Calculus I

Code	15962
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	-
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	1.1. Limited sets. Maximum, mínimum, supremum infimum. 1.2 topological notions. 1.2 Generalities on functions 2. Real functions of a real variable: limits and continuity 2.1 Examples of functions: exponential and logarithmic; trigonometric and respective inverses; hyperbolic functions 2.2 Limits 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 Extremes, concavity asymptotes 4. Integral calculus in R 4.1 Integral of Riemann 4.2 Fundamental Theorem of Integral Calculus 4.3 Immediate Primitives 4.4 primitives of rational functions 4.5 Primitive by parts and by substitution; 4.6 Geometric Applications of the integral calculus
Main Bibliography	– James Stewart, Cálculo, volume I, 7.ª Edição, Cengage Learning, 2013 Bibliografia Secundária: - Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 - H. Anton, I. Bivens, S. Davis, Cálculo, volume I, 8.ª Edição, Bookman, 2007 – Demidovitch, B., Problemas e exercícios de Análise Matemática, McGrawHill, 1977 - João Paulo Santos, Cálculo numa Variável Real, IST Press, 2012 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – Sarrico, C., Análise Matemática – Leituras e exercícios, Gradiva, 3.ª Edição, 1999
Teaching Methodologies and Assessment Criteria	The classes will be theoretical-practical. The teacher presents the concepts and the results and illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems. Autonomous work, consisting mainly in solving the exercises, is encouraged.
Language	Portuguese. Tutorial support is available in English.