# MATEMATICA GENERALE M - Z

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

**ANTONINO DAMIANO ROSSELLO**

## 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

Classroom is based on lectures. Definitions, theorems and relevant relations are introduced besides practical examples.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

1st PART

ELEMENTS OF MATHEMATICAL LOGIC: Statements (single). Connectives (compound statements). Open statements and quantifiers.

SET THEORY: Sets, subsets and their manipulation. Binary relations and functions. Real numbers and inequalities. Elementary trigonometry.

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

MATRICES AND DETERMINANTS: Definitions and classifications. Sum and product of matrices. Inverse and transpose matrices. 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.

2nd PART

ANALYTIC GEOMETRY: Cartesian coordinate system. Straight line equation.

NUMBER SYSTEMS: Lower bounds and upper bounds. Infimum and supremum of subsets of real numbers. Separate and contiguous sets. Elementary topology. Introduction to 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. Infinitesimal and infinite limits.

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. Nth 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 integrals and their geometric interpretation. Methods of integration.

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

## Textbook Information

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

A. Guerraggio | Matematica | Pearson | 2023 | 9788891931870 |

## Course Planning

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

1 | Maps/Functions: definition, injective, surjective and bijective applications, inverse function. | Guerraggio: Cap 1 and instructor's notes |

2 | Combinatorial calculus round 1: Dispositions, combinations, permutations and factorials. Binomial coefficient and its properties. | Guerraggio: Chap 4 and instructor's notes |

3 | Combinatorial calculus round 2: Dispositions, combinations and permutations with replacement and Binomial theorem. | Guerraggio: Chap 4 and instructor's notes |

4 | Matrices: definition, and classification (square, triangular and rectangular matrices; vectors; transpose of a matrix, diagonal; scalar, symmetric, submatrix). Sum of two matrices; product of a matrix by a scalar. | Guerraggio: Chaps 14, 15 and instructor's notes |

5 | Matrix calculus: product of two conformable matrices (Cayley); inverse matrix and its properties. Determinant of a matrix. | Guerraggio: Chaps 14, 15 and instructor's notes |

6 | More on Determinants: properties, Laplace Theorems, Binet Theorem, cofactor matrix, rank of a matrix and its properties, Pascal-Kronecker Theorem. | Guerraggio: Chaps 14, 15 and instructor's notes |

7 | Linear systems and their solutions: Method of Cramer; Rouché-Capelli Theorem. Homogeneous linear systems. | Guerraggio: Chaps 14, 15 and instructor's notes |

8 | Cartesian plane, distance between points in a plane. | Guerraggio: Chap 2 and instructor's notes |

9 | A glimpse of trigonometry: circumference and angles (radians and degrees); sine, cosine, tangent and cotangent, fundamental relations. | Guerraggio: Chap 2 and instructor's notes |

10 | Number systems: lower and upper bounds; infimum and supremum. Elementary topology of the real line: open ball; points of accumulation; interior points; boundary points; open and closed sets. | Guerraggio: Chap 3 and instructor's notes |

11 | Sequences and series: definitions, convergence and examples. | Guerraggio: Chaps 4, 5,11 and instructor's notes |

12 | Equation of a straight line. Intersecting straight lines; Parallel and perpendicular straight lines. Distance of a point from a straight line. | Guerraggio: Chaps 2, 3 and instructor's notes |

13 | Functions of a real variable: domain, codomain, graph of a function. | Guerraggio: Chaps 2, 3 and instructor's notes |

14 | Limits round 1: convergent and divergent function, fundamental theorems on limits | Guerraggio: Chap 5 |

15 | Limits round 2: properties of limits; indeterminate forms; special limits | Guerraggio: Chaps 5, 6 |

16 | Theorems on continuous functions (Bolzano, zero of a continuous function, Weierstrass). Points of discontinuity of the first, second and third type. | Guerraggio: Chap 6 |

17 | Monotone functions, inverse functions, composite functions, even and odd functions, periodic functions, infinitesimals and infinities and their comparison. | Guerraggio: Chaps 2, 3, 6, 8 |

18 | Derivative of a function: Difference quotient or Newton quotient and its geometric interpretation; definition of derivative, its geometric interpretation, Rules for calculating derivatives round 1. Points where there is no derivative. | Guerraggio: Chaps 7, 8 |

19 | Differentiable functions vs continuity functions. Differential. Rules for calculating derivatives round 2: composite and inverse functions. Nth derivatives. | Guerraggio: Chap 7 |

20 | Rolle Theorem, Cauchy Mean Value Theorem and Lagrange Mean Value Theorem. | Guerraggio: Chap 8 |

21 | De l'Hopital's theorem. Taylor's formula and Mac-Laurin's formula | Guerraggio: Chap 8 |

22 | Increasing and decreasing functions. Maxima and minima (local, global). Fermat's Theorem. Sufficient conditions for the existence of relative maxima and minima. Finding extremes of a function. | Guerraggio: Chap 8 |

23 | Convex functions and concave functions. Inflection points. Asymptotes. Elasticity of a function. | Guerraggio: Chaps 2, 8 |

24 | Indefinite integral: antiderivatives, definition and properties of indefinite integrals, immediate integrals. | Guerraggio: Chap 9 |

25 | Methods of indefinite integration: sum rule, integration by parts, partial fractions, integration by substitution. | Guerraggio: Chap 9 |

26 | Defined integral: Riemann’s definition, sufficient conditions for integrability, geometric interpretation. | Guerraggio: Chap 10 |

27 | Mean value theorem for definite integrals; Fundamental Theorem of Calculus (Torricelli-Barrow) | Guerraggio: Chap 10 |

28 | Functions of several variables: definition, continuity, partial derivative, gradient, free and constrained maxima and minima, Lagrange multipliers, Hessian. | Guerraggio: Chaps 13, 16 |

## Learning Assessment

### Learning Assessment Procedures

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

- What are the main set operations (union, intersection, Cartesian product, set difference, complementation) and which are their properties?
- What is a power set and what is its cardinality?
- Suppose you take the union of two finite sets: what is its cardinality (in terms of their intersection)?
- What is a function?
- What is a bijective function?
- What is an inverse function?
- What is a relation? What is a binary relation?
- What is a partition of a nonempty set?
- What is an equivalence relation?
- What is a partial (total)?
- What are dispositions, combinations and permutations (with or without replacement)?
- What is the binomial coefficient and which are its properties? (viz. symmetry, Stifel formula)
- What is the binomial theorem?
- What are matrices and their main operations?
- What the inverse of a given square matrix? Which are its properties?
- What is the determinant of a square matrix and which properties it satisfies?
- Tell the statement of the two theorems of Laplace.
- Tell the statement of the theorem of Binet
- What is the minor of a given matrix. What is the cofactor?
- What is an adjoint matrix and what are its properties?
- What is the rank of a matrix and what are its properties?
- Can you state the Kronecker-Pascal Theorem?
- When a linear system is possible, impossible, determined or undetermined?
- Can you state and prove the Cramer theorem?
- Can you state the Rouchè-Capelli theorem?
- Can you write the equation of straight line and can you sketch the parallelism and perpendicularity conditions between two straight lines?
- What are the supremum and the infimum, the minimum and the maximum of a given set of real numbers?
- When two numbers sets are separated and when are they contiguous?
- Give the definition of: interior point, boundary and accumulation points of a given set of real numbers.
- Explain what is an open and a closed subset of the real line. What its closure?
- What is a sequence of real numbers?
- What is the limit of a sequence of real numbers?
- Explain what is a monotone sequence of real numbers.
- What is a convergent (divergent) sequence of real numbers?
- What is a series of real numbers? When it is convergent?
- Define the following: domain, codomain, supremum, infimum, maximum, minimum and graph of a function of a real variable.
- What is the limit of a function? What is a convergent (divergent) function?
- Can you state the main theorems about limits of a function?
- What is a continuous function? What are the discontinuity point of a function?
- State the main theorems about continuous functions.
- What is a monotone function?
- What are odd, even and periodic functions?
- Explain how to compare infinite (infinitesimal) functions.
- What is the derivative of a function? What is its geometric interpretation?
- List the main types of points where a function does not have a derivative (e.g. tangent points)
- What is the relationship between differentiability and continuity of a function?
- List the main theorems about arithmetic operations on derivatives: sum rule, product rule, ratio of rule. What is the derivative of a composite function? What is the derivative of an inverse function?
- What is the differential of a function?
- What is the elasticity of a function?
- State and prove the following theorems: Rolle, Lagrange and Cauchy.
- What is the geometric interpretation of the Rolle and Lagrange theorems?
- Can you state the de l'Hopital theorems?
- What are the Taylor and MacLaurin series expansion of a function?
- When is a function said increasing at a point? What is a monotone function?
- Can you state and prove the main theorems about monotone functions?
- What is a local maximum point?
- Can you state and prove the Fermat theorem? What is its geometrical interpretation?
- State the theorems giving sufficient conditions for the existence of local maximum and minimum points.
- When is a function said to be convex (concave)?
- Can you state the main theorems about 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?
- Prove (or disprove) that the difference between two antiderivatives of a function is constant.
- What is an indefinite integral and what are its main properties?
- What is the integration by parts method?
- What is a definite integral (in the Riemann sense) and what is its geometrical interpretation? Whiat are the integrable functions?
- What are the properties of a definite integral?
- Can you state and prove the mean value theorem for definite integrals, together with its geometrical interpretation?
- Can you state and prove the Fundamental Theorem of Calculus (Torricelli-Barrow)?
- What is a function of several real variables?
- When a function of several variables is continuous at a point?
- What is the partial derivative of a function of several variables?
- What is the gradient of a function of several variables?
- What are the Lagrange multipliers? When are they used?

**VERSIONE IN ITALIANO**