Mathematics I

Knowledge and understanding: the student will learn some basic mathematical concepts and will develop the skills of calculation and manipulation of the most common objects of mathematics: among these, the sequences, the numerical series, the limits and the derivatives for  functions of one variable.

Applying knowledge and understanding: through examples related to applied sciences, the student will be able to appreciate the importance of mathematics in the scientific field,  not just as a discipline for its own sake, thus broadening their cultural horizons.

Making judgments: the student will be able to deal with some simple but significant methods of mathematics  with sufficient rigor to refine logical skills.

Communication skills: by studying Mathematical Analysis and putting themselves to the test through guided exercises and seminars,  students will learn to communicate with rigor and clarity both orally and in writing. Students will learn that using correct language is one of the most important means of communicating scientific language clearly, not only in mathematics.

Learning skills: students will be stimulated to deepen some topics through stimulating questions during the hours of practice.

Lectures in classroom. 

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programm planned and outlined in the syllabus.

PLEASE NOTE: Information for students with disabilities and / or SLI

To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.

It is also possible to contact the referent  of  CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA)  of the Department.

The student must have a thorough knowledge of the notions of Mathematics studied in the 5 years of high school.
In particular: Elements of Mathematical Logic, set theory, algebraic equations and inequalities, trigonometry.

• Sets of numbers. Real numbers. The ordering of real numbers. Completeness of R. Factorials and binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.
• Limits. Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences. Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their applications.
• Continuity. Continuous functions. Sequential characterization of the continuity. Points of discontinuity. Discontinuities for monotone functions. Properties of continuous functions (Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the inverse functions.
• Numerical series. Round-up on sequences. Numerical series. Series with positive terms. Alternating series. The algebra of series. Absolute and Conditional Convergence.

[1] Istituzioni di Matematica, Michiel Bertsch, Bollati Boringhieri.

1. A single in itinere written test is given (called test or part A) consisting of theoretical and practical questions concerning the part of the program treated up to that moment

2. The final exam consists of a written paper divided into two parts: part A (with the topics covered up to the in itinere test) and part B containing practical and theoretical questions concerning the part of the program treated after test A

3. Passing the in itinere test A allows the student to be exempted from completing the questions of part A contained in the final exam (thus increasing the time available for the sessions of the current Academic Year)

4. Those who have not passed the in itinere test A can also access the final exam, but in this case they will have to complete both the questions of part A and the questions of part B of the final exam.

5. The benefit of passing the in itinere test A remains valid until the end of the third exam session of the current Academic Year.

Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.

PLEASE NOTE: Information for students with disabilities and / or SLI

To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.

It is also possible to contact the referent  of  CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA)  of the Department.

Definitions of: upper bound, convergent sequence, one-to-one function, inverse function, first kind discontinuity, derivative, inverse matrix, ellipse.

Counterexamples: limited and non-convergent sequence, a function that admits finite upper bound but not the maximum.

Proofs: boundedness of convergent sequences, intermediate value theorem, derivative of a product theorem.

Continuous functions (knowledge and understanding, applying knowledge and understanding)

Numerical series (knowledge and understanding, applying knowledge and understanding).

Remarkable limits deduced from the Nepero number (knowledge and understanding, applying knowledge and understanding).