Università Roma Tre

Propositions. Logical operations with propositions. Logical implication. Sets. Operations with sets. Cartesian product. Applications. Injective and surjective applications. One-to-one correspondence. Inverse application.

Numeric numbers and sets: Natural numbers. Integer or relative numbers. Rational numbers. Real numbers and representation on the line. Bounded sets. Upper and lower extreme of sets of rational and real numbers. Intervals and neighborhoods. Accumulation, internal, isolated points. Open sets and closed sets.

Summations and products: Definition of summation. Properties. Special sums. Sum of the first n natural numbers. Arithmetic and geometric progressions and sum of their first n terms. Factorial.

Real functions of a real variable:

Definition of a real function of a real variable. The Euclidean plane and the graph of a function. Injective and surjective functions and graph. Even and odd functions. Increasing and decreasing functions. Concave and convex functions. Bounded functions. Composition function. Inverse function, monotonicity and invertibility, inverse function graph. Elementary functions. Functions with two laws. Transforming graphs. Domain of a function. Definition of a sequence.

Limits: Definition of limit. Convergence and divergence. Right limit and left limit. Vertical and horizontal asymptotes. Limit uniqueness theorem (w.p.). Sign permanence theorem in direct and inverse form (w.p.). Comparison theorem. Limit checks. Operations with limits. Indeterminate forms.

Infinitesimals and infinities: Definition of infinitesimal and infinite. Comparing infinitesimals and infinities. Order of infinitesimals and infinities. Propagation of the order. Computing limits with infinitesimals and infinities (w.p.).

Continuity and discontinuity: Definition of continuity. Limits and continuity. Classification of discontinuity points. Continuity of rational functions. Continuity of the inverse. Continuity of composition functions. Theorem of zeros (w.p.). Weierstrass theorem. Darboux's theorem (w.p.).

Differential calculus: Derivative of a function. Geometric interpretation. Derivability and continuity (w.p.) Points of non-derivability. Higher order derivatives. Derivatives of elementary functions. Rules of derivation. Chain rule. Derivative of the inverse function. Differential. First order approximation (w.p.). Taylor and McLaurin polynomial. Approximations of higher order. Stationary points. Local maxima and minima. First order necessary conditions for the existence of local maxima and minima. Fermat's theorem (w.p.). Rolle's theorem (w.p.). Lagrange theorem (w.p.). Lagrange theorem’s corollaries: zero-derivative functions (w.p.). Relations between monotonicity and derivative sign (w.p.). Local concave and convex functions. Relationship between the second order derivative and the concavity (w.p.). Points of inflection. Sufficient second order conditions for the existence of relative maxima and minima (w.p.). Sufficient conditions of order n for the existence of relative maxima and minima or inflections points (w.p.). De L'Hôpital theorem and application to limit calculus.

Graph of a function: Representation of the graph of a function on the Euclidean plane. Oblique asymptotes.

Linear algebra:

Vectors and vector spaces. Geometric representation of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Rank of a set of vectors. Matrices. Operations with matrices. Product rows by columns. Particular matrices. Transposed matrix. Determinant of a matrix of order n. Properties of the determinant. Rank of a matrix. Rank and linear independence of vectors. Systems of linear equations. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Integral calculus:

Primitive functions. Indefinite integral. Characterization of the set of primitives (w.p.). Properties of the indefinite integral. Integral of elementary functions. Integration by parts (w.p.). Integration by substitution (w.p.). Definite integral. Properties of the definite integral. Integral function. Integral mean theorem(w.p.). Fundamental theorem of integral calculus (w.p.). Corollary to Torricelli-Barrow's theorem: relationship between the definite integral and the indefinite integral (w.p.). Applications.

(w.p.) = “with proof”

1) all the material will be uploaded on Moodle
2)Mastroeni-Mazzoccoli-Vellucci
Esercizi di Matematica Generale
Esculapio Editore