Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 32

The table below represents the percentage $P(t)$ of Americans under the age of 18 in census years from 1950 to 2000.



$
\begin{array}{|c|c|c|c|c|}
\hline\\
t & P(t) & & t & P(t)\\
\hline\\
1950 & 31.1 & & 1980 & 28.0\\
1960 & 35.7 & & 1990 & 25.7\\
1970 & 34.0 & & 2000 & 25.7\\
\hline
\end{array}
$


a.) State what is the meaning of $P'(t)$ and its corresponding units.
b.) Construct a table of values of $P'(t)$
c.) Graph $P$ and $P'$
d.) How would it be possible to get more accurate values of $P'(t)$?

a.) The meaning of $P'(t)$ is the rate at which the percentage of population varies with respect to time.
Its unit is $\displaystyle \frac{\text{percent}}{\text{decade}}$
b.) Using the formula $\displaystyle P'(t) = \frac{P(t+h)-P(t)}{h}$ where $h$ is 10 since it represents a decade interval.

$
\begin{array}{|c|c|}
\hline\\
t & P'(t)\\
\hline\\
1950 & 0.46 \\
1960 & -0.17\\
1970 & -0.60\\
1980 & -0.23\\
1990 & 0\\
2000 & 0\\
\hline
\end{array}

\qquad

\begin{equation}
\begin{aligned}
P'(1950) & = \frac{P(1960)-P(1950)}{10}\\
P'(1950) & = \frac{35.7-31.1}{10}\\
P'(1950) & = 0.46 \frac{\text{percent}}{\text{decade}}
\end{aligned}
\end{equation}
$

c.)






d.) It would be possible to get more accurate values for $P'(t)$ by using a graphing device or a measuring tool
such as rule for its graph.