C9127 - MATEMATICA

Learning outcomes

At the end of the course, the student will have gained an understanding of and will be able to use the basic tools of mathematical analysis for functions of one real variable; moreover, the student will have acquired fundamental knowledge of matrix theory and linear algebra. In particular, the student will be able to: • compute limits, derivatives, and Taylor polynomials of a function; • analyze and graph the behavior of a function; • calculate definite and improper integrals of functions; • solve systems of linear equations using vectors and matrices; • identify linearly independent sets and subspaces; • compute inner products, norms, and orthogonal projections; • diagonalize square matrices

Course contents

The sets of numbers N, Z, Q, R, and their algebraic properties.

Real-valued functions of a real variable: basic definitions, elementary functions, and examples of mathematical models in Finance.

Discrete mathematics: sequences, limits of sequences, and series.

Limits of real-valued functions: properties and theorems on continuity, both local and global.

The concept of the derivative and its geometric interpretation. Properties of derivatives, differentiation rules, and theorems of Rolle, Lagrange (Mean Value Theorem), and Cauchy.

Unconstrained optimization of functions of one variable. Convexity, concavity, Taylor's theorem, and the Taylor–Maclaurin expansion.

The concept of the antiderivative and its properties. Methods of integration, the fundamental theorems of integral calculus, and improper integrals.

Finite-dimensional vector spaces; matrices, determinants, and rank.

Systems of linear equations; Cramer's rule and the Rouché–Capelli theorem.

Eigenvalues, eigenvectors, and their interpretation.

Introductory elements of unconstrained optimization in several variables. Gradient vector and Hessian matrix.

Readings/Bibliography

K. Sydsaeter, P. Hammond, A. Strom (a cura di D. La Torre), Metodi Matematici per l'Analisi Economica e Finanziaria, Pearson.

The book can also be found in English.

A booklet based on solved exam exercises will be downloadable from the Virtuale Platform.

Teaching methods

Active student participation during lectures is encouraged through the discussion of open-ended problems, whose solutions are developed and discussed collectively in class.

Students' learning is monitored through two complementary tools. At the end of each lecture, exercises related to the theoretical concepts covered are assigned, and their solutions are discussed at the beginning of the following class. In addition, at the end of each major topic, students are given a 30-question test to complete as part of their independent study.

Students are also invited to prepare short, optional case studies on the applications of mathematics in finance, based either on material provided by the instructor or on authoritative external sources. Each week, groups of two or three students present their case studies and discuss them with the class.

The weekly tutorial sessions provide an additional opportunity for students to consolidate and assess the knowledge and skills acquired during the lectures through guided problem-solving activities.

Assessment methods

The final assessment in Mathematics consists of two written midterm examinations, each graded on a 30-point scale.

The final written grade is calculated as the average of the two midterm examination scores. Students pass the written examination if this average is at least 18/30. Each midterm consists of 10–12 independent exercises, with the score assigned to each exercise clearly indicated on the examination paper. Students who are not satisfied with the result of the first midterm may retake it when the second midterm examination is held.

An oral examination is optional. If taken, the final course grade is calculated as the average of the written examination grade and the oral examination grade.

The written examinations are open-book. Students may consult their notes and textbooks during the examination but are not permitted to use programmable calculators.

Students who prepare and deliver clear and effective classroom presentations of case studies on the applications of mathematics in finance may be awarded up to two additional points. These points are added to the average of the two written examination scores.

The use of AI tools is prohibited in all assessments. Any use of AI tools constitutes a breach of academic integrity.

Students with specific learning disorders (SLD) or temporary or permanent disabilities are encouraged to contact the University's dedicated support office in good time (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will propose any appropriate accommodations for eligible students. These accommodations must be submitted to the course instructor for approval at least 15 days in advance. The instructor will assess their appropriateness, taking into account the intended learning outcomes of the course.

 

Teaching tools

Students are strongly encouraged to attend both the MATH&FIN crash course and the exercise-based course.

Office hours

See the website of Maria Letizia Guerra