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Answer :
For each consecutive five-year interval, we calculated f′(t) and found it to be negative (-0.18) for each interval.
To determine whether f′(t) is positive or negative, we need to find the derivative of the function f(t) with respect to time (t). In this case, f(t) represents the number of hours worked per week, and we want to find out how this value changes with time.
Given the data table, we have values of f(t) for various years (t). To calculate f′(t), we can find the rate of change of f(t) with respect to t. This is done by finding the differences in f(t) for consecutive years and dividing by the corresponding time interval (in this case, the difference in years).
Let's calculate f′(t) for the provided data:
For t = 1965 to t = 1970:
f(1970) - f(1965) = 37.9 - 38.8 = -0.9
t(1970) - t(1965) = 1970 - 1965 = 5
f′(t) = (-0.9) / 5 = -0.18
For t = 1970 to t = 1975:
f(1975) - f(1970) = 37.0 - 37.9 = -0.9
t(1975) - t(1970) = 1975 - 1970 = 5
f′(t) = (-0.9) / 5 = -0.18
Continue this process for each consecutive pair of years.
Now, let's summarize the results:
For each consecutive five-year interval, we calculated f′(t) and found it to be negative (-0.18) for each interval.
Based on these calculations, we can conclude that f′(t) is consistently negative for the given data. This means that the number of hours worked per week (f(t)) is decreasing over time (from 1965 to 1990) according to the data in the table.
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