#### To determine

**To explain:** The meaning of N′(t)

#### Explanation

**Given:**

N(t) is the number in thousands of minimally invasive cosmetic surgery procedures performed in the U.S. for various years.

**Explanation:**

N′(t) is the rate with which the number of minimally invasive cosmetic surgery procedures is changing with respect to time which are performed in the united states

The unit of N(t) is in thousands.

Thus, unit of N′(t) is thousands of surgeries per year.

#### To determine

**To construct:** The table of the estimated values of N′(t).

#### Answer

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

t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737

#### Explanation

**Formula used:** For small *h*,

limh→0N(t+h)−N(t)h≈N(t+h)−N(t)h

**Calculation:**

The value of N′(2000) is computed as follows,

Substitute h=2 in the formula,

N′(2000)=N(2000+2)−N(2000)2=N(2002)−N(2000)2=4897−55002=−301.5

Thus, N′(2000)=−301.5

The value of N′(2002) is computed as follows,

Substitute h=−2 in the formula,

N′(2002)=N(2000)−N(2002)−2=5500−4897−2=−301.5

Substitute h=2 in the formula,

N′(2002)≈N(2004)−N(2002)2=7470−48972=1286.5

Take the average of above two calculations of N′(2002)

N′(2002)=12(1286.5−301.5)=492.5

Thus, N′(2002)=492.5.

The value of N′(2004) is computed as follows,

Substitute h=−2 in the formula

N′(2004)=N(2002)−N(2004)−2=4897−7470−2=1286.5

Substitute h=2 in the formula,

N′(2004)≈N(2006)−N(2004)2=9138−74702=834

Take the average of above two calculations of N′(2004)

N′(2004)=12(1286.5+834)=1060.25

Thus, N′(2004)=1060.25.

Calculation of N′(2006) is as follows,

Substitute h=−2 in the formula,

N′(2006)=N(2004)−N(2006)−2=7470−9138−2=834

Substitute h=2 in the formula,

N′(2006)≈N(2008)−N(2006)2=10897−91382=879.5

Take the average of above two calculations of N′(2006)

N′(2006)=12(879.5+834)=856.75

Thus, N′(2006)=856.75.

Calculation of N′(2008) is as follows,

Substitute h=−2 in the formula,

N′(2008)=N(2006)−N(2008)−2=9138−10897−2=879.5

Substitute h=2 in the formula,

N′(2008)≈N(2010)−N(2008)2=11561−108972=332

Take the average of above two calculations of N′(2008)

N′(2008)=12(879.5+332)=605.75

Thus, N′(2008)=605.75.

Calculation of N′(2010) is as follows,

Substitute h=−2 in the formula,

N′(2010)=N(2008)−N(2010)−2=10897−11561−2=332

Substitute h=2 in the formula,

N′(2010)≈N(2012)−N(2010)2=13035−115612=737

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

N′(2010)=12(332+737)=534.5

Thus, N′(2010)=534.5.

Calculation of N′(2012) is as follows,

Substitute h=−2 in the formula,

N′(2012)=N(2010)−N(2012)−2=11561−13035−2=737

Thus, N′(2012)=737.

Thus, the required table will be:

t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737

#### To determine

**To sketch:** The graph of *N* and N′

#### Explanation

By hypothesis,

t2000200220042006200820102012N(t)5500489774709138108971156113035

From part (b),

t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737

**Graph:**

Use an online calculator to plot the N(t) on vertical axis and *t* on horizontal axis.

Use an online calculator to plot the N′(t) on vertical axis and *t* on horizontal axis.

#### To determine

**To explain:** The process to get more accurate values for N′(t).

#### Explanation

Since limh→0N(t+h)−N(t)h≈N(t+h)−N(t)h**,** the less time interval will be, the more accurate will be the data of N′(t).