Certainly! Here's a Calculus AB exam question on the topic of exponential growth and decay:

Question:

The population of a certain species of bird is modeled by the function P(t) = 5000 * e^(0.1t), where t represents time in years. Find the rate at which the population is increasing after 5 years.

Solution:

We are given the function P(t) = 5000 * e^(0.1t), where t represents time in years. We want to find the rate at which the population is increasing after 5 years, which corresponds to finding the derivative of the population function with respect to time t.

The derivative of P(t) with respect to t can be found using the chain rule, since we're differentiating a composite function involving exponential and linear functions.

We have:

P'(t) = 5000 * 0.1 * e^(0.1t) (using the chain rule)

Now, we need to find the rate of change after 5 years, i.e., P'(5):

P'(5) = 5000 * 0.1 * e^(0.1 * 5) = 500 * e^(0.5) ≈ 540.48

So, the rate at which the population of the bird species is increasing after 5 years is approximately 540.48 birds per year.

Therefore, the rate at which the population is increasing after 5 years is approximately 540.48 birds per year.

This is how the complete solution looks with the detailed explanation.