Calculus I

Code	11842
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Math A levels 10,11,12
Mode of delivery	Lectures and practical classes with face-to-face teaching.
Work placements	Not applicable.
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: a) Compute limits of functions of one variable. b) Investigate the continuity of functions of one variable. c) Compute the derivatives of functions of one variable. d) Apply the derivatives to compute maximums and minimums and to sketch graphs of functions . e) Compute antiderivatives of functions of one variable. f) Integrate functions of one variable. g) 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) Generalities about functions: the set of real numbers; real functions; composition of functions; the inverse of a function; exponential and logarithmic functions; trigonometric functions and their inverses; hyperbolic functions. 2) Limits and continuity: brief notions of topology; limits; asymptotes; continuity. 3) Differential Calculus: first order derivatives; theorems of Rolle, Lagrange and Cauchy; derivatives of higher order and Taylor formula; Cauchy rule; monotony and extremes; concavity and inflexion points. 4) Integral calculus: immediate primitives; primitives by parts; primitives by substitution; primitives of rational functions; Riemann integral; properties of integrable functions; Fundamental theorem of integral calculus; change of variables and integration by partes; geometric applications of calculus.
Main Bibliography	Main bibliography: - Apostol, T. M., Calculus , 2nd edition, Volume I, John Wiley & Sons, 1968. - Lang, S., A first course in Calculus, 5th edition, Undergraduate texts in Mathematics, Springer. - Stewart, J., Cálculo, Volumes I e II , Tradução da 6a edição norte-americana, CENGAGE Learning, 2010. - Thomas Jr.,G.; Weir, M.; Hass, J., Thomas’ Calculus Early Transcendentals (13th Edition) - Pearson (2013).
Language	Portuguese. Tutorial support is available in English.