ANALISI MATEMATICA 1 PARTE A

Knowledge and understanding

The course provides the basic knowledge of infinitesimal calculus for functions of one variable and contributes to developing inductive and deductive reasoning skills. Students will learn the fundamental mathematical concepts and develop the ability to compute and manipulate the most common objects of Mathematical Analysis, including numerical sequences and series and limits of functions of one variable.

Applying knowledge and understanding

Students will not simply learn individual concepts but will be able to apply the acquired knowledge to the mathematical modelling of classical problems arising from other scientific fields.

Making judgements

Students will be encouraged to deepen their knowledge independently and to work on exercises related to the topics covered. Seminars are planned to illustrate the topics studied, together with tutorials in which students can critically discuss with one another to find the correct solutions to the exercises.

Communication skills

Attending lectures and reading the recommended textbooks will help students become familiar with mathematical language. Through tutorials and seminars, they will learn to communicate the acquired knowledge rigorously and clearly, both orally and in writing. By the end of the course, students will have learned that mathematical language is a useful tool for clear scientific communication.

Learning skills

Students will be guided in refining their study method. In particular, through guided tutorials they will be able to independently approach new topics, recognising the prerequisites required to understand them.

The course is structured into 9 ECTS credits, of which 6 credits (42 hours) are dedicated to lectures and 3 credits (36 hours) to practice sessions.

The concepts and methods covered by the course will be presented through lectures (Direct Teaching). For each topic, the instructor will carry out an adequate number of tutorials. To develop judgement autonomy and communication skills, and to make participation in lectures more active and productive, some hours will be devoted to guided tutorials (Interactive Teaching), in which various exercises will be proposed. Students may work individually or in groups and discuss with one another.

In itinere Reinforcement Module (Trigonometry)

As part of the tutorial activities, an in-progress reinforcement module is scheduled, consisting of 5 additional hours beyond the ordinary workload, devoted to consolidating the trigonometry skills required among the course prerequisites (properties of trigonometric functions, solving trigonometric equations and inequalities). The module is aimed in particular at students who show gaps in these topics and takes the form of guided classroom exercises.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

Students must possess the specific mathematical knowledge provided by almost all secondary education pathways, including basic notions of arithmetic, algebra, Euclidean and analytic geometry, trigonometry, as well as logic and verbal comprehension. In particular, students are required to:

•     be able to operate with sets

•     know the number systems and, in particular, the algebraic and order properties of real numbers

•     know the definition and the main properties of power, exponential, logarithmic and trigonometric functions

•     be able to apply the algebraic and monotonicity properties of elementary functions to solve simple irrational, exponential, logarithmic and trigonometric equations and inequalities

A trigonometry reinforcement module is available in progress for students with gaps in this prerequisite (see the MS section).

Attendance is strongly recommended. To monitor and improve their preparation, students are strongly advised to also attend the supplementary activities and to make use of the instructor's office hours.

1.   Elements of set theory. Set operations and their properties. Functions. Domain, range and graph of a function. Injective, surjective and bijective functions. Infinite sets. Invertible functions. Composite functions. Binary relations. Equivalence and order relations. Ordered sets.

2.   Numbers sets. The set of natural numbers. Principle of induction. Relative integers. Rational numbers. Existence of irrational numbers. The set of real numbers: algebraic structure, ordering, completeness. Absolute value. Powers with natural and integer exponents. Existence and uniqueness of the n-th root. Solvability of the equation xⁿ=a. Powers with rational and real exponents. Logarithms. Density of ℚ and of the irrationals in ℝ. Bounded sets, bounds of a numerical set. The extended real line, intervals, neighbourhoods. Interior, exterior, boundary and accumulation points. Open and closed sets. Bolzano's Theorem. The set of complex numbers: algebraic and trigonometric forms, powers and roots.

3.   Real functions of a real variable. Domain, range and graph. Supremum and infimum of a function. Monotone, even, odd, periodic functions. Elementary functions and their qualitative graphs. Piecewise-defined functions. Finding the domain.

4.   Limits of functions and sequences. Definition of limit; limits of elementary functions; limits of sequences; one-sided limits. Uniqueness, sign-permanence and comparison theorems. Operations on limits; indeterminate forms. Bounded sequences and their bounds. Limits of monotone functions and sequences. Ratio test. Neper's number. Limit of a composite function. Subsequences. Bolzano-Weierstrass Theorem. Cauchy criterion. Limit superior and limit inferior. Sequences of arithmetic and geometric means. Sequentially compact sets. Infinitesimals and infinities. Asymptotes.

5.   Continuous functions. Definition and properties. Intermediate value and zero-existence theorems. Weierstrass' Theorem. Continuity of monotone and invertible functions; the functions arcsin x, arccos x, arctan x. Uniform continuity; Cantor's Theorem. Lipschitz functions.

6.   Numerical series. Character of a series; Mengoli's series, geometric, harmonic, telescoping series. Necessary condition for convergence. Series with non-negative terms: comparison, ratio, root, Raabe's and condensation tests. Generalized harmonic series; infinitesimal test. Absolutely convergent series; exponential series. Alternating series. Cauchy product of series; commutative and associative properties.

All topics covered are essential for acquiring a good knowledge of the subject and will be examined; for some theorems, the proof will not be required.

1.   P. Marcellini, C. Sbordone, Analisi Matematica 1, Zanichelli

2.   C. D. Pagani, S. Salsa, Analisi Matematica 1, Zanichelli

3.   G. Di Fazio, P. Zamboni, Analisi Matematica 1, Monduzzi Editore

4.   J. P. Cecconi, G. Stampacchia, Analisi Matematica, vol. 1, Liguori

5.   E. Giusti, Analisi Matematica 1, Bollati Boringhieri

Exercises

6.   M. Bramanti, Esercitazioni di Analisi Matematica 1, Esculapio

7.   T. Caponetto, G. Catania, Esercizi di Analisi Matematica 1, Culc

8.   P. Marcellini, C. Sbordone, Esercitazioni di Matematica, Vol. 1, Parte I e II, Liguori

9.   E. Giusti, Esercizi e complementi di Analisi Matematica, vol. 1, Bollati Boringhieri

10. G. Di Fazio, P. Zamboni, Eserciziari per l'Ingegneria, EdiSES

Assessment is carried out through a FINAL EXAMINATION, which always includes an oral interview covering the whole syllabus, together with a written test in either full or reduced form depending on whether the student has passed the ongoing test described below.

During the December teaching break, a written ONGOING TEST is held, consisting of two parts: A) theoretical questions, possibly multiple-choice; B) technical exercises, covering the topics taught up to a date announced by the instructor sufficiently in advance, so as to allow students an adequate period for revision before the test. The test is passed with a score ≥ 18/30 in each of the two parts.

Passing the December test exempts the student, at the final examination, only from the written exercises on the part of the syllabus covered by the test: exempted students take a reduced written test (3 exercises) on the remaining topics of the syllabus. The oral interview, however, always covers the whole syllabus, including the theoretical part assessed by the December test, regardless of its outcome.

Students who do not pass the December test take the final examination in its full form: a written test of 5 exercises covering the whole syllabus, in addition to the oral interview.

In all cases, admission to the oral interview requires a score ≥ 18/30 in the corresponding written test (full or reduced); the oral interview must take place within the same examination session as the written test. 

To take  the final examination, you must have booked on the SmartEdu portal. For any technical issues regarding your booking, please contact the Student Service Office.

Grading criteria

Not passed: the student has not acquired the basic concepts and is unable to solve the exercises.

18-23: basic concepts acquired, able to solve simple exercises; barely sufficient communication skills and ability to connect topics.

24-27: good command of the contents, solves more complex exercises with few errors; good communication and connection skills.

28-30 cum laude: full acquisition of the contents, solves the exercises completely and without errors; excellent communication, learning and connection skills.

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

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata — Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

•     Theoretical question (open answer) — State and prove the Bolzano–Weierstrass theorem for bounded real sequences.

•     Theoretical question — Give the definition of the absolute maximum of a function.

•     Technical exercise — Study the character of the series Σ (n≥1) [n / (n²+1)]^p as the real parameter p varies, using the asymptotic comparison test.

•     Technical exercise — Compute, if it exists, lim (x→0) [ (1 − cos x) / (x · sin x) ] using standard limits.