The population age 65 and over (in millions) of developed...

The population age 65 and over (in millions) of developed countries from 2005 through 2034 is projected to be $$ f(t)=3.567 t+175.2 \quad(5 \leq t \leq 35) $$ where $t$ is measured in years and $t=5$ corresponds to $2005 .$ On the other hand, the population age 65 and over of underdeveloped/emerging countries over the same period is projected to be $$ g(t)=0.46 t^{2}+0.16 t+287.8 \quad(5 \leq t \leq 35) $$ a. What does the function $D=g+f$ represent? b. Find $D^{\prime}$ and $D^{\prime}(10),$ and interpret your results. Source: U.S. Census Bureau, United Nations.

The population age 65 and over (in millions) of developed countries from 2005 through 2034 is projected to be
$$
f(t)=3.567 t+175.2 \quad(5 \leq t \leq 35)
$$
where $t$ is measured in years and $t=5$ corresponds to $2005 .$ On the other hand, the population age
65 and over of underdeveloped/emerging countries over the same period is projected to be
$$
g(t)=0.46 t^{2}+0.16 t+287.8 \quad(5 \leq t \leq 35)
$$
a. What does the function $D=g+f$ represent?
b. Find $D^{\prime}$ and $D^{\prime}(10),$ and interpret your results.
Source: U.S. Census Bureau, United Nations.

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Key Concepts

Function Modeling

Function modeling involves representing real-world phenomena using mathematical functions. In the context of population trends, functions are used to predict and analyze how the population changes over time, enabling researchers to forecast future values based on current and historical data.

Addition of Functions

The addition of functions is a method used to combine separate models into a single, overarching model. When dealing with different population groups, adding the functions corresponding to each group yields a comprehensive function that models the total population, thereby providing a complete picture of aggregate trends.

Derivative as a Rate of Change

The derivative represents the instantaneous rate of change of a function with respect to its independent variable. In population studies, the derivative indicates how quickly the population is increasing or decreasing at any given moment, based on the underlying model.

Interpreting the Derivative in Context

Interpreting the derivative in a real-world scenario involves explaining what the computed rate of change means for the model. This interpretation connects the abstract numerical value of a derivative to practical implications, such as understanding how fast the total population is growing at a specific time.

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