B: Algebra and functions: Use of functions in modelling, including consideration of limitations and refinements of the models.

The idea

Modelling uses functions to represent real-world situations, enabling predictions and explanations. You must evaluate how well a model fits reality. For example, a cooling cup of tea modelled by $T = 20 + 75e^{-0.05t}$ assumes the ambient temperature remains exactly 20°C. While it correctly models the cooling rate decreasing as temperature drops, a limitation is that it ignores factors like evaporation or draughts. Refinements could involve adjusting parameters, adding terms for fluctuating room temperatures, or restricting the valid domain (e.g., $t \ge 0$). You must state assumptions, interpret parameters in context, evaluate the model's limitations, and suggest sensible improvements.

What earns the marks

• State clearly what each variable and parameter represents in the real context, including units.
• Use the function to calculate values or solve equations within the model's valid domain.
• Identify at least one limitation, such as the model ignoring external environmental factors or failing for large values of the independent variable.
• Suggest a specific refinement, such as adding a term to account for a neglected factor or restricting the domain.
• Comment on the reasonableness of predictions by comparing them with given data or expected real-world behaviour.
• Explain how changing a parameter affects the model's output in context.

Common mistakes and how to put them right

• Ignoring the real-world context and treating the function purely algebraically.

  Put it right: Relate variables and parameters to the situation, explicitly stating units and physical meaning.

• Assuming the model is valid for all values of the variable, including negative time or extreme future values.

  Put it right: Identify the domain over which the model is intended and check whether predictions outside it are sensible.

• Misinterpreting exponential decay models by claiming they assume a 'constant cooling rate'.

  Put it right: Recognise that in models like $T = A + Be^{-kt}$, the rate of change is proportional to the difference from the ambient value; the rate itself decreases over time. State valid limitations like ignoring evaporation instead.

Exam tips

• Read the modelling question carefully and underline the context, variables, and any data given before calculating.
• When asked for limitations, think about assumptions made in deriving the function (e.g., constant ambient temperature) and how they might fail in reality.
• For refinements, suggest a concrete mathematical change to the function or its domain, rather than just stating 'make it more accurate'.