Using Mathematical Models

Video masterclass

Topic summary

Mathematical models are simplified representations of real-world situations, designed to help understand, analyse, and predict behaviours within a system. These models use mathematical equations and expressions to describe how variables interact and evolve over time.

Purpose of Mathematical Models

Mathematical models serve several key purposes:
  • They help explain complex systems or processes in a simplified way.
  • They allow predictions about future behaviour based on current or past data.
  • They support decision-making by simulating different scenarios and outcomes.

Constructing a Model

When constructing a mathematical model, a few steps are generally followed:
  • Identify the key variables involved in the system.
  • Determine the relationships between these variables (often through equations or functions).
  • Make necessary assumptions to simplify the system while maintaining its essential characteristics.
For example, to model the motion of an object, you might use the equation: \[ s = ut + \frac{1}{2} a t^2 \] Where:
  • \(s\) is the displacement.
  • \(u\) is the initial velocity.
  • \(a\) is the acceleration.
  • \(t\) is the time.

Limitations of Mathematical Models

All models are simplifications and, as such, have limitations:
  • They rely on assumptions that may not hold true in all real-world situations.
  • They may ignore variables that could affect the system.
  • They can only provide estimates or predictions, not exact results.

Testing and Refining Models

Once a model is created, it must be tested against real-world data to determine its accuracy. If the model’s predictions are not accurate, adjustments can be made to the model's assumptions or variables to improve its fit.

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