37292 - MATHEMATICS

Conoscenze e abilità da conseguire

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.

Contenuti

Course content - Calculus and Linear Algebra (90 hours)

Calculus (about 70 hours)

One-variable functions: basic definitions, graphs and elementary functions (linear, quadratic, polynomial, rational, irrational, power, exponential, logarithmic, absolute value). Odd and even functions. Composite functions. Inverse functions.

Limits and continuity.

Differentiation of one-variable functions: tangents and derivatives, rules of differentiation, chain rule, higher-order derivatives.

Implicit differentiation and economic examples, differentiation of the inverse function, linear and quadratic approximations, Taylor's formula, elasticities; continuity and differentiability, intermediate-value theorem, De L’Hôpital’s Rule.

Single-variable optimization: local and global extrema, stationary points and first-order condition, simple tests for extreme points, extreme points for concave and convex functions, second-order derivative and convexity, inflection points, study of the graph of a function, asymptotes.

Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate integrals, integration of rational functions, integration by parts, integration by substitution. Improper integrals.

Differential equations. First order differential equations. Linear, first order, autonomous differential equations. Steady state and convergence analysis. Linear, first order, non-autonomous differential equations. Differential equations with separable variables.

Sequences and series; convergence criteria; geometric series; Taylor's series. Sequences and series in financial mathematics.

Difference equations. Linear, first order, autonomous difference equations. Steady state and convergence analysis. Linear, first order, non autonomous, difference equations. Difference equations in financial mathematics.

Multi-variable calculus: partial derivatives with two variables, geometric interpretation; partial elasticities; chain rules, implicit differentiation along a level curve; functions of more variables, gradient, differentials and linear approximations; economic applications. Multi-variable optimization; maxima, minima and saddle points; tests based on second derivatives; constrained optimization and Lagrange multipliers.

Linear Algebra (about 20 hours)

Linear algebra: vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear maps and associated matrices, systems of equations, existence of solutions, cases of one solution and infinitely many solutions, Gaussian elimination, inverse of a matrix and Cramer's rule; eigenvalues and eigenvectors.

Testi/Bibliografia

R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, Pearson, 2018.

Chapters: preliminaries, 1, 2, 3, 4, 5, 6, 7.9, 9, 10, 12, 13

K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL. Essential Mathematics for Economic Analysis, 5th Edition. Pearson, 2016.

Chapters: 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18.

Teching material provided by the teacher

 

Metodi didattici

Class lectures

Modalità di verifica e valutazione dell'apprendimento

Written exam: students have to solve different exercises on the course topics. To each exercise a given maximum number of points is associated, and to get it the student has to solve correctly the exercise and all the steps must be justified. The theoretical maximum number of points atteinable in case of a perfect exam is 32.

The test assessment grid will be as follows:

· <18 insufficient

· 18-23 sufficient

· 24-27 average/good

· 28-30 very good

· 31-32 cum laude excellent/outstanding

Orario di ricevimento

Consulta il sito web di Fabio Gobbi