GENERAL MATHEMATICS
(objectives)
According to the Degree Course in Economics and Business Management (CLEGA), the course aims at enabling students to grasp the basic mathematical topics and tools needed in Economics and Firm Management modeling. Upon completion of this course, the student will be able to know the basics of mathematical analysis, of differential calculus and of linear algebra; articulate these notions in a conceptually and formally correct way; using adequately definitions, theorems and proofs understand the nature of mathematics as an axiomaticdeductive system; apply the fundamental theoretical results of mathematical analysis, of differential calculus and of linear algebra to the solution of problems and exercises; actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects and to solve assigned problems

Code

21210239 
Language

ITA 
Type of certificate

Profit certificate

Credits

7

Scientific Disciplinary Sector Code

SECSS/06

Contact Hours

56

Type of Activity

Elective activities

Group: A  C
Derived from

21210239 GENERAL MATHEMATICS in Economics and business administration L18 A  C CONGEDO MARIA ALESSANDRA, Capasso Armando
(syllabus)
GENERAL MATHEMATICS PROGRAM a.a 20212022 I CHANNEL
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 nonlimited 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. Multilaw 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. Righthand limit and lefthand 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 nonderivability. 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. Firstorder 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). TorricelliBarrow's theorem (with proof). Corollary to TorricelliBarrow'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. RouchéCapelli theorem. Homogeneous systems. Parametric systems
(reference books)
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/laureetriennali/matematicageneralenoiicanaledk/

Dates of beginning and end of teaching activities

From 02/10/2022 to 21/12/2022 
Delivery mode

Traditional

Attendance

not mandatory

Evaluation methods

Written test
Oral exam

Group: L  P
Derived from

21210239 GENERAL MATHEMATICS in Economics and business administration L18 L  P CENCI MARISA, MARTIRE ANTONIO LUCIANO
(syllabus)
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 nonlimited 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. Multilaw 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. Righthand limit and lefthand 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 nonderivability. 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. Firstorder 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). TorricelliBarrow's theorem (with proof). Corollary to TorricelliBarrow'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. RouchéCapelli theorem. Homogeneous systems. Parametric systems.
(reference books)
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 .

Dates of beginning and end of teaching activities

From 02/10/2022 to 21/12/2022 
Delivery mode

Traditional

Attendance

not mandatory

Group: Q  Z
Derived from

21210239 GENERAL MATHEMATICS in Economics and business administration L18 Q  Z CORRADINI MASSIMILIANO, MUTIGNANI RAFFAELLA
(syllabus)
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 nonlimited 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. Multilaw 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. Righthand limit and lefthand 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 nonderivability. 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. Firstorder 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). TorricelliBarrow's theorem (with proof). Corollary to TorricelliBarrow'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. RouchéCapelli theorem. Homogeneous systems. Parametric systems.
(reference books)
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.

Dates of beginning and end of teaching activities

From 02/10/2022 to 21/12/2022 
Delivery mode

Traditional

Attendance

not mandatory

Evaluation methods

Written test
Oral exam


