Mathematical Analysis I - F+EF+EB

Ano letivo: 2019-2020
Specification sheet

Specific details
course codecycle os studiesacademic semestercredits ECTSteaching language

Learning goals
Polar coordinates and its relation with cartesian coordinates. Study of curves defined by parametric equations.

Limits and derivatives of real functions.

Definition and understanding of the Riemann integral concept. Computation of integrals (by using different rules) and applications (areas between curves and volumes).

Solving differential equations.

Modeling and solving real problems that involve the new concepts.
1. Real functions of a real variable
1.1 Curves defined by parametric equations.
1.2 Polar coordinates
1.3 Hyperbolic functions

2. Differential calculus
2.1 Limits, continuity, asymptotes
2.2 Derivatives, tangents and rates of change
2.3 Implicit differentiation
2.4 Weirstrass theorem of extreme value, Fermat theorem, Rolle theorem, Lagrange mean value theorem
2.5 Maxima and minima values
2.6 Indeterminate forms and L'Hospital rule
2.7 Linear approximations and differentials
2.7 Newton method

3. Integral calculus
3.1 Antiderivatives
3.2 Integration rules: by parts, by substitution, by partial fractions
3.3 Definition and geometric interpretation of integral of Riemann
3.4 The fundamental theorem of calculus
3.5 Total variation theorem
3.6 Improper integrals
3.7 Integrals' applications

4. Differential equations of first order
4.1 Separable equations
4.2 Linear equations
4.3 Euler's method
Basic calculus
Generic skills to reach
. Competence in analysis and synthesis;
. Competence to solve problems;
. Critical thinking;
. Competence in autonomous learning;
. Competence in organization and planning;
. Competence in oral and written communication;
. Competence in applying theoretical knowledge in practice;
. Self-criticism and self-evaluation;
(by decreasing order of importance)
Teaching hours per semester
theory-practical classes45
total of teaching hours90

Sseminar or study visit---- %
Laboratory or field work---- %
Problem solving---- %
Synthesis work thesis---- %
Project---- %
Research work---- %
Mini tests---- %
Assessment Tests2 Frequências | 2 Midterm exams (100) %
Exam1 Exame | 1 Exam (100) %
Other---- %
---- %
---- %
assessment implementation in 20192020
Assessment Problem solving ? 0% to 50%. Mini-tests ? 0% to 50%. Midterm test ? 0% to 100%. Exam ? 0% to 100%.: 100.0%

Bibliography of reference

1. A. Araújo, Análise Matemática I, Notas de Curso, Coimbra, 2014.
2. J. Stewart, Cálculo, vol. I e II, Thomson Learning, 2001.


1. Elon Lages Lima, Curso de Análise, vol. 1 (11a edição), Projecto Euclides, IMPA, 2004.
2. J. Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, Lisboa, 1993.
3. H. Simmons, Cálculo com Geometria Analítica, McGraw-Hill do Brasil, São Paulo, 1987.
4. E.W. Swokowski, Cálculo com Geometria Analítica, McGraw-Hill do Brasil, São Paulo, 1987.
Teaching method
There are theoretical and theoretical-practical classes.

The theoretical classes are mainly expository, where each concept is introduced, if possible, in different ways (geometrically, numerically or algebraically). To facilitate the understanding of the concepts, many application examples are also described.

The theoretical-practical classes consist in exercises to be solved under the guidance of the professor. The students are also encouraged to solve problems autonomously.
Resources used