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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. 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, 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 January 20, 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.
Last updated on: 2021-10-20

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