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Calculus I

Calculus I

Code	16663
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	---
Learning outcomes	In this Curricular Unit it is intended that students acquire basic knowledge of Differential and Integral Calculus of functions of real variables. 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 the 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.
Main Bibliography	Main Bibliography: – James Stewart, Daniel Clegg, Saleem Watson – Cálculo, Volume 1, Cengage (2022) Additional Bibliography: – Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 – H. Anton, I. Bivens, S. Davis, Cálculo, volume I, 8.ª Edição, Bookman, 2007 – Adams, Robert Alexander_ Essex, Christopher - Calculus a complete course, Pearson (2018) – João Paulo Santos, Cálculo numa Variável Real, IST Press, 2012 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983
Teaching Methodologies and Assessment Criteria	Classes are both theoretical and practical. The teacher introduces key concepts and results, illustrating the theory with examples and applications. Students are encouraged to participate actively, engaging with the teacher and peers, reflecting on the topics, formulating and solving problems, and completing exercises. Independent work is also promoted. Assessment criteria 1. The assessment may be carried out during the course or in a final exam. 2. Knowledge assessment throughout the teaching and learning period will consist of two written tests, each lasting two hours and worth ten points. The tests will be held on 30 October and 16 December 2025. 3. Students who obtain a grade of 9.5 or higher in the assessments carried out during the teaching period will be exempt from the final exam. 4. Any attempt at fraud will result in failure of the course unit.
Language	Portuguese. Tutorial support is available in English.