Modeling Population Growth?

📌 CONCEPT: The differential equation dP/dt = kP models population growth when the rate of growth is proportional to the present size of the population.

📐 RULE / FORMULA: The general solution to this differential equation is P(t) = P0 * e^(kt), where P0 is the initial population and k is the growth rate.

💡 WORKED EXAMPLE: Given P0 = 100,000 and k = 0.02, we can estimate the population after 5 years as P(5) = 100,000 * e^(0.02*5) ≈ 100,000 * e^0.1 ≈ 113,295.

⚠️ COMMON MISTAKE: Students often forget to use the initial population (P0) in the general solution, leading to incorrect estimates of population growth.