Elements of Mathematical Analysis 1 A - E

Academic Year 2024/2025 - Teacher: Maria FANCIULLO

Expected Learning Outcomes

Aim of the course is to improve the knowledge of Calculus, learn the basic notions of Real Analysis, understand the concept of proof and use the common tools of Analysis. Students will also be prepared for future courses in Analysis. In particular the course objectives are:

Knowledge and understanding: students will learn calculus for one real variable functions.

Applying knowledge and understanding: by means of simple mathematical models, students will focus on the central role of Mathematics within science and not only as an abstract topic.

Making judgements: students will learn the concept of proof and basic techniques to prove a statement, formulating problems and solving them through rigorous reasonings.

Communication skills: 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 is the the key to clear scientific communication.

Learning skills: students will be stimulated to examine in depth some topics, thanks to individual activities or working in group.

Required Prerequisites

Students should have a drive through logical reasoning and already master the basics of numbers, operations, polynomials and their algebraic properties, inequalities of various types and their solutions. All these arguments will be anyway reviewed and recalled during the various “Corsi Zero” taught at the beginning of the academic year. Curiosity and attention are highly recommended.

Attendance of Lessons

Attendance to the lessons is mandatory according. Students are also encouraged to attend supplementary lessons and tutoring activities.

Detailed Course Content

1. Number sets and general notions about functions.

2. Limits of sequences and functions.

3. Continuous functions and their properties.

A detailed outline of the course will be given at the end of it.

A diary of the arguments taught will be provided on the Studium platform weekly. Any argument contained in the latter can be asked during the final exam.

Textbook Information

1. G. Anichini - G. Conti - M. Spadini, Analisi Matematica 1, 3a ed., Pearson

2. Lessons dictated in class by the teacher

Course Planning

	Subjects	Text References
1	Number sets and general notions about functions.	1,2
2	Limits of sequences and functions.	1,2
3	Continuous functions and their properties.	1,2

Learning Assessment

Learning Assessment Procedures

Learning will be constantly monitored through student interview in class.

The final exam consists of a written test to be taken in one of the scheduled exam sessions. It lasts 90 minutes and is made up of a theoretical part (T) and a practical part (E).

Part T consists of two questions, part E consists of two technical exercises.

Evaluation: in part T a maximum of ten points can be achieved, in part E a maximum of twenty points. The evaluation takes into account both correctness and clarity of presentation. To pass the written test it is necessary to obtain at least six points in part T and at least twelve points in part E.

The result of the test will be communicated within a few days of the test itself on the Studium portal.

However, the presence of the student will be necessary for the verbalization, who will be able to accept the grade obtained or choose to withdraw.

The following criteria will normally be followed to assign the grade:

not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, he is able to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents, his presentation and content connection skills are good, he solves the exercises with few errors.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; solves the exercises completely and without errors.

To participate in the exam you must have booked on the SmartEdu portal.

The final assessment of learning can also be carried out electronically, should conditions require it. In this case, it will consist of an oral interview, lasting a maximum of 20 minutes, including both theory questions and the carrying out of some exercises.

Examples of frequently asked questions and / or exercises

Part T:

· The statement, through correct formal language, of a Theorem

· The proof of a theorem or a proposition

· The precise definition of a mathematical object

· Exhibition of an example having certain required properties

· Exhibition of a counterexample to a mathematical statement

· Finding the true statement among false ones.

Part E:

· Determining extrema of a numerical set

· Producing the graph of a given function

· Determining the equation of the tangent line to a graph at one point

· Computing a limit, or a family of parameter dependant limits

· Verifying a certain asymptotic comparison

· Computing the limit of a recursively defined sequence

· Solving an equation over the complex field

· Verifying an inequality through graphical method

· Determining global and local extrema of a function

· Computing the derivative of an inverse function.

Degree Course in

Computer Science