Mathematical Analysis I

Academic Year 2023/2024 - Teacher: Giuseppa Rita CIRMI

Expected Learning Outcomes

The aim of the course is to give the basic  knowledges of infinitesimal, differential and integral calculus for real functions of one real variable. In particular the course objectives are:

Knowledge and understanding: the student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures among which sequences and series,  limits, derivatives and  integrals for functions of real variable.

Applying knowledge and understanding:

by means of examples related to applied sciences, students will focus on the central role of Mathematics within science and not only as an abstract topic. Furthermore, they will be able to apply the mathematical tools to some problems arysing from Physics.

Making judgements:
students will be stimulated, individually or in groups, to work on specific topics, developing exercises related on the field knowledge with greater independence. Seminars and lectures are scheduled to give students the chance to illustrate guided exercise on specific topics in order to share them with the other students and to find together the right solutions

Communication skills:
studying Mathematics and dedicating time to guided exercise and seminars, students will learn to communicate with clarity and rigour both, in the oral and written analysis. Moreover, students will learn that using a properly structured language means to find the key to a clear scientific  communication.


Learning skills:
students, in particular the more willing one, will be stimulated to examine in depth some topics, thanks to individual activities or working in team.




Course Structure

The principal concepts and learning outcomes will be structured by planning frontal lectures. Furthermore, to improve the making judgements and communication skills, students will dedicate time to guided exercises  and they can work in groups or individually .

The course is organized by lectures. There will be some team practices, during which students can work in groups or individually.

Students enrolled in Cinap are invited to meet the teacher before the exam.

Required Prerequisites

Basic mathematical knowledges, related to the field of Algebra, Geometry and Trigonometry, are requested (see art. 2.1. Regolamento didattico del CdS L-30 Fisica.)

Attendance of Lessons

Attendance of the lessons is mandatory (see Regolamento didattico del CdS).

Attendance of all   teaching activities is strongly recommended.

Detailed Course Content

The detailed program will be published at the end of the course.

The topics covered are:

- Set Theory. Real numbers. Numerical sets.

- Real functions of a real variable

- Numerical sequences and series

- Limits of real functions of a real variable

-Continuous functions

- Differential calculus and applications


- Linear differential equations

- Numerical series


All the above topics allow the student to acquire a good knowledge of the subject and will be the object of examination. 

Regular attendance and active participation to lessons and other activities are recommended to improve learning and to know how each topics will be presented.

Textbook Information


1. P. Marcellini, C. Sbordone, Analisi Matematica 1, Zanichelli

2. C.D.Pagani, S.Salsa, Analisi Matematica 1, Zanichelli.

3. J.P.Cecconi, G.Stampacchia, Analisi Matematica, volume 1, Liguori

4. E.Giusti,  Analisi Matematica 1, Bollati Boringhieri

5. N.Fusco, P.Marcellini, C. Sbordone, Analisi Matematica due, Liguori



6. M. Bramanti, Esercitazioni di Analisi Matematica 1, Esculapio

7. T. Caponetto, G. Catania, Esercizi di analisi Matematica 1, Culc.

8.. P. Marcellini, C. Sbordone, Esercitazioni di Matematica, Vol.1, Parte I e II, Liguori

9. E.Giusti, Esercizi e complementi di Analisi Matematica, volume primo, Bollati Boringhieri

Course Planning

 SubjectsText References
1Set theory ( 4 ore)Book 2 cap. 1 or Book  1 cap. 1
2Numerical sets (18 hours)Book 1 cap. 1 e 2 or Book  2 cap. 2
3One real variable functions( 4  hours) Book  2 cap. 4
4Limits of functions and sequences (20 hours)Book  2 cap. 2 or Book  1 cap. 3 and 4
5Continuous  functions( 4 hours)Book  1 cap 4 or Book  2 cap. 5
6Differential calculus ( 16 hours)Book  1 cap. 5, 6 and 10 or testo 2 cap. 6
7Integration ( 8 hours)Book  1 cap. 9 or Book  2 cap. 8
8Integration  ( 8hours)Book 1 cap. 8, or Book  2 cap.8
9First and second order ordinary differential equations ( 10hours)Book 1 cap.12
10Numerical series  ( 8 hours)Book 1 cap. 11or Book  2 cap. 8

Learning Assessment

Learning Assessment Procedures

The FINAL EXAM  consists in a written test  and  an oral exam. 

During the first semester there will be WRITTEN TEST , each of them  consisting in two theorical questions and two technical exercises. 

During February 2024 there will be an INTERMEDIATE WRITTEN TEST , consisting in two theorical questions and two technical exercises. 

Students who have already passed all the  2 written test  or the intermediate written test   could take the final exam studying only the remaining topics.

Final grades will be assigned taking into account the following criteria:

Rejected: Basic knowledges have not been acquired. The student is not able to solve simple exercises.

18-23: Basic knowledges have been acquired. The student solves simple exercises and has sufficient communications skills and making judgements.

24-27: All the  knowledges have been acquired. The student solves all the proposed exercises making few errors and has good communications skills and making judgements.

28-30 cum laude: All the knowledges have been completely acquired. The student applies knowledge and has excellent communications skills, learning skills and making judgements.