Mathematical Analysis II

Basic definitions of real-valued functions of a real variable: polynomial functions, exponential and logarithmic functions, trigonometric functions; calculation of derivatives (high school and Mathematical Analysis I program).

Applications of real-valued functions of a real variable (Mathematical Analysis I).

Calculation of determinants (Mathematical Analysis I).

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 mathematical knowledge of integration of functions, namely in its application in the calculation of areas of plane regions and in the resolution of differential equations.

Apply the study of real functions of two real variables in several problems, of which the optimization of functions stands out.

Chapter 1 – Primitivation

    1.1 Immediate primitives

    1.2 Primitivation by parts

    1.3 Primitivation of rational functions

    1.4 Primitivation by substitution

Chapter 2 – Differential equations

    2.1 Definitions

    2.2 Separable differential equations

    2.3 First order linear differential equations

Chapter 3 – Integral calculus

    3.1 Definite integral

    3.2 Application of the definite integral to the calculation of areas of plane regions

    3.3 Improper integrals

Chapter 4 – Real-valued function of two real variables

    4.1 Definitions

    4.2 Limits

    4.3 Continuity

    4.4 Partial derivatives

    4.5 Differentials. Calculation of approximate values    

    4.6 Free extrema of two variable functions

    4.7 Conditioned extremes. Application of the Lagrange multipliers method

NAO

Fundamental bibliography

Neves, Cidália, Support texts and pratical exercices available at NONIO, Author’s Edition.

Complementary bibliography

Amaro, A., Carvalho, M., Equações Diferenciais, ISCAC, 2007.
Azenha, A., Jerónimo, M. A., Elementos de Cálculo Diferencial e Integral em R e Rn, McGraw Hill, 1995.
Breda, A., Costa, J., Cálculo com funções de várias variáveis, McGraw-Hill, 1996.
Carvalho, M., Leite, J. Funções reais de duas variáveis reais, ISCAC, 2006.
Neves, C., Cálculo Integral, ISCAC, 2006.                                                                                                                                                                                                      Pires, C., Cálculo para Economia e Gestão, Escolar Editora, 2011.                                                                                                                                                     Saraiva ,M. A., Silva, M. A., Primitivação, Edições Asa, Rio Tinto, 1ª ed., 1990.                                                                                                                           Swokowski, E., Cálculo com Geometria Analítica, 2ª ed., vol. I,II, Makron Books, 1995.                                                                                                             Sydsæter, K., Hammond P., with Strøm, A., Essential mathematics for economic analysis, 4th ed., Pearson Education Limited, 2012.