and classical theorems. L'Hôpital's rule. Higher derivatives and convexity. Taylor formula. Function study.
INTEGRATION
Riemann integral. Primitives and integration techniques. Area of planar regions.
BASIC ORDINARY DIFFERENTIAL EQUATIONS
Generalites. Cauchy problems and qualitative studies. First order differential equations, separation of variables. Linear differential equations: generalities, the second order case with constant coefficients. 
Planned learning activities and teaching methods:

Blackboard lectures. 
Additional notes about suggested reading:

Possible references not included among the reference texts will be directly recommended in the classroom. 
Textbooks (and optional supplementary readings) 

Giusti, Enrico, Analisi matematica 1. Torino: BollatiBoringhieri, 2002.

Innovative teaching methods: Teaching and learning strategies
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)

