Mathematical Analysis I

Recommended Prerequisites

Basic calculus.

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

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.

Work Placement(s)

No

Syllabus

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.

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Continuous evaluation
2 Midterm exams : 100.0%

Final evaluation
Exam: 100.0%

Bibliography

Base

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

Complementar

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.