Modelling with exponential functions in SL AI (with a real dataset)

Modelling with Exponential Functions in SL AI (with a Real Dataset)

I want to talk about exponential functions today. Not just the theory, but how we use them to model real-world phenomena. In IB Maths SL AI, this topic is central. My students often find the initial setup daunting, but once they see a practical application, it clicks. I’ve seen this many times in my classroom. We move from abstract definitions to concrete, observable patterns. That is the goal of AI: to apply mathematics.

My aim here is to walk you through a real-world example. We will use a dataset, build an exponential model, and interpret its implications. This is the kind of problem you might face in Paper 2. Understanding the steps now will save you stress during exams.

Understanding Exponential Growth and Decay

Before we jump into data, let's establish what an exponential function looks like. Its general form is $f(x) = ab^x$ or $f(x) = ae^{kx}$. For SL AI, we primarily focus on $f(x) = ab^x$ for discrete data or $f(x) = ae^{kx}$ for continuous growth/decay. The key characteristic is that the rate of change is proportional to the function's value. This means things grow or decay faster as they get larger or smaller, respectively.

In $f(x) = ab^x$:

• $a$ is the initial value (when $x=0$).
• $b$ is the growth/decay factor. If $b > 1$, it's growth. If $0 < b < 1$, it's decay.
• $x$ is the independent variable, often time.

In $f(x) = ae^{kx}$:

• $a$ is again the initial value.
• $e$ is Euler's number, approximately $2.718$.
• $k$ is the continuous growth/decay rate. If $k > 0$, growth. If $k < 0$, decay.

Both forms are interconvertible, meaning you can express one in terms of the other. The IB often expects you to work with both, converting as needed. This shows a deeper understanding of the underlying principles.

Working with a Real Dataset: COVID-19 Cases

Let's use a real-world example: the early spread of COVID-19 cases in a country. I remember teaching this concept during the pandemic. My students could see the relevance immediately. We are going to model the cumulative number of cases. For this example, let's use hypothetical (but realistic) data for the first few weeks of an outbreak.

Here's our dataset:

Our goal is to find an exponential model of the form $y = ab^x$ that best fits this data. In the exam, you will use your GDC for this. I always remind my students to be familiar with their calculator's regression functions. It saves time and prevents errors. Specifically, you will use the exponential regression feature.

Steps on your GDC (TI-Nspire or Casio fx-CG50, for example):

1. Enter the data into two lists (e.g., L1 for Day, L2 for Cases).
2. Go to the statistics menu (e.g., "Stat Calc" or "Menu -> Statistics").
3. Select "Exponential Regression" (often ExpReg).
4. Specify your independent and dependent lists.
5. The calculator will output the values for $a$ and $b$, and often $r^2$ (coefficient of determination).

When I run this data through my GDC, I get approximately:

• $a \approx 10.05$
• $b \approx 1.206$
• $r^2 \approx 0.999$

So, our exponential model is $y = 10.05(1.206)^x$. The $r^2$ value being very close to $1$ indicates an excellent fit. This makes sense for early exponential growth phases of outbreaks.

Interpreting the Model and Making Predictions

Now we have a model: $y = 10.05(1.206)^x$. What does it tell us?

• The initial number of cases ($a$) is approximately $10.05$. This aligns with our first data point.
• The daily growth factor ($b$) is approximately $1.206$. This means the number of cases increases by about $20.6\%$ each day. This percentage is crucial for interpreting real-world implications.

We can use this model to make predictions. For example, what is the predicted number of cases on Day 25?

$y = 10.05(1.206)^{25}$

$y \approx 10.05 \times 98.45$

$y \approx 989.5$

So, our model predicts approximately $990$ cumulative cases on Day 25. This is extrapolation, which always carries more uncertainty than interpolation. The further you extrapolate, the less reliable your prediction becomes. My students often forget to mention this limitation in their answers. Always discuss limitations in Paper 3 for HL AI, but also in Paper 2 for SL AI where relevant.

We can also use the model to answer questions about specific thresholds. For example, on what day would the cumulative cases reach $5000$ based on this model?

$5000 = 10.05(1.206)^x$

First, divide by $10.05$:

$\frac{5000}{10.05} = (1.206)^x$

$497.51 \approx (1.206)^x$

Now, use logarithms to solve for $x$. I usually advise students to use the natural logarithm or $\log_{10}$ since those buttons are readily available on the GDC. Using $\ln$:

$\ln(497.51) = \ln((1.206)^x)$

$\ln(497.51) = x \ln(1.206)$

$x = \frac{\ln(497.51)}{\ln(1.206)}$

$x \approx \frac{6.21}{0.187}$

$x \approx 33.2$ days

So, according to this model, it would take approximately $33$ days for the cumulative cases to reach $5000$. Remember that we are dealing with discrete units (days, people), so rounding to a practical number makes sense. If asked for the first full day, it would be day $34$. These small details matter.

Limitations and Further Considerations

It's important to discuss the limitations of any model, especially exponential models for real-world phenomena. While an exponential model fits initial growth phases well, it rarely holds indefinitely. Consider the COVID-19 example: eventually, factors like social distancing, vaccinations, and herd immunity slow the spread. The growth rate changes, and the model would need adjustment or replacement with a more complex function, like a logistic model.

In IB SL AI, you might be asked to comment on these limitations. For instance, why would this model eventually become inaccurate? The answer lies in finite resources, saturation points, or external interventions. The model assumes an unconstrained environment, which is rarely the case in reality. This is where critical thinking comes into play, a skill the IB values highly. You can find more insights on applying critical thinking in your exam preparations on our Paper 2 SL AI guide.

Another consideration is the choice of the exponential form. If a problem states a continuous growth rate, using $y = ae^{kx}$ might be more appropriate. If it's discrete annual growth, $y = ab^x$ is better. Sometimes, you'll need to convert between them: $b = e^k$ and $k = \ln(b)$. Being flexible with these forms is part of mastering the topic.

Summary and Next Steps

Modelling with exponential functions is a fundamental skill in IB Maths SL AI. We started with the basic forms, then applied them to a real-world dataset of COVID-19 cases. We used a GDC to find the regression equation, interpreted the parameters, and made predictions. Finally, we discussed the crucial aspect of model limitations and their real-world implications.

My hope is that this practical example clarifies how these functions work beyond just abstract equations. It is about applying mathematics to understand the world. To further solidify your understanding, I recommend practicing with various datasets. Explore different scenarios: population growth, radioactive decay, compound interest. You can find more practice problems and detailed explanations in our study notes. Keep practicing those GDC skills and always remember to contextualise your answers. This will prepare you well for your exams.