Welcome to Modelling with Exponential Functions!

Ever wondered how scientists predict the spread of a virus, or how bankers calculate the interest on your savings? They use mathematical models. In this chapter, we are going to learn how to use exponential functions to describe things in the real world that grow or shrink very quickly.

Don't worry if this seems a bit abstract at first—we'll break it down into simple steps and use examples you can actually visualize. Let's dive in!


1. What is an Exponential Model?

In simple terms, an exponential model is used when the rate at which something changes depends on how much of it there is already.

Think about a rumor in a school: if only 2 people know it, it spreads slowly. If 500 people know it, it spreads incredibly fast because there are more people to tell it! This "snowball effect" is the heart of exponential growth.

The General Formula

Most exponential models in your OCR syllabus will look like this:
\( y = Ae^{kt} \)

Let's look at what each letter represents:

  • \( y \): The amount you have at any given time (e.g., the number of bacteria).
  • \( A \): The initial value (what you started with when \( t = 0 \)).
  • \( e \): Euler's number (approx. 2.718). We use this because its gradient properties make the math much easier!
  • \( k \): The growth constant. If \( k \) is positive, it’s growing. If \( k \) is negative, it’s decaying (shrinking).
  • \( t \): Time (seconds, years, days, etc.).

Quick Review: The graph of \( y = Ae^{kt} \) always crosses the y-axis at \( (0, A) \). If \( k > 0 \), the graph shoots upward. If \( k < 0 \), the graph curves downward toward the x-axis.


2. Why do we use the function \( e^{kx} \)?

You might ask: "Why can't we just use \( y = 2^x \)?". According to the syllabus (OCR Ref. 1.06b), we use \( e \) because of its gradient.

The derivative (gradient) of \( e^{kx} \) is \( ke^{kx} \). This means the rate of change is directly proportional to the function itself.

Analogy: Imagine a bank account that pays you more interest the more money you have in it. The "rate" of your wealth growing is tied to your current wealth. This is exactly what \( e^{kx} \) describes!

Key Takeaway: Exponential models are perfect for any situation where the "speed of change" depends on the "current size."


3. Exponential Growth vs. Exponential Decay

We can split our models into two main categories:

A. Exponential Growth (\( k > 0 \))

This is for things that get bigger and bigger, faster and faster.

  • Population Growth: The more people there are, the more babies are born.
  • Compound Interest: Interest is calculated on the total balance, so your money grows exponentially.

B. Exponential Decay (\( k < 0 \))

This is for things that shrink quickly at first, then slow down as they get smaller.

  • Radioactive Decay: Elements like Carbon-14 disappear over time.
  • Drug Concentration: When you take a painkiller, the amount in your blood is highest at the start and decays as your body processes it.

Did you know? This is why doctors tell you to take medicine every 4 or 8 hours. They are trying to top up the "exponential decay" before the concentration in your blood gets too low!


4. How to Solve Modelling Problems

Most exam questions will ask you to find either the initial amount, the growth constant \( k \), or a specific time \( t \).

Step-by-Step Guide to finding \( t \):

Suppose you have the equation \( 100 = 20e^{0.5t} \) and you need to find \( t \).

  1. Isolate the \( e \) part: Divide both sides by 20.
    \( 5 = e^{0.5t} \)
  2. Take the Natural Log (ln) of both sides: This "undoes" the \( e \).
    \( \ln(5) = \ln(e^{0.5t}) \)
  3. Simplify: Remember that \( \ln(e^x) = x \).
    \( \ln(5) = 0.5t \)
  4. Solve for \( t \): Divide by 0.5.
    \( t = \frac{\ln(5)}{0.5} \approx 3.22 \)

Common Mistake: Don't try to take the log before you've divided by the number in front of \( e \). Always get the \( e^{kt} \) term on its own first!


5. Limitations and Refinements

In the real world, nothing grows exponentially forever. A population of rabbits can't grow until there are more rabbits than atoms in the universe!

As a student of OCR Mathematics A, you need to be able to comment on the limitations of a model (OCR Ref. 1.06i):

  • Resource Limits: Animals run out of food or space.
  • External Factors: A change in interest rates or a new law might stop a financial model.
  • Refinement: We might need to adjust the model to include a "ceiling" or a maximum value that the graph cannot cross.

Quick Review Box:
- Growth: \( k \) is positive.
- Decay: \( k \) is negative.
- Initial Value: Set \( t = 0 \).
- Solving for time: Use natural logs (\( \ln \)).


Summary Checklist

Before you move on to practice questions, make sure you can:

  • Identify the initial value in an equation (it's the constant \( A \)).
  • Recognize whether a model is growth or decay based on the sign of \( k \).
  • Use natural logarithms to solve for the time variable \( t \).
  • Explain why \( e^{kx} \) is used (because the rate of change is proportional to the value).
  • Suggest limitations (e.g., populations cannot grow indefinitely due to lack of food).

You've got this! Exponential modelling is just about understanding the pattern of fast change. Keep practicing those log rearrangements, and the rest will fall into place.