# Modelling

#### 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.