Does caffeine build up if you drink coffee every few hours?

Why it makes a good exploration

Pharmacologists design dosing schedules with this model. It combines exponential decay with a geometric series that has a clean limit — and it is about something many students (and teachers) do every day.

The mathematics you'll need

• Exponential decay and half-life
• Geometric series and the sum to infinity for the steady state
• Piecewise (sawtooth) functions
• HL: the DE dC/dt = −kC between doses

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use published caffeine contents of drinks and a published range of caffeine half-lives (roughly 3–7 hours in healthy adults) and cite them.

• Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

1. Derive concentration after one dose.
2. Add doses and show the peaks form a geometric series.
3. Find the steady-state peak and trough.
4. Vary half-life and interval; find schedules that stay below a chosen level.
5. Reflect on individual variation and absorption time.

Showing personal engagement

• Model a typical day of your own or a family member's drinks.
• Compare coffee, tea and energy drinks.
• Explain why a late coffee affects sleep using your model.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Add an absorption phase (a two-compartment model) and compare peak timing with published data.