CALCULUS I

The MATHEMATICAL ANALYSIS I (a.k.a. CALCULUS 1) course is one of the core learning activities of the Engineering degree program, as it provides students with rigorous logical and mathematical tools to formulate, address, and solve problems they will encounter in all scientific disciplines. The specific educational objective of this course is to provide a rigorous foundation for the concept of limits for real functions of one real variable. This permits to define concepts like continuity, differentiation, and integration of functions, as well as the study of the behavior at infinity of numerical sequences and series.

1) Learn the basic features of the methodological rigor and the logical reasoning underlying the scientific method.
2) Learn the axiomatic foundation of mathematics and the main techniques for proving a mathematical result, such as the principle of induction. Understand the need for a rigorous foundation for mathematical tools, such as the formal language and the symbolic calculus. Become aware about the conceptual connections between set theory, arithmetic, algebra, and geometry.
Understand how these notions inform calculus, including the notions of limit, derivative, and integral.
3) Understand how the formal correctness of symbolic reasoning allows the use of complex concepts and technologies and justifies the results obtained in all scientific disciplines.

A prerequisite is a basic understanding of the contents of the first four years of the typical secondary school (elementary algebra and geometry, analytic geometry, functions, algebraic and transcendental equations and inequalities, properties and graphs of basic functions such as: linear functions, powers, polynomial functions, trigonometric functions, exponential functions, and logarithms).
Attendance at the PRECOURSE - GENERAL MATHEMATICS [CT0110] is recommended, especially for students who have not previously encountered the concepts of mathematical analysis in secondary school.

1. Elements of logic and set theory. Combinatorics.
2. Numerical sets. Induction principle. Irrationality of the square root of 2.
3. Ordered fields. Topology of the real line. Decimal expansions. Comparison of infinite sets. Uncountability of R. Completeness.
4. Functions and their properties. Invertibility. Transformations of function graphs. Symmetries. Elementary functions.
5. Infinitesimals and infinities. Hyperreal numbers. Limits of functions. Indeterminate forms and their solutions. Special limits.
6. Sequences. Hierarchies of infinities. Cauchy sequences and completeness. Numerical series. Convergence criteria.
7. Continuity. Bolzano-Weierstrass theorem. Theorems of Rolle, Lagrange, and Cauchy. Weierstrass theorem.
8. Derivative as the slope of the tangent and as a growth rate. Rules of differentiation. L'Hopital's theorem. Taylor's theorem and the computation of limits.
9. Riemann integration. Fundamental theorem of integral calculus. Elementary rules of integration. Integration by substitution. Integration by parts. Integration of rational functions. Non elementarily integrable functions. Generalized integrals. Volume and area of solids with rotational symmetry.

For each lecture, lecture notes or similar materials will be provided for individual study

Adopted books:
A. Marson, P. Baiti, F. Ancona, B. Rubino: Analisi matematica 1. Teoria e applicazioni, Carocci
M. Lanza de Cristoforis, Lezioni di Analisi Matematica 1, Esculapio
M. Bramanti, C. Pagani, S. Salsa: Analisi matematica 1, Zanichelli
P. Marcellini, C. Sbordone: Esercizi di matematica, Vol. 1 (Tomi 1-4), Liguori
S. Salsa, A. Squellati: Esercizi di analisi matematica 1, Zanichelli
G. De Marco, C. Mariconda, Esercizi di calcolo in una variabile, Zanichelli/Decibel
M. Bramanti: Esercitazioni di Analisi Matematica 1, Esculapio

The exam consists of a written test and a subsequent oral exam.
Written exam
The written exam, lasting approximately three hours, includes both theoretical questions (mainly definitions, statements of theorems, and short proofs) and exercises covering all the topics discussed in class. Evaluation will be based on the correctness of the exposition, the clarity and completeness of the justifications, the mastery of scientific terminology, and the ability to apply the tools of mathematical analysis.
Oral exam
Admission to and the structure of the oral exam depend on the results of the written test: it is mandatory for students who achieve a score of 23/30 or higher. It is optional for those who obtain a score between 18/30 and 22/30.
The oral interview will focus on solving exercises, as well as on definitions and proofs of the theorems presented in class.
Should they deem it necessary, the instructors reserve the right to request an oral integration of the written exam from any student.

written

The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

18-21: Basic understanding of the essential topics. The student can complete standard and purely mechanical exercises but presents significant theoretical gaps and struggles to understand proofs or set up problems that fall slightly outside standard patterns.
22-23: Fair command of the main calculation methods. The student correctly sets up and solves most exercises, despite encountering some computational inaccuracies. Theory is remembered mostly by rote: the student struggles to rigorously justify steps, apply theorems in non-elementary contexts, or connect different concepts together.
24-27: Good practical and theoretical understanding. Confidently handles the setup of complex problems covered in the syllabus. Correctly applies solving methods and understands the analytical and geometric meaning of theorems, but the rigor in justifying logical steps and proofs still lacks complete formal fluency.
28-29: Solid computational and theoretical competence. Can calculate, interpret, and justify results, skillfully combining analytical ability with the theoretical foundations of the course. Uses appropriate logical-mathematical terminology and presents theorems and concepts with excellent formal and structural clarity.
30: Masters the entire syllabus comprehensively and exhaustively. Structures answers impeccably, demonstrating a profound understanding of mathematical analysis concepts, combined with excellent expository skills in proofs and an almost total absence of relevant calculation errors in practical application.
30 cum laude: Exceptional analytical intuition and strong logical-critical thinking. Demonstrates absolute mastery of the subject, navigating between extreme mathematical rigor and practical applications with elegance and speed. Argues proofs and theoretical concepts brilliantly and autonomously, showing a maturity that goes well beyond the simple assimilation of notions.

Lectures: theory and exercises, use of the blackboard for the lectures.
Resources on prerequisite and integrative topics will be made available through the university moodle page.

Accommodation and support services for students with disabilities and students with specific learning impairments:
Ca’ Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding supportservices and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). In the case of disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.