Computer Science

Activity type

Formazione matematico-fisica

SSD

MATH-03/A

Year

1st year

Semester

2nd semester

CFU

6

Hours distribution

36 classroom hours, 24 training hours

Lecturers

FLAVIO D'ALESSANDRO
ANDREA TERRACINA

Activity type

Formazione matematico-fisica

SSD

MATH-03/A

Year

1st year

Semester

1st semester

CFU

6

Hours distribution

36 classroom hours, 24 training hours

Lecturers

VALERIANO AIELLO

Educational goals

The course aims to provide a basic knowledge of Calculus and Mathematical Analysis techniques, covering fundamental concepts such as functions, limits, derivatives, integrals, numerical and power series, Taylor expansions, and ordinary differential equations. Particular attention is devoted both to theoretical tools and to their main applications, such as the study of function graphs, maxima and minima problems, and the solution of elementary differential equations.

By the end of the course, the student will be able to compute limits of functions and sequences, apply differentiation rules to the local and global study of functions, use integration methods (integration by parts and substitution) to calculate definite and indefinite integrals, discuss the convergence of numerical series, power series, and Taylor expansions of elementary functions, as well as solve first-order ordinary differential equations (separable and linear) and second-order linear differential equations with constant coefficients, both homogeneous and non-homogeneous.


Knowledge and understanding

The student will acquire the basic notions and fundamental results of Calculus and Mathematical Analysis, learning essential techniques for computing derivatives, integrals, and limits, for studying the graphs of real functions of one variable, for analyzing the convergence of sequences and series, and for solving simple ordinary differential equations.


Applying knowledge and understanding

Lectures and exercises will enable the student to tackle and solve practical problems, progressively consolidating the acquired skills.


Making judgements

The student will develop the ability to interpret and analyze concrete situations that can be described mathematically, to use graphs as analytical tools, and to autonomously apply the acquired methods to new problems encountered during studies or in professional contexts.


Communication skills

The student will be able to express themselves correctly using mathematical language, understand scientific texts of moderate complexity, and summarize their main concepts.


Learning skills

The knowledge acquired will provide the foundation for further studies, both individually and within more advanced courses, fostering autonomy in exploring additional aspects of Mathematical Analysis.

Module I

General goals: to obtain a general knowledge of the basic techniques of Differential Calculus and of the standard applications to problems of maxima-minima of functions of a real variable and to the study of their graph.
Specific goals: at the end of the course the student will have acquired the notions and results of Calculus, with particular attention to the concept of function, limit of functions, and derivatives.
Knowledge and understanding: at the end of the course, students will master the basic notions of Differential Calculus, in particular the notions of function, limit and derivative.
Applying knowledge and understanding: students will be able to solve typical problems from Differential Calculus, such as the explicit calculation of derivatives, of maxima and minima of a function, and to plot an approximate graph of functions of a real variable.
Communication skills: students will be able to fully understand a scientific text of moderate complexity and relate to the essential ideas contained in it.
Learning ability: the notions and techniques learned will give students access to more advanced notions, either in a further course or in the form of self-study, concerning further aspects of Differential and Integral Calculus. Knowledge of the basic techniques of Differential Calculus and of the standard applications to problems of maxima-minima of functions of a real variable and to the study of their graph.

Module II


General goals: The purpose of the course is to provide the student with basic mathematical analysis tools such as : the theory of series, integral calculus and differential equations.
Specific goals: Students will be able to examine various techniques of integration and apply them to compute definite and indefinite integrals by using integration by substitution and the rule of integration by parts.Students will be able tocompute limits of sequences and discuss the convergence of number series, power seriesand Taylor series of elementary functions.
Knowledge and understanding: At the end of this course the students will be able to calculate integrals and the solutions of some first order differential equationsand the solutions of second order linear differential equations with constant coefficientsLearning activities consists of lectures and exercise sessions. The lecture aim to introduce fundamental concepts, to explain them by showing examples.
Applying knowledge and understanding: The exercise sessions aim to teach how to select and use calculation methods.
Critical and judgmental abilities: To be able to autonomously solve new problems, applying mathematical tools to phenomena or processes to be encountered in University studies or subsequent working activities.
Communication skills: To know how to communicate using properly mathematical language.
Learning ability: To be able to deepen autonomously some arguments introduced during the course.

Basic knowledge of mathematics

Module: MATHEMATICAL ANALYSIS I MODULE
Numerical series. Methods for establishing the character of a series. Power series. Radius of convergence, formula for successive derivatives. Examples of functions that can be written as power series.
Integration of functions of one real variabile. Fundamental theorem of integral calculus. Integration methods for indefinite and definite integrals.
Ordinary differential equations of first and second order (linear).


Module: MATHEMATICAL ANALYSIS II MODULE
- Real numbers, sequences of real numbers
- Limits of functions
- Continuous and differentiable functions
- Taylor polynomials

Module: MATHEMATICAL ANALYSIS I MODULE
Recommended textbook: C. Canuto, A. Tabacco, Analisi matematica 1, Pearson.
Lamberto Lamberti: Istituzioni di Matematica (the book will be made available online) (second module)
Lecture notes will be available on Moodle.


Module: MATHEMATICAL ANALYSIS II MODULE
Recommended textbook: C. Canuto, A. Tabacco, Analisi matematica 1, Pearson.
Lecture notes will be available on Moodle.

Module: MATHEMATICAL ANALYSIS I MODULE
C. Canuto, A. Tabacco, Analisi matematica 1, Pearson.
Lamberto Lamberti: Istituzioni di Matematica (the book will be made available online)


Module: MATHEMATICAL ANALYSIS II MODULE
N/D

Lessons in room

Not mandatory attendance (but strongly encouraged)

The evaluation exam consists of a written and an oral test. The latter may be optional depending on the result of the written test. The written tests can be replaced by two intermediate tests carried out in the middle and at the end of the course.

Examples of questions and exercises can be found on the e-learning pages of the two modules

Module: MATHEMATICAL ANALYSIS I MODULE

• The module is structured into sections on three main topics. One section covers numerical series, including power series expansions. This will take approximately 18 hours, including both theoretical and practical work.


• The second section  will be on integrals. This will introduce the concept of integrals in one variable and provide various methods for calculating a wide range of integrals. This will take approximately 24 hours and will include both theoretical and practical work.


• The final section will be on differential equations. It will primarily cover first- and second-order linear differential equations. This will take about 18 hours, and will cover both theory and practice in solving the exercises.

• Module Structure: The module is organized into three thematic sections, each lasting approximately 20 hours. One section is devoted to numerical sets (from natural numbers to real numbers), their properties, functions of one variable, and numerical sequences. Proof techniques by contradiction and by induction will also be taught.


• Module 2: The second module is devoted to the study of the limit of a function, the concept of continuity, and the fundamental results concerning this class of functions (Weierstrass Theorem and Intermediate Value Theorem).


• Module 3: The third module introduces the concept of the derivative of a function and presents the Taylor polynomial, a fundamental tool for approximating functions in a neighborhood of a point which is useful in the applications.