## Mathematics I — information on final exams

#### General rules

• The exam can be taken only by the students that are enrolled for Mathematics I in SIS.
• The exam comprises a written part and an oral part. The oral part is taken only after passing the written part.
• If the student fails an exam and has still attempts available, he/she has to repeat the whole exam, including the written part, regardless of the previous results.
• It is necessary to enrol for a particular exam using SIS.
• If a student enrolled for a particular exam does not take it, he/she can be excused only for serious reasons (e.g. health problems). In other cases the attempt is voided.

#### Rules for the written part

• The written part comprises three problems worth 50 points in total.
• 90 minutes are available for solving these problems.
• The three problems are as follows
• Compute the limit of a function (15 points).
• Investigate the function (monotonicity, convexity, limits in infinities or endpoints of the domain, local extrema) and draw its graph (20 points).
• Investigate the function in a neighborhood of a point (one-sided limits, one sided derivatives) or several points and draw its graph. (15 points)
• During the written part it is possible to use any literature (e.g. tables with formulas, textbooks, notes from lectures, etc.).
• During the written part it is forbidden to use any electronic devices (e.g. mobile phones, calculators or laptops).
• To pass the written part it is necessary to achieve at least 50 points altogether from the written part, homeworks and a midterm test. If this score is higher than 60, it is reduced to 60 for determining the final grade.

#### Rules for the oral part

• The oral part generally takes place the next day after successfully passing the written part.
• The student draws a set of questions at random. He/she can then prepare the answers during 40 minutes, not using anything except writing utensils. The answers are then presented to the examiner, who will assess them. The examiner can ask additional questions.
• Each set of the questions comprises the following (APPROXIMATE point value of each question is given in brackets):
• Definition with examples and counterexamples (and explanation) (8 points).
• Theorem with explanation (importance of assumptions, idea of proof). (12 points)
• Theorem with proof. (8 points)
• Application of definitions and theorems. (12 points)
• Exact formulation of the definition or the theorem is required in questions 1, 2, and 3. It is not necessary to use exactly the same words as in the lecture/script but the meaning must be the same. In particular, this means clear splitting to assumptions and conclusions, correct usage of quantifiers and mathematical symbols.
• In questions 1 and 2 student is asked to give explanation, which means to show (typically using pictures) the meaning of the definition or theorem on concrete examples, eventually to present counterexamples or idea of proof or show importance of the assumptions.
• In question 3, student has to explain to the examiner any mathematically correct proof that uses only theorems proved or formulated during the lecture earlier, or other theorems that the student can prove.
• In question 4, student has to explain in detail how to apply a definition or theorem to a particular problem. Mathematically exact arguments are required, exact formulation of definitions and theorems may be needed.
• The knowledge of all the key notions is a necessity. If anytime during the exam the student shows a substantial lack of the knowledge of any of the key notions, he/she automatically fails the exam.
• To pass the oral part it is necessary to achieve at least 20 points.

### Requirements for the oral part — list of definitions and theorems.

Small changes in the following list may appear during the semester.

#### Key notions

• supremum and infimum
• limit of a sequence
• neighbourhood of a point
• limit of a function
• continuity of a function at a point
• extremum of a function
• derivative of a function at a point
• convex and concave function

### Definitions

• a set bounded from above
• a set bounded from below
• upper bound
• lower bound
• bounded set
• supremum and infimum of a set of real numbers
• maximum of a set
• minimum of a set
• integer part of a number
• sequence bounded from above
• sequence bounded from below
• bounded sequence
• increasing sequence
• decreasing sequence
• non-increasing sequence
• non-decreasing sequence
• monotone sequence
• strictly monotone sequence
• finite limit of a sequence
• convergent sequence
• subsequence
• extended real line
• infinite limit of a sequence
• pre-image of a set under a mapping
• surjective mapping (onto)
• injective mapping (one-to-one)
• bijection
• increasing function
• decreasing function
• non-increasing function
• non-decreasing function
• monotone function
• bounded function
• even function, odd function
• periodic function
• neighbourhood of a point
• punctured neighbourhood of a point
• limit of a function
• limit of a function from the right/left
• continuity of a function at a point
• continuity of a function at a point from right/left
• continuity of a function on an interval
• extremum of a function
• local maximum
• local minimum
• general power - ab
• function arcsin
• function arccos
• function arctg
• function arccotg
• derivative of a function at a point
• derivative of a function at a point from the right/left
• tangent to a graph of a function
• convex and concave function
• strictly convex function
• strictly concave function
• n-th derivative of a function at a point
• inflection point
• asymptote of a function

### Theorems with proofs

• de Morgan rules (Theorem 1)
• existence of a supremum (Theorem 2)
• density of rational and irrational numbers (Theorem 6)
• uniqueness of a limit of a sequence (Theorem 7)
• limit of a sequence and boundedness of a sequence (Theorem 8)
• limit of a subsequence (Theorem 9)
• arithmetics of finite limits of sequences (Theorem 10, sum only)
• limits and ordering (Theorem 11)
• two policemen theorem for sequences (Theorem 12)
• limit of a product of a bounded and null sequence (Corollary 13)
• convergence criterion for sequences (Lemma 14)
• limit of a monotone sequence (Theorem 18)
• boundedness and a limit of a function (Theorem 21)
• limit of a function of the type "A/0" (Theorem 23)
• Heine theorem (Theorem 26)
• relation of the derivative and continuity (Theorem 38)
• arithmetics of derivatives (Theorem 39)
• derivatives of elementary functions
• necessary condition for a local extremum (Theorem 42)
• Rolle's theorem (Theorem 43)
• Lagrange theorem (mean value) (Theorem 44)
• sign of the derivative and monotonicity (Theorem 45)
• second derivative and convexity (Theorem 49)
• existence of an asymptote (Theorem 52)

### Theorems without proofs

• Archimedean property (Theorem 3)
• existence of an integer part (Theorem 4)
• existence of nth root (Theorem 5)
• arithmetics of limits of sequences (Theorem 10')
• limit of kth root of a sequence (Lemma 15)
• limit of a sequence of the type "A/0" (Theorem 16)
• supremum and limits (Lemma 17)
• Bolzano-Weierstraß theorem (IP) (Theorem 19)
• uniqueness of a limit of a function (Theorem 20)
• arithmetics of limits of functions (Theorem 22)
• continuity and arithmetic operations (Corollary)
• limits and inequalities for functions (Theorem 24)
• bounded times vanishing function (Corollary)
• limit of a compound function (Theorem 25)
• limit of a monotone function (Theorem 27)
• existence of the exponential (Theorem 28)
• existence of sine and cosine functions (Theorem 29)
• properties of the exponential, logarithm, sine, cosine, tangent, cotangent and their inverse functions
• continuity of a compound function on an interval (Theorem 30)
• Bolzano (intermediate value) theorem (Theorem 31) (IP)
• an image of an interval under a continuous function (Theorem 32)
• Heine for continuity on an interval (Theorem 33)
• extrema of continuous functions (Theorem 34) (IP)
• boundedness of a continuous function (Corollary 35)
• inverse function and continuity (Theorem 36)
• continuity of elementary functions (Corollary 37)
• derivative of a compound function (chain rule) (Theorem 40) (IP)
• derivative of an inverse function (Theorem 41) (IP)
• computation of one-sided derivative (Theorem 46)
• l'Hospital's rule (Theorem 47)
• characterization of convexity (Lemma 48)
• necessary condition for inflection (Theorem 50)
• sufficient condition for inflection (Theorem 51)