Università Roma Tre

GENERAL MATHEMATICS PROGRAM a.a 2019-2020
I CHANNEL

Logic, sets and numerical sets
Propositional logic. Propositions. Decidable propositions. Operations
logic between propositions. Logical implication. Need, su_cienza and necessity e
su_cienza. Tables of truth. Theorem. Methods for the proof of a theorem.
Join now. Operations between sets. Numerical sets:
natural numbers, relative numbers and rational numbers. Upper and lower extreme
of a numerical set. Axiom of Dedekind.
The set of real numbers. Real numbers and their representation on the line.
Limited and not limited sets. Maximum and minimum of a numerical set. intervals
and around. Elements of topology of the line: isolated, border points,
internal and accumulation. Open sets. Closed sets. A set is closed
sse contains all its accumulation points (c.d.).

Summation
Definition of summation. Sum of the first natural n (c.d.). Sum of
first n terms of a geometric progression (c.d.). Sum property.
Real functions of a real variable
Real functions of a real variable. The Cartesian plane and the graph of one
function. Injective and surjective functions. Invertible functions. Equal functions e
odd. Crescence and decrease. Link between monotony and injectivity. Function
reverse. Graph of the inverse function. Elementary functions. Transformations of
elementary functions. Composed function. Functions defined by multiple laws. Domain
of a function.

Sequences and numerical series
Converging, divergent and indeterminate sequences. Calculation of sequences limits.
Definition of numerical series. A necessary condition for convergence
of a series (c.d.). The character of the geometric series (c.d.). Series of Mengoli e
its value (c.d.).

Limits of real functions of real variable
Definition of finite or infinite limit at the finite and infinite. Right limit e
left limit. Vertical and horizontal asymptote. Oblique asymptote. Verification of
limit. Theorem of uniqueness of the limit (c.d.). Sign permanence theorem
in direct and in inverse form (c.d.). Rational operations on limits. Forms
indeterminate. Notable limits.

Infinitesimals and infinities
Definition of infinitesimal and infinity. Comparison between infinitesimals and between
endless. Order of infinitesimal and infinite. Deletion theorems.
Order propagation.

Continuity
Definition of continuity in a point. Continuity in a whole. Classification
of the points of discontinuity. Continuity of elementary functions. Continuity
of the composite function. Continuity of the defined function with multiple laws.
Zeros theorem for continuous functions. Weierstrass theorem. Theorem
of Darboux.

Differential calculation
Incremental report. Derivative of a function in a point and its meaning
geometric. Derivability implies continuity (c.d.). Points of non-derivability.
Derived function and derivatives of next order. Derivative of functions
elementary. The derivation rules. Derivative of composite functions. Theorem
by De L'Hopital and its application to indeterminate forms. Polynomial
of Taylor of order 1 and of order 2 (c.d.). The factorial of n: Taylor polynomial
of order n: Polynomial of Mc Laurin. Differential and its geometric meaning.
Theorem on the rest of first order (c.d.). Local highs and lows. Theorem
of Fermat (c.d.). Rolle's theorem (c.d.). Lagrange theorem (c.d.). corollaries
to the Lagrange theorem (c.d.). Relationship between the sign of the first derivative
and the growth / decrease of a function in an interval (c.d.). Concavity e
convexity in one point. Relationship between the sign of the second derivative and the convexity /
concavity of a function in an interval (c.d.). Points of inflection. Conditions
sufficient of order n for the existence of relative maxima and minima or inflections (c.d).
Concavity conditions and global convexity.

Integral calculation
Primitive of a function. Indefinite integral. Properties of primitives
(CD.). Property of the indefinite integral. Immediate indefinite integrals. Integration
for parts. Integration by substitution. Area subtended by one
curve. Upper and lower integral sum. Definition of definite integral
according to Riemann. Property of the definite integral. Integral function. Theorem
of the integral average (c.d.). Torricelli-Barrow theorem (c.d.). Corollary
to the Torricelli-Barrow theorem (c.d.).

Linear algebra
Vectors and their geometric representation. The space Rn: Product of a
vector for a scalar. Sum of vectors. Linear combination of vectors.
Linearly dependent and independent vectors. Matrices. Operations between matrices.
Product rows by columns. Determinant of a matrix. Reverse matrix.
Uniqueness of the inverse matrix (c.d.). Necessary condition for the existence of
inverse matrix (c.d). Sufficient condition for the existence of the inverse matrix.
Transposed matrix. Calculation of the inverse matrix. Rank of a matrix. The
principle of the edged minor. Systems of linear equations. Cramer's theorem.
Rouché-Capelli theorem. Homogeneous systems. Parametric systems.

MATEMATICA GENERALE- MARISA CENCI-EDIZIONI KAPPA

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

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

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.