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Let's break down the two parts of problem 2 in your image, which involves population growth modeled by the exponential function:

\[
P(t) = 1.2(1.05)^t
\]

Here, \(P(t)\) is the city's population (in tens of thousands), and \(t\) is the number of years since 2000.

### Part (a): Average Rate of Change
You are asked to examine what can be concluded about the **average rate of change** in the population between consecutive years as time increases.

1. The function \(P(t)\) is exponential, specifically in the form \(P(t) = A \cdot (1 + r)^t\), where \(r = 0.05\) is the growth rate (5% annual growth).
2. For an exponential function, the average rate of change between any two consecutive years will increase over time because the population is growing exponentially. The percentage increase each year is 5%, but because the population base is increasing, the absolute increase will get larger.

Conclusion: The **average rate of change increases over time** because the population grows exponentially. Each year, the city's population grows by 5% of the previous year's population, leading to a larger absolute increase as time progresses.

### Part (b): Instantaneous Rate of Change in 2010
You are asked to estimate the **instantaneous rate of change** of the population in 2010. To do this, we'll compute the derivative of \(P(t)\) and then evaluate it at \(t = 10\) (since 2010 is 10 years after 2000).

1. The derivative of \(P(t)\) with respect to \(t\) is:
\[
P'(t) = 1.2 \cdot \ln(1.05) \cdot (1.05)^t
\]
2. Substituting \(t = 10\) (for the year 2010):
\[
P'(10) = 1.2 \cdot \ln(1.05) \cdot (1.05)^{10}
\]

Let's calculate this derivative for \(t = 10\):

\[
P'(10) = 1.2 \cdot 0.04879 \cdot (1.62889)
\]
\[
P'(10) \approx 1.2 \cdot 0.07946 = 0.09535
\]

This gives us an instantaneous rate of change of about **0.09535**, which means the population is increasing by approximately **9,535 people per year in 2010** (since the population is measured in tens of thousands).

---

Would you like me to dive deeper into any part of this? Here are 5 related questions:

1. How does exponential growth differ from linear growth in terms of rate of change?
2. Why is the instantaneous rate of change important for understanding population dynamics?
3. How would you interpret the population growth rate if the base of the exponent was smaller than 1?
4. Can the average rate of change ever decrease in an exponential growth model?
5. How would the population model change if there were a yearly migration loss?

**Tip:** Exponential growth functions are commonly used to model populations because they capture the compounding effect of a growth rate applied over time.

A city's population (in tens of thousands) is modeled by the function P(t) = 1.2(1.05)^t, where t is the number of years since 2000. What can you conclude about the average rate of change in population between consecutive years as time increases? Estimate the instantaneous rate of change in population in 2010.

Explore the population growth model P(t) = 1.2(1.05)^t and learn how to calculate the average and instantaneous rate of change. Understand exponential growth and its impact on population dynamics. Ideal for students in Grades 11-12.

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