Calculus I

Code	15846
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Math A levels 10,11,12
Learning outcomes	This course constitutes an introduction to differential and integral calculus in IR. In the end on this curricular unit the student must be able to: - graphically represent and identify properties of a real function of a real variable - compute limits of functions of one variable - investigate the continuity of functions of one variable - compute the derivatives of functions of one variable - apply the derivatives to compute maximums and minimums and to sketch graphs of functions - compute antiderivatives of functions of one variable - integrate functions of one variable - apply the integral calculus to compute areas, to compute the length of curves and to compute the surface area and the volume of a solid of revolution
Syllabus	1) Real functions: generalities and examples. 1.1)The set of real numbers 1.2)Generalities about functions 1.3)Polynomial functions, rational functions and modulo function 1.4)Inverse function and composition of functions 1.5)Exponential function and logarithmic function 1.6)Trigonometric functions and their inverses 1.7)Hyperbolic functions 2)Real functions: limits and continuity 2.1)Brief notions of topology in R 2.2)Limits of real functions 2.3)Continuous functions 2.4)Infinity limits, limits at infinity, and asymptotes 3) Differential calculus in R 3.1) Derivatives, derivation rules and examples 3.2) Fundamental theorems of differential calculus 3.3) Applications of differential calculus 4)Integral calculus in R 4.1)Riemann integral: definition, properties, and examples 4.2)Fundamental theorem of integral calculus 4.3)Immediate primitives 4.4)Application to the calculus of areas of plane regions 4.5)Primitive and integration techniques 4.6) Other applications 4.7) Improper integral
Main Bibliography	Main bibliography: - Apostol, T.M., Cálculo, Vol. 1, Reverté, 1999 - Azenha, A., Jerónimo, M. A., Elementos de Cálculo Diferencial e Integral em R e Rn, McGraw-Hill, 1995 - Dias Agudo, F.R., Análise Real, Vol. I, Escolar Editora, 1994 - Demidovitch, B., Problemas e exercícios de Análise Matemática, Escolar Editora, 2010 - Campos Ferreira, J., Introdução à Análise Matemática, 11ªed., Fundação Calouste Gulbenkian, 2014 - Lima, E. L., Curso de Análise, Vol.1, Projecto Euclides, IMPA, 1989 - Lima, E. L., Análise Real, Vol.1, Coleção Matemática Universitária, IMPA, 2004 - Mann, W. R., Taylor, A. E., Advanced Calculus, 3rd ed., JW&Sons, 1991 - Sarrico, C., Cálculo Diferencial e Integral, Esfera do Caos, 2009 - Stewart, J., Cálculo, Volumes I e II , Tradução da 6a edição norte-americana, CENGAGE Learning, 2010 - Swokowski, E. W., Cálculo com Geometria Analítica, Vol.1, 2ªed, Makron Books, 1995
Teaching Methodologies and Assessment Criteria	In 2 of the 4 week curricular unit hours, there is a theoretical approach to the program subjects. In the other 2 week hours, students are encouraged to solve the assigned exercises. The learning evaluation is strongly based upon the realization of 2 written tests, as detailed. The valuation of attendance is included in the written tests since a quality presence in the classes necessarily means continuous work and dedication to the proposed exercises. The Minimum Grade for Admission to the Exam is established as the 6 values in the Teaching-Learning classification.
Language	Portuguese. Tutorial support is available in English.