• Natural numbers and the principle of induction; integer and rational numbers.
• Real numbers. Inequalities. Absolute value. Bernoulli’s inequality, Cauchy-Schwarz inequality, geometric mean – arithmetic mean inequality.
• Complex numbers. Real and imaginary parts, conjugate number, modulus and ar- gument, complex exponential, polar representation, de Moivre’s formula, roots of a complex number, logarithms.
• Elementary functions. Real functions of a real variable. Injective and bijective func- tions, inverse function. Monotonic, bounded, even, odd, periodic functions. Factoriza- tion of polynomials. Rational functions; partial fraction decomposition. Properties of the elementary functions.
• Sequences. Limit of a sequence. Domination hierarchy, equivalences, squeeze rule. Bounded and monotonic sequences.
• Series. The non-null test. Series with positive terms. Comparison test; limit compar- ison test. Absolute convergence. Ratio or D’Alembert test, root or Cauchy test, the Leibniz test for alternating series. Sum of series: telescopic series, series with rational terms.
• Limits of functions and continuity. Limits and inequalities, equivalences, domina- tion hierarchy. Bolzano’s theorem. Weierstrass’ extreme values theorem. Continuity of the inverse function.
• Differentiation. Derivative and continuity. Chain rule. Derivative of the inverse function. L’Hôpital’s rule. The mean value theorem. The intermediate value theorem for the differential. Differential and growth. Higher-order derivatives. Extreme values of funtions. Convex and concave functions. Young and Taylor’s formulas.
• Antiderivatives and integration. Methods of computation of antiderivatives. Rie- mann sums. Integrals and inequalities. Fundamental theorem of the integral calculus, Barrow’s rule, integration by parts, change of variable. Computation of areas, lengths, volumes, centers of gravity.
• Power series. Radius and interval of convergence. Continuity. Derivative. Higher- order derivatives. The general term formula. Antiderivatives of a power series. Power series expansion of elementary functions. 

Presencial sessions: continuously throughout the semestrer, during the tutorial hours and in concerted sessions.

Exams: written exams in the official periods (January-February and September). 

• BB Abellanas Rapun, Lorenzo. Teoría y problemas de métodos de cálculo / Lorenzo Abellanas Rapún, Alberto Galindo Tixaire . - [1a ed.] Madrid [etc] : McGraw-Hill, 1989
• BB Apostol, Tom M.. Calculus. Vol.1, Cálculo con funciones de una variable, con una introducción al álgebra lineal / Tom M. Apostol. - 2ª ed. reimp. Barcelona [etc.] : Reverté, cop. 2002
• BB Bradley, Gerald L.. Cálculo de una variable / Gerald L. Bradley, Karl J. Smith ; traducción, José Luis Vicente Córdoba ; revisión técnica, Pedro Paúl Escolano . - [1a. ed. en español] Madrid [etc.] : Prentice Hall, cop. 1998
• BB Curso práctico de cálculo y precálculo / Domingo Pestana...[et al.] Barcelona : Ariel, D.L. 2000
• BB Demidovich, B.P.. 5000 problemas de análisis matemático / B. P. Demidóvich ; traducido del ruso por Emiliano Aparicio Bernardo . - 5ª ed. Madrid : Paraninfo, 1993
• BB Fuertes García, Jesús. Problemas de cálculo infinitesimal / Jesús Fuertes García , Jesús Martínez Hernando Madrid [etc.] : McGraw-Hill, D.L. 2006
• BB Kaczor, Wieslawa J.. Problems in mathematical analysis. I, Real numbers, sequences and series / W.J. Kaczor, M.T. Nowak Providence, R.I : American Mathematical Society, cop. 2000
• BB Kaczor, Wieslawa J.. Problems in mathematical analysis. II, Continuity and differentiation / W.J. Kaczor, M.T. Nowak Providence, R.I. : American Mathematical Society, cop. 2001
• BB Maron, Isaak A.. Problemas sobre cálculo de una variable : elementos y teoría / I.A. Maron ; traducción de Emilio Romero Ros Madrid : Paraninfo, 1975