Analysis-I

This is a one-semester course (15 weeks), with two 90-minute lectures and two 90-minute tutorials per week. The course will cover the following subjects.

  1. Abstract foundations
  2. Real numbers
  3. Limit and continuity
  4. Differential calculus
  5. Applications
We shall use the following two textbooks:
  • V. A. Zorich, Mathematical Analysis-I, Springer, Berlin–Heidelberg, 2015.
  • W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book, New York, 1976.

All (mathematical) questions should be asked on this blog, to make sure that the answers are freely available to everybody. Note that MathJax is active, so that you can use Latex for formulas:
        $f(x)=\sum\limits_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k+o(|x-a|^k).$

Comments

  1. August 15
    Lecture: Elements of mathematical logic. Sets and operations on them
    Exercises: 2 of Section 1.1.5 and 1, 2, 3, 5, 6, 7 of Section 1.2.4 in Zorich.

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  2. August 16
    Lecture: Functions and their elementary properties (Sections 1.3.1-1.3.3 in Zorich)
    Exercises: 3, 4, 5, 6, 7 of Section 1.3.5 in Zorich.

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  3. August 18
    Lecture: Relations (Section 1.3.4)
    Exercises: 1, 2, 8a, 8c of Section 1.3.5

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  4. August 19
    Lecture: Cardinals (Sections 1.4.1 and 1.4.3)
    Exercises: 9 of Section 1.3.5 and 1, 5, 6 of Section 1.4.4

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  5. August 22
    Lecture: Axioms of real numbers and general properties (Section 2.1)
    Exercises: 5, 6, 7, 8 of Section 2.2.5

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  6. August 23
    Lecture: Mathematical induction and Archimedes principle (Sections 2.2.1-2.2.3)
    Exercises: 1a, 1b, 1e, 1f, 1g, 2, 3, 4, 13 of Section 2.2.5

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  7. August 25
    Lecture: Irrational numbers and Archimedes principle (Sections 2.2.2 and 2.2.3)
    Exercises: 1c, 1d, 19, 20 of Section 2.2.5

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  8. August 26
    Lecture: Approximation of irrational numbers (Sections 2.2.4)
    Exercises: 15, 16, 17, 18 of Section 2.2.5

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  9. Revision week (August 27-September 4)
    Read Theorem 1.19 and its proof in the Appendix (in Rudin)
    Solve Exercises: 11, 21, 23, 24 of Section 2.2.5 (in Zorich)

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  10. The first mid-term test will take place on August 30, from 10:00 to 11:30.

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  11. September 5
    Lecture: Completeness and its consequences (Section 2.3)
    Exercises: 1, 2, 3, 4 of Section 2.3.4 (in Zorich)

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  12. September 8
    Lecture: Countable and uncountable sets; Section 2.4
    Exercises: 1, 2, 3, 5a, 5b, 6, 7, 8 of Section 2.4.3

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  13. September 12
    Lecture: Limit of sequences and their properties; Section 3.1.1, 3.1.2, and 3.1.3 (a-b) in Zorich
    Exercises: 1-4 of the file I sent by email

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  14. September 15
    Lecture: Upper and lower limits, and the number $e$; Section 3.1.3 in Zorich
    Exercises: 1, 3, 6, 8 of Section 3.1.5

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  15. September 19
    Lecture: Series; Section 3.1.4 (a-b) in Zorich
    Exercises: 7, 8, 11 in Chapter 3 of Rudin (pages 78-79)

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  16. September 22
    Lecture: Series; Section 3.1.4 (b-c) in Zorich
    Exercises: 5a, 5b in Section 3.1.5 of Zorich and 12, 14 in Chapter 3 of Rudin (pages 79-80)

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  17. September 26
    Lecture: Definition and elementary properties of the limit of a function; Section 3.2 (a-b-c)
    Exercises: See the list of exercises sent by email.

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  18. September 29
    Lecture: Exponential and logarithmic functions; Section 3.2 (d)
    Exercises: Continue solving the exercises of the list sent earlier

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  19. Revision week (October 3-7)
    Construct the exponential and logarithmic function with all the details; Section 3.2.2 (d)
    Solve Exercise 1 in Section 3.2.5 of Zorich

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  20. The second mid-term test will take place on October 6, from 16:00 to 17:30

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  21. October 10
    Lecture: Limit over a base. Criteria for the existence of limit; Sections 3.2.3, 3.2.4 (a-b) Exercises: 2, 5 of Section 3.2.5; 602-609 from Demidovich

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  22. October 13
    Lecture: Limit of a monotonic function. Comparison of asymptotic behaviour of functions; Section 3.2.4 (c-d)
    Exercises: 3, 6, 8, 9, 10, 11 of Section 3.2.5

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  23. October 17
    Lecture: Continuous functions and their basic properties I; Sections 4.1 and 4.2
    Exercises: 1, 2, 3, 4, 5 of Section 4.2.3

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  24. October 20
    Continuous functions and their basic properties II; Sections 4.1 and 4.2
    Exercises: 6, 8, 9, 10 of Section 4.2.3

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  25. October 24
    Lecture: Differentiability of a function: interpretation and examples; Sections 5.1.2-5.1.5
    Exercises: 1, 2 of Section 5.1.6

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  26. October 27
    Lecture: Elementary applications in physics; van der Waerden example of nowhere differentiable continuous function; Sections 5.1.1-5.1.5
    Exercises: 3, 4, 5 of Section 5.1.6

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  27. October 31
    Lecture: The basic rules of differentiation; Section 5.2
    Exercises: 1, 2, 3 of Section 5.2.7

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  28. November 3
    Lecture: Applications in physics: Galilean and Lorentzian transformations, motion of a rocket, atmospheric pressure; Sections 5.2.5, 5.6.1, 5.6.2
    Exercises: 4, 5, 6, 7 of Section 5.2.7

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  29. Revision week (November 7-11): Read about radioactive decay and falling bodies (Sections 5.6.3 and 5.6.4)
    Solve Exercises 11, 12, 13 in Section 4.2.3 of Zorich

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  30. The third mid-term test will take place on November 12, from 9:45 to 11:45

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  31. November 14
    Lecture: Fermat, Rolle, Lagrange and Cauchy theorems; Sections 5.3.1 and 5.3.2
    Exercises: 1240, 1241, 1246.2, 1247, 1255, 1256, 1257, 1266, 1267 in Demidovich

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  32. November 17
    Lecture: Taylor formula; Section 5.3.3
    Exercises: 1381, 1387, 1388, 1393, 1397, 1403, 1406, 1413 in Demidovich; 1, 2, 4 of Section 5.3.4 in Zorich

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  33. November 21
    Lecture: Taylor formula, development of elementary functions and applications; Section 5.3.3 and 5.3.4
    Exercises: 3, 5, 6, 7b, 8b in Section 5.3.4 of Zorich

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  34. For today's class, read the following sections in Zorich:
    5.6.1: Motion of a body of variable mass
    5.3.3: Proposition 4 and its proof
    5.3.4: Exercises 9 and 11 (a-b)

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  35. November 24
    Lecture: Study of functions with the help of differential calculus; Sections 5.4.1 and 5.4.2
    Exercises: 1, 2, 3, 4 of Section 5.4.6; the examples of Section 5.6.3

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  36. November 28
    Lecture: Convex functions; Section 5.4.3
    Exercises: 5, 6, 8 of Section 5.4.6

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  37. December 1
    Lecture: Hôpital's rule; graphs of functions; Section 5.4.4 and 5.4.5
    Exercises: 9 of Section 5.4.6, Examples 27 and 28 of Section 5.4.5

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