Mathematical Modelling
- Mathematical modelling is a crucial skill in applied mathematics, allowing students to represent real-world phenomena using mathematical equations and functions.
- In the context of SL 2.6, students focus on developing, fitting, testing, and using theoretical models, particularly those covered in SL 2.5.
The Modelling Process
The modelling process typically involves several key steps:
- Identifying the problem
- Making assumptions and defining variables
- Formulating the model
- Solving the mathematical problem
- Interpreting the solution
- Validating the model
- Refining the model (if necessary)
It's important to remember that modelling is an iterative process. Often, the first attempt at creating a model may not be perfect, and refinements are necessary based on testing and validation results.
Developing and Fitting Models
Recognizing Appropriate Models
When presented with a real-world scenario, students must be able to recognize which theoretical model from SL 2.5 is most appropriate. This could include:
- Linear models: $y = mx + b$
- Quadratic models: $y = ax^2 + bx + c$
- Exponential models: $y = ab^x$ or $y = ae^{kx}$
- Sinusoidal models: $y = a\sin(bx + c) + d$
If data shows a constant rate of change, a linear model might be appropriate. If there's exponential growth or decay, an exponential model could be suitable.
Determining a Reasonable Domain
The domain of a model should reflect the realistic constraints of the situation being modelled.
ExampleWhen modelling population growth, the domain would typically start at $t=0$ (representing the initial time) and have an upper limit based on the timeframe being considered.
Finding Model Parameters
There are several methods to determine the parameters of a model:
- Solving Equations Simultaneously: This involves using known data points to create a system of equations and solving them using technology.
