Chapter 2.7, Problem 34E

a.

To determine

The meaning of $P\text{'}(t)$ and also the unit of $P\text{'}(t)$ .

Explanation of Solution

Given information:

  $P(t)$ is the percentage of Americans under the age $18$ of at time $t$

The table gives values of this function in census years from $1950\text{ to 2000}$

$t$	$P(t)$	$t$	$P(t)$
1950	31.1	1980	28.0
1960	35.7	1990	25.7
1970	34.0	2000	25.7

Calculations:

Here $P\text{'}(t)$ shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age $18$ with respect to time (in years). The unit of $P(t)$ will be Unemployed peer year.

The slope of the tangent is given the formula,

  $f\text{'}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

b.

To determine

The estimated values of $P\text{'}(t)$ in a tabulated manner.

Explanation of Solution

Given information:

  $P(t)$ is the percentage of Americans under the age $18$ of at time $t$

The table gives values of this function in census years from $1950\text{ to 2000}$

$t$	$P(t)$	$t$	$P(t)$
1950	31.1	1980	28.0
1960	35.7	1990	25.7
1970	34.0	2000	25.7

Calculations:

Here $P\text{'}(t)$ shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age $18$ with respect to time (in years) . The unit of $P(t)$ will be Unemployed peer year.

The slope of the tangent is given the formula,

  $f\text{'}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

t	$P(t)$	$P\text{'}(t)$	t	$P(t)$	$P\text{'}(t)$
1950	31.1		1980	28.0	-0.6
1960	35.7	0.46	1990	25.7	-0.23
1970	34.0	-0.17	2000	25.7	0

c.

To determine

The graph of $P(t)$ and $P\text{'}(t)$ .

Explanation of Solution

Given information:

  $P(t)$ is the percentage of Americans under the age $18$ of at time $t$

The table gives values of this function in census years from $1950\text{ to 2000}$

$t$	$P(t)$	$t$	$P(t)$
1950	31.1	1980	28.0
1960	35.7	1990	25.7
1970	34.0	2000	25.7

Calculations:

Here $P\text{'}(t)$ shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age $18$ with respect to time (in years) . The unit of $P(t)$ will be Unemployed peer year.

The slope of the tangent is given the formula,

  $f\text{'}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

t	$P(t)$	$P\text{'}(t)$	t	$P(t)$	$P\text{'}(t)$
1950	31.1		1980	28.0	-0.6
1960	35.7	0.46	1990	25.7	-0.23
1970	34.0	-0.17	2000	25.7	0

The graph of $P(t)$ is shown below.

  

The graph of $P\text{'}(t)$ is shown below.

  

d.

To determine

The ways to get more accurate values of $P\text{'}(t)$ .

Explanation of Solution

Given information:

  $P(t)$ is the percentage of Americans under the age $18$ of at time $t$

The table gives values of this function in census years from $1950\text{ to 2000}$

$t$	$P(t)$	$t$	$P(t)$
1950	31.1	1980	28.0
1960	35.7	1990	25.7
1970	34.0	2000	25.7

Calculations:

Here $P\text{'}(t)$ shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age $18$ with respect to time (in years) . The unit of $P(t)$ will be Unemployed peer year.

The slope of the tangent is given the formula,

  $f\text{'}(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

The value of $P\text{'}(t)$ can be more accurate if the time interval is less and more values of $P(t)$ is obtained.