Sure, let's work through the problem step by step to determine the rate of change of the fertility rate in 1979 for the given country. The number of years after 1969 is represented by [tex]\(x\)[/tex], and the fertility rate [tex]\(f(x)\)[/tex] is given by the quadratic equation:
[tex]\[ f(x) = 0.00925x^2 - 0.3385x + 3.875 \][/tex]
We are asked to find the rate of change of the fertility rate in 1979. To find the rate of change of the function [tex]\(f(x)\)[/tex] at a specific year, we need to calculate the derivative of [tex]\(f(x)\)[/tex] and then evaluate it at [tex]\(x\)[/tex] corresponding to the year 1979.
Step 1: Find the derivative of [tex]\(f(x)\)[/tex]
The derivative of the given quadratic function [tex]\(f(x)\)[/tex] is:
[tex]\[ f'(x) = \frac{d}{dx}[ 0.00925x^2 - 0.3385x + 3.875] \][/tex]
Using basic differentiation rules, we get:
[tex]\[ f'(x) = 2 \cdot 0.00925 x - 0.3385 \][/tex]
[tex]\[ f'(x) = 0.0185 x - 0.3385 \][/tex]
Step 2: Determine the value of [tex]\(x\)[/tex] for the year 1979
The year 1979 is 10 years after 1969. Therefore, in terms of our variable [tex]\(x\)[/tex]:
[tex]\[ x = 1979 - 1969 = 10 \][/tex]
Step 3: Evaluate the derivative at [tex]\(x = 10\)[/tex]
We now substitute [tex]\(x = 10\)[/tex] into the derivative [tex]\(f'(x)\)[/tex]:
[tex]\[ f'(10) = 0.0185 \cdot 10 - 0.3385 \][/tex]
[tex]\[ f'(10) = 0.185 - 0.3385 \][/tex]
[tex]\[ f'(10) = -0.1535 \][/tex]
Step 4: Round the result to three decimal places
When we round [tex]\(-0.1535\)[/tex] to three decimal places, it becomes:
[tex]\[ -0.154 \][/tex]
Thus, the rate of change of the fertility rate in 1979 is [tex]\(-0.154\)[/tex] births per woman per year, rounded to three decimal places.
