Learning objectives
The aim of the course is to provide students with the basic tools of Mathematical Analysis and Linear Algebra.
Course unit content
1. Real numbers.
Maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, nth roots of non-negative numbers;
rational and irrational numbers; intervals, distance.
Complex numbers.
2. An overview of linear algebra.
Vector spaces, linearly independent vecors, basis; matrix, determinant; linear operators;
systems of linear equations.
2. Functions.
One to one, surjective and bijective functions; inverse functions; graphs;
monotone functions; exponential and logarithmic functions; trigonometric functions.
4. Sequences and series.
Limits of sequences. Series with positive terms. Criteria for the convergence.
3. Limits.
Limits of functions with real values, uniqueness of the limit, limits of the restrictions;
limit of the sum, product, quotient of two functions; permanence of the sign,
comparison theorems; right and left limit; limits of monotone functions.
5. Continuous functions.
Continuity of real functions of a real variable, restrictions of continuous functions,
composition of continuous functions; sum, product, quotient of continuous functions;
examples of continuous functions; discontinuity, examples of discontinuous functions;
zero theorem; continuity and intervals; continuity and monotony;
continuity of inverse functions; Weierstrass theorem.
6. Differential calculus.
Incremental ratio, derivatives, right and left derivatives;
geometrical significance of the derivative; derivation rules:
derivatives of the sum, product, quotient of two functions;
derivatives of compound functions and inverse functions;
derivatives of elementary functions; relative maximums and minimums;
stationary points; relationship between monotony and sign of the derivative;
Rolle's theorem and Lagrange's theorem and their geometrical interpretation,
Cauchy's theorem and de l'Hopital's theorem; convex functions,
derivatives of convex functions,
relationship between convexity and sign of the second derivative;
study of local maxima and minima via the study of higher order derivatives.
7. Integrals.
Partitions of an interval; upper and lower integral, Integrability of continuous functions;
geometrical interpretation of the integral;
properties of integrals; mean of an integrable function; integrals on directed intervals;
fundamental theorem of integral calculus; primitives, indefinite integrals;
integration by parts and by substitution; integrals of rational functions.
8. Differential equations.
Bibliography
M. Bramanti, C.D. Pagani, S. Salsa, Matematica: calcolo infinitesimale e algebra lineare. Seconda edizione. Zanichelli, 2004
Assessment methods and criteria
The exam consists in a written part and an oral part in a different date.
