Requirements for the exam for Mathematics I, WS 2016/2017

Key notions

• supremum and infimum of a set of real numbers: [Ghorpade, Limaye: page 5]
• finite limit of a sequence: [Ghorpade, Limaye: page 44]
• 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: [Ghorpade, Limaye: page 4]
• a set bounded from below: [Ghorpade, Limaye: page 4]
• upper bound: [Ghorpade, Limaye: page 4]
• lower bound: [Ghorpade, Limaye: page 4]
• bounded set: [Ghorpade, Limaye: page 4]
• maximum of a set: [Ghorpade, Limaye: page 5]
• minimum of a set: [Ghorpade, Limaye: page 5]
• integer part of a number: [Ghorpade, Limaye: page 6]
• sequence bounded from above
• sequence bounded from below
• bounded sequence
• increasing sequence: [Ghorpade, Limaye: page 49]
• decreasing sequence: [Ghorpade, Limaye: page 49]
• non-increasing sequence: [Ghorpade, Limaye: page 49]
• non-decreasing sequence: [Ghorpade, Limaye: page 49]
• monotone sequence: [Ghorpade, Limaye: page 49]
• strictly monotone sequence: [Ghorpade, Limaye: page 49]
• convergent sequence: [Ghorpade, Limaye: page 44]
• subsequence: [Ghorpade, Limaye: page 55]
• 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: [Ghorpade, Limaye: page 12, P.1.12]
• existence of a supremum: [Ghorpade, Limaye: page 5, P.1.2]
• Archimedean property: [Ghorpade, Limaye: page 5, P.1.3]
• existence of an integer part
• density of rational and irrational numbers
• uniqueness of a limit of a sequence: [Ghorpade, Limaye: page 44, P.2.2(i)]
• limit of a sequence and boundedness of a sequence: [Ghorpade, Limaye: page 44, P.2.2(ii)]
• limit of a subsequence: [Ghorpade, Limaye: page 55]
• limit of a product of a bounded and null sequence
• limits and ordering: [Ghorpade, Limaye: page 46, P.2.4(i)]
• two policemen theorem for sequences: [Ghorpade, Limaye: page 47, P.2.5]
• limit of a sequence of the type "A/0"
• supremum and limits: [Ghorpade, Limaye: page 47, P.2.6]
• limit of a monotone sequence: [Ghorpade, Limaye: page 49, P.2.8]
• uniqueness of a limit of a function[Trench: page 35, T.2.1.3]
• 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: [Ghorpade, Limaye: page 45, P.2.3]
• Bolzano-Weierstraß theorem: [Ghorpade, Limaye: page 55]
• arithmetics of limits of functions: [Ghorpade, Limaye: page 56, P.2.15]
• 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