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Calculus I

Code 12803
Year 1
Semester S1
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements Not applicable
Mode of delivery Face to Face with recourse to e-learning.
Work placements N/A
Learning outcomes 1. To introduce the students in the main tools of differential and integral calculus.
2. To introduce the students in the main techniques for the integration of basic functions.
3.To implement the tools of differential and integral Calculus in the resolution of problems within Physics, Chemistry, and Biology.
At the end of the semester, the students should:
- know the basic definitions and main properties of functions
- know relevant families of functions
- know how to compute limits of functions of one variable
- know how to study the continuity of functions of one variable
- know how to compute the derivatives of functions of one variable
- know how to apply the derivatives to compute maximums and minimums and to sketch graphs of functions
- know how to compute antiderivatives of functions of one variable
- know how to integrate functions of one variable
- know how to apply the integral calculus to compute areas, to compute the volume of solids of revolution.
Syllabus 0. Generalities on the structure of the set of real numbers: geometric interpretation; topology in R
1. Functions of a real variable: general concepts, examples
2. Limits and continuity: 2.1. Limits; 2.2 Continuous functions; 2.3. Main properties of continuous functions
3. Differential Calculus on R: 3.1. Definition of derivative; 3.2 The derivative as a function; derivation rules; first order derivatives; 3.3 The Rolle Theorem; the Lagrange Theorem; the L’Hospital’s rule; 3.4 Applications to: computation of limits; the study of monotony and extremes; concavity and inflexion points; asymptotes
4. Integral calculus on R: 4.1 the Riemann integral; 4.2 Main properties of integrable functions; 4.3 The Fundamental Theorem of Calculus; 4.4 Applications to the calculus of areas and volumes
5. Techniques of integration.
Main Bibliography Main Reading

- Ferreira, Jaime Campos, Introdução à Análise Matemática, Fundação Caloust Gulbenkian, 1997.
- Lima, Elon Lages, Curso de Análise, Volume I, 1a edição, Projecto Euclides, IMPA, 2004.
- Stewart, James, Cálculo - Volume I, 7a edição, Pioneira Thomson Learning, 2013.
Teaching Methodologies and Assessment Criteria The curricular unit runs in theoretical-practical type of lessons. In the first part of the lesson, the professor
exposes the most relevant results and techniques, usually illustrated by examples. The second part of the
lesson is devoted to problem solving by the students. The students are proposed to solve a list of problems
from the adopted book.
Assessment criteria:
Three written tests will take place:
Test 1 (T1) - 14/11/2023 (evaluated for 10 points)
Test 2 (T2) - 03/01/2024 (evaluated for 10 points)
Test Global (TG) - 03/01/2024 (evaluated for 20 points)
Language Portuguese. Tutorial support is available in English.
Last updated on: 2023-10-04

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