Mathematics Ii
A.Y. 2025/2026
Learning objectives
The aim of the course is to teach the basic mathematical language and tools that allow a student to formulate, understand and solve a problem in mathematical analysis. Such problems concern for example: sequences, differential calculus for the study of the graph and of local extremes of a function, the study of limits with Taylor expansions and integral calculus.
Expected learning outcomes
The student must be able to identify and apply the mathematical tools needed to solve concrete problems of mathematical analysis. He or she must also be able to justify the solutions to such problems through theory and to present them correctly.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Numerical sequences.
Definitions and first properties. Definition of limiti a sequence. Bounded sequences. Operations with limits. Comparison theorems. Monotone sequences. Undetermined forms. Special limits.
Limits of functions and continuous functions.
Definition and first properties of limits of functions and of continuous functions. Types of discontinuities. Limits and continuity of the composition of functions. Some important theorems on continuous functions.
Derivatives and the study of functions.
Definition of derivatives. Computation of derivatives. Theorems of Fermat, Rolle, Lagrange and Cauchy and their consequences. Second and higher order derivatives. Applications to the study of functions. L'Hopital theorem and Taylor formula.
Integration
Definite integrals and method of exhaustion. Definition of integrable functions and classes of integrable functions. Properties of the definite integrals. Indefinite integrals. Fundamental theorem of integral calculus. Integration methods. Integration by parts and by substitution. Integration of rational functions.
The final program will we published at the end of classes on the MyAriel page of the course.
Definitions and first properties. Definition of limiti a sequence. Bounded sequences. Operations with limits. Comparison theorems. Monotone sequences. Undetermined forms. Special limits.
Limits of functions and continuous functions.
Definition and first properties of limits of functions and of continuous functions. Types of discontinuities. Limits and continuity of the composition of functions. Some important theorems on continuous functions.
Derivatives and the study of functions.
Definition of derivatives. Computation of derivatives. Theorems of Fermat, Rolle, Lagrange and Cauchy and their consequences. Second and higher order derivatives. Applications to the study of functions. L'Hopital theorem and Taylor formula.
Integration
Definite integrals and method of exhaustion. Definition of integrable functions and classes of integrable functions. Properties of the definite integrals. Indefinite integrals. Fundamental theorem of integral calculus. Integration methods. Integration by parts and by substitution. Integration of rational functions.
The final program will we published at the end of classes on the MyAriel page of the course.
Prerequisites for admission
There are no particular prerequisites except the basic notions of mathematics that are acquired in every high school. For example: ability to apply the elementary rules of algebra and arithmetic; familiarity with common numerical systems (natural, integer, rational and real); some elements of analytical geometry (Cartesian plane, lines, parabolas, hyperbolas, circles and ellipses); the properties of elementary functions (absolute value, polynomials, exponential and logarithm, trigonometric functions); equations and inequalities that involve such functions. It is advisable to have passed the exam of Mathematics I or at least to have learned its fundamental concepts. All the specific topics of the course are developed from the beginning without requiring previous knowledge by the student.
Teaching methods
Lectures and classwork. Notes and other material on MyAriel (lecture notes, link to the course Matematica Assistita, some quizes).
Teaching Resources
Textbook: P. Marcellini and C. Sbordone, Elementi di Analisi Matematica uno, Liguori, 2002.
Suggested exercise books: P. Marcellini e C. Sbordone, Esercitazioni di Matematica, primo volume, parte prima e parte seconda, Liguori, 2013 e 2017.
Exercises on the MyAriel page of the course.
Other exercise books: M. Amar e A.M. Bersani, Esercizi di Analisi Matematica I - Esercizi e richiami di teoria, Edizioni La Dotta, 2014
Suggested exercise books: P. Marcellini e C. Sbordone, Esercitazioni di Matematica, primo volume, parte prima e parte seconda, Liguori, 2013 e 2017.
Exercises on the MyAriel page of the course.
Other exercise books: M. Amar e A.M. Bersani, Esercizi di Analisi Matematica I - Esercizi e richiami di teoria, Edizioni La Dotta, 2014
Assessment methods and Criteria
The exam consists of a written test where students will be asked to solve some exercises on the topics of the course and to answer questions about tthe theory. The written test lasts 2 hours.
The exam will be scored with a maximum of 30, and it is passed if the score is greater than or equal to 18/30.
The final results of the exam or the partial results of the midterms will be comunicated on SIFA via the UNIMIA portal.
More details on the exam can be found on the MyAriel website of the course.
The exam will be scored with a maximum of 30, and it is passed if the score is greater than or equal to 18/30.
The final results of the exam or the partial results of the midterms will be comunicated on SIFA via the UNIMIA portal.
More details on the exam can be found on the MyAriel website of the course.
MAT/03 - GEOMETRY - University credits: 5
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 4
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 4
Practicals: 36 hours
Lessons: 48 hours
Lessons: 48 hours
Professors:
Benedikter Niels Patriz, Matessi Diego
Professor(s)
Reception:
by appointment via e-mail
office 1024 (first floor, Via Cesare Saldini 50)