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 axiomatic-deductive 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
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 axiomatic-deductive 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
Curriculum
Canali
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
teacher profile teaching materials
Introduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Only Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Programme
1) Logic, sets and numerical setsIntroduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Attendance
Class attendance is not mandatory, but strongly recommendedType of evaluation
Ongoing evaluation approach - To stimulate an active attitude throughout the course, the students’ behaviour during the classes and willingness to participate in learning activities are taken into account in the evaluation process. For the same reason, two midterm tests are scheduled during the course. Each test lasts one hour and consists of exercises covering specific parts of the exam syllabus. The exam consists in a written test, lasting 2 hours, and in an oral test. The written test consists of exercises, multiple choice questions, theoretical questions regarding the whole program and a question regarding theory. The oral test consists in one or more questions, including proofs of the main results/theorems. teacher profile teaching materials
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.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Programme
1) Logic, sets and numerical setsPropositional 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Type of delivery of the course
Teaching activities include, in addition to lectures, lasting 2 hours three times a week, also a exercises session, lasting 2 hours, once a week. Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature. Class attendance is not mandatory, but strongly recommended.Attendance
Class attendance is not mandatory, but strongly recommended.Type of evaluation
The exam will consist of a written test, lasting 2 hours, and in a theory test. The written test will consist of exercises, individuation of true / false propositions with short justification, theoretical questions regarding the whole program and a question regarding theory. The theory test will consist of one or more questions throughout the program including proofs of the main results/theorems.Canali
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
teacher profile teaching materials
Introduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Only Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
Programme
1) Logic, sets and numerical setsIntroduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Attendance
Class attendance is not mandatory, but strongly recommendedType of evaluation
Ongoing evaluation approach - To stimulate an active attitude throughout the course, the students’ behaviour during the classes and willingness to participate in learning activities are taken into account in the evaluation process. For the same reason, two midterm tests are scheduled during the course. Each test lasts one hour and consists of exercises covering specific parts of the exam syllabus. The exam consists in a written test, lasting 2 hours, and in an oral test. The written test consists of exercises, multiple choice questions, theoretical questions regarding the whole program and a question regarding theory. The oral test consists in one or more questions, including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
teacher profile teaching materials
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.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,
Programme
1) Logic, sets and numerical setsPropositional 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Type of delivery of the course
Teaching activities include, in addition to lectures, lasting 2 hours three times a week, also a exercises session, lasting 2 hours, once a week. Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature. Class attendance is not mandatory, but strongly recommended.Attendance
Class attendance is not mandatory, but strongly recommended.Type of evaluation
The exam will consist of a written test, lasting 2 hours, and in a theory test. The written test will consist of exercises, individuation of true / false propositions with short justification, theoretical questions regarding the whole program and a question regarding theory. The theory test will consist of one or more questions throughout the program including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,
Canali
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
teacher profile teaching materials
Introduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Only Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
Programme
1) Logic, sets and numerical setsIntroduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Attendance
Class attendance is not mandatory, but strongly recommendedType of evaluation
Ongoing evaluation approach - To stimulate an active attitude throughout the course, the students’ behaviour during the classes and willingness to participate in learning activities are taken into account in the evaluation process. For the same reason, two midterm tests are scheduled during the course. Each test lasts one hour and consists of exercises covering specific parts of the exam syllabus. The exam consists in a written test, lasting 2 hours, and in an oral test. The written test consists of exercises, multiple choice questions, theoretical questions regarding the whole program and a question regarding theory. The oral test consists in one or more questions, including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
teacher profile teaching materials
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.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,
Programme
1) Logic, sets and numerical setsPropositional 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Type of delivery of the course
Teaching activities include, in addition to lectures, lasting 2 hours three times a week, also a exercises session, lasting 2 hours, once a week. Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature. Class attendance is not mandatory, but strongly recommended.Attendance
Class attendance is not mandatory, but strongly recommended.Type of evaluation
The exam will consist of a written test, lasting 2 hours, and in a theory test. The written test will consist of exercises, individuation of true / false propositions with short justification, theoretical questions regarding the whole program and a question regarding theory. The theory test will consist of one or more questions throughout the program including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,
Canali
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R A - C CONGEDO MARIA ALESSANDRA,
teacher profile teaching materials
Introduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Only Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
Programme
1) Logic, sets and numerical setsIntroduction to propositional logic. 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. 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. Sum of the first n terms of a geometric progression. 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. Monotone functions. Link between monotonicity and injectivity. Inverse function. Inverse function graph. Elementary functions. Transformations of elementary functions. Composite function. Multi-law defined functions. Domain of a function.
4) 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. Theorems on sign permanence. Rational operations on limits. Indeterminate forms. Notable limits.
5) Infinites and infinitesimals
Definition of infinitesimal and infinite. Comparison between infinitesimal and infinite. Order of infinitesimal and infinite. Cancellation theorems. Order propagation.
6) Continuity
Definition of continuity at a point. Continuity over a set. Classification of discontinuity points. Continuity of elementary functions. Continuity of composite functions. Continuity of functions defined by several laws. Theorem of zeros for continuous functions. Weierstrass's theorem. Darboux's theorem.
7) Differential calculus
Difference quotient. Derivative of a function at a point and its geometrical meaning. Differentiability implies continuity. Points of non-differentiability. Derivative function and derivatives of subsequent order. Derivative of elementary functions. The rules of derivation. Derivative of composite functions. De L’Hopital's theorem and its application to indeterminate forms. Taylor polynomial of order 1. Differential and its geometric meaning. First-order error theorem. Local maxima and minima. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Corollaries to Lagrange's theorem. Sign of the first derivative vs monotonicity of a function over an interval. Concavity and convexity. Sign of the second derivative vs convexity/concavity of a function over an interval. Inflection points.
8) Integral calculus
Primitive of a function. Indefinite integral. Properties of primitives. Property of the indefinite integral. Immediate indefinite integrals. 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. Torricelli-Barrow's theorem. Corollary to Torricelli-Barrow's theorem.
9) Linear algebra
Vectors and their geometric representation. Product of a vector for a scalar. Vectors addition. The vector space R^n: 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. Necessary condition for the existence of the inverse matrix. 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Attendance
Class attendance is not mandatory, but strongly recommendedType of evaluation
Ongoing evaluation approach - To stimulate an active attitude throughout the course, the students’ behaviour during the classes and willingness to participate in learning activities are taken into account in the evaluation process. For the same reason, two midterm tests are scheduled during the course. Each test lasts one hour and consists of exercises covering specific parts of the exam syllabus. The exam consists in a written test, lasting 2 hours, and in an oral test. The written test consists of exercises, multiple choice questions, theoretical questions regarding the whole program and a question regarding theory. The oral test consists in one or more questions, including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R L - P LAMPARIELLO LORENZO,
teacher profile teaching materials
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.
Only: Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Appendix 4.
Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,
Programme
1) Logic, sets and numerical setsPropositional 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.
Core Documentation
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.
Reference Bibliography
Knut Sydsæter, Peter Hammond, Arne Strøm & Andrés Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2012. Only : Chapter1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 15, Chapter 16.Type of delivery of the course
Teaching activities include, in addition to lectures, lasting 2 hours three times a week, also a exercises session, lasting 2 hours, once a week. Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature. Class attendance is not mandatory, but strongly recommended.Attendance
Class attendance is not mandatory, but strongly recommended.Type of evaluation
The exam will consist of a written test, lasting 2 hours, and in a theory test. The written test will consist of exercises, individuation of true / false propositions with short justification, theoretical questions regarding the whole program and a question regarding theory. The theory test will consist of one or more questions throughout the program including proofs of the main results/theorems.Mutuazione: 21210239 MATEMATICA GENERALE in Economia e gestione aziendale L-18 R Q - Z CORRADINI MASSIMILIANO,