Faculdade de Ciências e Tecnologia

Mathematical Analysis I



Academic unit

Faculdade de Ciências e Tecnologia


Departamento de Matemática



Teacher in charge

José Maria Nunes de Almeida Gonçalves Gomes, Paula Alexandra da Costa Amaral

Weekly hours


Total hours


Teaching language



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

The students should acquire not only calculus capabilities fundamental 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.


The student must master the mathematical knowledge lectured until the end of Portuguese High School.

Subject matter

  1. Basic topology of the real numbers.
    1.1 Neighborhood of a Point. Interior, Exterior, Frontier, Isolated, Adherent and and Accumulation Point.
    1.2 Open, Closed, Limited and Compact Set.

    2. Mathematical induction and sequences
    2.1 Mathematical Induction.
    2.2 Limit of a sequence. Algebra of  Limits. Subsequences and Sublimits. Theorem of Squeeze Sequences. Bolzano-Weierstrass theorem and other fundamental theorems. Cauchy''s sequences.

    3. Limits and Continuity in R
    3.1 Convergence  according to Cauchy and according to Heine. Algebra of  Limits.
    3.2 Continuity of a Function in a Point and in a Set. Prolonging a Function by Continuity. Bolzano''s theorem and Weierstrass''s theorem. Continuity of the Composite Function and Continuity of the Inverse Function. Inverse Trignometric Functions.

    4. Differential Calculus in R
    4.1 Derivative Definition. Physical and Geometric Interpretation. Differentiability. Algebra of Derivatives. Derivative of the Composite and Derivative of the Inverse Function. Derivatives of Inverse Trignometric Functions. Rolle''s Theorem, Lagrange''s Theorem. Derivative and Monotony. Darboux''s Theorem and Cauchy''s Theorem. Indeterminations and Cauchy’s Rule.
    4.2 Taylor''s theorem and applications to the study of extremes and concavities.

5. Integration in R

5.1 Primitives. Primitives by Parts. Primitives by Substitution. Primitives of Rational Functions. Primitives of Irrational Functions and Transcendent Functions.
5.2 Integral of Riemann. Theorem of the Mean Value. Fundamental Theorem of Integral Calculus. Barrow Rule. Integration by Parts and integration by Substitution. Application to the Calculus of Areas.


Adopted text

  1. Ana Alves de Sá e Bento Louro, Cálculo Diferencial e Integral em

Recommended Bibliography

  1. Alves de Sá, A. e Louro, B. - Cálculo Diferencial e Integral em , Exercícios Resolvidos, Vol. 1, 2, 3
  2. Anton, H. - Cálculo, um novo horizonte, 6ª ed., Bookman, 1999
  3. Campos Ferreira, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
  4. Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
  5. Larson, R.; Hostetler, R.; Edwards, B. - Calculus with Analytic Geometry, 5ª ed., Heath, 1994
  6. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996

Teaching method

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

Practical classes consist in the resolution of  application exercises for the methods and results presented in the theoretical classes.

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


In order to be evaluated, the student must attend all problem solving sessions (up to a maximum of 3 unjustified absences).


Evaluation Methods:

1-Continuous evaluation

The continuos evaluation consists on three tests during the semester. One of the tests may be improved in the final examination date. The final grade is the average of the grades of the three tests. The student is aproved if the final grade is greater or equal than 9,5.

Each test lasts for 1h 30min. 

2-Final exam evaluation.

The student is aproved if the grade of the final exam is greater or equal than 9,5.

The final exam lasts for 3h.


Calculatory devices are not authorized in tests or exams.