Learning objectives
The object of the course is to familiarize the students with the basic language of and some fundamental theorems in the theory of real valued functions of one rela variable, focusing on fundamental notions as sequences, continuos functions, derivatives, integrals .
Prerequisites
- - -
Course unit content
Calculus.
Full programme
1) Real numbers and functions of one real variable.
Axiomatic theory of real numbers.
Set theory.
Natural, Integres and Ratinal numbers.
Functions.
Linear functions.
Power, trigonometric, exponential and logarithmic functions .
Mathematical induction.
Maximum, minimum, infimum and supremum.
2) Sequences.
Sequences: definition and examples.
Limit of sequences.
Bounded sequences.
Special limits.
3) Continuos functions.
Functions and limits.
Continuos functions: examples and basic properties.
Singularities.
Limit of functions and limit of sequences.
Weierstrass Theorem
4) Derivatives.
Derivatives: definition and examples.
Geometrical meaning of derivatives: tangent lines.
Rules for calculation of derivatives.
Derivatives of elementary functions.
Derivatives of higher order.
5) Fundamental Theorems of differentiable functions.
Rolle, Lagrange and Cauchy Theorems and consequences.
Extremal points. Local maximum and local minimum of functions.
Convex functions.
Taylor formula.
6) Theory of Riemann integration.
Notation. The Riemann integral.
Basic properties of Riemann integrable functions.
Definite integrals: geometrical interpretation.
Mean Value Theorems for Integrals.
Fundamental theorem of calculus. Fundamental formula of calculus. indefinite integrals. Methods of integrations.
Bibliography
P. Marcellini, C. Sbordone: Calcolo, Liguori Editore, P. Marcellini, C. Sbordone: Esercitazioni di matematica, Volume 1, parte prima e seconda, Liguori Editore.
Teaching methods
Theoretical lectures and sessions of oral and written exercises.
Assessment methods and criteria
Written and oral exams.
Other information
- - -
