### Mathematical Analysis I NMAI054

**Time and place:** Tuesday at 12:20 in S10 (lecture) and Wednesday at 15:40 in T5 (tutorial)

**Tutorial organization and credit: ** At each tutorial a set of homework will be issued, due for the next tutorial. You are expected to attempt to solve all homework and write your solution of each part on a separate sheet of paper. At the next tutorial, one part of the homework will submitted to be marked for credit and the solution of the remaining homework will be discussed. Any homework submitted later than at the beginning of the tutorial will score 0%. If you are unable to attend the tutorial and want to submit homework, you can to submit all parts of homework before the tutorial.
To obtain credit from the tutorial, you need to meet the following three conditions

- at most three absences (with exceptions for serious medical and personal reasons),
- score at least 50% in the test at the end of the term,
- to score average 60% of credit from homework, three lowest marks/undelivered homeworks will not be considered into this average.

**Test from tutorials: 14.1. 2019 at 9:30 in S4 (building at Malostranske namesti). Time: 90 minutes. Topics: limits of sequences, convergence of series, limits of functions, derivatives. Any written/printed materials allowed, no electronics. ** Solution of the test.

**Test from tutorials 2nd attempt: 18.1. 2019 at 10:30 in S8 (building at Malostranske namesti). Same conditions and topics. ** There will be one more opportunity to pass the test after this one, but it will be the last one.

**Test from tutorials 3rd attempt: 22.1. 2019 at 13:30 in S10 (building at Malostranske namesti). Same conditions and topics. **

**Exam:** Exam will be oral with time for written preparation. You must obtain credit from the tutorial to be able to take the exam. You must register in SIS to take the exam.
Exam description and list of exam requirements.

Handrwritten lecturenotes from lectures 1-3. Handrwritten lecturenotes from lecture 4. and tutorials (note: we skipped proofs during tutorials). List of basic limits that can be used without further justification. For recommended study materials see the syllabus, you might also find useful notes by Aleš Pultr or some parts of Introduction to Mathematical Analysis by John E. Hutchinson. In both cases, beware of typos. Collection of questions and exercises, some with solutions, I highly reccommend: Margit Gémes, Zoltán Szentmiklóssy: Mathematical Analysis - Exercises I. Another collection of problems (with solutions) you migt find useful.

Homework and reviewing the material from the course is supposed to take you about 3 hours a week (might vary depending on your background from highschool). If it takes significantly more, consult your progress with me or with your mathematics mentor (Katerina Altmanova).

**Covered material:**

- 2./3.10.: Introduction, logic, types of proofs. Irrationality of square root of 2. Bernoulli inequality. Tutorial and homework problems.Solutions.
- 9./10.10.: Set operations. Supremum, infimum. Types of numbers. Tutorial problems. Homework problems. Solutions.
- 16./17.10.: Axioms of real numbers. Homework problems.
- 24./25.10.: Sequences, their properties and limits. Tutorial and homework problems. Solutions.
- 30.10.: Immatriculation - no lecture.
- 31.10.: Limits. Homework problems.
- 6.11.: Dean's day - no lecture.
- 7.11.: Limits. Homework problems.
- 13.11.: Extended arithmetic of limits. Bolzano-Weierstrass theorem. Cauchy sequences. Lecturenotes.
- 14.11.: Limits with (1+1/n)^n. Little o notation, comparing speeds of growth: log < polynomial < exponential < factorial < n^n. Homework problems.
- 20.11.: Limes superior and inferior. Series - definition and examples. Lecturenotes.
- 21.11.: Limit of a recursively defined sequence. Sums of geometric series. Homework problems. Solutions of Problems 2 and 3 fom Homework 8.
- 27.11.: Necessary condition for convergence of a series. Cauchy condition for series. Comparison test. Ratio test. Lecturenotes.
- 28.11.: Root test. Determining convergence using tools from the lecture. Homework problems.
- 4.12.: Series with negative terms: absolute convergence, Leibniz criterion. Rearranging series. Exponential function, sinus, cosinus. Lecturenotes.
- 5.12.: Series with negative terms: absolute and conditional convergence. Homework problems.
- 11.12.: Limit of a function. Continuity. Arithmetics of limits of functions. Lecturenotes.
- 12.12.: Limit of a composed functions. Exercises and homework.
- 18.12.: Uniqueness and other properties of limits of functions. Heine theorem. Continuous functions. Darboux theorem. Continuous function has extremes on a compact interval. Derivative - definition. Lecturenotes.
**Reading assignment:**notes by Aleš Pultr, Chapter VI.2-3 and VII.1. - 19.12.: Limits. Cyclometric functions. Rules for calculating derivatives. Homework. Learn methods of calculating derivatives by hand, even though computers can do it easily for you: this derivative calculator provides also detailed calculaction and graph of the function. Use it for practicing!
- 8.1.: One sided derivatives, calculating them as limits of derivative. Derivative and continuity. Mean value theorems. L'Hospital rule. Higher order derivatives. Taylor polynomial, its remainder. Taylor series. Second derivative and convexity/concavity. Lecturenotes.
- 9.1.: Necessary and sufficient condition for inflection point. Sketching graphs of functions. Using L'Hospital rule. Tutorial problems.