Mathematical modelling and problem solving in applied contexts
What this domain tests
Modelling tasks wrap the other three domains in a real context: a tank, a population, a drug concentration, a production process. You must choose a function type, fit its parameters to given data, work with it, and - this is where the marks are - interpret every result back in the situation, including units and the limits of the model.
The modelling cycle
- Simplify: name the variables, state the assumptions (constant rate, proportional growth, no saturation).
- Mathematise: choose linear, exponential (a·e^(kx) or a·b^x), logistic (saturating), trigonometric (periodic) or polynomial.
- Solve with the tools of Analysis.
- Interpret: derivative = rate with unit "per time"; integral = accumulated quantity (volume, energy, cost).
- Validate: does the result make sense, and is the domain of the model respected?
Fitting and interpreting
- Exponential decay with half-life T: k = ln2/T. Growth with doubling time T uses the same formula with k > 0.
- Two data points determine a and k in a·e^(kx): divide the two equations to eliminate a, then take ln.
- A model is valid only where its assumptions hold; extrapolating far beyond the data is a standard source of lost marks.
- Round with respect to the context (whole parts, cents, days) but keep intermediate values unrounded.
Worked example
A drug decays exponentially with half-life 6 hours; the initial dose gives 80 mg in the blood. Model: m(t) = 80·e^(−kt) with k = ln2/6 ≈ 0.1155 per hour. When has 90% decayed, i.e. m(t) = 8 mg? 80·e^(−kt) = 8 → e^(−kt) = 0.1 → −kt = ln 0.1 → t = ln10/k ≈ 2.3026/0.1155 ≈ 19.9. Answer: after about 20 hours. Cross-check: 90% decay lies between three and four half-lives (18 h leaves 12.5%, 24 h leaves 6.25%), so 19.9 h is plausible.
Traps
- No Antwortsatz and no unit - a correct number alone often earns only the method mark.
- Mixing up the base: take k from the half-life, not from a percentage per hour.
- Using the model outside its stated domain (negative times, saturation regions).
- Rounding k to 0.12 too early; the error is amplified in the exponent.
- Treating a discrete quantity (number of items) with a continuous model without saying so.