## Lecture notes

##### 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.

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##### Unit 3. Sequences.
Sequence of real numbers. Limit of a sequence. Number e. Indeterminacies. Asymptotic comparison of sequences.

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##### Unit 4. Series.
Series of real numbers. Series of nonnegative numbers. Alternating series. Telescoping series.

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##### Unit 5. Limit of a Function.
Concept and definition. Algebraic properties. Asymptotic comparison of functions.

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##### Unit 6. Continuity.
Definition, properties and continuity of elementary functions. Discontinuities. Continuous functions in closed intervals.

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##### Unit 7. Derivatives.
Concept and definition. Algebraic properties. Derivatives and local behaviour.

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##### 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.

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##### Unit 9. Primitives.
Definition. Immediate primitives. Integration by parts. Primitives of rational functions. Change of variable.

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