Università Roma Tre

GENERAL MATHEMATICS PROGRAM a.a 2024-2025
CHANNELS A-C e D-K

1) Logic, sets and numerical sets
Propositional logic. Propositions. Decidable propositions. Logical operations between propositions. Logical implication. Necessity, sufficiency and necessity and sufficiency. Truth tables. Theorem. Methods for proving a theorem. Sets. Operations between sets. Numerical sets:
natural numbers, relative numbers and rational numbers. Infimum and supremum of a numerical set. Irrationality of square root of 2 (with proof). Dedekind's axiom. The real numbers set. Real numbers and their representation on the straight line. Limited and non-limited sets. Maximum and minimum of a numerical set. Intervals and boundary of a point. Elements of line topology: isolated, boundary, internal and accumulation points. Open sets. Closed sets. A set is closed if and only if it contains all its accumulation points.
2) Sums
Definition of summation. Sum of the first natural n (with proof). Sum of the first n terms of a geometric progression (with proof). Summation properties.
3) Real functions of real variable
Real functions of real variable. The Cartesian plane and the graph of a function. Injective and surjective functions. Invertible functions. Even and odd functions. Increasing and decreasing functions. Link between monotony and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Compound function. Multi-law defined functions. Domain of a function.
4) Sequences and numerical series
Converging, diverging and indeterminate sequences. Limits of sequences. Definition of numerical series. Necessary condition for convergence
of a series (with proof). The character of the geometric series (with proof). Mengoli series and its value (with proof).
5) Limits of real functions of real variable
Definition of limits. Right-hand limit and left-hand limit. Vertical and horizontal asymptote. Oblique asymptote. Theorem on uniqueness of limit. (with proof). Theorems on sign permanence (with proof). Rational operations on limits. Indeterminate forms. Notable limits.
6) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems (with proof). Order propagation.
7) Continuity
Definition of continuity in a point. Continuity in a set. Classification of discontinuity points. Continuity of elementary functions. Continuity
of the compound function. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem (with proof).
8) Differential calculus
Difference quotient. Derivative of a function in a point and its geometric meaning. Derivability implies continuity (with proof). Points of non-derivability.
Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of compound functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1 and order 2 (with proof). The factorial of n. Taylor polynomial
of order n. Mc Laurin polynomial. Differential and its geometric meaning. First-order error theorem (with proof). Local maxima and minima. Fermat's theorem (with proof). Rolle's theorem (with proof). Lagrange's theorem (with proof). Corollaries to Lagrange's theorem (with proof). Relationship between the sign of the first derivative and the increasing / decreasing functions in an interval (with proof). Concavity and convexity in a point. Relationship between the sign of the second derivative and the convexity / concavity of a function in an interval (with proof). Flex points. Global concavity and convexity conditions.
9) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives (with proof). Property of the indefinite integral. Immediate indefinite integrals. Integration by parts (with proof). Area subtended by a curve. Superior and inferior integral sum. Definition of definited Riemann integral. Property
of the definite integral. Integral function. The integral mean value theorem (with proof). Torricelli-Barrow's theorem (with proof). Corollary to Torricelli-Barrow's theorem (with proof).
10) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space Rn: Linear combination of
vectors. Linearly dependent and independent vectors. Matrices. Matrix addition. Matrix multiplication. Determinant of a matrix. Matrix
inverse. Uniqueness of the inverse matrix (with proof). Necessary condition for the existence of the inverse matrix (with proof). Sufficient condition for the existence of the inverse matrix. Transposed matrix. Rank of a matrix. Linear equation systems. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems

Carl P. Simon, Lawrence Blume, Mathematics for economists, W.W. Norton and Company, 2010.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
http://disa.uniroma3.it/didattica/lauree-triennali/matematica-generale-n-o-ii-canale-d-k/

1) Logic, sets and numerical sets
Propositional logic. Propositions. Decidable propositions. Logical operations between propositions. Logical implication. Necessity, sufficiency and necessity and sufficiency. Truth tables. Theorem. Methods for proving a theorem. Sets. Operations between sets. Numerical sets:
natural numbers, relative numbers and rational numbers. Infimum and supremum of a numerical set. Irrationality of square root of 2 (with proof). Dedekind's axiom. The real numbers set. Real numbers and their representation on the straight line. Limited and non-limited sets. Maximum and minimum of a numerical set. Intervals and boundary of a point. Elements of line topology: isolated, boundary, internal and accumulation points. Open sets. Closed sets. A set is closed if and only if it contains all its accumulation points.
2) Sums
Definition of summation. Sum of the first natural n (with proof). Sum of the first n terms of a geometric progression (with proof). Summation properties.
3) Real functions of real variable
Real functions of real variable. The Cartesian plane and the graph of a function. Injective and surjective functions. Invertible functions. Even and odd functions. Increasing and decreasing functions. Link between monotony and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Compound function. Multi-law defined functions. Domain of a function.
4) Sequences and numerical series
Converging, diverging and indeterminate sequences. Limits of sequences. Definition of numerical series. Necessary condition for convergence
of a series (with proof). The character of the geometric series (with proof). Mengoli series and its value (with proof).
5) Limits of real functions of real variable
Definition of limits. Right-hand limit and left-hand limit. Vertical and horizontal asymptote. Oblique asymptote. Theorem on uniqueness of limit. (with proof). Theorems on sign permanence (with proof). Rational operations on limits. Indeterminate forms. Notable limits.
6) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems (with proof). Order propagation.
7) Continuity
Definition of continuity in a point. Continuity in a set. Classification of discontinuity points. Continuity of elementary functions. Continuity
of the compound function. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem (with proof).
8) Differential calculus
Difference quotient. Derivative of a function in a point and its geometric meaning. Derivability implies continuity (with proof). Points of non-derivability.
Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of compound functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1 and order 2 (with proof). The factorial of n. Taylor polynomial
of order n. Mc Laurin polynomial. Differential and its geometric meaning. First-order error theorem (with proof). Local maxima and minima. Fermat's theorem (with proof). Rolle's theorem (with proof). Lagrange's theorem (with proof). Corollaries to Lagrange's theorem (with proof). Relationship between the sign of the first derivative and the increasing / decreasing functions in an interval (with proof). Concavity and convexity in a point. Relationship between the sign of the second derivative and the convexity / concavity of a function in an interval (with proof). Flex points. Global concavity and convexity conditions.
9) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives (with proof). Property of the indefinite integral. Immediate indefinite integrals. Integration by parts (with proof). Area subtended by a curve. Superior and inferior integral sum. Definition of definited Riemann integral. Property
of the definite integral. Integral function. The integral mean value theorem (with proof). Torricelli-Barrow's theorem (with proof). Corollary to Torricelli-Barrow's theorem (with proof).
10) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space Rn: Linear combination of
vectors. Linearly dependent and independent vectors. Matrices. Matrix addition. Matrix multiplication. Determinant of a matrix. Matrix
inverse. Uniqueness of the inverse matrix (with proof). Necessary condition for the existence of the inverse matrix (with proof). Sufficient condition for the existence of the inverse matrix. Transposed matrix. Rank of a matrix. Linear equation systems. Cramer's theorem. Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

Carl P. Simon, Lawrence Blume, Mathematics for economists, W.W. Norton and Company, 2010.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.