Calculus I

Code	14630
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Do not exist.
Mode of delivery	Face-to-face
Work placements	N/A.
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. Real Numbers: order relations; absolute value. 2. real functions of real variable: definition and examples; composition of functions; inverse of a function; graphic representation; exponential and logarithmic function; trigonometric and inverse functions; hyperbolic functions; continuous functions and fundamental properties. 3. Differential calculation in R: definition of derivative; derivation rules; derived function; higher-order derivatives; Rolle, Lagrange, Cauchy and Taylor theorems; calculation of limits; monotony and local extremes; inflection points and concavities; optimization problems; asymptotes; Taylor and Maclaurin polynomials. 4. Integral calculation in R; primitivation; Riemann integral: definitions and examples; properties of integrable functions; fundamental theorem of calculus; integration by substitution and integration by parts; geometric applications. 5. Numerical Series and Power Series: convergence criteria; Power series; Taylor series.
Main Bibliography	Main Bibliography: Alberto Simões, Apontamentos de Cálculo I, UBI. F. R. Dias Agudo, Análise Real, Vol. I, Escolar Editora, 1989. J. Campos Ferreira, Introdução à Análise Matemática, 6ª Edição, Fundação Calouste Gulbenkian, 1995. Tom M. Apostol, Cálculo I, Editorial Reverté, 1994. Supplementary Bibliography: James Stewart, Cálculo - 5ª edição, volume 1 e volume 2, CENGAGE Learning, 2008. Lang, S., A first course in Calculus, 5th edition,Undergraduate texts in Mathematics, Springer.
Teaching Methodologies and Assessment Criteria	The evaluation will be periodic. There is a minimum score of 6 values for the student to be admitted to the exam. They will carry out 3 practical tests (T1, T2, T3). The quotation for the various tests will be: First test (T1): 8 values. Second test (T2): 8 values. Third test (T3): 4 values. The teaching-learning (CEA) classification will be: CEA = T1 + T2 + T3 The final classification (CF) will be: CF = Not allowed if CEA <6 CF = E if 6 <= CEA <10 CF = CEA if CEA >= 10 being E the exam classification.
Language	Portuguese. Tutorial support is available in English.