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
Language	Portuguese. Tutorial support is available in English.