## 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