Requirements for the exam for Mathematics I, WS 2014/2015

Key notions

• supremum and infimum of a set of real numbers
• finite limit of a sequence
• infinite limit of a sequence
• 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
• 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

Definitions

• a set bounded from above
• a set bounded from below
• upper bound
• lower bound
• bounded set
• 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
• convergent sequence
• subsequence
• extended real line
• 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
• punctured neighbourhood of a point
• local maximum
• local minimum
• general power - a^b
• function arcsin
• function arccos
• function arctg
• function arccotg
• strictly convex function
• strictly concave function
• n-th derivative of a function at a point
• inflection point
• asymptote of a function

Theorems with easier proofs

• Cauchy inequality
• existence of a supremum
• Archimedean property
• existence of an integer part
• density of rational and irrational numbers
• uniqueness of a limit of a sequence
• limit of a sequence and boundedness of a sequence
• limit of a subsequence
• limit of a product of a bounded and null sequence
• limits and ordering
• two policemen theorem for sequences
• limit of a sequence of the type "A/0"
• supremum and limits
• limit of a monotone sequence
• uniqueness of a limit of a function
• boundedness and a limit of a function
• continuity and arithmetic operations
• limits of functions and inequalities
• two policemen for functions
• Heine theorem
• limit of a monotone function
• an image of an interval under a continuous function (including "no holes" lemma)
• boundedness of a continuous function
• exponential function and its properties
• relation of the derivative and continuity
• necessary condition for a local extremum
• Lagrange theorem (mean value)
• sign of the derivative and monotonicity
• characterisation of convex functions

Theorems with harder proofs

• existence of nth root (proof only for n=2)
• arithmetics of finite limits of sequences
• Bolzano-Weierstraß theorem
• arithmetics of limits of functions
• limit of a compound function
• Bolzano (intermediate value) theorem
• extrema of continuous functions
• inverse function and continuity
• properties of the logarithm
• arithmetics of derivatives
• derivative of a compound function (chain rule)
• derivative of an inverse function
• Rolle's theorem
• computation of one-sided derivative
• second derivative and convexity
• necessary condition for inflection
• sufficient condition for inflection

Theorems without proof

• de Morgan rules
• arithmetics of limits of sequences
• limit of a function of the type "A/0"
• continuity of a compound function on an interval
• the function sine
• l'Hospital's rule
• existence of an asymptote