# 75229 - Mathematics (B.A.)

### Course Unit Page

• Credits 12

• SSD SECS-S/06

• Language Spanish

• Campus of Bologna

• Degree Programme First cycle degree programme (L) in Business and Economics (cod. 8965)

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.  ## Learning outcomes

At the end of the course the student will be capable of using the techniques of Linear Algebra; furthermore he will have acquired a working knowledge of First Year Calculus, together with the related applications in Finance and Economics.

## Course contents

Sequences and limits.

Definition of sequence. Convergence and divergence. Limit of a sequence. Limit of a function. Geometrical interpretation of a limit. Calculate lateral limits. Finding the limit of a function algebraically.

Derivative.

Functions and continuity. Definition of the derivative. Geometric interpretation of the derivative. Differentiable functions. Stationary and extreme points. Differentiation rules and associated theorems. High order derivatives. Taylor expansion. Second derivative condition. Concavity and extremizing a function. Applications of the derivative. Study and interpretation of graphs of functions. Optimization.

Integrals.

Riemann’s sums. Definition of integral. Geometric interpretation of the integral. Proper integrals. Anti-derivation. Fundamental theorem of calculus. Improper integrals. Integration techniques: integration by substitution, integration by parts. Application of integrals.

Matrices.

Definition of matrix. Matrices classification. Rank. Operations between matrices. Elementary transformations of a matrix. The inverse matrix. Determinants. The adjoint of a matrix.

System of linear equations.

Relation between matrices and linear systems. Solution set and geometric interpretation. Algorithms for solving SLE: Gauss, Gauss-Jordan, inverse.

Vector spaces.

Definition of vector space. Sub-spaces and their properties. Linear combination and linear independence. Inner product. Basis and dimension. Change of basis. Orthonormal bases.

Eigenvalues and eigenvectors.

Definition of eigenvalues and eigenvectors of matrices. Characteristic polynomial.

Applications of linear algebra.

Sequences and series.

Sequences and series. Convergence.

Multivariate functions.

Definition of function of several variables. Vector-valued functions and graphs in space.

Differentiation.

Limit and continuity. Partial derivatives. Directional derivative. Multivariate continuity. Gradient. Total derivative. Chain rule. Laplacian.

Optimization.

Extremum points: maximum, minimum and saddle points, Lagrange’s multipliers for constrains extrema.

Applications of multi-variable calculus.

Cálculo de una variable. J. Stewart, 7ma. edición, Cengage Learning Editores.

Cálculo de varias variables. J. Stewart, 7ma. edición, Cengage Learning Editores.

Calculus (vol. 1 y 2). T. M. Apostol. Wiley.

Calculus. M. Spivak. Editorial Reverté.

Álgebra lineal con aplicaciones. S. Grossman, Ed. McGraw Hill.

Introducción al álgebra lineal. Howard Antón, Editorial Limusa.

## Teaching methods

Lectures, exercises, and multimedia tools

## Assessment methods

Multiple choice exams, exercises in class and for homework, midterm projects.

An oral examination could be carried out in exceptional circumstances (if decided by the University).

## Teaching tools

Digital notes, lists of exercises and TPs, videos

## Office hours

See the website of Rocio Angelica Bermudez Ramos