Elements of Mathematical Analysis 1 A - E

The objectives of the course are the following:

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 limits and derivatives for functions of real variable.

Applying knowledge and understanding: by means of examples related to applied sciences, the student will learn the central role of Mathematical Analysis within science and not only as an abstract topic. This will expand his cultural horizon.

Making judgements: the student will tackle with rigour some simple meaningful methods of Mathematical Analysis. This will sharpen his logical ability. Many proofs will be exposed in an intuitive and schematic way, to make them more usable also to students that are not committed to Mathematics.

Communication skills: By studying Mathematics and doing guided exercitations, the student will learn to communicate with clarity and rigour both, verbally and in writing. The student will learn that the use of a properly structured language is the key point to clear and effective scientific, and non-scientific, communication.

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

Traditional (teacher up front) lessons.

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 programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

Note: Information for students with disabilities and/or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco).

1. 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.
2. 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.
3. 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.
4. Differential Calculus. The derivative. Derivatives of the elementary functions. Rules of differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle, Lagrange and Cauchy. Consequences of Lagrange's Theorem. De L'Hoôpital Rule. Monotone functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function. Recurrences. Numerical methods: Newton method and Secant Method.

The proofs related to the chapter and the contents marked with * are optional.

1. S. Motta, M.A. Ragusa, A. Scapellato – Mathematical Methods and Models – Libreria CULC (2021) [in press].
2. C. Canuto, A. Tabacco – Mathematical Analysis I – Springer (2015).

The final exam consists of a written test which is graded out of thirty. After passing the written test, the student can ask to take a
supplementary oral exam that contributes to the formulation of the final grade, expressed out of thirty.
The registration of the exam will take place only after passing the written test and the possible oral exam.

Note: The verification of learning can also be carried out electronically, should the conditions require it.