Example 2: Number of U.S. Farms

Example 2: Number of U.S. Farms

The table below shows the number N of farms in the US, in thousands, at five year intervals between 1950 and 1975.

For each of the five-year intervals shown, we can compute the change in value and the average rate of change in value for N(t):

1950 – 55: DN = 4654 – 5648 = – 994 thousand farms,
DN/Dt = – 994/5 = – 198.8 thousand farms per year;

1955 – 60: DN = 3963 – 4654 = – 691 thousand farms,
DN/Dt = – 691/5 = – 138.2 thousand farms per year;

1960 – 65: DN = 3356 – 3963 = – 607 thousand farms,
DN/Dt = – 607/5 = – 121.4 thousand farms per year;

1965 – 70: DN = 2949 – 3356 = – 407 thousand farms,
DN/Dt = – 407/5 = – 81.4 thousand farms per year;

1970 – 75: DN = 2767 – 2949 = – 182 thousand farms,
DN/Dt = – 182/5 = – 36.4 thousand farms per year;

We see that the change in value DN in each interval is negative, indicating that the function is decreasing. The size (absolute value) of the average rates of change DN/Dt are also decreasing, indicating that the decrease in N is slowing over time. Graphically, a function which is decreasing at a decreasing (absolute) rate is concave up, as a plot of the data shows:

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1950 – 55:	DN = 4654 – 5648 = – 994 thousand farms, DN/Dt = – 994/5 = – 198.8 thousand farms per year;
1955 – 60:	DN = 3963 – 4654 = – 691 thousand farms, DN/Dt = – 691/5 = – 138.2 thousand farms per year;
1960 – 65:	DN = 3356 – 3963 = – 607 thousand farms, DN/Dt = – 607/5 = – 121.4 thousand farms per year;
1965 – 70:	DN = 2949 – 3356 = – 407 thousand farms, DN/Dt = – 407/5 = – 81.4 thousand farms per year;
1970 – 75:	DN = 2767 – 2949 = – 182 thousand farms, DN/Dt = – 182/5 = – 36.4 thousand farms per year;