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.
For a linear model $y=mx+b$, if we know points $(1, 3)$ and $(2, 5)$, we can set up:
$$3=m(1)+b$$
$$5=m(2)+b$$
Solving these simultaneously gives $m=2$ and $b=1$.
- Consideration of Initial Conditions: This is particularly useful for exponential and sinusoidal models.
In an exponential decay model $y = ae^{-kt}$, if we know the initial value $y(0) = 100$, then $a = 100$.
- Substitution of Points: This method involves plugging known points into the general form of the model.
For a quadratic model $y = ax^2 + bx + c$, if we know points (0, 1), (1, 6), and (2, 15), we can set up:
$1 = a(0)^2 + b(0) + c$ $6 = a(1)^2 + b(1) + c$ $15 = a(2)^2 + b(2) + c$
Solving these gives $a = 2$, $b = 3$, and $c = 1$.
TipWhen solving systems of equations, use technology like graphing calculators or computer algebra systems to handle more complex calculations efficiently.
Testing and Reflecting on the Model
Assessing Appropriateness and Reasonableness
After developing a model, it's crucial to assess whether it makes sense in the context of the problem.
ExampleIf a population model predicts negative values or unrealistically large values, it may not be appropriate for the given situation.
NoteJustifying Model Choice
Students should be able to explain why they chose a particular model based on:
- The shape of the data
- Properties of the curve
- The context of the situation
If data shows a clear periodic pattern, a sinusoidal model might be justified. If there's a constant percentage increase, an exponential model could be more appropriate.
Using the Model
Interpreting and Making Predictions
Once a model is developed and validated, it can be used to:
- Interpret current data points
- Make predictions about future values
- Interpolate between known data points
If we have a linear model for ice cream sales based on temperature, $S = 20T - 100$ (where $S$ is sales in dollars and $T$ is temperature in Celsius), we can predict sales for any given temperature within the model's domain.
Dangers of Extrapolation
Common MistakeA common mistake is to use a model to predict values far outside the range of the original data. This is called extrapolation and can lead to unreliable results.
ExampleFor instance, while our ice cream sales model might work well for temperatures between 10°C and 35°C, using it to predict sales at 50°C or -10°C could give unrealistic results.
Key Notes
- While fitting models using regression is covered in topic 4, students at SL are not expected to perform non-linear regressions.
- Students should be able to set up and solve up to three linear equations in three variables using technology.
The skills developed in this topic are fundamental to many areas of applied mathematics and are crucial for students planning to pursue fields such as engineering, economics, or data science.
