Analytical Methods in Engineering I
Academic Year 2026/2027 - Teacher: RITA TRACINA'Expected Learning Outcomes
The course aims to provide students with the foundational concepts of single-variable differential and integral calculus, and numerical series, developing their ability to critically analyze fundamental topics and enhancing their formal reasoning skills.
In accordance with the Dublin Descriptors specified in the Degree Programme's Tuning Matrix, students will develop:
· Knowledge and understanding (A): Knowledge and understanding of the fundamental concepts and theorems of mathematical analysis and analytic geometry for single-variable functions (elements of topology, limits, continuity, differentiability, geometric interpretation of function properties, integration, and numerical series).
· Applying knowledge and understanding (B): Ability to operationally apply mathematical analysis and analytic geometry concepts to solve quantitative problems (computing limits, sketching function graphs in the Cartesian plane, differential calculus, geometric meaning of derivatives and integrals, analyzing the convergence of numerical series).
· Making judgements (C): Ability to make reasoned methodological choices, critically analyzing logical-deductive steps, the validity of mathematical proofs, and the correctness of analytical and numerical results.
· Communication skills (D): Ability to present analytical procedures, geometric representations, and theoretical proofs clearly and rigorously, using appropriate mathematical, symbolic, and formal language.
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· Learning skills (E): Mastery of logical deduction methods and essential cognitive tools required to autonomously learn and continuously update knowledge on mathematical models and quantitative techniques in subsequent engineering courses.
Course Structure
Lectures and exercises in the classroom.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
The course contents contribute to the achievement of the 2030 Agenda Goals (4, 9, and 11), as outlined below for each module:
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 *. (UN 2030 Agenda Reference – Goal 4: Quality Education, establishing the foundations of formal language and mathematical rigor).
Limits and continuous functions
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. 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. (UN 2030 Agenda Reference – Goal 4: Quality Education, fostering critical thinking and quantitative modeling).
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. (UN 2030 Agenda Reference – Goal 9: Industry, Innovation and Infrastructure, providing optimization and analytical tools essential for technological and engineering innovation).
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. (UN 2030 Agenda Reference – Goal 11: Sustainable Cities and Communities, offering cumulative calculation tools essential for physical and environmental modeling).
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.