Question

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 1993 to 2002. $$\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 1993 & {6.9} & {1998} & {4.5} \\ {1994} & {6.1} & {1999} & {4.2} \\ {1995} & {5.6} & {2000} & {4.0} \\ {1996} & {5.4} & {2001} & {4.7} \\ {1997} & {4.9} & {2002} & {5.8} \\ \hline\end{array}$$ (a) What is the meaning of $U^{\prime}(t) ?$ What are its units? (b) Construct a table of values for $U^{\prime}(t).$

Name: Problem 33, Chapter 2: Calculus Early Transcendentals
Uploaded: 2021-08-11T14:43:06-08:00
Duration: 1 min 8 s
Channel: Carson Merrill
Description: 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 1993 to 2002. t U(t) t U(t) 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 (a) What is the meaning of U^'(t) ? What are its units? (b) Construct a table of values for U^'(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 1993 to 2002.
$$\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 1993 & {6.9} & {1998} & {4.5} \\ {1994} & {6.1} & {1999} & {4.2} \\ {1995} & {5.6} & {2000} & {4.0} \\ {1996} & {5.4} & {2001} & {4.7} \\ {1997} & {4.9} & {2002} & {5.8} \\ \hline\end{array}$$
(a) What is the meaning of $U^{\prime}(t) ?$ What are its units?
(b) Construct a table of values for $U^{\prime}(t).$

Calculus Early Transcendentals

James Stewart 6th Edition

Chapter 2, Problem 33

Instant Answer

Solved by Expert Carson Merrill

08/11/2021

Step 1

In this context, $U(t)$ is the unemployment rate at time $t$. Therefore, $U^{\prime}(t)$, the derivative of $U(t)$ with respect to $t$, represents the rate of change of the unemployment rate at time $t$. In other words, it tells us how quickly the unemployment Show more…

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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 1993 to 2002. $$\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 1993 & {6.9} & {1998} & {4.5} \\ {1994} & {6.1} & {1999} & {4.2} \\ {1995} & {5.6} & {2000} & {4.0} \\ {1996} & {5.4} & {2001} & {4.7} \\ {1997} & {4.9} & {2002} & {5.8} \\ \hline\end{array}$$ (a) What is the meaning of $U^{\prime}(t) ?$ What are its units? (b) Construct a table of values for $U^{\prime}(t).$

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Key Concepts

Derivative

The derivative of a function measures the instantaneous rate of change of the function with respect to its variable. In the context of U(t), it indicates how the unemployment rate is changing at a particular time, capturing both the direction and the speed of that change.

Rate of Change

The rate of change provides a numerical value that represents the change in the unemployment rate per unit time. For U(t), this means quantifying how many percentage points the unemployment rate is increasing or decreasing per year.

Finite Difference Approximation

Finite difference approximation is a technique used to estimate the derivative of a function using discrete data points. By calculating the difference between successive values of U(t) over corresponding time intervals and dividing by the length of the interval, one obtains an approximate value for U'(t), which is useful when the analytical form of U(t) is not available.

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