30062 - MATEMATICA - MODULO 1 (GENERALE) / MATHEMATICS - MODULE 1 (GENERAL)

The aim of this course is to give students the basic mathematical knowledge and instruments that are necessary to cope with the quantitative study of problems in Economics, Finance and Management. In order to reach this aim, it is first of all necessary that students understand which are the internal structures and the essential procedures of Mathematics, and that they get to comprehend the nature of Mathematics as an axiomatic-deductive system.

• Structures. The set R: real numbers, operations, properties. The set R^n: vectors, operations, properties.
• Functions. Composite function, inverse function. Real functions of one real variable: domain, maxima/minima, convexity, other properties. Real functions of n real variables: domain, maxima/minima, convexity, other properties.
• Sequences of real numbers: definition and properties. Limits of sequences and their computation.
• Number series. Series with non-negative terms, series with terms of indefinite sign.
• Limits and continuity for functions of one or n real variables.
• One-variable differential calculus. Difference quotient, derivative. Differentiability. Differentiation rules. Fermat's and Lagrange's Theorems. Higher-order derivatives. De L'Hopital Theorem. Taylor formula. Optimization and convexity conditions.
• N-variable differential calculus. Partial derivatives and gradient. Differentiability. Unconstrained extrema, optimization conditions. Constrained extrema, Lagrangean function.
• Linear algebra. Subspaces. Linear dependence and independence. Basis and dimension of a subspace. Matrices and their operations. Linear functions and applications: definition, properties, representation. Determinant, rank and inverse matrix. Linear systems: discussion and structure of the solutions, solution.

• Know the fundamental notions of mathematical analysis, of differential calculus, and of linear algebra.
• Articulate these notions in a conceptually and formally correct way, using adequate definitions, theorems, and proofs.
• Understand the nature of mathematics as an axiomatic-deductive system.

• Apply the fundamental theoretical results of mathematical analysis, of differential calculus and of linear algebra to the solution of problems and exercises.
• Actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects and to solve assigned problems.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature.

Students are evaluated on the basis of a written exam (in some cases, an oral part is included). This exam may be taken in one of the two following ways.

• It can be split in two partial exams (October, January/February), which may take place either in a traditional form or online depending on the social and sanitary conditions, and two additional assessments (September, November), which take place online. Each of the two partial exams contains both open-answer questions and closed-answer questions; each one covers one half of the course syllabus; each one weighs for approximately forty percent of the final grade. Each of the two additional assessments contains only closed-answer questions; each one covers only a limited part of the course syllabus; and each one weighs for approximately ten percent of the final grade. If the social and sanitary conditions are such that a session of the second partial exam has to take place online, only for students whose final grade is larger than 24/30 and in order to guarantee a stronger reliability of their results the exam will include an oral part. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. In particular, closed-answer questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises. While open-answer questions (and - whenever included - the oral part) mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
  • The ability to actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.  
• It can be taken as a single general exam, which may take place either in a traditional form or online depending on the social and sanitary conditions. This exam contains both open-answer questions and closed-answer questions, covers the entire course syllabus, and can be taken in one of the four general sessions which are scheduled in the academic year (the two regular sessions in January and February, or the two make-up sessions in June and August/September). If the social and sanitary conditions are such that a session of the general exam has to take place online, only for students whose final grade is larger than 24/30 and in order to guarantee a stronger reliability of their results the exam will include an oral part. This way is mainly meant for students who have withdrawn from the four partials procedure or could not follow it. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. In particular, closed-answer questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises. While open-answer questions (and - whenever included - the oral part) mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs. 
  • The ability to actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.

We take a special care to adjust the raw grades assigned in each exam, to obtain final grades whose distribution follows as closely as possible the normal distribution of grades that is recommended by Università Bocconi.

• S. CERREIA VIOGLIO, M. MARINACCI, E. VIGNA, Principles of Mathematics and Economics, Milano (draft version available as a pdf file).
• Integrative teaching materials.