# Calculus I

### Presentation

The curricular unit of Calculus I introduces the essential concepts and tools for differential and integral calculus of real-valued functions of a real variable. It is intended to extend the mathematical training acquired in secondary education by developing the capacity for abstraction and logical reasoning in formulating and solving problems, formalizing them with rigor and precision, but without neglecting the virtues of intuitive thinking. The main objective is to provide an introduction to Mathematical Analysis, showing the rigorous side of Calculus and establishing the theoretical foundations for further studies in areas of Science and Engineering.

### Part of this Programme

Environmental Engineering

### Level of Qualification|Semesters|ECTS

| Semestral | 5

### Year | Type of course unit | Language

1 |Mandatory |Português

### Code

ULHT39-705

### Recommended complementary curricular units

Basic language about mathematical logic, set theory and functions.

### Prerequisites and co-requisites

n/a

### Professional Internship

Não

### Syllabus

1. Sets and functions

2. Algebraic and topological properties of real numbers

3. Natural numbers and method of mathematical induction

4. Sequences

5. Real-valued functions of a real variable: limits, continuity, differentiability and integrability

### Objectives

- Consistently apply the language of sets and functions in formulating and solving problems in Mathematics

- Solve equations and inequations in the set of real numbers

- Apply the method of mathematical induction in the demonstration of P(n) properties

- Understand the definitions of bounded sequence, monotone sequence and limit of a sequence

- Calculate limits of sequences

- Understand the definitions of limit, continuity and differentiability of real-valued functions of a real variable

- Calculate limits of real-valued functions of a real variable

- Know the techniques of primitivation by parts, by substitution and of rational functions

- Know how to apply Barrow's formula and the fundamental theorem of Calculus

### Teaching methodologies and assessment

The curricular unit is taught in theoretical-practical classes. The assessment is continuous and consists of three tests during the academic period. The continuous assessment grade is calculated as the arithmetic mean of the three tests. If the continuous assessment grade is equal to or greater than 9.5, the student passes the curricular unit. Otherwise, the student is admitted to an exam to be held during the appeal period. Regardless of the grade obtained in each test or in the exam, the student can be called for an oral exam in order to confirm the grade in front of the teacher.

### References

- SANTOS, J. P. (2012). Cálculo Numa Variável Real. IST Press.

- SÁ, A. A., & LOURO, B. (2020). Cálculo Diferencial e Integral em R (2ª edição). Matcubo.

### Office Hours

Durante a primeira semana de aulas será fixado o horário de atendimento aos alunos.