IA idea · Differential equations & dynamics
Does caffeine build up if you drink coffee every few hours?
Research question
With a first-order elimination model, what steady-state caffeine level is reached with regular doses, and how does the answer depend on half-life and dosing interval?
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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
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.
A possible outline
- Derive concentration after one dose.
- Add doses and show the peaks form a geometric series.
- Find the steady-state peak and trough.
- Vary half-life and interval; find schedules that stay below a chosen level.
- Reflect on individual variation and absorption time.
Pitfalls that cost marks
- Treating the half-life as fixed for everyone.
- Ignoring absorption (instant-dose assumption).
- Giving medical advice — keep it to the mathematics.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Exponential decay and half-life; Geometric series and the sum to infinity for the steady state |
| AA HL | Good fit | Exponential decay and half-life; Geometric series and the sum to infinity for the steady state |
| AI SL | Good fit | Exponential decay and half-life; Geometric series and the sum to infinity for the steady state |
| AI HL | Good fit | Exponential decay and half-life; Geometric series and the sum to infinity for the steady state |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: treating the half-life as fixed for everyone; and ignoring absorption (instant-dose assumption). Say how each one affects your answer.
Use of mathematics (E)
SL: exponential decay and half-life; geometric series and the sum to infinity for the steady state; piecewise (sawtooth) functions, each explained in your own words and carried out correctly.
HL adds: the DE dC/dt = −kC between doses, each explained in your own words and carried out correctly.
How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Add an absorption phase (a two-compartment model) and compare peak timing with published data.
Extending it for HL
This idea already has HL mathematics in it: the DE dC/dt = −kC between doses. Make that the centre of the exploration and use it with full rigour.
See a complete IA, marked
Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Exponential decay and half-life, geometric series and the sum to infinity for the steady state, piecewise (sawtooth) functions and hL: the DE dC/dt = −kC between doses.
Where can I get data for this IA?
Use published caffeine contents of drinks and a published range of caffeine half-lives (roughly 3–7 hours in healthy adults) and cite them. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “Does caffeine build up if you drink coffee every few hours?” filled in: the question, the maths and the data sources.
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