Continuum Mathematics
A.Y. 2023/2024
Learning objectives
The aim of the course is to provide the basic mathematical tools, both from a conceptual and from a calculus point of view, which are essential to successfully attend a university undregraduate program in a scientific area. The course should also provide the required mathematics prerequisites for the other courses of the program.
Expected learning outcomes
At the end of the course, students should prove to have a sufficient knowledge of basic mathematics, that includes the main properties of sets, of the main number sets, in particular of real numbers, of functions between sets, of elementary functions, of combinatorics and of complex numbers. Also, she/he should know the basic results in the theory of differential and integral calculus for functions of one real variable. Finally, at the end of the course students should be able to apply the theoretical results to solve elementary problems and exercises and in
particular they should be able to tackle the following kinds of problems: computation of limits of sequences or functions, analysis of the continuity of a function, computation of derivates, study of the qualitative graph of a function, computation of the Taylor polynomial and expansion, computation of definite and indefinite integrals.
particular they should be able to tackle the following kinds of problems: computation of limits of sequences or functions, analysis of the continuity of a function, computation of derivates, study of the qualitative graph of a function, computation of the Taylor polynomial and expansion, computation of definite and indefinite integrals.
Lesson period: First 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
First semester
Course syllabus
Real numbers and real functions.
The set of real numbers. Maximum, minimum, supremum, infimum. Elementary properties of functions. Elementary functions. Basics of combinatorics. Complex numbers.
Limits of sequences.
Definitions and first properties. 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 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 course web page https://lrondimc.ariel.ctu.unimi.it
The set of real numbers. Maximum, minimum, supremum, infimum. Elementary properties of functions. Elementary functions. Basics of combinatorics. Complex numbers.
Limits of sequences.
Definitions and first properties. 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 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 course web page https://lrondimc.ariel.ctu.unimi.it
Prerequisites for admission
There are no particular prerequisites except the basic notions of mathematics that can be acquired in any secondary high school. All the topics of the course are developed from the very beginning and students are not required to know anything about them in advance.
Teaching methods
Lectures and classwork.
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 course web page: https://lrondimc.ariel.ctu.unimi.it
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 course web page: https://lrondimc.ariel.ctu.unimi.it
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 the 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 written test may be replaced by two midterm tests. The first midterm test usually takes place in the second half of November, the second at the same time with exam sessions in the January-February exam period. Structure and rules of the midterm tests are similar to those of the written tests, except for their duration.
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 Ariel 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 written test may be replaced by two midterm tests. The first midterm test usually takes place in the second half of November, the second at the same time with exam sessions in the January-February exam period. Structure and rules of the midterm tests are similar to those of the written tests, except for their duration.
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 Ariel website of the course.
MAT/01 - MATHEMATICAL LOGIC
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
Practicals: 48 hours
Lessons: 64 hours
Lessons: 64 hours
Educational website(s)
Professor(s)