Calculus I

Code	15067
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.