Mathematical Analysis II A


Obtain a global view of the presented matter. Proficiency over each subject that allows for  solving integration problems in a variable, as well as problems of infinite series. Relate the subjects presented with other subjects given in related disciplines. Get an idea of ​​concrete applications of these topics.

General characterization





Responsible teacher

Rogério Ferreira Martins


Weekly - 6

Total - 88

Teaching language



Familiarity with sequences, continuity and differentiability of an introductory course of mathematical analysis.


Anton, H., Bivens, I., Davis, S., Calculus, John Wiley & Sons, 10ª edition, 2012.

Apostol, T, Calculus, Vol. 1, John Wiley & Sons, 1967.

Campos Ferreira, J.,  Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 10ª edição, 2011.

Demidovitch, B., Problemas e Exercícios de Análise Matemática, Mir, Moscovo, 6ª edição, 1987.

Elon Lages Lima, Análise Real, Volume 1, IMPA, 10ª edição, 2009.

M. Figueira, Fundamentos de Análise Infinitesimal, Textos de matemática, Dep. de matemática da FCUL, 1996.

Mattuck, A. Introduction to Analysis, Prentice Hall, 1999.

Santos J.P., Cálculo Numa Variável Real, IST Press, Lisboa, 2012.

Sarrico, C., Análise Matemática, Gradiva, 8ª edição, 2011.

G. Simmons, Calculus with Analytic Geometry, 2nd edition, McGraw Hill, 1996.

G. Strang, Calculus I, OpenStax, 2018.

Strogatz, S., Infinite Powers - How Calculus Reveals the Secrets of the Universe, Houghton Mifflin Harcourt, 2019.

Teaching method

Half the class time will be dedicated to the presentation of the subjects, along with illustrative examples. The other half will be dedicated to solving and analysing problems and exercises. Students will be required to prepare exercises in advance that will be presented on the board to the class, with subsequent group discussion.

Evaluation method

The Continuous Assessment of the course is composed of:

Theoretical-Practical Assessment: two written tests, each lasting one and a half hours, to be carried out during the semester. Each test will be classified between 0 and 8 points

Summative Assessment: homework assignment of solved exercises. In each class, two exercises will be chosen that students must hand out. In each class, out of all the students who handed out the exercises, two students will be chosen at random to present it to the class. The final grade of this component, between 0 and 4, will be assigned by the teacher based on these resolutions. In this evaluation component, more than presenting a correct resolution, the student''''s work will be valued.

Frequency: to obtain frequency, a student must hand solved exercises  in more than 10 classes.

A student who fulfills the frequency criterion of the previous paragraph, will have a final classification by continuous Assessment equal to T1 + T2 + AS, rounded to the nearest integer. Where T1 and T2 are the final grades for the first and second tests, respectively and AS is the summative assessment grade. The student will be approved, by continuous assessment, if this grade is greater than or equal to 10 points.

Appeal Exam:

Students who have not obtained approval by Continuous Assessment and who have obtained Frequency, can present themselves to an appeal exam. This is a written exam, lasting 3 hours, which evaluates the totality of the contents taught in the course. The exam is divided into two parts, each classified from 0 to 8 points, whose assessed material corresponds, respectively, to the first and second tests. The final grade will be T1 + T2 + AS, where T1 and T2 are the final grades of the first and second part. The student is approved if this grade is greater than or equal to 10 points.

Alternatively, at this stage, the student may take only one part of the exam, the duration of which will in this case be equal to that of the tests (1h and 30m). In this case, the final grade will be calculated according to the Continuous Assessment rules, replacing the classification of the repeated test with the new classification. The student must choose what type of assessment he intends to carry out before seeing the exam.

Grade Improvement:

An improvement of grade, my be requested by students approved in the curricular unit, upon the fulfilment of all the rules imposed by FCT NOVA, taking the Appeal Exam in full.

The Regulamento de Avaliação de Conhecimentos da Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, revised on January 16, 2018, applies in any omitted point.

Subject matter

1. Primitives. Primitivivation by parts and substitution. Primitives of rational, irrational and transcendent functions.

2. Integral. Fundamental theorem. Mean value theorem. Barrow formula. Areas of planar regions.

3. Improper integrals. Convergence criteria. Area of unbounded domains.

4. Series of real numbers. Series with non-negative terms. Alternating series. Absolute convergence. Product of series.

5. Series of functions. Pontual and uniform convergence. Power series. Taylor series.


Programs where the course is taught: