# Mathematical Analysis I

Module DIDATTICA FRONTALE

**Academic Year 2022/2023**- Teacher:

**Pietro ZAMBONI**

## Expected Learning Outcomes

The aim of the course of Analisi Matematica I is to give the basic skills real and complex numbers,

differential and integral calculus for real functions of one real variable.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

**Knowledge and understanding**: The student will learn some basic concepts of Mathematical

Analysis and will develop both computing ability and the capacity of manipulating some common

mathematical structures, as complex numbers, limits, derivatives and integrals for real functions of

one real variable, numerical series.The student will be able to apply the acquired**Applying knowledge and understanding**:

knowledge in the basic processes of mathematical modeling of classical problems arising from

Physics.**Making judgements**: The student will be stimulated to autonomously deepen his/her knowledge

and to carry out exercises on the topics covered by the course. Constructive discussion between

students and constant discussion with the teacher will be strongly recommended so that the

student will be able to critically monitor his/her own learning process.The frequency of the lessons and the reading of the recommended books**Communication skills**:

will help the student to be familiar with the rigor of the mathematical language. Through constant

interaction with the teacher, the student will learn to communicate the acquired knowledge with

rigor and clarity, both in oral and written form. At the end of the course the student will have

learned that mathematical language is useful for communicating clearly in the scientific field.The student will be guided in the process of perfecting his/her study method. Inparticular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.**Learning skills**:

## Course Structure

The concepts and methods covered by the course will be presented through lectures. For each topic the teacher will carry out an adequate number of exercises. To develop independent judgment and communication skills, and to make participation in lessons more active and fruitful, guided exercises will be held in a few hours, in which various exercises will be proposed. Students will be able to work individually or in groups and confront each other.

## Required Prerequisites

Ability to argue and communicate, orally and in writing. Knowing how to identify, describe and work with sets. Recognize hypotheses and theses of a theorem. Recognize whether a condition is necessary or sufficient. Knowing how to deny a proposition and understand an absurd reasoning. Understanding the difference between examples and counterexamples. Know the numerical sets and, in particular, the algebraic and ordering properties of real numbers. Know the definition, the graph and the main properties of the power, exponential, logarithmic and trigonometric functions. Knowing how to apply the algebraic and monotonic properties of the fundamental functions for the solution of simple irrational, exponential, logarithmic and trigonometric equations and inequalities. Know the equations or inequalities of simple geometric places (straight line, half plane, circumference, circle, ellipse, hyperbola, parabola). Know the main trigonometric formulas.## Attendance of Lessons

Attendance of lessons is required## Detailed Course Content

**Sets of numbers**. Real numbers. The ordering of real numbers. Completeness of R. Factorials and

binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian

coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.-
Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and**Limits**.

sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and

exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a

sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences.

Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their

applications. **Continuity**. Continuous functions. Sequential characterization of the continuity. Points of

discontinuity. Discontinuities for monotone functions. Properties of continuous functions

(Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the

inverse functions.**Differential Calculus**. The derivative. Derivatives of the elementary functions. Rules of

differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle,

Lagrange and Cauchy. Consequences of Lagrange's Theorem. De L'Hôpital Rule. Monotone

functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function.

Recurrences.Areas and distances. The definite integral. The Fundamental Theorem of Calculus.**Integrals**.

Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts.

Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial

fractions. Strategy for integration. Impropers integrals. Applications of integration.**Numerical series**. Round-up on sequences. Numerical series. Series with positive terms.

Alternating series. The algebra of series. Absolute and Conditional Convergence. The Integral Test

and Estimates of Sums.

## Textbook Information

- Di Fazio G., Zamboni P., Analisi Matematica 1, Monduzzi Editoriale.
- Di Fazio G., Zamboni P., Eserciziari per l'Ingegneria, Analisi Matematica 1, EdiSES.
- D'Apice C., Manzo R. Verso l'esame di Matematica, vol. 1 e 2, Maggioli editore.
- Caponetto T., Catania G., Esercizi di Analisi Matematica I, vol 1 e 2, CULC

## Course Planning

Subjects | Text References | |

1 | Sistemi numerici. | Testo 1 cap. 2, Testo 2 cap. 1, Testo 3 vol. 1, cap. 1 e 2. |

2 | Limiti delle funzioni reali di una variabile reale. | Testo 1 cap. 3, Testo 2 cap. 2, Testo 3 vol. 1, cap. 4. |

3 | Calcolo differenziale. | Testo 1 cap. 5, Testo 2 cap. 3, Testo 3 vol. 1, cap. 5 e 6. |

4 | Integrazione secondo Riemann. | Testo 1 cap. 7, Testo 2 cap. 5, Testo 3 vol. 2, cap. 1 e 2. |

5 | Serie numeriche. | Testo 1: Cap. 6 e 7. Testo 2: Cap. 4. Testo 4: Cap. 3. |

## Learning Assessment Procedures

The final exam consists of a written test and a subsequent oral test. The written test consists of four exercises. Candidates who have obtained a score greater than or equal to 18/30 in the written test will access the oral test.

During the suspension of the lessons of the first semester (February 2023), an ongoing test will take place, which focuses on the following topics:

Set theory - Numeric sets - Real functions - Limits of functions and sequences - Continuous functions - Derivation

The ongoing test consists of a written test consisting of two parts:

A) theoretical questions

B) technical exercises

Candidates who score greater than or equal to 18/30 in each of parts A and B pass the ongoing test.

Students who pass the ongoing test access the end-of-course test, which will take place in the first exam session (June-July 2023) and will focus on the remaining part of the program.

The end-of-course test consists of a written test and an oral interview. The written test consists of 4 exercises. Candidates who have obtained a score greater than or equal to 18/30 in the written part will access the oral examination.

In the first semester, three Verifications will take place on the following topics:

Set theory - numerical sets (first verification)

Real functions - Limits of functions and sequences (second verification)

Continuous functions - Derivation (third verification)

Each verification consists of a written test consisting of two parts:

A) theoretical questions

B) technical exercises

The candidates who obtain a score greater than or equal to 18/30 in each of the parts A and B pass the verification.

Students who pass the three verifications are admitted to the end-of-course test.

Students who pass 2 of the 3 verifications can retrieve the failed verification on the same date on which the ongoing test will take place.

## Examples of frequently asked questions and / or exercises

Uniqueness Theorem, Monotonic functions, Theorem of existence of zeros, Weierstrass theorem, Derivability implies continuity, Fermat's theorem, Characterization of increasing functions, Root and ratio theorems, Leibnitz theorem, Riemann integrability condition, Integrability of continuous and monotonic functions, Integrable functions in an improper and generalized sense.

The student will be able to find examples of exam exercises on Studium.