# 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.