Introduction: The Power of Compounding
Welcome to one of the most exciting parts of A Level Mathematics! Have you ever wondered why a rumor spreads so fast, or why interest on a bank account grows quicker over time? This happens because of exponential growth. Conversely, exponential decay explains why medicine leaves your system or how radioactive materials lose their energy.
In this chapter, we are going to look at how to model these real-world situations using the number \( e \). Don’t worry if the formulas look a bit intimidating at first; we’ll break them down into simple steps that anyone can follow!
1. What Makes it "Exponential"?
In most math problems, things change at a steady rate (like walking at 3 miles per hour). However, in the real world, many things change based on how much is already there.
Imagine a pond with a single lily pad that doubles every day. On day 1 there is 1, day 2 there are 2, day 3 there are 4, then 8, 16, and so on. The rate of growth is getting faster because there are more lily pads to produce new ones! This is the core of exponential growth.
The Key Relationship
The syllabus (Ref: E9) tells us an important rule: If the gradient (the rate of change) of a graph is proportional to its y-coordinate, the result is an exponential graph. Mathematically, we write this as: \( \frac{dy}{dt} = ky \) where \( k \) is a constant. If \( k \) is positive, it grows. If \( k \) is negative, it decays.
Quick Review: - Growth: The more you have, the faster it increases. - Decay: The less you have, the slower it decreases.
2. The General Model: \( y = Ae^{kt} \)
To solve problems, we use a standard formula. You will see this everywhere in your exams:
\( y = Ae^{kt} \)
Let's break down what these letters actually mean:
- \( y \): The amount you have at any given time.
- \( A \): The initial amount (the value of \( y \) when time \( t = 0 \)).
- \( e \): Euler's number (approx. 2.718). We use this because its gradient is exactly the same as its value!
- \( k \): The growth constant (positive for growth, negative for decay).
- \( t \): Time.
Analogy: Think of \( A \) as the "starter motor" of a car and \( k \) as the "accelerator." \( A \) tells you where you started, and \( k \) tells you how fast you're picking up speed.
Key Takeaway: Whenever a question mentions a "constant relative rate" or "growth proportional to the size," reach for the \( y = Ae^{kt} \) formula!
3. Step-by-Step: Solving Decay Problems
Radioactive decay is a classic exam favorite. Let's look at how to handle a "Half-Life" problem.
Example: A radioactive substance has a half-life of 10 years. If we start with 100g, how much is left after 25 years?
Step 1: Identify your starting value (\( A \)). We start with 100g, so \( A = 100 \). Our formula is now \( y = 100e^{kt} \).
Step 2: Use the "half-life" to find \( k \). In 10 years (\( t = 10 \)), the amount \( y \) will be half of 100, which is 50. \( 50 = 100e^{k \times 10} \) \( 0.5 = e^{10k} \) Take the natural log (\( \ln \)) of both sides: \( \ln(0.5) = 10k \) \( k = \frac{\ln(0.5)}{10} \approx -0.0693 \)
Step 3: Solve for the target time. Now use \( t = 25 \) in your completed formula: \( y = 100e^{-0.0693 \times 25} \) \( y \approx 17.7g \)
Don't worry if this seems tricky at first! The most common mistake is forgetting that for decay, your \( k \) value must be negative. If your substance is growing instead of shrinking, check your signs!
4. Real-World Applications (Syllabus E11)
The MEI syllabus expects you to apply these models to specific areas. Here is what you need to know for each:
Continuous Compound Interest: In finance, money grows exponentially if interest is calculated every single second. Tip: If the interest rate is 5%, then \( k = 0.05 \).
Drug Concentration: When you take a pill, the concentration in your blood is highest at the start and then decays over time as your body processes it. Did you know? Doctors use these math models to decide how many hours you should wait between doses!
Population Growth: Bacteria in a petri dish or humans in a city initially grow exponentially because more people means more births.
5. Limitations and Refinements
Math models are great, but they aren't perfect. The syllabus (E11) wants you to be able to critique a model.
Why might exponential growth stop? - Resources: A population cannot grow forever because it will run out of food or space. - External Factors: A disease or a predator might change the growth rate. - Long-term values: Often, a model will have a "carrying capacity" or a limit it cannot cross (an asymptote).
Key Takeaway: If an exam question asks "Why might this model be unrealistic after 100 years?", talk about physical limits like space, food, or resources.
6. Summary and Quick Review
Common Mistakes to Avoid: - Units: Make sure your time (\( t \)) matches the units of \( k \). If \( k \) is "per year," \( t \) must be in years. - Calculator Error: When using \( e^{kt} \), make sure the entire \( kt \) is in the exponent (use brackets if needed!). - The Inverse: Remember that \( \ln \) is the "undo" button for \( e \). Use it to get the variable down from the power.
Memory Aid: "A-K-T" - A is the Amount at the start. - K is how Kwickly it grows. - T is the Time passed.
Final Quick Check:
1. If \( k > 0 \), is it growth or decay? (Answer: Growth)
2. What is the value of \( y \) when \( t = 0 \) in the equation \( y = 500e^{kt} \)? (Answer: 500)
3. What is the gradient of \( e^{3x} \)? (Answer: \( 3e^{3x} \))
You've got this! Exponential growth and decay is just a fancy way of saying "the more you have, the more you get (or lose)." Practice a few half-life and interest problems, and you'll be a master in no time.