Test Questions for University Differential Equations: Exponential Models

📚 Quick Study Guide

• 📈 Exponential Growth: A quantity increases at a rate proportional to its current value. The model is given by $y(t) = y_0e^{kt}$, where $y_0$ is the initial value, $k > 0$ is the growth constant, and $t$ is time.
• 📉 Exponential Decay: A quantity decreases at a rate proportional to its current value. The model is given by $y(t) = y_0e^{-kt}$, where $y_0$ is the initial value, $k > 0$ is the decay constant, and $t$ is time. Note the negative sign in the exponent.
• ☢️ Half-Life: The time it takes for a quantity to reduce to half of its initial value. If $T$ is the half-life, then $e^{-kT} = \frac{1}{2}$, so $T = \frac{\ln 2}{k}$.
• 🌡️ Newton's Law of Cooling: The rate of change of an object's temperature is proportional to the difference between its own temperature and the ambient temperature (the temperature of its surroundings). The model is given by $\frac{dT}{dt} = k(T - T_a)$, where $T$ is the object's temperature, $T_a$ is the ambient temperature, and $k$ is a constant.
• 💰 Continuously Compounded Interest: The amount of money accumulates continuously at a given interest rate. The formula is $A = Pe^{rt}$ where A is the final amount, P is the principal, r is the annual interest rate, and t is the number of years.

🧪 Practice Quiz

1. What is the general form of an exponential growth model?
   1. $y(t) = y_0e^{-kt}$
   2. $y(t) = y_0e^{kt}$
   3. $y(t) = y_0 + kt$
   4. $y(t) = y_0 - kt$
2. A bacterial culture doubles every 3 hours. If the initial population is 100, what is the population after 6 hours?
   1. 200
   2. 300
   3. 400
   4. 800
3. The half-life of a radioactive substance is 10 years. How much of a 200g sample will remain after 30 years?
   1. 100g
   2. 50g
   3. 25g
   4. 12.5g
4. Newton's Law of Cooling is modeled by the differential equation $\frac{dT}{dt} = k(T - T_a)$. What does $T_a$ represent?
   1. The initial temperature of the object
   2. The ambient temperature
   3. The rate of cooling
   4. The time elapsed
5. A cup of coffee cools from 90°C to 60°C in 10 minutes in a room at 20°C. What is the temperature after 20 minutes?
   1. 30°C
   2. 40°C
   3. 45°C
   4. 50°C
6. If $y(t) = 5e^{-0.2t}$ represents exponential decay, what is the decay constant?
   1. 5
   2. -5
   3. 0.2
   4. -0.2
7. An investment of $1000 is made with continuous compounding at an annual interest rate of 5%. How much will the investment be worth after 10 years?
   1. $1500
   2. $1648.72
   3. $1628.89
   4. $1050