# ANALISI MATEMATICA 1 PARTE A

**Academic Year 2023/2024**- Teacher:

**Maria Alessandra RAGUSA**

## Expected Learning Outcomes

At the end of the course the student will acquire both theoretical and practical knowledge on the main contents of the course.
1. Knowledge and understanding - Knowledge and understaning: The student will be able to understand and assimilate the definitions and main results of the basic mathematical analysis, for real functions of a real variable.

2. Ability to apply knowledge and understanding - Applying Knowledge and understaning: The student will be able to acquire an appropriate level of autonomy in theoretical knowledge and in the use of basic analytical tools.

3. Making judgments - Making judgments: Ability to reflect and calculate. Ability to apply the notions learned to solving problems and exercises.

4. Communication skills - Communication skills: Ability to communicate the knowledge acquired through an adequate scientific language.

5. Learning skills - Learning skills: Ability to deepen and develop the knowledge acquired. Ability to critically use tables and analytical and computer tools of symbolic computation.

## Course Structure

Direct Instruction.

The lessons are integrated with exercises related to the topics covered by the course and will take place in the classroom. It should also be noted that there are 49 hours of lessons (typically, these are theory) and 24 hours of other activities (typically, these are exercises).

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

NOTE: **Information for students with disabilities and/or SLD **

To guarantee equal opportunities and in compliance with the laws in
force, interested students can ask for a personal interview in order to
plan any compensatory and / or dispensatory measures, based on the
didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco).

## Required Prerequisites

The student must have a thorough knowledge of the notions of Mathematics studied in the five years of high school. In particular: Elements of Mathematical Logic, set theory, algebraic equations and inequalities, trigonometry.

## Attendance of Lessons

## Detailed Course Content

1. Sets and Logic. Basic concepts on sets, elementary logic.

2. The numbers. Natural, relative, rational and real numbers. Continuity axiom of real numbers. Lower and upper bounds of a numerical set. Absolute value and its properties. Radicals, powers, logarithms. Principle of induction. Complex numbers.

3. Functions of one real variable. Function concept. Bounded, symmetric, monotone, periodic functions. Elementary functions. Compound functions and inverse functions.

4. Limits and continuity. Numerical sequences. Definition of limit. Fundamental theroemes on limits. Calculation of limits. The number of Napier. Comparisons and asymptotic estimates. Limits of functions, continuity, asymptotes. Fundamental theorems on limits of functions. Calculation of limits. Notable limits. Comparisons and asymptotic estimates. Graph of a function. Fundamental properties of continuous functions.

5. Sequences and numerical series. Definition of succession. Limits of sequences. Extracted sequences. Definition of series. Examples of numerical series. Fundamental theorems on series. Series with non-negative terms. Series with terms of variable sign. Notable numerical series.

## Textbook Information

[1] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 2015.

## Course Planning

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

1 | Sets and Logic. | [1] Chapter 1. |

2 | The numbers. | [1] Chapter 1. |

3 | Functions of one real variable. | [1] Chapter 4. |

4 | Limits and continuity. | [1] Chapter 4. |

5 | Sequences and numerical series. | [1] Chapter 3. |

## Learning Assessment

### Learning Assessment Procedures

The final exam consists of a written test and an interview, both rated out of thirty. Passed the written test lo student must undergo an interview that contributes to the formulation of the final grade, expressed in thirty. The registration of the exam will take place only after passing the interview.

NOTE: The learning assessment can also be carried out electronically, should the conditions require it.

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

All the topics mentioned in the program can be requested during the exam.

The frequency of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of the questions that may be proposed during the exam.
An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the concepts exposed (definitions, propositions, theorems, corollaries).

The main types of exercises are as follows:

- Search for the extrema of a numerical set.
- Exercises on complex numbers (algebraic manipulations, writing complex numbers in algebraic, trigonometric and exponential form).
- Calculation of limits of sequences and functi
- ons. Study of the character of a numerical series.
- Study of the continuity of real functions of a real variable.