UNIT 1: SEQUENCES AND SERIES OF NUMBERS.
The real line, intervals, inequalities, absolute value, sets in the real line and in the plane. Mathematical induction. Sequences of numbers, main notions, limits of sequences, recurrent sequences. Stirling formula and Stolz test. Series of numbers, main notions. Tests for convergence for series of positive numbers, absolute and conditional convergence. Leibniz¿s test.
UNIT 2: LIMITS AND CONTINUOUS FUNCTIONS.
Elementary functions, composition of functions, inverse function. Polar coordinates and sketch of graphs of functions. Limits of functions, definition, main theorems. Evaluation of limits. Continuous functions, properties and main theorems.
UNIT 3: DIFFERENTIAL CALCULUS IN ONE VARIABLE
Differentiation of functions: definition, differentiation rules, interpretation. Bernoulli-L¿Hôpital rule. Main theorems on differentiation. Extrema of functions. Optimization problems with constraints. Convexity and asymptotes. Graph of functions. Taylor polynomial and series: definition, main theorems. Evalution of limits with Taylor polynomial. Convergence domain for a Taylor series.
UNIT 4: INTEGRATION
Antiderivatives, integration rules, integration by parts and by decomposition in simple fractions. Integration by substitution and other methods to evaluate integrals. Definite integral and the fundamental theorem of calculus. Applications of integration: areas, volumes and length. Physical applications of the definite integral.