Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 31

The table below represents the unemployment rate $U(t)$ in percent of the US labor force from 1993 to 2002.



$
\begin{array}{|c|c|c|c|c|}
\hline\\
t & U(t) & & t & U(t)\\
\hline\\
1993 & 6.9 && 1998 & 4.5\\
1994 & 6.1 && 1999 & 4.2\\
1995 & 5.6 && 2000 & 4.0\\
1996 & 5.4 && 2001 & 4.7\\
1997 & 4.9 && 2002 & 5.8\\
\hline
\end{array}
$



a.) State what is the meaning of $U'(t)$ and its corresponding units.



b.) Contruct a table of values for $U'(t)$


$\quad$ a.) The meaning of $U'(t)$ is the rate at which the percentage of the unemployed varies with
respect to time. Its unit is $\displaystyle \frac{\text{percent}}{\text{year}}$.

$\quad$ b.) Using the formula $\displaystyle U'(t) = \frac{U(t+h)-U(t)}{h}$ where $h$ is 1.


$
\quad
\begin{array}{|c|c|}
\hline\\
t & U'(t)\\
\hline\\
1993 & -0.8 \\
1994 & -0.5\\
1995 & -0.2\\
1996 & -0.5\\
1997 & -0.4\\
1998 & -0.3\\
1999 & -0.2\\
2000 & 0.7\\
2001 & 1.10\\
2002 & 0\\
\hline
\end{array}
\qquad
\begin{equation}
\begin{aligned}
U'(1993) &= \frac{U(1994)-U(1993)}{1}\\
U'(1993) &= \frac{6.1-6.9}{1}\\
U'(1993) &= -0.8 \frac{\text{percent}}{\text{year}}
\end{aligned}
\end{equation}
$