LEARNING OUTCOMES

The course aims at providing the student with basic operative knowledge on differential and integral calculus for functions of one and two real variables, with some attention to mathematical rigour. Some of the founding elements of mathematical modeling are developed in the second half of the course, such as the elementary theory of ordinary differential equations.

TEACHING METHODS

Lectures and practice

SYLLABUS/CONTENT

Functions of one real variable. Real numbers, the oriented real line. The Cartesian plane, graphs of elementary functions. Operations on functions and their graphical interpretation. Monotonicity. Composition and inversion. Powers, exponentials and logarithms. Supremum and infimunm. Sequences and series: the basic notions and examples. Limits of functions. Infinitesimal and infinite functions. Continuous functions and their local and global, derivative, derivation rules. Derivatives of elementary functions. Sign of derivatives in the study of monotonicity and convexity. The classical theorems of Rolle, Cauchy, Lagrange and de l'Hôpital. Taylor expansions and applications to critical points. Definite and indefinite integrals.

 Functions of two (or more) real variables. Continuity, directional and partial derivatives, gradient.  Differentiability and tangent plane. Level sets. Local minima and maxima: second order derivatives and the Hessian. Schwartz’s theorem.

Differential equations. Separation of variables.  The existence and uniqueness theorem for the Cauchy problem. Linear first and second order differential equations: solving methods. General solution for the linear equation.

RECOMMENDED READING/BIBLIOGRAPHY

A. Bacciotti, F. Ricci, Lezioni di Analisi Matematica 1 e 2, Levrotto & Bella, 1991.

C. Canuto, A. Tabacco, Analisi Matematica 1 e 2, Springer-Verlag Italia, 2003.

F.De Mari, Dispense di Analisi Matematica 1, http://www.dima.unige.it/~demari/DIDA.html