# Mathematical Analysis I

## Objectives

Domain of the basic techniques required for the Mathematical Analysis of real functions of real variable.

The students should acquire not only calculus skills, mandatory to the acquisition of some of the knowledge lectured in Physics, Chemistry and other Engineering subjects, but also to develop methods of solid logic reasoning and analysis.

Being a first course in Mathematical Analysis, it introduces some of the concepts which will be deeply analyzed and generalized in subsequent courses.

## General characterization

##### Code

11504

##### Credits

6.0

##### Responsible teacher

Ana Luísa da Graça Batista Custódio, Paula Alexandra da Costa Amaral

##### Hours

Weekly - 5

Total - 60

##### Teaching language

Português

### Prerequisites

The student must master the mathematical knowledge lectured until the end of Portuguese high school teaching.

### Bibliography

**Recommended Bibliography**

- Ana Alves de Sá e Bento Louro, Cálculo Diferencial e Integral em R
- Jaime Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
- Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
- Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Inc., 1999
- Rod Haggarty, Fundamentals of Mathematical Analysis, Prentice Hall, 1993

### Teaching method

Theoretical classes consist in a theoretical exposition of subjects, illustrated by application examples.

Practical classes consist in the resolution of some application exercises for the methods and results lectured in the theoretical classes, as well as support for exercises solved by the students in autonomous work.

Any questions or doubts will be addressed during the classes, during the weekly sessions specially programmed to attend students or in individual sessions previously scheduled between professors and students.

### Evaluation method

**Frequency**

Frequency is granted to any student who attends at least 2/3 of the classes taught in the practical session in which they are enrolled. Absences are counted from the first day of classes, not from the first day the student enrolls in the practical session.

Students with the status of working students and, according to point 4 of Article 6 of the Assessment Regulations of the Faculty of Sciences and Technology of the Universidade Nova de Lisboa, students who obtained frequency in the second semester of the academic year 2022/23 are exempt from obtaining it. However, it is recommended that students exempt from obtaining frequency attend classes with the same regularity as students not exempt from obtaining it.

Only students with frequency will receive a final grade in the course unit.**Registration for In-Person Exams (Tests, Retakes, Exams)**

With the aim of optimizing the resources of FCT NOVA (facilities, teaching staff, and non-teaching staff), only students properly registered for a specific exam through the CLIP page of the course unit can participate in any in-person exam. They must also bring a blank exam booklet, writing materials, and an official identification document with a recent photograph.

Any exam for the course unit must be completed by the student themselves, without any reference materials or the use of any computing calculation devices.**Continuous Assessment**

The continuous assessment of the course unit is carried out using a Theoretical-Practical Assessment. The Theoretical-Practical assessment includes two in-person tests, each lasting 1 hour and 30 minutes.

Let T1 and T2 be the scores for each of the two tests on a scale of 0 to 20, all values rounded to the tenths place. A student will have a final grade of 0.5 T1 + 0.5 T2, rounded to the nearest whole number. The student will pass the course unit if this grade is equal to or greater than 10. Otherwise, the student will have failed the course unit due to continuous assessment.**Resit Period**

Students who have failed in the continuous assessment may take a resit exam, which will be held during the resit period and will last for 3 hours, covering all the material taught, and will be graded on a scale of 0 to 20, rounded to the nearest whole number. The student will pass the course unit if this grade is equal to or greater than 10. Otherwise, the student will have failed the course unit.**Grade Defense**

All students with a final grade equal to or greater than 18 (through continuous assessment or during the resit period) may, if they wish, participate in a grade defense exam. Not taking this exam will result in a final grade of 17 for the course unit.**Grade Improvement**

Students who have passed the course unit may request a Grade Improvement, following the procedure described in Article 22 of the Assessment Regulations of the Faculty of Sciences and Technology of the Universidade Nova de Lisboa, dated November 17, 2020. If they have obtained a pass grade for the course unit in the current semester or in the previous academic year, the student may take an improvement exam during the resit period. The exam will last for 3 hours, and the material assessed will correspond to all the material taught. The final grade will be the higher of the grade obtained previously and the grade obtained in the improvement exam, on a scale of 0-20 rounded to the whole numbers.

All students with a provisional grade equal to or greater than 18 may, if they wish, participate in a grade defense exam. Not taking this exam will result in a provisional grade of 17 for the course unit.

## Subject matter

**1. Topology - Mathematical Induction - Sequences**

Basic topology of the real numbers. Order relation.

Mathematical induction.

Generalities about sequences. Convergence of a sequence and properties for calculus of limits. Subsequences. Bolzano-Weierstrass theorem.

**2. Limits and Continuity**

Generalities about real functions of real variable. Convergence according to Cauchy and Heine. Calculus properties.

Continuity of a function at a given point. Properties of continuous functions. Bolzano theorem. Continuity and reciprocal bijections. Weierstrass theorem.

**3. Differenciability**

Generalities. Fundamental theorems: Rolle, Lagrange and Cauchy. Calculus techniques for limits. Taylor theorem and applications.

**4. Indefinite Integration**

Introduction. Indefinite integration by parts. Indefinite integration by substitution. Indefinite integration of rational functions.

**5. Riemann Integration**

Introduction. Fundamental theorems. Definite integration by parts and by substitution. Some applications.

Improper integration.

## Programs

Programs where the course is taught:

- Bachelor in Biomedical Engineering
- Bachelor in Civil Engineering
- Bachelor in Materials Engineering
- Bachelor in Micro and Nanotechnology Engineering
- Bachelor in Environmental Engineering
- Bachelor in Industrial Engineering and Management
- Bachelor in Electrical and Computer Engineering
- Bachelor in Engineering Physics
- Geological Engineering
- Bachelor in Computer Science and Engineering
- Bachelor in Mechanical Engineering
- Bachelor in Chemical and Biological Engineering
- Applied Mathematics to Risk Management