Chapter 2, Problem 46RE

(a)

To determine

To find:The value of $F\text{'}\left(1950\right),F\text{'}\left(1965\right)$ , and $F\text{'}\left(1987\right)$ .

Answer to Problem 46RE

The value of $F\text{'}\left(1950\right),F\text{'}\left(1965\right)$ , and $F\text{'}\left(1987\right)$ is $F\text{'}\left(1950\right)\approx 0.1\text{children/year}$ , $F\text{'}\left(1965\right)\approx -0.2\text{children/year}$ , $F\text{'}\left(1987\right)\approx 0\text{children/year}$ .

Explanation of Solution

Given information:

The total fertility rate at time $t$ , denoted by $F\left(t\right)$ is shown in figure (1).

  

Figure (1)

Calculation:

Using the above graph the function $y=F\left(t\right)$ , where $t$ is measured in years and $y$ is measured in children.

The derivative of $F$ at a number measures the slope of the graph $a$ .

Calculate the value of $F\text{'}\left(1950\right)$ , from graph appears that to pass through the points $\left(1945,2.5\right)$ and $\left(1955,3.5\right)$ .

  $\begin{array}{c}F\text{'}\left(1950\right)\approx \frac{3.5-2.5}{1955-1945}\\ =0.1\text{children/year}\end{array}$

Similarly, estimate the derivative at the other given values of $t$ by estimating the slope of the graph.

  $\begin{array}{l}F\text{'}\left(1965\right)\approx -0.2\text{children/year}\\ F\text{'}\left(1987\right)\approx 0\text{children/year}\end{array}$

Therefore,the value of $F\text{'}\left(1950\right),F\text{'}\left(1965\right)$ , and $F\text{'}\left(1987\right)$ is $F\text{'}\left(1950\right)\approx 0.1\text{children/year}$ , $F\text{'}\left(1965\right)\approx -0.2\text{children/year}$ , $F\text{'}\left(1987\right)\approx 0\text{children/year}$

(b)

To determine

To find:The meaning of derivatives.

Answer to Problem 46RE

The meaning of derivatives is discussed in calculation.

Explanation of Solution

Given information:

The total fertility rate at time $t$ , denoted by $F\left(t\right)$ is shown in figure (1).

  

Figure (1)

Calculation:

The derivatives measure the rate of change of the total fertility rate relative to time.

  $F\text{'}\left(1950\right)\approx 0.1\text{children/year}$ means that the average number of children born to each woman was increasing at a rate of $0.1\text{children/year}$ in $1950$ .

  $F\text{'}\left(1965\right)\approx -0.2\text{children/year}$ means that the average number of children born to each woman was decreasing at a rate of $0.\text{children/year}$ in $1965$ .

  $F\text{'}\left(1987\right)\approx 0\text{children/year}$ means that the average number of children born to each woman was not changing in $1987$ .

Therefore, the meaning of derivativesis discussed in calculation.

(c)

To determine

To find:The reason for the values of these derivatives.

Answer to Problem 46RE

The success and stability of the economy was likely a factor in the varying average number of children born to each woman.

Explanation of Solution

Given information:

The total fertility rate at time $t$ , denoted by $F\left(t\right)$ is shown in figure (1).

  

Figure (1)

Calculation:

The success and stability of the economy was likely a factor in the varying average number of children born to each woman.

Therefore, the success and stability of the economy was likely a factor in the varying average number of children born to each woman.