Unit 1. The Real Line.
Ordered fields. Number systems. Absolute value, bounds and intervals.
Unit 2. Real Functions.
Definition and basic concepts. Elementary functions. Operations with functions.
Unit 3. Sequences.
Sequence of real numbers. Limit of a sequence. Number e. Indeterminacies. Asymptotic comparison of sequences.
Unit 4. Series.
Series of real numbers. Series of nonnegative numbers. Alternating series. Telescoping series.
Unit 5. Limit of a Function.
Concept and definition. Algebraic properties. Asymptotic comparison of functions.
Unit 6. Continuity.
Definition, properties and continuity of elementary functions. Discontinuities. Continuous functions in closed intervals.
Unit 7. Derivatives.
Concept and definition. Algebraic properties. Derivatives and local behaviour.
Unit 8. Taylor Expansions.
Landau’s small o notation. Taylor’s polynomial. Calculating limits with Taylor expansions. Remainder and Taylor’s theorem. Taylor series. Numerical approximations.
Local behaviour of functions. Function graphing.
Unit 9. Primitives.
Definition. Immediate primitives. Integration by parts. Primitives of rational functions. Change of variable.
Unit 10. Fundamental Theorem of Calculus.
Riemann’s integral. Properties of the integral. Riemann’s sums. Fundamental Theorem of Calculus.
Unit 11. Geometric Applications of Integrals.
Area of flat figures. Area of flat figures in polar coordinates. Volumes. Length of curves.