Requirements for the exam for Mathematics II, SS 2014/2015

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
• convergence of a sequence in Rⁿ
• 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
• 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
• 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
• critical point 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
• rational function
• Riemann integral

Theorems with easier proofs

• properties of the Euclidean metric
• properties of open sets
• characterisation of convergence of sequences in Rⁿ
• characterisation of closed sets
• properties of closed sets
• boundedness of the closure
• continuity and level sets
• necessary condition for a local extremum
• continuity of functions of class C¹
• super-level sets of concave functions
• extremum of a concave function of class C¹
• uniqueness of an extremum
• super-level sets of quasi-concave functions
• structure of the set ∫f(x)dx
• linearity of antiderivative
• integration by substitution
• integration by parts

Theorems with harder proofs

• Heine theorem
• characterisation of compact sets in Rⁿ (proof in R²)
• on attaining extrema of functions
• weak Lagrange theorem
• tangent hyperplane theorem
• implicit function theorem
• Lagrange multipliers theorem in R²
• integral with variable upper limit

Theorems without proof

• properties of the interior and the closure
• continuity and arithmetic operations
• continuity and composition of functions
• derivative of a compound function
• interchanging of partial derivatives
• implicit functions theorem
• Lagrange multipliers theorem with more constraints
• relation of concavity and continuity
• characterisation of concave functions of class C¹
• existence of an antiderivative
• existence of the Riemann integral
• Riemann integral over subintervals
• linearity of the Riemann integral
• Riemann integral and inequalities
• Newton-Leibniz formula