# MATEMATICA GENERALE A - L

**Academic Year 2024/2025**- Teacher:

**SALVATORE CORRENTE**

## Expected Learning Outcomes

- KNOWLEDGE AND UNDERSTANDING: Attending the classroom, students get acquainted with the essential mathematical toolkit as applied to economics and business management. Emphasizing economic, financial and business applications helps motivate certain mathematical topics and as a byproduct this enable students acquire reinforcing intuition in economic applications. The needed mathematical sophistication help students tackle problems from other courses as well as professional issues and specialist studies to be handled.
- APPLYING KNOWLEDGE AND UNDERSTANDING: Students should acquire enough mathematical skill to handle quantitative problems and then solve them properly. As a part of the mathematical reasoning, students face definitions and essential theorems some of which are proved to reinforce their understanding. The mathematical machinery developed is illustrated through solved problems, e.g. profit maximization, cost minimization and utility maximization as well as other problems from economics. This applying knowledge helps students improve their understanding and their problem-solving skill.
- MAKING JUDGMENTS: Students are encouraged to properly learn the mathematical skill, avoiding merely use of formulas and sterile patterns. After developing a quantitative model, students are requested to judge them from different perspectives such as their mathematical “elegance” and powerful as well as their computational complexity.
- COMMUNICATION SKILLS: Students are encouraged to communicate the learned key concepts and techniques using appropriate mathematical language. Problems are essential to the learning process, then students are also encouraged to ask why a result is true, or why a problem should be tackled using particular hypotheses.
- LEARNING SKILLS: Modern economics and business students are expected to attend courses which make increasing mathematical demands. Mastering the relevant mathematical tools students acquire enough quantitative skill to access the literature that is most relevant to their undergraduate study as well as their ongoing professional experience.

## Course Structure

Lectures will be
face-to-face and there will be practical exercises in which it will be shown
how to apply the notions introduced during the lectures

## Required Prerequisites

Although not mandatory, knowledge of the following concepts is strongly recommended: arithmetic operations and their properties; prime numbers, prime factorization, greatest common divisor and least common multiple; fractions and their manipulation; powers, roots and logarithms; monomials, polynomials and polynomial factorization; first and second order equations and inequalities; equations and inequalities involving rational fractions; equations and inequalities in absolute value; Euclidean geometry (circumferences, polygons, Pythagorean theorem).

## Attendance of Lessons

Attendance at classes is highly recommended.

## Detailed Course Content

**1 ^{st} PART**

APPLICATIONS/FUNCTIONS: Injective, surjective and bijective functions. Inverse function. .

COMBINATORICS: Dispositions, combinations and permutations. Binomial theorem. Binomial coefficients.

MATRICES AND DETERMINANTS: Definitions and classifications. Sum and product of matrices. Inverse matrix. Determinant and its property. Rank of a matrix.

LINEAR SYSTEMS: Linear systems. Cramer theorem and Cramer’s rule. Rouché-Capelli Theorem. Solution of parameterized systems.

**2 ^{nd} PART**

ANALYTIC GEOMETRY: Cartesian coordinate system. Straight line equation. Elementary trigonometry.

NUMBER SYSTEMS: Lower bound and upper bound. Infimum and supremum of subsets of real numbers. Separate and contiguous sets. Outlines of topology. Notes on sequences and series.

FUNCTIONS OF A REAL VARIABLE: Definitions, classifications, graphs of functions and their plots. Composite functions and inverse functions. Limits: definitions and theorems. Continuous functions. Infinitesimals and infinities.

DERIVATIVES AND DIFFERENTIALS: Definitions, properties and their geometric interpretation. Derivatives of elementary functions. Calculating derivatives and differentials: the sum rule; the product rule; the quotient rule. Derivatives of composite and inverse functions. N-th order derivatives and differentials. Main theorems on differentiable functions.

APPLICATIONS OF DIFFERENTIAL CALCULUS: Taylor’s and Mac Laurin’s series and related theorems. Indeterminate forms/limits. Monotonic functions, convex functions, local and global extrema, inflection points, asymptotes. Elasticity of a function.

INTEGRALS: Indefinite integral and antiderivatives. Definite integral and their geometric interpretation. Methods of integration.

FUNCTIONS OF SEVERAL VARIABLES: Continuity. Partial derivatives. Gradient. Free maxima and minima. Hessian matrix. Constrained maxima and minima. Lagrangian function.

## Textbook Information

- S. Corrente, S. Greco, B. Matarazzo, S. Milici, "Matematica Generale", Giappichelli Editore, Torino, 2021.

Author | Title | Publisher | Year | ISBN |
---|---|---|---|---|

S. Corrente, S. Greco, B. Matarazzo, S. Milici | Matematica Generale | Giappichelli | 2021 | 9788892141711 |

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Application/function: definition, injective, surjective, bijective function, inverse function. | Corrente, Greco, Matarazzo e Milici: Chapter 1 |

2 | Combinatorial calculus: Dispositions, combinations, permutations and factorials. Binomial coefficient and its properties. | Corrente, Greco, Matarazzo e Milici: Chapter 5 |

3 | Combinatorial calculus: Dispositions, combinations and permutations with replacement. Binomial theorem. | Corrente, Greco, Matarazzo e Milici: Chapter 5 |

4 | Matrices: definition, classification: square and rectangular matrices; vectors; transposed, diagonal, scalar, symmetrical, extracted, complementary. The sum of two matrices; the product of a scalar and a matrix. | Corrente, Greco, Matarazzo e Milici: Chapter 6 |

5 | Matrix calculus. Cayley multiplication, inverse matrix and its properties. Determinant of a matrix. | Corrente, Greco, Matarazzo e Milici: Chapter 6 |

6 | Determinant of a matrix round 2: properties, Laplace theorems, Binet Theorem, cofactor matrix, rank of a matrix and its properties, Pascal Theorem. | Corrente, Greco, Matarazzo e Milici: Chapter 6 |

7 | Linear systems and their solutions: Method of Cramer, Rouché-Capelli theorem, homogeneous linear systems. | Corrente, Greco, Matarazzo e Milici: Chapter 7 |

8 | Cartesian plane, distance between points in a plane. | Corrente, Greco, Matarazzo e Milici: Chapter 8 |

9 | A glimpse of trigonometry: circumference and angles (radians and degrees); sine, cosine, tangent and cotangent, fundamental relations. | Corrente, Greco, Matarazzo e Milici: Chapter 9 |

10 | Number systems: lower and upper bounds; Infimum and Supremum; Separate and contiguous sets. Elementary topology of the real line: open ball, interior points, boundary points points of accumulation - cluster points, open sets, closed sets. | Corrente, Greco, Matarazzo e Milici: Chapter 10 |

11 | Sequences and Series: definitions, convergence and examples. | Corrente, Greco, Matarazzo and Milici: Chapter 10 |

12 | Equation of a straight line. Intersecting straight lines. Parallel and perpendicular straight lines. Distance of a point from a line. | Corrente, Greco, Matarazzo e Milici: Chapter 11 |

13 | Functions of a real variable: domain, codomain, graph of a function. | Corrente, Greco, Matarazzo e Milici: Chapter 12 |

14 | Limit round 1: convergent and divergent function, fundamental theorems on limits | Corrente, Greco, Matarazzo e Milici: Chapter 12 |

15 | Limits round 2: Properties of limits, indeterminate forms, speciallimits | Corrente, Greco, Matarazzo e Milici: Chapter 12 |

16 | Theorems on continuous functions (Bolzano, zero of a continuous function, Weierstrass). Points of discontinuity of the first, second and third type. | Corrente, Greco, Matarazzo e Milici: Chapter 12 |

17 | Monotone functions, inverse functions, composite functions, even and odd functions, periodic functions, infinitesimals and infinities and their, comparison. | Corrente, Greco, Matarazzo e Milici: Chapter 12 |

18 | Derivative of a function: Difference quotient or Newton quotient and its geometric interpretation; definition of derivative, its geometric interpretation, Rules for calculating, round 1 derivatives, points of non-derivability. | Corrente, Greco, Matarazzo e Milici: Chapter 13 |

19 | Derivability and continuity of a function. Differential. Rules for calculating, round 2. Derivatives of composite and inverse functions. Nth derivatives. | Corrente, Greco, Matarazzo e Milici: Chapter 13 |

20 | Rolle theorem, Cauchy mean value theorem and Lagrange mean value theorem. | Corrente, Greco, Matarazzo e Milici: Chapter 14 |

21 | De l'Hopital's theorem. Taylor's formula and Mac-Laurin's formula | Corrente, Greco, Matarazzo e Milici: Chapter 14 |

22 | Increasing and decreasing functions. Maxima and minima. Fermat's Theorem. Sufficient conditions for the existence of relative maxima and minima. Finding extremes of a function. | Corrente, Greco, Matarazzo e Milici: Chapter 14 |

23 | Convex functions and concave functions. Inflection points. Asymptotes. Elasticity of a function. | Corrente, Greco, Matarazzo e Milici: Chapter 14 |

24 | Indefinite integral: antiderivatives, definition and properties of indefinite integral, immediate integrals. | Corrente, Greco, Matarazzo e Milici: Chapter15 |

25 | Methods of indefinite integration: sum rule, integration by parts, partial fractions, integration by substitution. | Corrente, Greco, Matarazzo e Milici: Chapter 15 |

26 | Defined integral: Riemann,'s definition, sufficient conditions for integrability, geometric interpretation. | Corrente, Greco, Matarazzo e Milici: Chapter 15 |

27 | Definite integral: properties of the definite integral, mean value theorem for definite integrals, Fundamental theorem of calculus. | Corrente, Greco, Matarazzo e Milici: Chapter 15 |

28 | Functions of several variables: definition, continuity, partial derivative, gradient, free minima and maxima, Hessian matrix, constrained maxima and minima, Lagrangian multipliers. | Corrente, Greco, Matarazzo e Milici: Chapter 16 |

## Learning Assessment

### Learning Assessment Procedures

### Examples of frequently asked questions and / or exercises

- What is a
function?
- What is a bijective function?
- What is an inverse function?
- What are
simple and with repetition dispositions, combinations and
permutations?
- What is the binomial coefficient and which are its properties? (Symmetry property and Stifel formula)
- What is the Newton binomial theorem?
- What is a matrix and which are the main operations can be done on them?
- What is submatrix of a given matrix?
- What is the inverse matrix of a given matrix? Which are its properties?
- Can you enunciate the two Laplace theorems?
- Can you enunciate the Binet theorem?
- What is a minor of order r of a given matrix? What is the complement of a minor of a given matrix? What is the cofactor of the elements of a given matrix?
- What is the adjoint matrix and which are its properties?
- What is the rank of a matrix and which are its properties?
- Can you enunciate the Kronecker-Pascal theorem?
- What is a linear form?
- When a linear system is possible, impossible, determined or indetermined?
- Which are the equivalence principles for linear systems?
- Can you enunciate and prove the Cramer theorem?
- Can you enunciate the Rouchè-Capelli theorem?
- Can you write the straight line equation and can you prove the parallelism and perpendicularity conditions between two straight lines?
- What are the supremum and the infimum of a given numbers set? Which are their properties? What are the minimum and the maximum of a given numbers set?
- When two numbers sets are separated and when are they contiguous?
- What are the interior, boundary and accumulation points of a given numbers set?
- What is the closure of a given numbers set?
- When is a set said closed or open?
- What is a sequence?
- What is the limit of a sequence?
- When a sequence is said monotone?
- What is a convergent sequence? What is a divergent sequence?
- What is a serie?
- What is a convergent serie? What is a divergent serie?
- What are the domain, codomain, supremum, infimum, maximum, minimum and diagram of a function?
- What is the limit of a function? When is a function said convergent for x tending towards x0?
- When is a function said divergent?
- Can you enunciate the main theorems about limits?
- What is a continuous function? What are the point of discontinuity of a function? How many types of them do exist?
- Can you enunciate the main theorems about continuous functions?
- When is a function said monotonic?
- What are odd, even and periodic functions?
- What is the comparison between infinitesimals? What is the comparison between infinities?
- What is the derivative of a function? Which is its meaning from the geometrical point of view?
- What are corners, cusps and vertical tangents?
- Which is the relationship between differentiability and continuity of a function?
- Can you enunciate the theorems related to the derivative of the sum of two functions, of the product of two functions, of the ratio of two functions and of the composed functions?
- What is the differential of a function and which is its mathematical meaning?
- What is the elasticity of a function?
- Can you enunciate and prove the Rolle, Lagrange and Cauchy theorems?
- Which is the geometrical interpretation of the Rolle and Lagrange theorems?
- Can you enunciate the de l'Hopital theorems?
- What are the Taylor and MacLaurin formula? What are they used for?
- When is a function said increasing in a point?
- Can you enunciate and prove the main theorems related to increasing functions in a point?
- What is a relative maximum point?
- Can you enunciate and prove the Fermat theorem? Which is its geometrical interpretation? Can the theorem be inverted?
- Can you enunciate the theorems giving sufficient conditions for the existence of relative maximum and minimum points?
- When is a function said convex? When is a function said concave?
- Can you enunciate the main theorems related to convex and concave functions?
- What is an inflection point?
- What is an asymptote? How many types of asymptotes do exist?
- What is a primitive of a function?
- Can you prove that the difference between two primitive of a function is a constant?
- What is the indefinite integral and which are its main properties?
- What is the integration by parts method?
- What is the definite integral and which is its geometrical interpretation? Which are the integrable functions?
- Which are the properties of the definite integral?
- Can you enunciate and prove the mean value theorem? Which is its geometrical interpretation?
- Can you enunciate and prove the Torricelli-Barrow theorem?
- Why is the Torricelli-Barrow theorem said "fundamental theorem of the integral calculus"? Which is the link between indefinite integral and definite integral?
- What is a function of several variables?
- When a function of several variables is continuous in a point?
- What is the partial derivative of a function of several variables?
- What is the gradient of a function of several variables?
- What is the Hessian matrix?
- What is the Lagrangian function? When is it used?

**VERSIONE IN ITALIANO**