## Lecture notes

##### Unit 1. The Real Line.
Ordered fields. Number systems. Absolute value, bounds and intervals.

1.2. Videos unit 1.  (YouTube)

##### Unit 2. Real Functions.
Definition and basic concepts. Elementary functions. Operations with functions.

(PDF)

2.2. Videos unit 2.  (YouTube)

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

(PDF)

3.2. Videos unit 3.  (YouTube)

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

(PDF)

4.2. Videos unit 4. (YouTube)

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

(PDF)

5.2. Videos unit 5. (YouTube)

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

(PDF)

6.2. Videos unit 6. (YouTube)

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

(PDF)

7.2. Videos unit 7. (YouTube)

##### 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.

(PDF)

8.2. Videos unit 8. (YouTube)

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

(PDF)

9.2. Videos unit 9. (YouTube)

##### 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.

Last modified: Friday, 15 July 2022, 1:11 PM