#### To determine

**To Explain:** The meaning of U′(t) and its units.

#### Explanation

**Given:**

U(t) is the unemployment rate.

**Calculation:**

U′(t) is the rate with which the unemployment rate is changing with respect to time.

The unit of U(t) is unemployed percent.

Thus, unit of U′(t) is percent unemployed per year.

#### To determine

**To Construct:** The table of the estimated values of U′(t).

#### Answer

The table of the estimated values of U′(t) will be:

t2003200420052006200720082009201020112012U′(t)−0.5−0.45−0.45−0.250.62.351.9−0.2−0.75−0.8

#### Explanation

**Formula used:**

For small *h*, limh→0U(t+h)−U(t)h≈U(t+h)−U(t)h**.**

**Calculation:**

The value of U′(2003) is computed by using the formula as follows,

U′(2003)=U(2003+h)−U(2003)h

Substitute h=1 in the above expression,

U′(2003)≈U(2004)−U(2003)1=U(2004)−U(2003)=5.5−6=−0.5

Thus, U′(2003)≈−0.5.

The value of U′(2004) is computed by using the formula as follows,

For h=−1,

U′(2004)=U(2003)−U(2004)−1=6−5.5−1=0.5−1=−0.5

For h=1,

U′(2004)≈U(2005)−U(2004)1=U(2005)−U(2004)=5.1−5.5=−0.4

Take the average of above two calculations of U′(2004),

U′(2004)=12(−0.5−0.4)=−0.45

Thus, U′(2004)=−0.45.

The value of U′(2005) is computed by using the formula as follows,

For h=−1,

U′(2005)=U(2004)−U(2005)−1=5.5−5.1−1=−0.4

For h=1,

U′(2005)≈U(2006)−U(2005)1=U(2006)−U(2005)=4.6−5.1=−0.5

Take the average of above two calculations of U′(2005),

U′(2005)=12(−0.5−0.4)=−0.45

Thus, U′(2005)=−0.45.

The value of U′(2006) is computed by using the formula as follows,

For h=−1,

U′(2006)=U(2005)−U(2006)−1=5.1−4.6−1=−0.5

For h=1

U′(2006)≈U(2007)−U(2006)1=U(2007)−U(2006)=4.6−4.6=0

Take the average of above two calculations of U′(2006),

U′(2006)=12(−0.5)=−0.25

Thus, U′(2006)=−0.25.

The value of U′(2007) is computed by using the formula as follows,

For h=−1

U′(2007)=U(2006)−U(2007)−1=4.6−4.6−1=0

For h=1

U′(2007)≈U(2008)−U(2007)1=U(2008)−U(2006)=5.8−4.6=1.2

Take the average of above two calculations of U′(2007),

U′(2007)=12(1.2)=0.6

Thus, U′(2007)=0.6.

The value of U′(2008) is computed by using the formula as follows,

For h=−1

U′(2008)=U(2007)−U(2008)−1=4.6−5.8−1=1.2

For h=1

U′(2008)≈U(2009)−U(2008)1=9.3−5.8=3.5

Take the average of above two calculations of U′(2008),

U′(2008)=12(3.5+1.2)=2.35

Thus, U′(2008)=2.35.

The value of U′(2009) is computed by using the formula as follows,

For h=−1

U′(2009)=U(2008)−U(2009)−1=5.8−9.3−1=3.5

For h=1

U′(2009)≈U(2010)−U(2009)1=9.6−9.3=0.3

Take the average of above two calculations of U′(2009),

U′(2009)=12(3.5+0.3)=1.9

Thus, U′(2009)=1.9.

The value of U′(2010) is computed by using the formula as follows,

For h=−1

U′(2010)=U(2009)−U(2010)−1=9.3−9.6−1=0.3

For h=1

U′(2010)≈U(2011)−U(2010)1=8.9−9.6=−0.7

Take the average of above two calculations of U′(2010),

U′(2010)=12(0.3−0.7)=−0.2

Thus, U′(2010)=−0.2.

The value of U′(2011) is computed by using the formula as follows,

For h=−1

U′(2011)=U(2010)−U(2011)−1=9.6−8.9−1=−0.7

For h=1

U′(2011)≈U(2012)−U(2011)1=8.1−8.9=−0.8

Take the average of above two calculations of U′(2011),

U′(2011)=12(−0.8−0.7)=−0.75

Thus, U′(2011)=−0.75.

The value of U′(2012) is computed by using the formula as follows,

For h=−1

U′(2012)=U(2011)−U(2012)−1=8.9−8.1−1=−0.8

Thus, U′(2012)=−0.8.

Thus, the required table will be:

t2003200420052006200720082009201020112012U′(t)−0.5−0.45−0.45−0.250.62.351.9−0.2−0.75−0.8