Introduction to Modelling

  • Modelling involves creating mathematical models to describe real-world situations.
  • A mathematical model is a mathematical representation, often in the form of a function, of a system or process that helps to predict or understand its behaviour.
  • The process of modelling usually involves making assumptions, simplifying situations and dealing with a certain amount of uncertainty.

Steps in Modelling

  • A typical problem-solving cycle in modelling includes: identify problem, formulate a model, solve the model, validate the model and interpret the model results.
  • The process is iterative, meaning it is often necessary to revise and refine the model based on the results of validation or when new information becomes available.

Use of Functions in Modelling

  • Functions serve as the primary tool in creating mathematical models due to their ability to represent relationships.
  • The choice of a particular type of function (such as linear, quadratic, exponential etc.) depends on the nature and details of the situation being modelled.

Goodness of Fit

  • The concept of goodness of fit refers to the level of agreement between observed data and the predictions made by a model.
  • Techniques like least-squares regression are often employed to optimise the ‘fit’ of a model to data

Validity and Limitations of Models

  • The validity of a model is judged by the extent to which the assumptions and simplifications made accurately represent the system being modelled.
  • A model can be valid for some purposes and not for others. Understanding the limitations of a model is equally important to prevent misuse.

Example Modelling Problems

  • Modelling problems might include predicting population growth using exponential functions, analysing motion under gravity using quadratic functions, or interpreting periodic phenomena using trigonometric functions.
  • Each of these examples require different functions and assumptions to create a model that can accurately answer specific questions or predict certain outcomes.