EN
IT
Prerequisiti
Regole elementari di calcolo che coinvolgono numeri, frazioni e polinomi, rappresentazione dei punti nel piano attraverso coordinate cartesiane, nozioni di base riguardanti rette e parabole e la loro rappresentazione nel piano, metodi di risoluzione di equazioni e disequazioni di primo e secondo grado, definizione e proprietà fondamentali dei logaritmi, delle esponenziali e delle funzioni trigonometriche. Tutti gli argomenti nell'elenco sopra sono inclusi tra gli argomenti dei corsi preparatori di matematica in modo che tutti gli studenti possano attraversare una rapida revisione di tali concetti durante la partecipazione ai corsi preparatori. Tuttavia, una revisione veloce potrebbe non essere sufficiente per gli studenti che hanno una base debole o addirittura molto debole in matematica. In questo caso, si incoraggia vivamente gli studenti a impegnarsi anche prima dell'inizio dei corsi preparatori al fine di recuperare il livello scolastico italiano ordinario. Qualsiasi libro di testo scolastico può essere utilizzato per rivedere gli argomenti sopra menzionati.
Prerequisites
Elementary rules of computation involving numbers fractions and polynomials, representation of points in the plane through Cartesian coordinates, basic notions concerning lines and parabolas and their representation in the plane, methods of resolution of first and second order equations and inequations, definition and basic properties of logarithms, exponentials and trigonometric functions. All the arguments in the above list are included
among the topics of the math Pre-courses so that all students can go through a quick review of those notions while attending the Pre-courses. However, a quick review may be
not enough for all students who have a weak- or even very weak -background in mathematics. In this case, students are strongly encouraged to work hard even before the beginning of Pre-courses in order to catch up with the ordinary Italian school level. Any school textbook can be used for reviewing the above mentioned topics.
Programma
**Argomento 1**
Numeri reali, funzioni elementari e grafici
**Argomento 2**
Successioni e limiti
**Argomento 3**
Ricorrenza, modelli a tempo discreto: esponenziali e logaritmi, scale logaritmiche
**Argomento 4**
Derivate: regole e applicazioni, tasso di cambiamento in modelli applicati
**Argomento 5**
Ottimizzazione: massimi e minimi, convessità, tracciamento di curve
**Argomento 6**
Integrazione: aree, antiderivate, Teorema Fondamentale del Calcolo
**Argomento 7**
Equazioni differenziali e modelli di crescita: punti di equilibrio, stabilità
**Argomento 8**
Calcolo multivariabile: derivate parziali, ottimizzazione
Program
- Topic 1 Real numbers, elementary functions and graphs
- Topic 2 Sequences and limits
- Topic 3 Recurrence, discrete time models: exponentials and logarithms, log scales
- Topic 4 Derivatives: rules and applications, rate of change in applied models
- Topic 5 Optimization: maxima and minima, convexity, curve sketching
- Topic 6 Integration: areas, antiderivatives, Fundamental Theorem of Calculus
- Topic 7 Differential equations and growth models: equilibrium points, stability
- Topic 8 Multivariable calculus: partial derivatives, optimization
Testi Adottati
Lawrence D. Hoffmann, Gerald L. Bradley, Dave Sobecki, Michael Price: Applied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition, ed. Mc Graw-Hill, 2012-2013.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Books
Lawrence D. Hoffmann, Gerald L. Bradley, Dave Sobecki, Michael Price: Applied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition, ed. Mc Graw-Hill, 2012-2013.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Bibliografia
Appunti forniti dal docente sulle equazioni differenziali.http://www.mat.uniroma2.it/~porretta/notes-porretta2.pdf
Bibliography
Further readings: Notes given by teacher on differential equations : http://www.mat.uniroma2.it/~porretta/notes-porretta2.pdf
Modalità di svolgimento
In presenza.
Teaching methods
In-classs teaching.
Regolamento Esame
Sono previsti due esami parziali, ciascuno dei quali contribuisce fino al 30% della valutazione finale, a condizione che l'esame venga superato nella sessione invernale, per gli studenti frequentanti. Alla fine del periodo di lezioni, si svolge un esame scritto finale, che contribuisce fino al 40% della valutazione.
Exam Rules
Two mid-term examinations are given, each make up to 30% of the final grade, provided the exam is passed in the winter session, for attending students. At the end of the lecture period, a final written examination is done, which make up to 40% of the grade.
Updated A.Y. 2019-2020
Updated A.Y. 2019-2020
Course Description
One of the things we learned from 20th century is that complex phenomena arising from social and life sciences cannot be deeply analyzed and understood unless we make use of the universal language of mathematics. This goes back to Galileo's intuition that universe be written in mathematical symbols, a principle nowadays applied to human sciences as well as to natural ones. The purpose of this course is to introduce students to those basic notions in mathematics which are essential to describe, understand and analyze possibly different models of quantitative phenomena. Main concepts and tools of differential and integral calculus are taught in order that students become familiar with functions of real variables, notions of growth, limits, rate of change, optimization, time evolution, all of them being necessary to approach mathematical models in applied sciences as well as to pursue further studies in probability and statistics. Students are expected to learn the main concepts, to practice with basic tools of calculus and to understand the use of mathematical language in applied models of real life.
Teaching Method
The course is essentially taught in the traditional way through classroom lectures; this is necessary since the mathematical language needs to be presented in action on the blackboard, where theory and practice become intrinsically linked. Additional tutorials will be devoted to those students who need extra practice as well as specific care because they lack of background. At the beginning of the year, this group of students is identified from an evaluation test at the end of the Math pre-courses. During the semester, as the program goes further, periodic evaluations are scheduled to monitor the group of students who need to follow the additional tutorials.
Schedule of Topics
Topic 1 Real numbers, elementary functions and graphs
Topic 2 Sequences and limits
Topic 3 Recurrence, discrete time models: exponentials and logarithms, log scales
Topic 4 Derivatives: rules and applications, rate of change in applied models
Topic 5 Optimization: maxima and minima, convexity, curve sketching
Topic 6 Integration: areas, antiderivatives, Fundamental Theorem of Calculus
Topic 7 Differential equations and growth models: equilibrium points, stability
Topic 8 Multivariable calculus: partial derivatives, optimization, integration
PLEASE FIND MORE INFORMATION ON THE COURSE ON THE SYLLABUS IN THE TEACHING MATERIAL SECTION
Updated A.Y. 2018-2019
Updated A.Y. 2018-2019
Course Description
One of the things we learned from 20th century is that complex phenomena arising from social and life sciences cannot be deeply analyzed and understood unless we make use of the universal language of mathematics. This goes back to Galileo's intuition that universe be written in mathematical symbols, a principle nowadays applied to human sciences as well as to natural ones. The purpose of this course is to introduce students to those basic notions in mathematics which are essential to describe, understand and analyze possibly different models of quantitative phenomena. Main concepts and tools of differential and integral calculus are taught in order that students become familiar with functions of real variables, notions of growth, limits, rate of change, optimization, time evolution, all of them being necessary to approach mathematical models in applied sciences as well as to pursue further studies in probability and statistics. Students are expected to learn the main concepts, to practice with basic tools of calculus and to understand the use of mathematical language in applied models of real life.
Teaching Method
The course is essentially taught in the traditional way through classroom lectures; this is necessary since the mathematical language needs to be presented in action on the blackboard, where theory and practice become intrinsically linked. Additional tutorials will be devoted to those students who need extra practice as well as specific care because they lack of background. At the beginning of the year, this group of students is identified from an evaluation test at the end of the Math pre-courses. During the semester, as the program goes further, periodic evaluations are scheduled to monitor the group of students who need to follow the additional tutorials.
Schedule of Topics
Topic 1 Real numbers, elementary functions and graphs
Topic 2 Sequences and limits
Topic 3 Recurrence, discrete time models: exponentials and logarithms, log scales
Topic 4 Derivatives: rules and applications, rate of change in applied models
Topic 5 Optimization: maxima and minima, convexity, curve sketching
Topic 6 Integration: areas, antiderivatives, Fundamental Theorem of Calculus
Topic 7 Differential equations and growth models: equilibrium points, stability
Topic 8 Multivariable calculus: partial derivatives, optimization, integration
PLEASE FIND MORE INFORMATION ON THE COURSE ON THE SYLLABUS IN THE TEACHING MATERIAL SECTION
