Calculus I

Code	15751
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	-
Learning outcomes	This Course Unit is intended for students to acquire and apply basic knowledge of Differential and Integral Calculus of real functions of real variable. At the end of this course students should be able to: - 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 length of curves and to compute areas of surfaces and volumes of solids generated by revolution.
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 asymptotes 3. DIFFERENTIAL CALCULUS Definition, rules and examples Fermat, Rolle, Lagrange and Taylor theorems Cauchy's Rule Applications 4. INTEGRAL CALCULUS Antiderivatives. Definition, properties and technics of integration Definition and properties of Riemann integral Fundamental Theorem of Calculus Applications
Main Bibliography	Main Bibliography: – James Stewart, Daniel Clegg, Saleem Watson – Cálculo, Volume 1, Cengage (2022) Additional Bibliography: – Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 – H. Anton, I. Bivens, S. Davis, Cálculo, volume I, 8.ª Edição, Bookman, 2007 – Adams, Robert Alexander_ Essex, Christopher - Calculus a complete course, Pearson (2018) – João Paulo Santos, Cálculo numa Variável Real, IST Press, 2012 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983
Language	Portuguese. Tutorial support is available in English.