ISLA Santarém 1

Mathematics I

• ApresentaçãoPresentation

  This curricular unit presents the basic concepts and techniques of Mathematical Analysis.
• ProgramaProgramme

  1. Real functions of a real variable. 2. Limits and Continuity: Notion of limit. Limit definition. Lateral limits. Limit calculation theorems. Indeterminacy in the calculation of limits. Definition of continuity. 3. Differential Calculus: Definition of derivative. Geometric interpretation of derivative definition. Differentiability and Continuity. Derivation rules. Derived from the implicit function. Successive derivatives. Cauchy's rule and L'Hôpital's rule. Indeterminacy in the calculation of limits. Applications of derivatives. 4. Integral Calculus: Primitives. Integration rules. Integration by parts. Integration by substitution. Integration of rational functions. Definite integral. Fundamental theorem of calculus. Applications of integral calculus: areas and volumes. Improper integrals. 5. Numerical Series: Definitions and Generalities. Successions.Series and convergence. Integral criterion. Comparison of series. Alternate series. Quotient and root criteria. Power series.
• ObjectivosObjectives

  O1. Provide students with the knowledge to use algebraic expressions. O2. Solve equations and inequalities and apply this knowledge to real practical problems. O3. Use real variable functions and their properties to model real situations and solve problems. Competences: C1. Master and apply the techniques of derivation and integration in IR. C2. Model and solve optimisation problems for differentiable functions. C3. Calculate the sum of geometric and reducible series and develop some functions in power series. C4. Apply integral calculus in IR to solve problems.
• BibliografiaBibliography

  Brokate, M., Manchanda, P., & Siddiqi, A. H. (2019). Calculus for Scientists and Engineers. Springer Singapore. Carneiro, C. E., Prado, C. P., & Salinas, S. R. (2020). Introdução elementar às técnicas do cálculo diferencial e integral. Instituto de Física. 3a edição. São Paulo (Brasil). Macedo, M. (2022).  Funções Reais de Várias Variáveis Reais. Universidade Católica Editora. Pires, G. (2022). Cálculo Diferencial e Integral em Rn. IST Press.
• MetodologiaMethodology

  Synchronize remotely: 1. Expository methodology: address the concepts of different themes, following the demonstrative method in practical exemplification of the content. In person: 2. Practical application: carrying out a series of exercises inside and outside the classroom, inducing active and autonomous participation from students. 3. Active methods, using Motivation strategies and Active Pedagogical Methodologies such as Flipped Classrooms and Gamification. Autonomous: 4. Consolidation of the content taught in class through research for additional information on the different topics covered and development of additional work proposed by the teacher through the Moodle platform. The teacher provides feedback (Tutorial Guidance – OT) on the results obtained by the student in solving the proposed problems, in person in a classroom context or remotely in asynchronous mode via the Moodle platform.
• LínguaLanguage

  Português
• TipoType

  Semestral
• ECTS

  6
• NaturezaNature

  Mandatory
• EstágioInternship

  Não
• AvaliaçãoEvaluation

  Descrição	Data limite	Ponderação
  Teste de avaliação	12-11-2025	50%
  Teste de avaliação	14-01-2026	50%