Ateneo - UPO

• Number sets.
• Real functions of one real variable: definition and properties, operations.
• Differential calculus and its applications.
• Introduction to integral calculus.

• Salinelli E., Appunti di Matematica, Giappichelli, 2021 (theory)
• Salinelli E., Esercizi svolti di Matematica, III edizione, Giappichelli, 2020 (exercises)
• D'Ercole R., Precorso di Matematica, Pearson, 2011 (math refresher)

The course is worth 8 ECTS, corresponding to 60 hours of direct teaching. Overall, approximately 40 hours consist of lecture-based teaching (DE) and 20 hours of interactive teaching (DI), with flexible proportions within each lesson.

The course aims at helping the students to acquire the knowledge and skills of mathematics, related to the differential and integral calculus in a real variable, necessary for the study and analysis of simple economic-business models and for the understanding of the tools of basic statistics.

• Elementary set theory, number sets.
• Equations and inequalities (linear and quadratic, rational, radical, logarithmic, exponential), systems.
• Introductory elements of analytic geometry (Cartesian plane, points, lines).

Mastering the prerequisites is necessary for all students and, for those who wish to attend lectures, should be completed before classes begin. As no dedicated remedial course is provided, students are advised to refer to the recommended textbook and/or the free online course Matematica di base (https://lms.federica.eu) to get up to speed. An open-access self-assessment test to check your preparation on the prerequisites is available on the learning portal (DIR).

On the web page of the course at the URL: www.dir.uniupo.it any useful information on the course and learning resources can be found. More precisely:

• self-assessments: self-assessment tests are available divided by topics, useful for verifying the level of learning achieved;
• complementary learning material: examples and/or extra material; review test for each macro-topic;
• samples of unfolded exercises which are more frequently assigned at exams.

There will be 20 additional hours of laboratory during the academiv year to allow the active verification of one's level of preparation.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services- students-physical-or-learning-disabilities.

Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a mandatory written test and an optional oral test.

The written test consists of 12 multiple-choice questions to be completed in 50 minutes. For each question, there are 5 options, only one of which is correct. Every answer provided must be accompanied by an explanation/justification.

Each correct and well-justified answer is worth 1 point. Any incorrect answer, unanswered question, or answer provided without justification (or with an incorrect justification) is worth 0 points.

The final score is converted into a grade as follows:

• points 12, grade 25
• points 11, grade 24
• points 10, grade 22
• points 9, grade 21
• points 8, grade 20
• points 7, grade 19
• points 6, grade 18
• points <6, fail

The questions cover the course topics and aim to verify the skills achieved in applying acquired knowledge to solve moderately complex problems. The written exam is finalized only after a potential confirmation oral interview, should the Professor deem it necessary to clarify the submitted written test.

The oral exam is open to students who achieve a passing mark (greater than 18/30) on the written exam and allows them to earn a maximum score of 8/30. Students must first demonstrate the ability to identify any errors made in the written exam, pinpointing the correct approaches and providing a clear explanation. In addition, a score of up to 5 points requires demonstrating a sufficiently broad knowledge and understanding of the course contents that, in particular, enables the student to establish connections between different topics. To achieve a score of 6 to 8 points, students must be able to independently apply their knowledge and understanding of the course content to engage in an in-depth discussion on critical aspects related to the subjects covered, using clear and rigorous language.

The overall grade is calculated by summing the scores obtained in the written exam and in the optional oral exam, up to a maximum of 30 cum laude. The award of cum laude is conditional upon the ability to independently solve questions (exercises or more theoretical questions) of above-average difficulty.

Registration for the written exam via the student portal is mandatory. Once the maximum number of exam attempts permitted by the Academic Regulations is reached, the system will not allow further registrations.

On the educational portal (DIR), a learning path structured around self-assessments is available; upon completion, students can access a written exam simulator.

Further details about the exam can be found on the course DIR page.

Introductory concepts

• The set of real numbers. Real line, intervals, neighborhoods; introductory elements of topology.
• Ordered sets in R: upper bounds, lower bounds, bounded sets, supremum, infimum, maximum, minimum.

Real functions of a real variable

• Definition of a function: natural domain, image, preimage (inverse image), graph. Elementary functions, piecewise functions.
• Properties: injective, surjective, and bijective functions; even and odd functions; monotonic functions; concave and convex functions; extrema (extreme points).
• Operations on functions, composite functions, inverse functions.

Limits and continuity

• Definition of a limit (brief overview); limits of elementary functions; uniqueness of the limit.
• Evaluation of limits: algebra of limits, indeterminate forms, notable limits, limits of sums of powers.
• Continuous functions and related theorems.
• Asymptotes: vertical, horizontal, and oblique (slant) asymptotes; definitions and their determination.

Elements of differential calculus in one variable

• Differentiable functions, derivative, tangent line.
• Calculation of derivatives: derivatives of elementary functions, algebra of derivatives, derivatives of composite functions (chain rule).
• Derivative function, higher-order derivatives.
• Theorems of differential calculus and their application to the study of monotonicity, convexity, finding extrema, and evaluating limits.

Introduction to integral calculus

• Antiderivatives, indefinite integral.
• Evaluation of indefinite integrals: immediate integrals, decomposition method, integration by parts, integration by substitution.
• Application of finding antiderivatives to the calculation of definite integrals.
• Integral functions and their study.
• Introduction to improper integrals (brief overview).