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In this problem, the population growth of a city is modeled using an exponential function. The function provided is \( p(t) = 130(1 + 12 e^{-0.02 t})^{-1} \), where \( t \) is the number of years since 2010. This model helps us understand how the population changes over time.
Exponential models in population growth are common because they can predict how populations increase or decrease naturally. Population changes often start quickly and then slow down as resources become limited or as birth and death rates stabilize. By examining this function, we can predict the number of people present at any specific time, which is crucial for planning city resources and facilities.

The garbage truck requirement function describes the relationship between the population of the city and the number of garbage trucks needed. This function, \( g(p) = 2p - 0.001p^3 \), shows us how the demand for garbage removal equipment scales with population size.
The linear term \( 2p \) indicates that more people naturally require more service—specifically, more garbage trucks. The cubic term \(-0.001p^3\) represents the idea that as the population grows very large, efficiency improvements or shared services may mean fewer trucks are needed per person.
This function aids city planners in ensuring that resources are efficiently allocated, avoiding either a surplus of trucks or a shortage.

Differentiation is a calculus technique used to determine how a function changes at any given point. It is crucial for understanding rates of change.
For the population function \( p(t) \), differentiation allows us to find \( \frac{dp}{dt} \). This tells us how the population's size is changing per year—either increasing or decreasing. Calculating \( \frac{dp}{dt} \) involves the chain rule because the function is composed of layers, such as exponential and algebraic operations.
Similarly, differentiating the garbage truck function \( g(p) \) with respect to \( p \) helps us determine how the need for trucks changes as population grows, calculated as \( \frac{dg}{dp} \).
Understanding differentiation provides clear insight into dynamics and trends, assisting in making informed decisions.

The population function \( p(t) \) is the mathematical expression used to represent the population of a city over time. It accounts for factors such as birth rate, death rate, and migration to predict changes.
For this particular function, \( t \) represents the time in years from 2010, and the formula predicts the population’s size in thousands. For instance, in 2012, two years after 2010, \( t = 2 \). By plugging this into the function, we calculate the population at that specific time.
This function serves as a foundational tool for planning—the basis for understanding demographic changes that affect economic, social, and urban development policies.

The chain rule is a fundamental differentiation technique used when dealing with composite functions—functions within functions. It provides a way to handle complex expressions like our population function \( p(t) \), where variables are nested.
In our scenario, the chain rule is employed to differentiate \( p(t) \) with respect to \( t \). It involves breaking down the function into separate parts, differentiating each, and then combining the results.
To compute \( \frac{dp}{dt} \), we express \( p(t) \) as \( 130(1 + 12 e^{-0.02t})^{-1} \). Here, \( 1 + 12 e^{-0.02t} \) represents one layer, and its derivative is part of calculating the overall rate of change \( \frac{dp}{dt} \). This rule is equally applied when evaluating the rate of change of garbage truck requirements concerning time, emphasizing its versatility and importance in calculus.