Calculus I

Recommended Prerequisites

Mathematics A from the Portuguese High School Curriculum.    

Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasized. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

Learning Outcomes

Provide the students with basic knowledge of differential and integral calculus for real functions as well as the fundamental concepts in the study of  numerical sequences and series, and sequences and series of functions. The student who successfully completes this course will be able to:

1. Compute limits of functions beyond those studied in High School;

2. Compute derivatives and primitives of elementary functions;

3. Use the Fundamental Theorem of Calculus to compute areas and lengths;

4. Compute limits of sequences and series;

5. Determine the convergence interval of a power series;

6. Compute the Taylor polynomial and the Taylor series of a function.   

Work Placement(s)

No

Syllabus

. Functions of a real variable

I.1 Trigonometric and hyperbolic functions and their inverses

I.2 Limits, continuity and differentiability of functions of a real variable

 

II. Integration

II.1 Primitives

II.2 Riemann integral and applications

II.3 Improper integrals

 

III. Series

III.1 Sequences and number series

III.2 Convergence criteria

III.3 Power series (Taylor formula and Tayor series).

Head Lecturer(s)

António José Esteves Leal Duarte

Assessment Methods

Continuous assessment
2 or more midterm exams: 100.0%

Final assessment
Exam: 100.0%

Bibliography

Stewart, J. Cálculo, Volumes I e II., Cengage Learning, (tradução da 8ª edição norte-americana), 2017.

Edwin “Jed” Herman & Gilbert Strang (entre outros), Calculus, volumes 1 e 2, OpenStax, 2018.

Carvalho e Silva, J. Princípos de Análise Matemática Aplicada, McGraw-Hill, Lisboa, 1999.

Swokowski, E. Cálculo com geometria analítica Vol I e Vol II, Makron Books, 1995

Baptista, M.,Silva, A. Matemática – Equações Diferenciais e Séries, 2ª edição, Edições Sílabo, 2005.