## 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. The principle of induction.

2. An overview of linear algebra.

Vector spaces, linearly independent vecors, basis; matrix, determinant; linear operators; systems of linear equations. Lines and planes in the space.

3. 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 their convergence.

5. 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.

6. 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; zeroes of continuous functions defined in an interval; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

7. 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 composite functions and inverse functions; derivatives of elementary functions; delative maxima and minima; stationary points; connection between the monotonicity and the 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, connection between the convexity and the sign of the second order derivative; study of local maxima and minima via the study of the derivatives.

8. 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 oriented intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions.

9. Ordinary differential equations.

Separable differential equations; first-order linear differential equations with variable coefficients; linear differential equations of order n with constant coefficients.

## 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 different dates.