Ateneo - UPO

The course covers the main topics of mathematical analysis for real-valued functions of a real variable. Specifically, the key topics are: sets, functions, limits of real-valued functions of a real variable, continuity, and differential and integral calculus.

Bramanti, Pagani, Salsa, ``Analisi Matematica 1 con elementi di geometria e algebra lineare'', Ed. Zanichelli, 2014.

This course aims to provide students with the fundamental knowledge of mathematical analysis required for application areas that utilize advanced mathematical tools. The course covers the basics of differential and integral calculus for real-valued functions of a real variable.

Knowledge of the fundamentals of algebra and trigonometry is required.

The course is structured around lectures comprising both theoretical content and exercises. Each topic is introduced via a general discussion designed to make the material as accessible as possible. Next, the fundamental concepts of the topic are presented, followed by examples to clarify their meaning. The third stage covers the statements of key theorems and their proofs. The final part is dedicated to exercises. Active participation is encouraged through questions that also aim to gauge the level of difficulty students experience in following the lectures. The university platform (www.dir.uniupo.it) is used to provide supplementary teaching materials, such as lecture notes, various exercises, and past written exam papers.

In addition to the suggested books, further material for the preparation of the exam will be provided during the development of the course. It is made up of the slides used during the course of the last year and of the written exams of the previous years.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage:
https://uniupo.it/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa
Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test. This test typically comprises 4–6 exercises covering various course topics, along with one or more questions on the theoretical material. The main topics that may appear in the exercises include:
a) calculation of derivatives;
b) calculation of limits;
c) calculation of integrals and antiderivatives;
d) complete analysis of a function's graph;
e) analysis of a function's continuity and differentiability, including the classification of points of discontinuity or non-differentiability;
f) calculation of composite and inverse functions;
g) exercises involving sets of real numbers, determining the supremum, infimum, maximum, and minimum;
h) discussion of a theoretical topic, including its meaning and potential practical applications, key definitions stated in precise and rigorous mathematical language, illustrative examples, and major results accompanied by their proofs.
The marks awarded for each exercise vary depending on the type and difficulty of the exercise.
The mark for each exercise is indicated next to the problem statement.

To pass the exam, a minimum score must be achieved on the written test. In cases where a fully passing grade is not achieved on the written test, a final oral exam is held to assess the student's actual level of preparation.
The oral exam is optional for students who have already achieved a fully passing grade on the written test.
In all cases, the oral exam covers the following points:
a) understanding of any errors made in the written test;
b) discussion of a theoretical topic of the student's choice;
c) an additional, optional question on a topic chosen by the instructor.
If the answers and the language used are satisfactory, the final grade may be higher than the grade obtained on the written test. However, the final score cannot exceed the written test score by more than 7 points. The final grade may be lower than the written exam grade; specifically, if the written exam performance was not fully sufficient, an unsatisfactory oral exam would result in failing the course.

Achieving an excellent grade requires substantial knowledge of the topics combined with excellent proficiency in calculations; furthermore, a satisfactory discussion of the theoretical material is virtually mandatory for achieving excellence.

1. Fundamentals of set theory and functions: sets, set operations, functions, function composition, injectivity, surjectivity, bijectivity, invertibility, cardinality of a set.

2. Limits: limit of a real-valued function of a real variable, one-sided limits (left and right), limits and algebraic operations, Squeeze Theorem, existence theorem for limits of monotonic functions, change-of-variable theorem for limits.

3. Continuity: continuity of real-valued functions of a real variable, continuity and algebraic operations, continuity of composite functions, sign preservation theorem, Weierstrass Theorem (Extreme Value Theorem), Intermediate Value Theorem (including the Zero Theorem), continuity of the inverse function.

4. Derivatives and applications: derivatives and algebraic operations, derivative of a composite function, differentiability of the inverse function, relative maxima and minima, monotonicity and the sign of the first derivative, L'Hôpital's rules, convex and concave functions, inflection points and their relationship to the sign of the second derivative.

5. Riemann integration: integrability of sums and products of integrable functions, Mean Value Theorem for integrals, integral functions, Fundamental Theorem of Calculus, integration by substitution and by parts.

To achieve the minimum level of proficiency, students are required to demonstrate:

(Knowledge)

- knowledge of the fundamental concepts of set theory and functions;

- knowledge of the main elementary functions, specifically trigonometric, exponential, and logarithmic functions;

- knowledge of the key concepts and results of differential and integral calculus for real-valued functions of a real variable.

(Competencies)

- the ability to apply the fundamental concepts of differential and integral calculus;

- the ability to identify, within a specific application area, the appropriate mathematical analysis concept required for the context at hand.

(Transversal Skills)

- the ability to use correct mathematical terminology within an applied project;

- the ability to grasp the underlying meaning of material from sources such as textbooks and articles, even when advanced mathematical language is employed.

Achievement of an advanced level: students must demonstrate a high degree of mastery of the subject matter, both in terms of problem-solving skills and knowledge of the key theoretical results of mathematical analysis.