Chapter 5
Functions and their Infinitesimal Properties

 5.1 Definition of Functions
 5.2 Difference Quotient and Differential Quotient
 5.3 Derivatives of a Few Fundamental Functions
  5.3.1 Powers and Polynomials
  5.3.2 Exponential Function
  5.3.3 Trigonometric Functions
  5.3.4 Rules for the Differentiation of combined Functions
  5.3.5 Derivatives of further Fundamental Functions
 5.4 Series expansion, Taylor Series
  5.4.1 Coefficients of the Taylor Series
  5.4.2 Approximation Formulas for simple Functions
  5.4.3 Derivation of Formulas and errors bounds for numerical differentiation
  5.4.4 Interactive Visualization of Taylor expansions
 5.5 Graphical Presentation of Functions
  5.5.1 Functions of 1 to 3 Variables
  5.5.2 Functions of four variables: World line in the Theory of Relativity
  5.5.3 General Properties of Functions y = f(x)
  5.5.4 Exotic Functions
 5.6 The Limiting process for Obtaining the Differential Quotient
 5.7 Derivative and Differential Equations
 5.8 Phase Space Diagrams
 5.9 Integral
  5.9.1 Definition of Anti-derivative via its Differential Equation
  5.9.2 Definite Integral and Initial Value
  5.9.3 Integral as Limit of a Sum
  5.9.4 The Definition of the Integral due to Riemann
  5.9.5 Lebesgue Integral
  5.9.6 Rules for the Analytical Integration
  5.9.7 Numerical Integration Methods
  5.9.8 Error Estimate for Numerical Integration
 5.10 Series Expansion (2): the Fourier Series
  5.10.1 Taylor Series and Fourier Series
  5.10.2 Determination of the Fourier coefficients
  5.10.3 Visualizing the Calculation of Coefficients and Spectrum
  5.10.4 Examples of Fourier Expansions
  5.10.5 Complex Fourier Series
  5.10.6 Numerical Solution of Equations and iterative Methods