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Calculus I

Code 10341
Year 1
Semester S1
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements Does not have.
Mode of delivery Theoretical/Practical
Work placements Not applicable.
Learning outcomes
In this Curricular Unit the students will obtain the basic knowledge of Differential and Integral Calculus of real-valued functions of a real variable.

At the end of this Curricular Unit the students should
- solve inequalities involving rational expressions and absolute values;
- determine domains and sketch the graph of functions;
- compute limits of functions;
- study the continuity of functions;
- compute derivatives of functions;
- know how to approximate functions by Taylor's polynomials;
- apply the derivatives to compute maximums and minimums and sketch the graph of functions;
- integrate functions;
- apply integrals to compute plane areas, to compute the length of curves and to compute areas of surfaces and volumes of solids generated by revolution;
- determine whether a numerical series is convergent or divergent;
- compute the interval of convergence of a power series.
Syllabus
1. GENERALITIES AND EXAMPLES OF FUNCTIONS
Real numbers
Generalities about functions
Inverse and composition of functions
Polynomial, rational, absolute value, exponential, logarithmic, trigonometric, trigonometric inverse and hyperbolic functions

2. LIMITS AND CONTINUITY
Topological notions
Limits
Continuity
Bolzano and Weierstrass theorems
Infinite limits, limits at infinite and assymptotes

3. DIFFERENTIAL CALCULUS
Definition, rules and examples
Fermat, Rolle, Lagrange and Taylor theorems
Cauchy's Rule
Applications

4. INTEGRAL CALCULUS
Definition and properties of Riemann integral
Fundamental Theorem of Calculus
Antiderivatives
Applications
Techniques of antidifferentiation and of integration

5. SEQUENCES AND SERIES
Sequences
Convergent and divergent series
Comparison, limit, D'Alembert, Cauchy and Leibniz tests
Absolute convergence
Power series
Interval of convergence of a power series
Taylor 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
Teaching Methodologies and Assessment Criteria
The curricular unit is organized in theoretical-practical lessons. At the beginning of each lesson the teacher makes a theoretical presentation of the subject and then the students solve exercises from worksheets provided by the teacher.

The evaluation will consist in two tests, each of one with a maximum value of 10 points.

Designating the result of the first test by T1 and the result of the second test by T2, the final grade will be calculated as follows:
- if T1 + T2 is less than 18.5 points, the final grade will be the rounding of T1+T2;
- if T1 + T2 is greater than or equal to 18.5 points, the student must do an oral examination; the result of the oral exam, that we will designate by PO, will be between 0 and 20 points; the final grade will be the rounding of max {18, (T1 + T2 + PO) / 2}.

The students with a final grade greater than or equal to 10 points will pass the course.

All students will be admited to exam.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2023-09-29

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