Analytical Methods in Engineering I
Academic Year 2025/2026 - Teacher: RITA TRACINA'Expected Learning Outcomes
The aim of the course is to provide students with the basic concepts of differential calculus of single-variable functions, to develop their ability to critically elaborate on fundamental topics, and to enhance their reasoning skills. These learning outcomes are aligned with the United Nations 2030 Agenda for Sustainable Development, in particular with Goal 4 (Quality Education), as they contribute to strengthening the mathematical competencies essential for higher education, research, and professional growth. Furthermore, they are consistent with Goal 9 (Industry, Innovation and Infrastructure), as they support the development of scientific foundations useful for technological innovation, and with Goal 11 (Sustainable Cities and Communities), as they provide students with educational tools necessary for a conscious and responsible participation in the sustainable development processes of communities.
Course Structure
Lectures and exercises in the classroom.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
Elements of set theory. Numerical sets. Topology elements
Cartesian product. Definition of function. Special functions. Succession. Composed function. Injective and surjective functions. Inverse function. Internal, external, border, accumulation, isolated points. The extended line R *.
Limits
Real functions of real variable. Positivity and symmetries. Limited functions. Monotone functions. Definition of limit. Theorem of uniqueness of the limit. Right and left limit. Theorem of the permanence of the sign. Operations with function limits. Infinitesimal and infinite. Asymptotes. Limit of a succession. The number "e", some notable limits. Cauchy convergence criterion.
Continuous functions
Definition of continuity. Operations on continuous functions. Points of discontinuity. Discontinuity of monotonic functions. Basic properties of continuous functions over a range. Theorem of existence of zeros and intermediate values. First and second Weierstrass theorem. Elementary functions: rational functions; algebraic, exponential and logarithmic functions; hyperbolic functions and their inverse; trigonometric functions and their inverse.
Differential calculus
Definition of derivative. Derivability and continuity. Right derivative, left derivative. Operations with derivatives. Differential. Local extremes. Fermat's theorem. Theorem of Lagrange. Consequences of the Lagrange theorem. De L'Hôpital theorems and applications. Taylor's formula. Concave and convex functions. Determination of the nature of stationary points. Determining the graph of a function.
Integrals of functions of one variable
Definition of integral according to Riemann and geometric meaning. Classes of integrable functions. Properties of the integral: additivity; homogeneity; monotony; average theorem; additivity to the integration interval. Integral function. The fundamental theorem of integral calculus. Undefined integral. Rules for integration by decomposition, by parts and by substitution. Improper integrals.
Numerical series
Definition of numerical series and first properties. Cauchy criterion. Series with non-negative terms. Convergence criteria. Convergence and absolute convergence. Leibniz criterion. Operations on the series.
Textbook Information
1) C.D. Pagani, S. Salsa - Analisi Matematica I - Zanichelli
2) S. Salsa, A. Squellati - Esercizi di Analisi matematica vol. 1- Zanichelli
3) G. Zwirner, Esercizi di Analisi Matematica I, CEDAM
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Elementi di teoria degli insiemi. Insiemi numerici. Elementi di topologia. | 1 |
| 2 | L'operazione di limite | 1 |
| 3 | Funzioni continue | 1 |
| 4 | Calcolo differenziale | 1 |
| 5 | Integrali di funzioni di una variabile | 1 |
| 6 | Serie numeriche | 1 |
Learning Assessment
Learning Assessment Procedures
The examination consists of a written test and an oral test. The written test involves solving exercises with justification, organized into three sections (functions, integrals, series). The oral test covers the theory in its entirety.
To take the written test, students must register by the deadlines set for each exam session exclusively online through the student portal.
During the written test, only a non-programmable calculator is allowed. The use of any other tools, books, or personal notes is prohibited.
To access the oral test, students must achieve at least 5/10 in each section (even if obtained in different exam sessions).
If the overall score is ≥18/30, with at least 5/10 in each section (even if obtained in different exam sessions), the student may choose not to take the oral test. In this case, the final grade is calculated as the average between the written test score and 18 (rounded up), with a maximum of 24/30.
If the student also takes the oral test, the final grade is determined by an overall assessment of both tests, without applying arithmetic averages, taking into account knowledge, application, connections, and clarity of exposition.
An outstanding oral performance can lead to a high final grade even if the written test is not excellent, provided that the student clearly demonstrates solid and comprehensive preparation during the oral exam, so that the written test can be considered an isolated and non-representative instance. The assignment of the final grade will take the following parameters into account:
Qualitative description of the oral exam (in addition to a sufficient written test) | |
Not passed | Lacks minimum knowledge of the main contents and has very poor or no ability to use the specific terminology. Unable to independently apply acquired knowledge. |
18–21 | Essential and fragmented knowledge. Mechanical application with limited connections. Very basic expression, not always comprehensible. |
22-24 | Fair but incomplete knowledge. Correct applications to standard cases with few connections. Simple but understandable expression. |
25–26 | Good but not in-depth knowledge. Correct application of methods in familiar contexts; connections only between closely related topics. Clear expression with adequate language skills. |
27–28 | Solid and well-structured knowledge. Correct applications even in non-standard contexts. Appropriate and accurate connections. Clear and fluent expression. |
29–30 | Complete and in-depth knowledge. Confident and consistent applications. Cross-disciplinary connections. Rigorous and effective expression. |
30 cum laude | Excellence in all respects. Strong abstraction ability, personal insights, impeccable and independent presentation. |
Information for students with disabilities and/or specific learning disorders (SLD):
In order to ensure equal opportunities and in compliance with current legislation, interested students may request a personal meeting to plan possible compensatory and/or exemption measures, based on the learning objectives and their specific needs.
Students may also contact the CInAP (Center for Active and Participatory Integration – Services for Disabilities and/or SLD) representative of the Department.