Mathematical Analysis
Academic Year 2022/2023 - Teacher: Maria Alessandra RAGUSAExpected Learning Outcomes
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 numerical sequences, numerical series, 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.
Course Structure
Traditional (teacher up front) lessons.
Required Prerequisites
Basic knowledge of arithmetic, algebra, analytic geometry, trigonometry.Attendance of Lessons
Attendance to the courses is compulsory. In addition to the cases already provided for by the Regulations, partial or total motivated exemptions from attendance can be recognized, by means of a specific resolution of the Degree Course Council, upon presentation of a motivated request recognized by the Council and if the conditions exist, agreed with the teachers in charge of the courses concerned, to activate the necessary forms of supplementary didactic support, suitable for guaranteeing adequate preparation of the student. Attendance is understood to have been acquired if the student has attended at least 60% of the curricular hours provided for by the discipline. The student who has not acquired the attendance of the courses foreseen by his own training course, in the previous course year, is regularly enrolled in the following year, without prejudice to the obligation to attend the courses for which he has not obtained the certificate of attendance . At the end of the 3 years of regular enrollment, the student is enrolled as out of course with the obligation to obtain the certificate of attendance of the courses in accordance with the prerequisite principle of the same. According to the provisions of article 27 of the RDA and in the Regulations for the recognition of the status of student worker, student athlete, student in difficulty and student with disabilities (D.R.n.1598 of 2/5/2018) for working students , student athletes, students in difficult situations (RDA, art.27), whose status is duly certified, the Degree Program includes: - the reduction of the obligation to attend, up to a maximum of 20%; - the possibility of taking the exams in the extraordinary sessions reserved for repeating and out-of-course students; - specific didactic support activities to be agreed with the teachers of the individual disciplines. The methods of conducting the courses and the relative verification of attendance are delegated to the organizational autonomy of the teachers holding the courses.
Detailed Course Content
- 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.
- 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'Hôpital Rule. Monotone functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function. Recurrences. Numerical methods: Newton method and Secant Method.
- Combinatorics, Statistics and Probability. Arrangements, permutations, combinations, simple and with repetition. Definition of classical probability and frequentist probability. Conditional probability. Mode, mean and median. Hints to hypothesis tests. Applications.
- Integrals. Areas and distances. The definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts. Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial fractions. Strategy for integration. Impropers integrals. Applications of integration.
Textbook Information
- C. Canuto, A. Tabacco – Mathematical Analysis I – Springer-Verlag Italia, Milano, 2015.
- J. Stewart, D. Clegg, S. Watson – Calculus. Early Transcendentals – Ninth Edition, Cengage Learning, Boston, USA, 2021.
- Lecture notes.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Insiemi numerici | Testi: 1, 2 |
| 2 | Funzioni reali di una variabile reale | Testi: 1, 2 |
| 3 | Successioni numeriche | Testi: 1, 2 |
| 4 | Serie numeriche | Testi: 1, 2 |
| 5 | Funzioni continue | Testi: 1, 2 |
| 6 | Calcolo differenziale | Testi: 1, 2 |
| 7 | Calcolo combinatorio, statistica e probabilità | Testi: 1, 2, 3 |
| 8 | Calcolo integrale per funzioni reali di una variabile reale | Testi: 1, 2 |
Learning Assessment
Learning Assessment Procedures
The average learning of the students will be periodically evaluated through guided exercises in the classroom.
The final exam consists of a written test and an interview. The interview is accessed once the written test
has been passed. Both the written test and the interview will be evaluated out of thirty. The evaluation of
the written test partially affects the formulation of the final grade. The registration of the exam will
take place only after passing the interview.
N.B .: The verification of learning can also be carried out electronically, should the conditions require it.