Sets; Real Numbers; Sequences and Series; Continuity; Derivatives; Antidifferentiation and Integration

Formulate and solve complex problems in diverse contexts, applying logical, abstract and quantitative reasoning.

Construct and analyse mathematical models applied to engineering, management, finance, economics, physics, biology, computer science, healthcare and other scientific fields.

Develop autonomy and critical thinking in the context of new technologies, methodologies and tools, demonstrating the ability to keep up to date, solve problems and adapt to diverse technological and professional contexts.

Act in an ethical and socially responsible manner in technological contexts, recognising the impact of solutions on society and promoting inclusive and sustainable practices.

Apply fundamental properties of sets and real numbers to solve problems in Mathematical Analysis.

Determine the convergence or divergence of numerical sequences and series, applying appropriate criteria and justifying the procedures.

Calculate limits of real functions of a real variable and analyse the continuity of functions at points and intervals.

Apply differential calculus to the study of real functions, including derivatives, monotonicity, extrema, concavity and simple optimisation problems.

Calculate antiderivatives, understand the Riemann integral and apply fundamental properties of the integration of real-valued functions of a real variable, including integration techniques and improper integrals in elementary cases.

Solve mathematical problems by presenting clear, rigorous and properly substantiated reasoning.

Apply concepts of Mathematical Analysis to simple modelling situations, critically interpreting the results and recognising the ethical and responsible role of the use of digital tools and AI.

1 Sets and Real Numbers

2 Sequences and Series

3 Limits and Continuity

4 Derivatives and Applications

5 Antiderivatives and Applications

Required Bibliography:

LIMA, E.L. Análise Real, Vol 1, funções de uma variável. Coleção Matemática Universitária, IMPA, 2014.

Further Reading:

SANTOS, João Paulo. Cálculo Numa Variável Real. Coleção Ensino da Ciência e da Tecnologia, n.º 49. IST Press.

DEMIDOVICH, B. et al. Problemas e Exercícios de Análise Matemática. McGraw Hill/Mir, Amadora/Moscovo, 1999.

LIMA, E.L. Curso de Análise - Volume 1. Projeto Euclides, IMPA, 15ª edição, 2019.

TAO, Terence, Analysis I, Texts and Readings in Mathematics Vol. 37, 3rd Edition, Springer, 2016.

E-learning.

Assessment

Assessment in this curricular unit has been designed to support student learning throughout the semester, valuing not only the results achieved but also the process of knowledge construction. Assessment activities aim to promote critical reflection, the practical application of content, active participation, and the development of skills relevant to students’ academic and professional pathways.

Where applicable, assessment may include different modes and instruments, such as individual and collaborative activities, written assignments, projects, presentations, participation in discussion forums, or assessment tests, in accordance with the Curricular Unit Plan.

The assessment criteria, planned activities, and their respective weighting will be presented at the beginning of the curricular unit, ensuring transparency and enabling students to organise their learning pathway in an informed manner.

To learn more about the principles and guidelines for assessment within the Universidade Aberta Pedagogical Model, please consult: https://portal.uab.pt/modelo-de-ensino/