Welcome to Modelling with Exponentials!
Ever wondered how a social media post goes "viral," how a cup of coffee cools down, or why your savings account grows faster over time? These are all real-world examples of exponential functions. In this chapter, we’ll learn how to use these powerful mathematical tools to predict the future (at least, mathematically speaking!).
Don't worry if you find the word "exponential" a bit intimidating. By the end of these notes, you'll see that it’s just a way of describing things that grow or shrink at a rate that depends on how much is already there. Let's dive in!
1. What is an Exponential Model?
An exponential model is used when the rate of change of a quantity is proportional to the quantity itself. Think of it like a snowball rolling down a hill: the bigger it gets, the more snow it picks up, which makes it grow even faster!
The General Formula
In your OCR A Level course, you will mostly see models in this form:
\(V = Ae^{kt}\)
Let’s break down what each letter means (don't worry, they are just placeholders!):
- \(V\): The value at a specific time (e.g., population, temperature, or money).
- \(A\): The initial value. This is the amount you start with when time \(t = 0\).
- \(e\): Euler’s Number (approx. \(2.718\)). This is a special constant used because its gradient is the same as its value, making it perfect for modelling.
- \(k\): The growth constant. This determines how fast things change.
- \(t\): Time (usually in years, days, or seconds).
Quick Review: If you see a formula like \(P = 500e^{0.2t}\), you immediately know that the starting population was 500.
2. Growth vs. Decay
There are two main types of exponential models you need to recognize:
Exponential Growth (\(k > 0\))
This happens when the amount is increasing. The graph starts flat and shoots up like a rocket!
Example: A colony of bacteria doubling every hour.
Exponential Decay (\(k < 0\))
This happens when the amount is decreasing. The graph starts high and curves down toward zero (but never quite touches it).
Example: The amount of caffeine in your blood after a cup of tea, or radioactive decay.
Memory Aid: Think of k as the "kindness" of the growth. If \(k\) is positive, your bank account is "growing" (Growth). If \(k\) is negative, your battery life is "decaying" (Decay).
Quick Takeaway: If the number in the power is positive, it's getting bigger. If it's negative, it's getting smaller.
3. Working with the Natural Number \(e\)
You might ask: "Why use \(e\) instead of just a normal number like 2 or 10?"
Did you know? The number \(e\) is special because the function \(y = e^x\) is its own derivative. This means the gradient (how fast it’s changing) is exactly the same as the y-value (how much is there). This is why it shows up so often in nature!
Finding the Gradient
According to the syllabus (1.06b), the gradient of \(e^{kx}\) is \(ke^{kx}\).
In modelling terms, if your population is \(P = Ae^{kt}\), the rate of change is:
\(\frac{dP}{dt} = kP\)
This is a fancy way of saying: "The more people there are, the faster the population grows."
4. Step-by-Step: Solving a Modelling Problem
Most exam questions follow a similar pattern. Let's look at how to tackle them. Don't worry if this seems tricky at first; it's all about following the steps!
Step 1: Find the Initial Value (\(A\))
Usually, the question tells you the starting amount. If it says "A car is worth £20,000 when new," then \(A = 20,000\).
Step 2: Find the Growth Constant (\(k\))
You’ll be given a second piece of information (e.g., "After 2 years, the car is worth £15,000").
Substitute \(V = 15,000\), \(A = 20,000\), and \(t = 2\) into the formula:
\(15,000 = 20,000e^{2k}\)
Step 3: Use Logarithms to solve for \(k\)
- Divide by \(A\): \(0.75 = e^{2k}\)
- Take the Natural Log (ln) of both sides: \(\ln(0.75) = 2k\)
- Solve for \(k\): \(k = \frac{\ln(0.75)}{2} \approx -0.144\)
Step 4: Answer the specific question
Now that you have the full formula \(V = 20,000e^{-0.144t}\), you can find the value at any time \(t\), or find when the car will be worth a certain amount.
Common Mistake to Avoid: When using your calculator, don't round your value of \(k\) too early! Keep the full number in your calculator's memory to ensure your final answer is accurate.
5. Limitations of Exponential Models
While exponential models are great, they aren't perfect. In the real world, things can't grow forever!
Analogies:
- Population: A population of rabbits can't grow exponentially forever because they will eventually run out of food or space.
- Temperature: A hot cup of coffee won't keep cooling down to \(-100^{\circ}C\); it will stop at room temperature.
Refining the Model
Sometimes, we add a constant to the model to represent a limit. For example:
\(T = 20 + Ae^{-kt}\)
In this case, as time goes on, the \(Ae^{-kt}\) part goes to zero, and the temperature \(T\) settles at 20 (the room temperature).
Summary Point: Always check if your model makes sense for very large values of \(t\). If it predicts a billion rabbits in a small garden, the model needs refinement!
Quick Review Box
The Formula: \(y = Ae^{kt}\)
A: Starting amount (at \(t = 0\))
k > 0: Growth
k < 0: Decay
To solve for a power: Use the \(\ln\) button on your calculator!
Final Takeaway
Modelling using exponential functions is simply the art of turning real-world patterns into the equation \(V = Ae^{kt}\). Whether it's money in a bank or the spread of a flu, the steps are always the same: find your starting value, find your rate, and use your logs to solve. You've got this!