Mathematical Analysis I

Solving equations and inequations (secondary education)

Basic definitions of real-valued functions of a real variable: polynomial functions, exponential and logarithmic functions, trigonometric functions; derivatives (secondary education)

Classes are taught in a theoretical-practical regime.
The expository methodology is used for the presentation of the concepts of the curricular unit, supported by the realization of exercises to apply the transmitted concepts. Some of these exercises are based on real-life practical problems.

Goals:

Provide knowledge of the wide variety of problems that can be solved using mathematical concepts and results.

Foster logical/deductive reasoning and mental calculation.

Encourage the use of analytical methods in solving concrete problems through the application of acquired knowledge.

Skills:

Use some notions of Logic and Set Theory that support the algorithmic construction and research in databases.

Ability to apply mathematical knowledge of matrix calculus, namely in solving systems of linear equations.

Apply the knowledge of real functions of real variable in several problems such as function optimization, among others.

Chapter 1 – Logic

1.1 Propositions and logical operators

1.2 Propositional expressions and quantifiers

Chapter 2 – Sets

2.1 Generalities

2.2 Real number intervals

2.3 Operations with sets

2.4 Cartesian products, binary relations and functions

Chapter 3 – Matrix Calculus

3.1 Matrix definition. Generalities

3.2 Operations with matrices. Invertible matrices

3.3 Condensation of matrices. Gauss-Jordan elimination method

3.4 Systems of linear equations

3.5 Determinants

Chapter 4 – Real-valued functions of a real variable

4.1 Generalities

4.2 Composition of functions. Inverse function

4.3 Limits and Continuity

4.5 Exponential function and logarithmic function

4.6 Inverse trigonometric functions

4.7 Derivatives. Applications of derivatives

4.8 Study of functions

4.9 Taylor and Mac-Laurin polynomials

NAO

Fundamental bibliography

Neves, Cidália, Support texts and practical exercises available at NONIO, Author’s Edition, 2022.

Complementary bibliography

Amaro, M. F., Funções reais de variável real, ISCAC, 2008.

Carvalho, M., Cálculo Matricial, ISCAC, 2006.

Leite, J., Rocha, M. C., Larguinho, M., Conjuntos, ISCAC, 2007.

Martins, P. C., Neves, C., Leite, J., Carvalho, M. e Amaro, A., Matemática I, ISCAC, 2006.

Azenha, A., Jerónimo, M. A., Elementos de Cálculo Diferencial e Integral em R e Rn, McGraw Hill, 1995.

Ferreira, J. C., Elementos de Lógica Matemática e Teoria dos Conjuntos, IST, 2001.

Giraldes, E., Fernandes, V., Smith, M. P., Álgebra Linear e Geometria Analítica, McGraw Hill, 1995.

Pires, C., Cálculo para Economia e Gestão, Escolar Editora, 2011.

Swokowski, E., Cálculo com Geometria Analítica, 2ª ed., vol. 1, Makron Books, 1995.

Sydsæter, K., Hammond P., with Strøm, A., Essential mathematics for economic analysis, 4th ed., Pearson Education Limited, 2012.