Welcome to Exponential Growth and Decay!
In this chapter, we explore one of the most powerful tools in mathematics. Have you ever wondered how a viral video spreads across the internet, how interest builds up in a savings account, or how doctors know how long a medicine stays in your system? All of these are real-world examples of exponential growth and decay.
Don't worry if this seems tricky at first—we are going to break it down step-by-step. By the end of these notes, you'll be able to model everything from rabbit populations to radioactive waste!
1. What is an Exponential Model?
Most things in the world don't change at a constant, steady rate (like adding 2 every hour). Instead, many things change at a rate proportional to their current size. The bigger a population is, the faster it grows. The more of a drug is in your blood, the faster your body processes it.
In your MEI syllabus, we use the special number \(e\) (roughly 2.718) to model this. The standard formula you will use is:
\(y = Ae^{kt}\)
Let's break down what these letters actually mean:
- \(y\): The amount or value at a specific time.
- \(A\): The initial value (the amount you have at the very start, when \(t = 0\)).
- \(e\): The mathematical constant (use the \(e^x\) button on your calculator!).
- \(k\): The growth constant. This determines how fast things are changing.
- \(t\): Time (usually in seconds, years, or hours).
Quick Review: Growth vs. Decay
How do you know if something is getting bigger or smaller just by looking at the equation?
- Exponential Growth: If \(k > 0\) (positive), the value increases. Think of a snowball rolling downhill!
- Exponential Decay: If \(k < 0\) (negative), the value decreases. Think of a hot cup of tea cooling down.
Key Takeaway: The value of \(A\) is always the starting point, and the sign of \(k\) tells you if it's growing or shrinking.
2. Why use \(e\)?
You might ask: "Why can't we just use \(y = 2^x\)?"
The reason we use \(e\) is because of its unique gradient. As noted in your syllabus (Ref E9), the gradient of \(e^{kx}\) is \(ke^{kx}\). This means the rate of change is directly linked to the function itself. This makes it the "natural" choice for modeling growth that depends on current size.
Analogy: Imagine a bank that doesn't just give you interest once a year, but every single tiny fraction of a second. That "continuous" growth is exactly what \(e\) represents!
3. Solving Growth and Decay Problems
Most exam questions follow a similar pattern. You are usually given some information and asked to find the value of \(k\) first, then use it to predict a future value.
Step-by-Step Guide:
- Identify your variables: Write down what \(A\), \(t\), and \(y\) are from the text.
- Substitute: Plug these into \(y = Ae^{kt}\).
- Solve for \(k\): You will usually need to use Natural Logarithms (\(\ln\)) to get the \(k\) down from the power. Remember that \(\ln(e^x) = x\).
- Answer the question: Now that you have the full formula, plug in the new time or value requested.
Common Mistake: Forgetting to include the minus sign for decay. If the problem says "decaying at a rate of...", your \(k\) value must end up being negative!
4. Real-World Applications
The syllabus (Ref E11) specifically mentions several areas where you might see these models applied:
Continuous Compound Interest
If money grows "continuously," we use \(V = Pe^{rt}\), where \(P\) is the principal (start money) and \(r\) is the interest rate. It's the same formula, just different letters!
Radioactive Decay
Radioactive substances vanish over time. We often talk about Half-life (the time it takes for half the substance to disappear).
Example: If a substance has a half-life of 10 years, then after 10 years, \(y = 0.5A\).
Drug Concentration
When you take ibuprofen, the amount in your blood is highest at the start and then decays exponentially as your kidneys filter it out.
Population Growth
Bacteria in a petri dish are the classic example. With unlimited food and space, they grow exponentially. However, this leads us to an important point...
Did you know? If a single bacterium divided every 20 minutes without stopping, in just two days it would weigh more than the entire Earth! This is why we need to look at limitations.
5. Limitations and Refinements
The syllabus requires you to "consider limitations and refinements" of these models. Simple exponential growth (\(y = Ae^{kt}\)) assumes that things can grow forever. In reality, this isn't possible.
Why models might be limited:
- Resources: Populations run out of food or space.
- Environment: Disease or predators might slow down growth.
- Context: A cooling cup of tea won't keep getting colder forever; it stops when it reaches room temperature.
Finding Long-Term Values
Sometimes you are asked what happens as \(t\) becomes very large (as \(t \to \infty\)).
- For decay (\(k < 0\)): \(e^{kt}\) gets closer and closer to zero.
- For growth (\(k > 0\)): \(e^{kt}\) goes toward infinity (unless the model is refined).
Key Takeaway: Always check if your mathematical answer makes sense in the "real world." If your model says there are 10 trillion rabbits on a small island, the model probably needs a "refinement"!
Quick Review Box
The Formula: \(y = Ae^{kt}\)
Finding k: Use \(\ln\) to solve for the power.
Growth: \(k\) is positive. Decay: \(k\) is negative.
Initial Value: This is \(A\) (when \(t=0\)).
Long-term: If decaying, \(y \to 0\).
Common Pitfalls to Avoid
- Units: Ensure your time (\(t\)) matches the units of \(k\). If \(k\) is "per year," \(t\) must be in years.
- Rounding: Do not round your value of \(k\) too early! Keep the exact value in your calculator (or use 4 or 5 decimal places) until the very end. Early rounding can lead to big mistakes in exponential sums.
- The "A" value: Don't assume \(A\) is always given. Sometimes you have to solve for \(A\) and \(k\) using simultaneous equations if you are given two different points in time.
Keep practicing! Exponential models might feel strange because they move so fast, but once you master the \(\ln\) button on your calculator, you'll find they are very predictable.