IB Maths: Modelling Exponential Growth & Decay FAQ

It's tested through GDC-heavy Paper 2 questions asking you to form, fit or interpret a model like $N=A{e}^{kt}$, plus regression tasks in the calculator and statistics sub-topics. According to the IB Mathematics guide (first exams 2021, still current for AA and AI), exponential functions sit under Topic 2 (Functions) and Topic 5 (Calculus) in HL.

You need the general model $N(t)=N_0{e}^{kt}$, where $k>0$ means growth and $k<0$ means decay, plus $\ln$ laws to solve for $t$ or $k$. Half-life and doubling-time problems use $N(t)=N_0{\left(\frac{1}{2}\right)}^{t/h}$ as an equivalent form.

A typical Paper 2 question: a radioactive sample decays from 80g to 20g in 12 days; find $k$ and the amount remaining after 30 days using $N(t)=80e^{kt}$. You solve $20=80e^{12k}$ for $k$, then substitute $t=30$ into the model.

The core model is identical in both, but AI leans harder on GDC regression and real-world fitting (population, depreciation), while AA expects you to also differentiate the model analytically and connect it to differential equations at HL. Both SL courses stop at algebraic manipulation and basic calculus.

Yes — it's one of the most reliable IA topics because it gives you a clear model to fit, test and criticise, which maps neatly onto the personal engagement and reflection criteria. I've marked dozens of IAs on cooling curves, population growth and drug elimination; the ones that score well go beyond fitting a curve and actually critique the model's limitations.

Yes, at HL you're expected to recognise that $\frac{dN}{dt}=kN$ is the differential equation whose solution is $N=N_0{e}^{kt}$, and to solve it by separation of variables. This connects Topic 5 (Calculus) directly to the exponential models from Topic 2.

The biggest one is confusing the sign of $k$ — writing decay as $e^{+kt}$ with a negative $k$ hidden inside, then losing track when asked to interpret it. A close second is rounding $k$ too early, which throws off every answer that follows.

Set $N(t)=\frac{1}{2}{N}_0$ (or $2N_0$ for doubling), substitute into $N_0{e}^{kt}$, cancel $N_0$, then take natural logs to isolate $t$. The result is $t=\frac{\ln (0.5)}{k}$ for half-life, which is always positive when $k$ is negative.

Usually it's entering the data in the wrong list order, forgetting to clear old lists first, or picking the wrong regression type — logarithmic and power regression give similarly shaped curves but very different equations. Always check the $r^2$ value the calculator gives you as a sanity check before writing anything down.

Master the algebra of forming the model from two data points, practise both the analytic (logs) and GDC (regression) routes, and always sanity-check your answer against the context — does a population of 40 million after 3 years actually make sense? Examiners reward answers that interpret the model, not just calculate it.

Targeted practice matters more than re-reading notes for this topic — you need repeated exposure to the 'find A and k from two data points' pattern until it's automatic. On RevisionPrep, the Topical Worksheets for Functions and Calculus include exponential modelling questions graded by difficulty, alongside worked Revision Notes.

It's not conceptually difficult — most students grasp the idea of growth and decay quickly — but it becomes a scoring risk when combined with logs, calculus or regression, especially under exam time pressure. Students who struggle usually have a gap in log laws from Topic 1, not in the exponential concept itself.