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

Foundations: real numbers; sequences and limits; functions and continuity

Differentiation: Derivatives and its geometrical interpretation. Left and right derivative. Continuity of differentiable functions. Differentiation rules and theorems (Rolle, Lagrange, Cauchy). Local and global extrema of a function. Stationary points and extrema. Higher order derivatives. Second order derivative and convexity. Taylor's expansion. Higher order derivatives criterion for maxima and minima. Indeterminate forms and De l'Hôpital's theorems.The study of the graph of a function. Asymptotes.

Integrals: Partitions. The Riemann integral and its geometrical interpretation as area. Primitives and indefinite integrals. First and second fundamental theorems of calculus. Integration by parts. Integration by substitution. Improper integrals.

Linear algebra: Vector spaces. Matrices and their properties. Rank and determinant. Systems of equations. Existence of solutions. Case of one solution and of infinitely many solutions. Triangular systems. Linear maps, associated matrices, matrix of basis change. Invertible linear maps, image, kernel, Dimension Theorem. Eigenvectors, eigenvalues, characteristic polynomial, diagonalizability. Symmetric matrices, Spectral Theorem (over the real numbers).

Series: Sequences and series. Telescopic series, geometric series, harmonic series. Convergence criteria: comparison test, root test, ratio test, absolute convergence.

Multi-variable calculus: Limits and continuity for scalar functions of more than one variable. Limits and continuity. Directional derivatives. Partial derivatives. Gradient. Derivative of composite functions. Total derivative. Differentiability. Continuity of differentiable functions. Extremum points: maxima, minima and saddle points. Second derivatives test. Lagrange's multipliers for constrained extrema.

Testi/Bibliografia

Tom M. Apostol, Calculus (vol. 1 and 2) Wiley

Robert A. Adams, Christopher Essex , Calculus, a complete course. Pearson.

M. Spivak, Calculus, 4th edition, Publish or Perish

Metodi didattici

Lezioni frontali alla lavagna e strutturate in parti teoriche, esempi ed esercitazioni.

Modalità di verifica e valutazione dell'apprendimento

Esame scritto finale. Gli studenti del primo anno possono sostenere esami intermedi e il voto finale sarà una media degli esami sostenuti.

 

Strumenti a supporto della didattica

Uso occasionale del proiettore per l'illustrazione dello studio di funzioni e simulazioni al computer.

Orario di ricevimento

Consulta il sito web di Fabrizio Lillo

Consulta il sito web di Andrea Santi