Mathematics II — information on final exams

General rules

• The exam can be taken only by the students that are enrolled for Mathematics II 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
  • Find global extrema (resp. supremum, infimum) of a function of several variables (20 points).
  • Matrix computations (system of linear equations, determinant, inverse matrix) (15 points).
  • Antiderivative of Riemann integral of a function of one variable. (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.

Key notions from Mathematics I

• supremum and infimum of a set of real numbers
• limit of a sequence
• limit of a function
• continuity of a function at a point
• derivative of a function at a point

Key notions

• open ball
• open set
• closed set
• function continuous on a set
• compact set
• maximum/minimum of a function on a set
• local maximum/minimum of a function with respect to a set
• partial derivative
• function of class C¹
• convex set
• concave function
• matrix
• product of two matrices
• determinant
• antiderivative

Definitions

• the Euclidean metric
• interior point of a set
• interior of a set
• boundary point of a set
• boundary of a set
• closure of a set
• bounded set
• convergence of a sequence in Rⁿ
• a function continuous with respect to a set
• a function continuous at a point
• strict maximum/minimum of a function on a set
• strict local maximum/minimum of a function with respect to a set
• local maximum/minimum
• strict local maximum/minimum
• limit of a function at a point
• tangent hyperplane
• gradient of a function
• partial derivative of the second order
• function of class C^k and C^∞
• convex function
• strictly concave/convex function
• quasiconcave/quasiconvex function
• strictly quasiconcave/quasiconvex function
• square matrix
• sum of matrices
• multiple of a matrix
• invertible matrix
• inverse of a matrix
• linear combination of vectors
• non-trivial linear combination of vectors
• linearly independent vectors
• linearly dependent vectors
• rank of a matrix
• row echelon form of a matrix
• elementary row operation
• transformation of a matrix
• upper triangular matrix
• coefficient matrix of a system of equations
• augmented matrix of a system of equations
• rational function
• partition
• refinement of a partition
• Riemann integral

Theorems with proofs

• properties of open sets (Theorem 2)
• on attaining extrema of functions (Theorem 14)
• necessary condition for a local extremum (Theorem 16)
• sufficient condition for a maximum (Corollary 29)
• super-level sets of concave functions (Theorem 30)
• super-level sets of quasi-concave functions (Theorem 31)
• uniqueness of an extremum (Theorem 32)
• properties of matrix multiplication (Theorem 35)
• operations with invertible matrices (Theorem 37)
• rank of invertible matrices (Theorem 41)
• determinant of a transformed matrix, part (i),(iii),(iv) (Theorem 44)
• determinant of a triangular matrix (Theorem 45)
• determinant and invertibility (Theorem 46)
• solutions of a transformed system (Proposition 49)
• solvability of nxn system (Theorem 51)
• structure of the set ∫f(x)dx (Theorem 57)
• linearity of antiderivative (Theorem 59)
• integration by parts (Theorem 61)
• Newton-Leibniz formula (Theorem 72)

Theorems without proof

• properties of the Euclidean metric (Theorem 1)
• characterisation of convergence of sequences in Rⁿ (Theorem 3)
• characterisation of closed sets (Theorem 4)
• properties of closed sets (Theorem 5)
• properties of the interior and the closure (Theorem 6)
• boundedness of the closure (Theorem 7)
• continuity and arithmetic operations (Theorem 8)
• Heine theorem (Theorem 10)
• continuity and level sets (Theorem 11)
• characterisation of compact sets in Rⁿ (proof in R²) (Theorem 12)
• limit of a composite function (Theorem 15)
• tangent hyperplane theorem (Theorem 18)
• continuity of functions of class C¹(Theorem 19)
• derivative of a compound function (Theorem 20)
• interchanging of partial derivatives (Theorem 21)
• implicit function theorem (Theorem 22) (IoP)
• Lagrange multiplier theorem (Theorem 23)
• implicit functions theorem (Theorem 24)
• Lagrange multipliers theorem with more constraints (Theorem 25)
• relation of concavity and continuity (Theorem 26)
• characterisation of concave functions of class C¹ (Theorem 27)
• characterisation of strictly concave functions of class C¹ (Theorem 28)
• basic properties of sum of matrices and their scalar multiples (Proposition 34)
• properties of matrix transformation (Theorem 38) (IoP)
• representation of a matrix transformation (Theorem 39) (IoP)
• transformation to the identity matrix (Lemma 40) (IoP)
• cofactor expansion (Theorem 42)
• sum of two matrices that differ in one row (Lemma 43) (IoP)
• determinant of product of two matrices (Theorem 47)
• Rouché - Fontené Theorem(Theorem 50) (IoP)
• Cramer's rule (Theorem 52)
• existence of an antiderivative (Theorem 58)
• integration by substitution (Theorem 60)
• polynomial division (Lemma 62)
• factorization of a polynomial into monomials (Theorem 63) (IoP)
• roots of polynomial with real coefficients (Theorem 64) (IoP)
• factorization of a polynomial with real coefficients (Theorem 65)
• existence of the Riemann integral (Theorem 70) (IoP)
• integral with variable upper limit (Theorem 71) (IoP)
• integration by parts for Riemann integral (Theorem 73)
• substitution for Riemann integral (Theorem 74)

Theorems and definitions not required

• continuity and composition of functions (Theorem 9)
• empty theorem (Theorem 13)
• weak Lagrange theorem (Theorem 17)
• sufficient condition for a convex function in R² (Theorem 33)
• properties of transpose of a matrix (Theorem 36)
• determinant of a transpose (Theorem 48)
• definiteness of a diagonal matrix (Proposition 53)
• neccesary condition for definiteness (Proposition 54)
• Sylveser's criterion (Theorem 55)
• Hessian matrix and convexity (Theorem 56)
• Decomposition to partial fractions (Theorem 66)
• critical point of a function
• transpose of a matrix
• PD, PSD, ND, NSD, ID matrix
• Riemann integral over subintervals (Theorem 67)
• linearity of the Riemann integral (Theorem 68)
• Riemann integral and inequalities (Theorem 69)