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.


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