a. To understand the concept of real number and its implications, mainly the concept of limit. b. To understand and manipulate series of real numbers. c. To identify functions, their dependence on variables and their basic properties (monotony, parity, continuity, differentiability). d. To master the basic operations of Calculus: limits, derivatives, integrals and Taylor expansions. e. To interpret the derivative as rate of variation of a function, and the integral as an area. f. To understand the Taylor polynomial as the best polynomial local approximation for a sufficiently smooth function, and to apply that approximation to simple cases. g. To be able to graph simple functions. h. To be able to solve simple optimization problems.

Part I: Real Numbers and Functions Chapter 1: The Real Line 1.1 Ordered Fields 1.2 Number Systems 1.3 Absolute value, bounds, and intervals Chapter 4: Real Functions 2.1 Definition and basic concepts 2.2 Elementary functions 2.3 Operations with functions Part II: Sequences and Series Chapter 3: Sequences 3.1 Sequences of real numbers 3.2 Limit of a sequence 3.3 Number e 3.4 Indeterminacies 3.5 Asymptotic comparison of sequences Chapter 4: Series 4.1 Series of real numbers 4.2 Series of nonnegative terms 4.3 Alternating series 4.4 Telescopic series Part III: Differential Calculus Chapter 5: Limit of a Function 5.1 Concept and definition 5.2 Algebraic properties 5.3 Asymptotic comparison of functions Chapter 6: Continuity 6.1 Definition, properties, and continuity of elementary functions 6.2 Discontinuities 6.3 Continuous functions in closed intervals Chapter 7: Derivatives 7.1 Concept and definition 7.2 Algebraic properties 7.3 Derivatives and local behaviour Chapter 8: Taylor expansions 8.1 Asymptotic comparison of functions 8.2 Taylor¿s polynomial 8.3 Calculating limits 8.4 Remainder and Taylor¿s theorem 8.5 Taylor series 8.6 Numerical approximations 8.7 Local behaviour of functions 8.8 Function graphing Part IV: Integral Calculus Chapter 9: Primitives 9.1 Integration by parts 9.2 Primitives of rational functions 9.3 Change of variable Chapter 10: Fundamental Theorem of Calculus 10.1 Riemann¿s integral 10.2 Properties of the integral 10.3 Riemann¿s sums 10.4 Fundamental theorem of calculus Chapter 11: Geometric Applications of Integrals 11.1 Area of flat figures 11.2 Area of flat figures in polar coordinates 11.3 Volumes 11.4 Length of curves

The methodology will be the usual one for classes in the classroom, writing on the blackboard, with the occasional help of some resources on-line to illustrate some graphic or computational aspects of the course. Also, the classroom notes will be uploaded in Aula Global at the end of each chapter, along with the problem sheets that will be solved and discussed in the small groups.