Contents of the lecture: Mathematical Methods

Introduction

• motivation
• functions in economics
• recap of school knowledge
• important sets
• sigma sign
• fractional arithmetic
• percentage calculation
• roots, powers, logarithms
• absolute values
• transforming equations, p-q formula, factorising
• transforming inequations
• functions, inverse functions
• numerical sequences
• limits of sequences and functions, l'Hôpital's rule
• composite functions
• continuity

• functions of one variable:
• differentiability
• tangent line
• differential
• elasticity

Fundamentals of Financial Mathematics

• geometric series
• interests
• present values
• loans, annuities, credit periods

Linear Algebra

• matrices and vectors
• basic matrix algebra
• inverse matrix
• linear independence
• rank of a matrix
• linear map
• solving linear equations

Differentiable functions of several variables

• graphic representation, level curves
• continuity
• partial differentiability
• partial elasticity
• partial differential
• total differentiability
• tangent function
• total differential
• homogeneous functions, Euler's theorem
• vector-valued function, Jacobi matrix
• derivative rules, theorem of total derivative, chain rule
• implicit functions and their derivatives
• marginal rate of substitution
• elasticity of substitution
• higher order partial derivatives, Hessian matrix

Extrema of functions of several variables

• relative extrema
• convexity und concavity
• global extrema
• extrema at the boundary
• extrema under constraints:
• method of elimination
• method of Lagrange multipliers
• envelope theorem

Integral Calculus

• basics and economic examples
• integration by parts
• integration by substitution
• improper integrals
• integral calculus for functions of two variables
• integral calculus in the calculus of probability

Differential and difference equations (only until summer term 2018)

• applications in economics
• homogeneous and inhomogenous linear differential equations of order one
• separable differential equations
• linear difference equations