## MATHEMATICS I

None outside the program of the TOLC entrance exam. Namely:

Integer and rational numbers. Decimal representation of a number.

Linear and quadratic algebraic equations (in one variable).

Polynomials. Manipulation of polynomials.

Cartesian coordinates. Graphs of real functions. Equation of lines

in the plane.

Logarithms.

Trigonometric functions.

Written and oral exam. The written exam consists of problems. Both parts are mandatory.

The exam will be held in presence or online, depending on the evolution of the pandemic.

The course mainly deals with basic techniques (rather

than formal proofs) in Linear Algebra, Vector Algebra and One-variable Calculus. Students will get a grasp of

basic concepts in Analysis, such as limits, continuity,

differentiability, local and asymptotic behaviour of elementary functions.

Linear systems and eigenvectors are introduced from a geometric viewpoint. Complex numbers are introduced in the usual form a+ib to highlight similarity with vector sum, but the use of polar form in trigonometric

sums and integrals is also shown.

Analysis and Linear Systems will be developed further in the Course "Matematica II".

Real powers, modulus, roots.

Polynomials. Elements of combinatorics and binomial theorem.

Cartesian and parametric equations. Sums and products of vectors.

Linear systems. Matrices and determinants. Independence, bases. Projections. Matrix of an operator. Eigenvectors.

Complex numbers. Euler Formula. Complex roots.

Functions and graphs. Invertible functions.

Convergent sequences and series.

Theorems about limits. Properties of continuous functions.

Differentiation. Mean value Theorem. If the sign of f' is constant

in an interval, then f is monotonic in that interval.

f'' and convexity. Criteria for maxima and minima in stationary points.

Taylor expansions.

Definite and indefinite integrals. Fundamental Theorem of Calculus.

Integration by change of variables and by parts.

Examples of improper integrals.

Bramanti--Pagani--Salsa: Matematica. Calcolo infinitesimale e algebra

lineare

(ZANICHELLI)

OR

Bramanti--Pagani--Salsa: Analisi matematica 1 con elementi di algebra lineare

(ZANICHELLI)

In presence.

48 hours face to face; 32 hours of problems class.

Exercises and some course notes will be made available

through the course website.

Readiness to offer individual assistance also in a foreign language to incoming students:

Yes (English, French)

Availability of supporting material and bibliographic references also in a foreign language

Yes (English, French)

Readiness to accept examination of incoming students also in a foreign language

Yes (English, French)