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

Real functions of a real variable: definition and properties. Elementary functions: linear, quadratic, power functions with integer or fractional exponents, exponential, logarithmic, and trigonometric functions. Quasi-elementary functions. Examples of mathematical models in finance.

Discrete mathematics: combinatorial calculus and sequences of real numbers. Limits of sequences. Real numerical series; arithmetic series, geometric series, and harmonic series. Series with positive terms and convergence criteria. Geometric series in finance.

Definition of limits for real functions: properties and theorems. Infinitesimals and infinities. Continuous functions: operations and properties. Local theorems for continuous functions. Global theorems for continuous functions: Bolzano, Weierstrass, and Darboux.

The derivative and its geometric meaning. Algebra of derivatives. Derivatives of composite functions. Unconstrained optimization for functions of one variable. First-order conditions for the existence of maxima and minima. Theorems on differentiable functions: Rolle, Lagrange, and Cauchy. L'Hôpital's theorem. Concavity and convexity. Second-order conditions for the existence of maxima and minima. Taylor's and Taylor-MacLaurin's formulas.

Definition of the integral and its properties. Conditions for integrability. The fundamental theorem of integral calculus. Antiderivatives (primitive functions) and definite integrals. Integration by parts and by substitution. Improper integrals of unbounded functions or functions defined on unbounded intervals.

n-dimensional vector space; subspaces, linear independence, generators of a vector space, bases, and dimension. Linear maps between vector spaces. Matrix algebra. Determinants of matrices and Laplace expansion. Properties of the determinant. The inverse matrix and its properties. Rank and its properties.

Functions of several variables. Partial derivatives and differentiability. Gradient vector. Hessian matrix. Unconstrained optimization for functions of several variables.

Linear systems; vector and matrix representation. Linear systems of n equations in n unknowns and Cramer's rule. Linear systems of m equations in n unknowns and the Rouché–Capelli theorem. Homogeneous systems and kernel.

Eigenvectors and eigenvalues of a real square matrix. Characterization of the eigenvalues of a matrix, characteristic polynomial.

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.

The oral examination consists of two parts: a mandatory part, in which the written examination is discussed and reviewed, and an optional part, covering the topics included in the course syllabus. The final grade is calculated as the average of the grade obtained in the written examination and the grade obtained in the oral examination (mandatory part and, if applicable, optional part).

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