Calculus I

Code	9086
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Non applicable
Mode of delivery	Face-to-face.
Work placements	Non applicable.
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 and 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 computation of areas and volumes 5. Techniques of integration.
Main Bibliography	- Stewart, J., Cálculo, Volume 1, 7a edição, CENGAGE Learning, 2013. - 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.
Language	Portuguese. Tutorial support is available in English.