The unemployment rate U(t) varies with time. The table (from...

The unemployment rate $U(t)$ varies with time. The table (from the Bureau of Labor Statistics) gives the percentage of unemployed in the US labor force from 1999 to 2008 . $$\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 1999 & {4.2} & {2004} & {5.5} \\ {2000} & {4.0} & {2005} & {5.1} \\ {2001} & {4.7} & {2006} & {4.6} \\ {2002} & {5.8} & {2007} & {4.6} \\ {2003} & {6.0} & {2008} & {5.8} \\ \hline\end{array}$$ (a) What is the meaning of $U^{\prime}(t) ?$ What are its units? (b) Construct a table of estimated values for $U^{\prime}(t)$ .

The unemployment rate $U(t)$ varies with time. The table
(from the Bureau of Labor Statistics) gives the percentage
of unemployed in the US labor force from 1999 to 2008 .
$$\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 1999 & {4.2} & {2004} & {5.5} \\ {2000} & {4.0} & {2005} & {5.1} \\ {2001} & {4.7} & {2006} & {4.6} \\ {2002} & {5.8} & {2007} & {4.6} \\ {2003} & {6.0} & {2008} & {5.8} \\ \hline\end{array}$$
(a) What is the meaning of $U^{\prime}(t) ?$ What are its units?
(b) Construct a table of estimated values for $U^{\prime}(t)$ .

Key Concepts

Interpretation of the Derivative

The derivative U?(t) represents the instantaneous rate of change of the unemployment rate with respect to time. It tells us how quickly the unemployment percentage is increasing or decreasing at any specific moment, providing insight into the trend of unemployment dynamics over the given time period.

Units of the Derivative

Since U(t) is measured in percentage points and t is measured in years, the derivative U?(t) has units of percentage points per year, indicating the change in the unemployment rate per one-year interval.

Finite Difference Approximation

When working with discrete data from a table, the derivative is often estimated using finite difference approximations. This involves calculating the change in the unemployment rate between consecutive years and dividing by the change in time (typically one year), which provides an approximate value for U?(t) at those intervals.

Transcript

00:01 Given the table that's provided here, we first want to look at what are the units of u -prim of t.

00:08 And so because the table gives us u of t, which is the unemployment rate or the percentage, i should say, the unemployment percentage each year of unemployed u .s.

00:22 Labor force, then u .s.

00:26 Will show us the rate of unemployment, the rate of per year, the rate of percent, percentage.

00:56 So whether or not that percentage is increasing or decreasing at that given time, basically.

01:03 Because we're just given all positive values, but the rate, of course, could be positive or negative per year.

01:14 All right.

01:16 Another way to say that is like the change in the unemployment, unemployment over the change in time.

01:30 So a couple different ways to do that.

01:33 And then, of course, the units there will be years in the bottom, change in time.

01:45 So that's in years.

01:46 And then the top is in percentage of unemployed.

01:58 Okay, so let's jump into it, given the table that we are given here from 1999 to 2008, we're going to choose two values in between.

02:09 And so to find out the rate of change for the year 2000, for instance, we're going to look at the year before and the year after it.

02:16 And we're going to take 4 .7 minus 4 .2, and that'll give us 0 .5 is the change in percentage, divided back by the number of years, which is two.

02:29 And so that would give us 0 .25.

02:33 So there was an increase in the year 2000.

02:37 The rate of unemployment was increasing by 0 .25 percentage amounts.

02:45 And then in 2001, we used the value after it, 5 .8, and the value before it, 4.

02:51 And so we do 5 .8 minus 4, so that gets us 1 .8 divided by, we went from 2000 to 2002 to estimate 2001.

03:02 So divide that again by two.

03:03 Each of these is going to be divided by two because we're looking two years apart.

03:08 And then that will give us, of course, positive 0 .9.

03:12 And so let's keep going here.

03:14 In 2002, we use 6 .0, 4 .7...