- Shortage of Nurses The projected number of nurses (in...

- Shortage of Nurses The projected number of nurses (in millions) in year $t$ from 2000 through 2015 is given by $$ N(t)=\left\{\begin{array}{ll} 1.9 & \text { if 0 } \leq t<5 \\ -0.0004 t^{2}+0.038 t+1.72 & \text { if 5 } \leq t \leq 15 \end{array}\right. $$ where $t=0$ corresponds to 2000 . The projected number of nursing jobs (in millions) over the same period is $$ J(t)=\left\{\begin{array}{ll} -0.0002 t^{2}+0.032 t+2 & \text { if } \leq t < 10 \\ -0.0016 t^{2}+0.12 t+1.26 & \text { if } 10 \leq t \leq 15 \end{array}\right. $$ a. Find the rule for the function $G=J-N$ giving the gap between the supply and the demand of nurses from 2000 through $2015 .$ b. How fast was the gap between the supply and the demand of nurses changing in 2008 ? In $2012 ?$ Source: U.S. Department of Health and Human Services.

- Shortage of Nurses The projected number of nurses (in millions) in year $t$ from 2000 through 2015 is given by
$$
N(t)=\left\{\begin{array}{ll}
1.9 & \text { if 0 } \leq t<5 \\
-0.0004 t^{2}+0.038 t+1.72 & \text { if 5 } \leq t \leq 15
\end{array}\right.
$$
where $t=0$ corresponds to 2000 . The projected number of nursing jobs (in millions) over the same period is
$$
J(t)=\left\{\begin{array}{ll}
-0.0002 t^{2}+0.032 t+2 & \text { if } \leq t < 10 \\
-0.0016 t^{2}+0.12 t+1.26 & \text { if } 10 \leq t \leq 15
\end{array}\right.
$$
a. Find the rule for the function $G=J-N$ giving the gap between the supply and the demand of nurses from 2000 through $2015 .$
b. How fast was the gap between the supply and the demand of nurses changing in 2008 ? In $2012 ?$
Source: U.S. Department of Health and Human Services.

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by different expressions over different intervals of the domain. They are useful for modeling situations where a rule or behavior changes at specified values, such as time periods. In this context, both the nurse supply and nursing job functions are given as piecewise functions to reflect distinct trends over different time intervals.

Function Subtraction

Function subtraction involves creating a new function by subtracting one function from another. This is used to model the gap between two quantities—in this case, the difference between the number of nursing jobs and the number of nurses. This process allows us to construct a function that directly represents the imbalance or difference between supply and demand.

Differentiation

Differentiation is a fundamental concept in calculus that computes the derivative of a function, which represents the rate at which the function's value is changing with respect to changes in the independent variable. In this problem, differentiation is used to determine how quickly the gap between supply and demand is increasing or decreasing over time.

Rate of Change Analysis

Rate of change analysis involves evaluating the derivative of a function at specified points to understand how rapidly the measured quantity is changing. In this case, calculating the derivative of the gap function at specific years (2008 and 2012) provides insight into the dynamics of the nurse shortage, indicating whether the situation is improving or deteriorating over time.

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