Analyse I (anglais)

Topics for lecture number (in parenthesis is marked chapters in text): 

1. Introduction, numbers, sets, fields, ordered sets (Chap 1).
2. Rational and real numbers, bounds, inf/sup, completeness (Chap 1).
3. Infinity, absolute values, triangle inequalities, complex numbers (Chap 1, Appendix A).
4. Complex numbers, roots (Chap 1). Sequences and limits (Chap 2), proof by induction (Appendix B).
5. Sequences, limits, geometric sequence, limits of rationals, special limits, sandwich theorem (Chap 2).
6. Guess limits, quotient criteria, infinity, monotone sequences, number e, number pi, subsequences (Chap 2).
7. Lim sup/Lim inf, Cauchy sequence and convergence, Bolzano-Weierstrass theorem (Chap 2).
8. Review and examples for sequences (Chap 2).
9. Euler's sequence, number e (Chap 2), Series, Cauchy criterion, absolutely convergent series, non-negative elements, comparison criteria, alternating series, Leibniz' criteria (Chap 3).
10. Series, d'Alemberts criteria, Cauchy's criteria (Chap 3). Review session
11. Functions of one real variable, graph, surjective/injective/bijective, identity, function composition, inverse function, restriction/extension, absolute value, bounded functions, inf/sup, local/global extrema (Chap 3)
12. Functions, inverse functions, monotone functions, even/odd functions, limits, definitions and properties of functions, Sandwich theorem (Chap 4)
13. Limits of functions, sign function, one-sided limits, limits of composite functions, infinite limits and limits at infinity, limits of monotone functions, asymptote (Chap 4)
14. Limits of functions, limits of composite functions, infinite limits and limits at infinity, limits of monotone functions, asymptote, continuous functions (Chap 4).
15. Continuous functions, properties of continuos functions, composite functions, mean value theorem (Chap 4).
16. Continuous functions, fix points, continuous monotone function, inverse functions (Chap 4).
17. Differentiation of functions of one real variable, properties and rules (Chap 5).
18. Differentiation, extrema, Rolle's theorem, mean value theorem, Lipschitz continuity (Chap 5).
19. Mock exam - Chap 1-4, Appendix A+B.
20. Differentiation, monotone functions, l'hospitals rule, Taylor expansion (Chap 5)
21. Review of Mock Exam, extrema, inflection points, convex functions (Chap 5).
22. Banach fixed point theorem, extrema, asymptotes, convex and concave functions (Chap 5).
23. Darboux and Riemann integrals. Fundamental theorem of calculus, anti-derivatives, basic rules of integration (Chap 7).
24. Integration, change of variables, integration by parts (Chap 7)
25. Integration, rational functions, generalized integrals (Chap 7)
26. Power series and radius of convergene (Appendix C), some review
27. Review session