Calculus I

Code	8476
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	---
Mode of delivery	Face-to-face
Learning outcomes	With this Curricular Unit it is intended that students acquire basic knowledge of Differential and Integral Calculus of real functions of real variable. At the end of this Curricular Unit the student should be able to: 1) Compute limits of real functions of real variable; 2) Study the continuity of real functions of real variable; 3) Compute derivatives of real functions of real variable; 4) Apply the derivatives to the computation of maxima and minima of real functions of real variable; 5) Compute primitives and integrals of real functions of real variable; 6) Use integral calculus to determine areas and volumes of surfaces generated by revolution, as well as length of plain curves.
Syllabus	1. Real functions of a real variable: generalities and examples 1.1 The set of real numbers 1.2 Generalities on functions 1.3 Examples of functions: exponential and logarithmic; trigonometric and respective inverses; hyperbolic functions 2. Real functions of a real variable: limits and continuity 2.1 Limits 2.2 Asymptotes 2.3 Continuous Functions 3. Differential calculus in R 3.1 Definition of derivative and examples 3.2 Derivation rules 3.3 Theorems of Rolle, Lagrange, and Cauchy 3.4 Higher order derivatives and Taylor formula 3.5 Applications to the computation of limits 3.6 Monotonicity and local extremes; concavity and inflection points 4. Integral calculus in R 4.1 Immediate Primitives 4.2 Primitive by parts and by substitution; primitives of rational functions 4.3 Integral of Riemann 4.4 Fundamental Theorem of Integral Calculus 4.5 Variable change and integration by parts 4.6 Application of integral calculus to the computation of areas and volumes 5. Series
Main Bibliography	Main bibliography – Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 – Stewart, J., Calculus (International Metric Edition), Brooks/Cole Publishing Company, 2008 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983 Additional bibliography – Dias Agudo, F.R., Análise Real, Vol. I, Escolar Editora, 1989 – Demidovitch, B., Problemas e Exercícios de Análise Matemática, McGrawHill, 1977 – Lang, S., A First Course in Calculus, Undergraduate texts in Mathematics, Springer, 5th edition – Lima, E. L., Curso de Análise, Vol. 1, Projecto Euclides, IMPA, 1989 – Lima, E. L., Análise Real, Vol. 1, Colecção Matemática Universitária, IMPA, 2004 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – J. P. Santos, Cálculo numa Variável Real, IST Press, 2013 – Sarrico, C., Análise Matemática – Leituras e exercícios, Gradiva, 3.ª Ed., 1999
Language	Portuguese. Tutorial support is available in English.