# MATHEMATICS 1 — WS 2017/18

Office hours:
• upon request (see my timetable and suggest me some days and times via e-mail)
• my office is in Sokolovská 83, 2nd floor, behind the glass door opposite to the staircase

## News

Exam D. Problems and handwritten solutions. One student passed the exam, 9 in total.

Exam C. Problems and handwritten solutions. Four students passed the exam, 8 in total.

Exam B. Problems and handwritten solutions. One student passed the exam.

Exam A. Problems and handwritten solutions. Three students have passed the exam.

The exams are held
• written parts on Mondays 8.1., 15.1., 22.1., 5.2. and 12.2. at 9:00 - 10:30 AM.
• oral parts on the following Tuesdays.
• 8.1. and 9.1. in Opletalova.
• All other dates in Sokolovska 83, Prague 8 (the building where is my office), the written part in room K3 (together with course ODE2), the oral part in K2 (together with the czech course Matematika 1). Both rooms are in the 2nd floor.

## Basic information

• midterm test: 10 points
• homeworks: 20 points (10 homeworks, 2 points each)
• final exam: 90 points (50 points written exam, 40 points oral exam)
Students will be admitted to the oral exam only if the score from the first part (midterms, homeworks and written exam together) is at least 50 points. To pass the exam successfully, at least 20 points from the oral exam are required.

Grading: The total score is obtained as the sum of the points from the oral part and the first part, where the score of the first part is reduced to 60 if it exceeds 60. The final grade depends on the total score as follows.
• 70-75.5 points ... "E"
• 76-81.5 points ... "D"
• 82-87.5 points ... "C"
• 88-93.5 points ... "B"
• 94-100 points ... "A"
Final Exam takes part in the examination period at the end of the semester. Students have three attempts to pass the final exam. It consists of a written part and oral part.
• Written part. Students have 90 minutes to solve problems on limit of a function, derivatives, investigation of a function. Lecture notes and other materials are allowed, electronic devices are prohibited. A sample test is available here.
• Oral part follows typically the day after the written exam. The oral part tests understanding the definitions and theorems and ability to apply them. Each student should prepare answers within approximately 40 minutes. During the oral part only pencil and paper are allowed. Then the student should present answers and should answer additional questions. Here is a sample question.
See exam details for more details concerning both parts, as well as general exam rules and the requirements (list of required definitions, theorems and proofs).

## Lecture

Here is the beamer presentation to download and the list of definitions and theorems for printing (last update November 15, 2017).

Preliminary plan of lectures:
• 1. lecture: INTRODUCTION - sets, statements and predicates, quantifiers
• 2. lecture: methods of proofs
• 3. lecture: REAL NUMBERS, infimum, supremum
• 4. lecture: Supremum theorem
• 5. lecture: integer part, Archimedes property, roots, density of Q
• 6. lecture: SEQUENCES, LIMIT, uniqueness of limit
• 7. lecture: arithmetics of limits, ordering, sandwich theorem
• 8. lecture: infinite limit, extending theorems to infinite limits
• 9. lecture: limit of monotone sequence, Theorem of Bolzano and Weierstrass
• 10. lecture: FUNCTION, LIMIT of function, continuity, one-sided and infinite limits
• 11. lecture: arithmetics of limits, division by positive zero
• 12. lecture: limit of composite function
• 13. lecture: Heine Theorem, limit of monotone function, intermediate value theorem
• 14. lecture: image of interval, attaing of extrema, continuity of inverse function, ELEMENTARY FUNCTIONS, logarithm
• 15. lecture: properties of logarithm, exponential, trigonometric functions
• 16. lecture: inverse trigonometric functions, DERIVATIVE, arithmetics of derivatives
• 17. lecture: derivative of composite function, inverse function, elementary functions, neccessary condition for local extremum
• 18. lecture: Rolle and Lagrange Theorem, sign of derivative and monotonicity
• 19. lecture: L'Hospital rule
• 20. lecture: Examples to L'Hospital rule
• 21. lecture: convex and concave function,
• 22. lecture: inflexions
• 23. lecture: investigation of functions
• 24. lecture: asymptotes

## Seminar and exercises

Homeworks with solutions.
HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10.

Recomended problems.

Some further problems to practice from the web pages of Kristyna Kuncova:
Some past exam problems (from previous years) you can find here and here (click on year numbers). However, these problems can be more difficult than the midterm limit.

Preliminary plan:
• 1. week: statements and predicates
• 2. week: suprema and infima
• 3. week: limits of sequences, rational functions, roots, exponentials, factorials
• 4. week: limits of sequences, sandwich theorem, integer parts
• 5. week: limits of sequences, midterm test
• 6. week: limits of functions, trics from sequences
• 7. week: limit of composite functions, elementary functions
• 8. week: functions f^g, Heine theorem
• 9. week: continuity and derivatives
• 10. week: derivatives
• 11. week: investigation of functions
• 12. week: investigation of functions