Real-world examples of exponential growth and decay (calculus)

📚 Quick Study Guide

• 🌱 Exponential Growth: Occurs when the rate of increase is proportional to the current value. The general formula is $y = Ae^{kt}$, where:
  • $y$ is the final amount
  • $A$ is the initial amount
  • $k$ is the growth rate (positive)
  • $t$ is the time
• ☢️ Exponential Decay: Occurs when the rate of decrease is proportional to the current value. The general formula is $y = Ae^{-kt}$, where:
  • $y$ is the final amount
  • $A$ is the initial amount
  • $k$ is the decay rate (positive)
  • $t$ is the time
• 🧪 Half-Life: The time it takes for half of a substance to decay. It's related to the decay constant $k$ by the formula: $t_{1/2} = \frac{\ln(2)}{k}$.
• 📈 Applications of Exponential Growth:
  • Population growth
  • Compound interest
  • Spread of diseases
• 📉 Applications of Exponential Decay:
  • Radioactive decay
  • Drug metabolism
  • Cooling of an object
• 💡 Key Calculus Concepts: Derivatives are used to find the rate of change. Integrals can be used to find the total amount over a time period.

Practice Quiz

1. What is the general formula for exponential growth?
   1. A) $y = Ae^{-kt}$
   2. B) $y = Ae^{kt}$
   3. C) $y = A + kt$
   4. D) $y = A - kt$
2. A population of bacteria doubles every 3 hours. If the initial population is 100, what is the population after 6 hours?
   1. A) 200
   2. B) 300
   3. C) 400
   4. D) 800
3. What is the formula for half-life ($t_{1/2}$) in terms of the decay constant $k$?
   1. A) $t_{1/2} = \frac{k}{\ln(2)}$
   2. B) $t_{1/2} = k\ln(2)$
   3. C) $t_{1/2} = \frac{\ln(2)}{k}$
   4. D) $t_{1/2} = -k\ln(2)$
4. A radioactive substance decays according to the formula $y = Ae^{-0.02t}$, where $t$ is in years. What is the approximate half-life of the substance?
   1. A) 34.66 years
   2. B) 0.02 years
   3. C) 50 years
   4. D) 100 years
5. The rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature. This is an example of:
   1. A) Exponential growth
   2. B) Exponential decay
   3. C) Linear growth
   4. D) Constant rate
6. A bank account earns interest compounded continuously at a rate of 5% per year. If the initial deposit is $1000, how much will be in the account after 10 years?
   1. A) $1500
   2. B) $1648.72
   3. C) $1050
   4. D) $1628.89
7. Which of the following is NOT an application of exponential growth?
   1. A) Population growth
   2. B) Compound interest
   3. C) Radioactive decay
   4. D) Spread of diseases