Mathematics I
Academic Year 2026/2027 - Teacher:
MARIA FANCIULLO
Expected Learning Outcomes
Knowledge and understanding: the student will learn some basic mathematical concepts and will develop the skills of calculation and manipulation of the most common objects of mathematics: among these, the sequences, the numerical series, the limits and the derivatives for functions of one variable.
Applying knowledge and understanding: through examples related to applied sciences, the student will be able to appreciate the importance of mathematics in the scientific field, not just as a discipline for its own sake, thus broadening their cultural horizons.
Making judgments: the student will be able to deal with some simple but significant methods of mathematics with sufficient rigor to refine logical skills.
Communication skills: by studying Mathematical Analysis and putting themselves to the test through guided exercises and seminars, students will learn to communicate with rigor and clarity both orally and in writing. Students will learn that using correct language is one of the most important means of communicating scientific language clearly, not only in mathematics.
Learning skills: students will be stimulated to deepen some topics through stimulating questions during the hours of practice.
PLEASE NOTE: Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.
It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.
Course Structure
Lectures in classroom.
Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programm planned and outlined in the syllabus.
PLEASE NOTE: Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.
It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.
Required Prerequisites
Any student must have a thorough knowledge of the notions of Mathematics studied during the school.
In particular: Elements of Mathematical Logic, set theory, algebraic equations and inequalities, trigonometry.
Attendance of Lessons
Mandatory attendance
Detailed Course Content
1. Sets and Logic. Basic concepts of sets, elementary logic.
2. Notions of Arithmetic and Algebra. Scientific notation of real numbers. Fractions, powers, logarithms, absolute value and their properties. Simplification of algebraic expressions. Factorization of polynomials. Equations and inequalities.
3. Introduction to Set Theory. Definition of a set. Empty set. Subset. Union, intersection, and difference between sets. Distributive properties. Numerical sets. Natural, integer, rational, and real numbers. Sets bounded below and above. Minimum, maximum, lower bound and upper bound. Infimum and supremum and their properties.
4. Sequences. Definition of a numerical sequence. Arithmetic, geometric, and harmonic sequences. Convergent sequences, positively and negatively divergent sequences, oscillating sequences. Bounded sequences. Monotonic sequences. Limits of sequences. Proof of convergence and divergence of elementary sequences via the definition. Boundedness of convergent sequences. Algebra of limits. Notable limits. Comparison theorems.
5. Real functions of a real variable. Definition of a function and of the graph of a function. Injective, surjective, and bijective functions. Even, odd, and periodic functions. Composite functions and inverse functions. Monotonic functions. Bounded functions. Absolute minimum and maximum points. Relative (local) minimum and maximum points. Examples of functions: linear functions, the identity function, exponential functions, logarithmic functions, the modulus (absolute value) function. Operations between functions.
6. Trigonometric functions and Euclidean vectors. Angles: radians and degrees. Definition of sine, cosine, tangent. Properties of trigonometric functions. Trigonometric equations. Inverse functions: arcsine, arccosine, arctangent. Trigonometric identities. Trigonometric functions and triangles. Operations with vectors.
7. Continuous functions. Definition of the limit of a function. Various theorems on function limits. Continuity of a function at a point. Operations between continuous functions. Weierstrass theorem. Zero existence theorem. Intermediate value theorem. Points of discontinuity. Asymptotes of a function's graph: horizontal, vertical, and oblique.
8. Differential calculus. Definition of the first derivative of a function at a point and its geometric interpretation. Relationship between continuity and differentiability, with related counterexamples. Derivatives of elementary functions. Differentiation rules. Chain rule for composite functions. Differentiation theorem for inverse functions. Theorems of differential calculus: Fermat's theorem, Rolle's theorem, Lagrange's (mean value) theorem, Cauchy's theorem. L'Hôpital's theorem. Increasing and decreasing functions: necessary conditions, sufficient conditions. Determination of relative and absolute minima and maxima. Higher-order derivatives. Taylor's formula. Concavity, convexity, and inflection points. Function analysis (curve sketching).
9. Matrices and linear systems. Definition of a matrix. Operations with matrices and their properties. Identity matrix. Transpose matrix. Inverse matrix. Determinant. Systems of linear equations. Eigenvalues and eigenvectors.
10. Introduction to combinatorics, probability, and statistics. Combinatorial calculus. Random phenomena. Probability calculus. Measures of central tendency. Organization of statistical data.
Contribution of the course to the goals of the 2030 Agenda for Sustainable Development
Goal 4: QUALITY EDUCATION
Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.
Goal 5: GENDER EQUALITY
Achieve gender equality and empower all women and girls.
Goal 8: DECENT WORK AND ECONOMIC GROWTH
Promote sustained, inclusive, and sustainable economic growth, full and productive employment, and decent work for all.
Teaching method: lecture-based instruction (frontal lessons).
Course Planning
| | Subjects | Text References |
|---|
| 1 | Topic 1 | [1] |
| 2 | Topic 2 | [1] |
| 3 | Topic 3 | [1] |
| 4 | Topic 4 | [1] |
| 5 | Topic 5 | [1] |
| 6 | Topic 6 | [1] |
| 7 | Topic 7 | [1] |
| 8 | Topic 8 | [1] |
| 9 | Topic 9 | [1] |
| 10 | Topic 10 | [1] |
Learning Assessment
Learning Assessment Procedures
1. A first in itinere written test is given consisting of theoretical and practical questions concerning the first part of the program
2. The final exam consists of a written paper divided into two parts: first part (with the topics covered up to the in itinere test) and second part containing practical and theoretical questions concerning the part of the program treated after the first test. After the comunication of the written exam results, students, who received a grade of 18/30 or higher, may request to take a short oral exam. The oral exam can raise the grade obtained on the written test, but it could also lower it, and if none of the questions asked are answered accurately and correctly, it may result in the cancellation of the result achieved on the written test, requiring the student to retake it.
3. Passing the first in itinere test allows the student to be exempted from completing the questions of first part contained in the final exam
4. The benefit of passing the first in itinere test remains valid until the end of the third exam session of the current Academic Year.
As a rule, marks will be assigned according to the following scheme
- not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.
- 18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, she/he is able to solve simple exercises
- 24-27: the student demonstrates good mastery of the course contents, her/his presentation and content connection skills are good, she/he solves the exercises with few errors
- 28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; she/he solves the exercises completely and without errors.
Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.
PLEASE NOTE: Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.
It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.
Examples of frequently asked questions and / or exercises
Definitions of: upper bound, convergent sequence, one-to-one function, inverse function, first kind discontinuity, derivative, inverse matrix, ellipse.
Counterexamples: limited and non-convergent sequence, a function that admits finite upper bound but not the maximum.
Proofs: boundedness of convergent sequences, intermediate value theorem, derivative of a product theorem.
Continuous functions (knowledge and understanding, applying knowledge and understanding)
Numerical series (knowledge and understanding, applying knowledge and understanding).
Remarkable limits deduced from the Nepero number (knowledge and understanding, applying knowledge and understanding).
