Ateneo - UPO

Sets, functions, limits for real functions of one real variable, continuity, differential and integral calculus for real functions of one real variable.

Suggested book for the theoretical part:
Bramanti, Pagani, Salsa, “Analisi Matematica 1 con elementi di geometria e algebra lineare”, Ed. Zanichelli.
As an alternative: Bramanti, Pagani, Salsa, “Analisi Matematica 1”, Ed. Zanichelli.
Suggested book for exercises:
Salsa, Squellati: "Esercizi di Analisi Matematica 1", Ed. Zanichelli.

1. Knowledge and Understanding:
- State and define with rigor the fundamental concepts of calculus (e.g., limits, continuity, differentiability, and integrability) for functions of a real variable.
- Present the proofs of the fundamental theorems of calculus (e.g., the Mean Value Theorem), demonstrating a clear understanding of their hypotheses and conclusions.

2. Applying Knowledge and Understanding:
-Compute limits, derivatives, and integrals (both definite and indefinite) by correctly applying the rules and techniques covered in the course (e.g., substitution, integration by parts).
- Analyze and qualitatively sketch the graph of a real-valued function (identifying monotonicity, concavity, and local/absolute extrema).
- Solve both standard and non-standard problems and exercises, selecting the most appropriate mathematical tools from those introduced during the course.

Basic notions of algebra and trigonometry.

The class is organized in specific lectures, that will cover both theory and excercises. In both situations we will look for the active participation and the interaction of the students and among them, also through focused questions on the current arguments.

In addition to the suggested books for the theoretical part and the exercises, further material for the preparation of the exam will be provided during the development of the course.

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://www.uniupo.it/en/services/services-students-physical-or-learning-disabilities
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 written exam will test both knowledge and understanding of the theory (theorems, proofs, definitions) and the ability to solve exercises. The oral exam is at the discretion of the instructor if additional assessment elements are needed.

A passing grade is achieved if the student knows the definitions and theorem statements and is able to use the basic tools learned during the course (e.g., derivatives and integrals) to solve simple exercises. Excellence is achieved if the student masters mathematical formalism, correctly presenting definitions, statements, and proofs. Furthermore, excellence requires the ability to solve non-standard and non-mechanical exercises."

Sets and operations between sets; definition of function, composition between functions, injectivity, surjectivity, bijectivity, invertibility, cardinality of a set.
Limit of a real function of one real variable; right limit and left limit; limits and algebraic operations; Comparison Theorem; existence of the limit for a monotone function; change of variable.
Continuity for real functions of one real variable; continuity and algebraic operations; continuity of a composition; Sign Permanence Theorem; Weierstrass Theorem; Intermediate Values Theorem; continuity of the inverse function.
Differential Calculus for real functions of one real variable; derivatives and algebraic operations; derivative of a composition; differentiability of the inverse function.
Relative maxima and minima; monotonicity of a function and sign of the first order derivative; de l'Hopital Theorems; convex and concave functions, flex points and sign of the second order derivative.
Riemann integral; integrability of sums and products of integrable functions; Mean Value Theorem; integral functions; Fundamental Theorem of Integral Calculus; integration by parts and change of variable formula.

In addition to achieving the Learning Objectives (knowledge and understanding of theory and the ability to apply it to exercises), the other expected outcomes are as follows:

Making Judgements:
- Independently select the most effective resolution strategy for a given problem, weighing the pros and cons of different analytical approaches.
- Evaluate the consistency and correctness of the results obtained through logical verification or qualitative estimates.

Communication Skills:
- Present definitions, statements, and proofs with logical rigor and appropriate technical terminology.
- Articulate and justify, both in writing and orally, the logical-mathematical steps taken during problem-solving, making them clear to an expert interlocutor.

Learning Skills:
- Develop an autonomous study method that allows for the further exploration of mathematical analysis topics or the application of these concepts in other scientific contexts.