Five years ago, the government of a Pacific Island country...

Five years ago, the government of a Pacific Island country launched an extensive propaganda campaign aimed toward curbing the country's population growth. According to the Census Department, the population (measured in thousands of people) for the following 4 years was $$ P(t)=-\frac{1}{3} t^{3}+64 t+3000 $$ where $t$ is measured in years and $t=0$ corresponds to the start of the campaign. Find the rate of change of the population at the end of years $1,2,3,$ and $4 .$ Was the plan working?

Five years ago, the government of a Pacific Island country launched an extensive propaganda campaign aimed toward curbing the country's population growth. According to the Census Department, the population (measured in thousands of people) for the following 4 years was
$$
P(t)=-\frac{1}{3} t^{3}+64 t+3000
$$
where $t$ is measured in years and $t=0$ corresponds to the start of the campaign. Find the rate of change of the population at the end of years $1,2,3,$ and $4 .$ Was the plan working?

Key Concepts

Polynomial Function

A polynomial function is an expression made up of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents. Such functions are common in practical modeling scenarios because they are smooth and continuous, making them suitable for representing gradual changes such as population growth.

Model Interpretation

Interpreting the results of a derivative in a modeled scenario involves understanding whether the population is growing or declining. By evaluating the derivative at different time points, one can assess the effectiveness of interventions or policies over time, determining if they are achieving the intended effect of, for example, curbing population growth.

Derivative

The derivative of a function represents the instantaneous rate of change of that function with respect to its independent variable. In many contexts, such as population dynamics, it is used to determine how quickly a population is increasing or decreasing at a specific moment or over a small interval of time.

Rate of Change

The rate of change in the context of a real-world model like population growth describes how the number of individuals in the population is changing over time. It is typically computed by taking the derivative of the model function and then evaluating it at specific time points to understand trends or the effects of policies.

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