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

Paula Alexandra da Costa Amaral

Hours

Weekly - 5

Total - 70

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, Escolar Editora, 2022
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

Attendance

Attendance is granted to any student who attends at least two-thirds of the practical classes for which they are registered. Absences are counted from the first day of classes, not from the first day the student registers for the practical session.

Students with worker-student status and, according to point 4 of article 6 of the Knowledge Assessment Regulations of the Faculty of Science and Technology of the New University of Lisbon, students who obtained attendance in the second semester of the 2023/24 academic year (Registered SUFICIENT) are exempt from obtaining attendance. However, it is recommended that students exempted from obtaining attendance attend classes with the same regularity as students not exempted from obtaining attendance.

Only students with attendance or those exempt from obtaining it will receive a final grade in the course.

Registration for In-Person Exams (Tests and Exams)

To optimize the resources of NOVA FCT (facilities, teaching staff, and non-teaching staff), only students properly registered through the course''s CLIP page may present themselves for any in-person exam. They must also bring a blank exam notebook, writing materials, and an official identification document with a recent photo.

Any exam for the course must be taken by the student themselves, without consulting any materials and without using any electronic calculation devices.

Continuous Assessment

The continuous assessment of the course is conducted through Theoretical-Practical Assessment, which includes two in-person tests, each lasting 1.5 hours.

Let T1 and T2 be the grades for each of the two tests, expressed on a scale of 0 to 20 points, rounded to the nearest tenth. 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 if this final grade is greater than or equal to 10 points. Otherwise, the student will have failed the course by continuous assessment.

Resit Period

Students who fail by continuous assessment may take the resit exam, which will last 3 hours.

The final grade for the student in the resit period will be obtained exclusively from the exam grade. The student will pass the course if this final grade is greater than or equal to 10 points. Otherwise, the student will have failed the course.

Grade Review

All students with a final grade of 18 points or higher (by continuous assessment or in the resit period) may, if they wish, take a grade review exam. Not taking this exam implies a final grade of 17 points for the course.

Grade Improvement

Students who pass the course may request a Grade Improvement in the resit period, according to the procedure described in Article 22 of the Knowledge Assessment Regulations of the Faculty of Science and Technology of the New University of Lisbon.

All students with a provisional grade of 18 points or higher may, if they wish, take a grade review exam. Not taking this exam implies a provisional grade of 17 points for the course.

The final grade for the course after the grade improvement will be the higher of the provisional grade and the current grade of the student.

Subject matter

1. Topology - Mathematical Induction - Sequences

Basic topology of the real numbers.

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: