Mathematical Analysis I

Recommended Prerequisites

N.A.

Teaching Methods

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.

Learning Outcomes

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.
Study of curves defined by parametric equations.
Polar coordinates and its relation with Cartesian coordinates.
Modeling and solving real problems that involve the new concepts.

Work Placement(s)

No

Syllabus

1- Real functions of a real variable
a) Limits, continuity, asymptotes
b) Derivatives, tangents and rates of change
c) Differentiation rules
d) Implicit differentiation
e) Hyperbolic functions
f) Linear approximations and differentials
2- Application of Differentiation
a) Maxima and Minima values
b) Rolle theorem, Fermat theorem, mean value theorem, extreme value theorem
c) Graphics of functions
d) Indeterminate forms and L'Hospital rule
e) Newton method
3- Integral of Riemann
a) Definition and geometric interpretation
b) Antiderivatives
c) The fundamental theorem of calculus
d) Indefinite integrals and the net change theorem
4- Integrals' applications: areas and volumes.
5- Integration rules (by parts, by substitution, by partial fractions). Improper integrals.
6- Differential equations of first order (separable equations and linear equations). Euler method.
7- Curves defined by parametric equations.
8- Polar coordinates

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Evaluation
Exam: 100.0%

Bibliography

STEWART, J. (2001). Cálculo. 4ª ed. São Paulo: Pioneira. Vol. 1 e Vol.2.
CARVALHO e SILVA, J. (1994). Princípos de Análise Matemática Aplicada. Lisboa: McGraw-Hill.
FERREIRA, Campos J. (1993). Introdução à Análise Matemática. Fund. Calouste Gulbkenkian.