Learning objectives
Through the course the students are supposed to acquire the fundamental concepts of the mathematical analysis and the calculus knowledge.
Prerequisites
Abilty to hand mathematical expressions and to resolve mathematical equalities and inequalities.
Course unit content
Basic notions of set theory and mathematical logic. Real numbers. Real functions of a real variable and their properties. Limit, continuity and Riemann integral. Brief introduction to statistic theory.
Full programme
1.Basic notions of mathematical logic. Basic notions of set theory.
2.Integers, rational numbers, irrational numbers. Upper bound, maximum, least upper bound (supremum). The completeness axiom. Functions and terminology concerning functions. Composite functions. One-to-one functions and inverses. Elementary functions and their diagrams : absolute value, rational, exponential, logarithmic, power with real exponent, trigonometric.
3.Limit of a function. One-side limit of a function. Properties of the limits of functions. Continuous functions. Theorems concerning continuous functions on an interval.
4.Definition of derivative. Derivatives and continuity. Algebra of derivatives. The chain rule. One-sided derivatives and infinite derivatives. Zero derivatives and local extrema. Rolle's theorem. The mean-value theorem for derivatives. Higher order derivatives. Taylor's formula with remainder. Convexity of a function. Diagram of a function.
5.Definition of the Riemann integral. Linear properties. Integration by parts. Change of variable in a Riemann integral. Mean value for the Riemann integral. The integral as a function of the interval. Fundamental theorems of integral calculus. Generalized integrals and comparison theorems.
6.Differential equations. Cauchy problems. Linear first order equations. Separable variables equations. Second oreder linear equations with constant coefficients. 7. Basic notions of probability and statistic.
7. Complements : (a) complex numbers (b) matrices and systems (c) ellipse, parabola, hyperbole.
Bibliography
P. Marcellini, C.Sbordone, "Calcolo", Ed. Liguori.
P. Marcellini, C. Sbordone, "Esercitazioni di matematica" Ed. Liguori.
Marco Abate, "Matematica e Statistica - Le basi per le scienze della vita", McGraw-Hill
Teaching methods
Frontal lessons
Assessment methods and criteria
Written test followed by oral test.
Other information
None
