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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
  • gradient
  • 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 equation

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

Combinatorics

Mathematical Methods

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