Infinitesimal Analysis I

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

 12th year course of mathematics- Secondary School.

Teaching Methods

 TP classes are expositive where a rigorous treatment of calculus is presented with concepts and proofs. We begin with detail and accuracy, accompanied by illustrative examples, applications and adequate historical notes. In PL classes, some exercises are presented, ranging from routine to very difficult. Active participation of students in class discussions, and individual and team work is encouraged. Extra projects are posted in the webpage of the course, in order to stimulate and promote the best students. This homework will be discussed with the instructor during office hours.

Learning Outcomes

The program is designed in order that the studens can fulfill gaps in their basic calculus background from high school, right at the beginning of the semester.  We  progress on a steady basis into new concepts and results in real analysis, giving them enough background to understand and be successful in more advanced courses. At the end of the semester, the student is expected to have acquired knowledge about real-valued functions, both theory and techniques, but also to be able to use correctly a formal language and a high level of rigor to communicate mathematics at the level of what is taught in this uc.

Work Placement(s)



Foundations of mathematics – Propositional and predicate calculus. Methods and techniques of proof. Sets, relations, order, functions and cardinality.

The Real Line – The system of real numbers as a complete ordered field. The topology of the real line. Sequences of real numbers.

Differential Calculus (of real functions of one variable) – Functions and limits. Continuity. Differentiable functions. Indeterminate forms. Implicit differentiation. Taylor polynomials and Taylor´s Theorem, applications.

Head Lecturer(s)

Amílcar José Pinto Lopes Branquinho

Assessment Methods

Exam(100%) or Midterm exame(60%)+ Test(20%)+Team work(5%)+Assiduity and participation/behaviour in class 5%)+Problem resolving report (10%): 100.0%


E. Lages Lima, Curso de Análise, Vol. 1, 7ªed., IMPA, 1992.


M. T. Martins, Tópicos Fundamentais da Matemática, ed. Dep. Matemática, Série A, nº3, 1999.


M. Spivak, Calculus, 3th ed, Cambridge Univ. Press, 2008.


J. Stewart, Calculus, vols I e II, 4th ed., Thomson, 2001