Calculus I

Code	8622
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	There is no entry requirement.
Mode of delivery	Presential classes
Work placements	Not applicable
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	Main bibliography: – James Stewart, Cálculo, volume I, 7.ª Edição, Cengage Learning, 2013 Secondary Bibliography: - 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 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983
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. Assessment Criteria: 1. Evaluation may be done during the class period or in a final exam. 2. The evaluation of knowledge throughout the teaching-learning period will be periodic and will consist of two written tests, each lasting two hours and rated at ten (10) points, to be held on December 2, 2021 and February 3, 2022. 3. Students who have obtained a score of 9.5 or higher in the assessment carried out throughout the teaching activities will be exempt from the final exam. 4. Any attempt at fraud will result in failure in the Calculus I course.
Language	Portuguese. Tutorial support is available in English.