Lecturer: Marco Luigi Bernardi  

Course name: Calculus I (Mathematical analysis I)
Course code: 500115
Degree course: Ingegneria Elettronica e delle Telecomunicazioni, Ingegneria Industriale
Disciplinary field of science: MAT/05
University credits: CFU 9
Course website: n.d.

Specific course objectives

The course is aimed at providing the basic knowledge of calculus (differential, integral) for real functions of one real variable, together with an introduction to ordinary differential equations. Lectures will be mainly focused on the comprehension of notions (definitions, results), although some proofs will still be detailed. Examples and exercises will be presented. By the end of the course the Students are expected to be able to handle correctly and without hesitation limits, derivatives, function graphs, integrals, differential equations, and the corresponding theoretical facts.

Course programme

Preliminaries.
Recalls and complements on: set theory, mathematical logic, real numbers. Complex numbers: algebraic, trigonometric, and exponential form. Operations on complex numbers; algebraic equations on the complex field.

Functions, Limits, Continuity.
Functions: definitions, graphs; invertible functions; odd and even functions; monotone functions; periodic functions; operations on functions; nested functions. Elementary functions and corresponding graphs. Limits of functions: definitions, operations on limits. Continuous functions. Discontinuity points and their classification. Global properties of continuous functions.

Differential Calculus in one real variable and Applications.
Derivative of a function: definition and properties, applications in Geometry and Physics. Derivation rules and calculus. Fundamental theorems of differential calculus. Primitives and indefinite integrals. Successive derivatives. Function study: extrema, monotonicity, convexity. De l'Hopital rules.

Integral Calculus.
Definite integrals: definitions and basic properties, applications in Geometry and Physics. Fundamental theorems of integral calculus. Integration techniques. Improper integrals.

Ordinary Differential Equations.
Introduction to ordinary differential equations. The Cauchy problem. Separation of variables. Linear ordinary differential equations of the first order. Linear ordinary differential equations of the second order with constant coefficients. Linear systems of ordinary differential equations.

Course entry requirements

Mathematics: the required prerequisites for enrollment into the Engineering Faculty.

Course structure and teaching

Lectures (hours/year in lecture theatre): 45
Practical class (hours/year in lecture theatre): 45
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

C. Canuto, A. Tabacco. Analisi Matematica I (terza edizione). Casa Editrice Springer, Milano, 2008.

Testing and exams

Finals consist in a written (exercises) and an oral test (theory plus possibly exercises). Both written and orals have to be passed within the same finals session. In order to be admitted to the oral test, a specific minimum of points has to be obtained in the written test.