MATHEMATICAL INSTITUTIONS I

Course objectives

1) Knowledge and understanding At the end of the course, the students will know and understand: a) the basic facts about numerical sets; b) the idea of limit for sequences of real numbers; c) the idea of limit for functions of one real variable; d) the idea of continuous, differentiable, integrable function; e) the idea of approximations of functions of one real variable using polynomials; f) the idea of ordinary differential equation of first and second order. 2) Applying knowledge and understanding At the end of the course, the students will be able to: a) apply the basic topological properties of the real line; b) calculate limits of sequences and of functions; c) establish both qualitative and quantitative properties of functions of one real variable (monotonicity, minima and maxima...) in bounded and unbounded intervals; d) calculate integrals of functions of one real variable defined on intervals; e) calculate approximate values of functions of one real variable; f) solve linear ordinary differential equations of first and second order. 3) Making judgements During the lessons, several exercise sheets will be distributed to the students, as well as auto-evaluation tests. Thanks to the autonomous resolution of the exercises, and the subsequent correction in the classroom, the students will acquire both the ability to evaluate their knowledge and the ability to tackle similar problems. 4) Communication skills The written form of the exercises, assigned either during lessons or during the written test, and the oral exam will allow the students to evaluate their skill in correctly communicating the knowledges acquired during the course. 5) Learning skills At the end of the course the students will be able to generalize to more complex cases the basic knowldeges of mathematical analysis; such skill is acquired by means of several "theoretical-type" exercises assigned during the course.

Channel 1
FRANCESCO BEI Lecturers' profile

Program - Frequency - Exams

Course program
1) The set of real numbers and its properties. 2) Elementary functions and their properties 3) Sequences and series of real numbers 4) Limits of functions and continuity 5) The derivative. Local and global maxima and minima. Rate of divergence. De L’Hopital’s thorem. 6) Taylor’s expansion of the elementary functions. 7) Computation of area. Riemann integral. Integral and primitive functions: the foundamental theorem of Calculus. Methods of integration. 8) Complex numbers. 9) Linear differential equations (of first and second order, with constant coefficients).
Books
M Bramanti, C.D. Pagani, S. Salsa "Matematica - Calcolo infinitesimale e algebra lineare" Zanichelli Editore
Frequency
Strongly recommended
Exam mode
The evaluation is based on a written and oral exam. The last one can be granted depending on the result of the written part. The written exam can be substituted by two intermediate partial exams given after the first half of the course and after the end of the course.
Lesson mode
Lectures and exercise sheets
RUGGERO BANDIERA Lecturers' profile
Channel 2
MARCELLO PONSIGLIONE Lecturers' profile
Channel 3
VINCENZO NESI Lecturers' profile

Program - Frequency - Exams

Course program
Preparation must first of all take care of the ability to pass a written test (or correspondingly the in itinere tests). To this end, it is advisable to: a) carry out the weekly exercises available on the teaching moodle site; b) follow tutorials and, if deemed useful, attend the weekly reception; c) to ask questions in class, without any regard for contextual topics: ignore your possible shyness, don't worry about possible comments from colleagues (if someone dares to mock a colleague or colleague, my reaction will be prompt and firm). Topics covered include the following. Prerequisites (the zero sheet must be done correctly by the whole class) 1a) The concept of function, which is of enormous importance 1b) The qualitative properties of functions (injectivity, invertibility) and on the real straight line monotony 2) An intuitive version of the real straight line and functions defined on real numbers 3) A formal version of the real straight line (completeness) 4) Functions defined on natural numbers (successions) and their limits 5) Limits and continuity of functions defined on subsets of the real straight line 6) The incremental ratio and the derivative: kinematic and geometric interpretations 7) Some theorems on continuous functions and derivable functions: Weierstrass, intermediate values, Rolle, Lagrange, Cauchy 8) Local and global maxima and minima for functions defined on finite and unbounded subsets of the real line 9) Taylor polynomials of order one and two with many applications. 10) The integral. Geometric significance when the function is positive 11) The properties of the integral: additivity, monotonicity, linearity, mean theorem, fundamental theorem of calculus 12) Linear differential equations with constant coefficients of the first and second order, homogeneous with references to the non-homogeneous case.
Prerequisites
As approved in due course by the CAD of Chemistry, the teaching aims to fill, in the first two weeks, the gaps of students and students who have studied little mathematics in the institutes from which they come. The initial level assumes only that actually carried out by a medium-level professional institution in Rome.
Books
At the moment, you can consult web page 24-25, which will be updated as the course starts. The material is free of charge and available at https://elearning.uniroma1.it/course/view.php?id=7629
Frequency
Attendance is recommended for those who can do it. It makes it much easier to pass the examination. However, anyone and everyone can take the in-person tests. There are no exceptions. If you can take the exam with me, it does not matter if you are a first second or nth year, you can take the in-person tests and you are incraggiate to do so. Leave it to the teacher to assess your preparation. For those who cannot attend assiduously, I suggest a) consult the moodle page on a weekly basis; b) try to do the exercises assigned weekly c) for any difficulties or even just to make faster progress, I recommend attending the online receptions. You can do this from the centre of Rome but also from the outskirts of Cagliari.
Exam mode
Three in-progress tests will take place. Maximum score 32 on each test. If the sum of the three scores is higher than 48, there will be no need to take the final written test. Otherwise, there will be a written test at each ordinary or extraordinary session. The oral test consists of an interview in which the tests already taken (in-progress or written) are mainly discussed
Lesson mode
Lectures. All lecture materials will be available on the wew incontruzione page. For the time being we refer to the previous year's page https://elearning.uniroma1.it/course/view.php?id=7629
Channel 4
PIERO ANTONIO D'ANCONA Lecturers' profile

Program - Frequency - Exams

Course program
1) The real numbers. 2) Elementary functions. 3) Limits of functions, continuity. Sequences and series. 4) The derivative. Local and global maxima and minima. L’Hopital’s thorem. 5) Taylor’s expansion of the elementary functions, expression and estimates for the remainder. 6) The Riemann integral. The fundamental theorem of Calculus. Methods of integration. 7) Complex numbers. 8) Linear differential equations (of first order, of second order with constant coefficients).
Prerequisites
A basic knowledge of algebra and trigonometry is required, comparable to the typical knowledge of students at the end of high school.
Books
The course’s reference text is available and can be freely downloaded from the course's eLearning webpage. The text is also accompanied by supplementary readings, various exercise sheets, and collections of sample exam problems.
Frequency
Following the course is not mandatory, but strongly encouraged (also taking into account the two tests performed during the course period).
Exam mode
The evaluation exam consists of a written and an oral test. The latter may be optional depending on the result of the written test. The written test can be replaced by two intermediate tests carried out in the middle and at the end of the course.
Lesson mode
The lessons will be in person
VITO CRISMALE Lecturers' profile
  • Lesson code10592899
  • Academic year2025/2026
  • CourseChemical Sciences
  • CurriculumSingle curriculum
  • Year1st year
  • Semester1st semester
  • SSDMAT/05
  • CFU12