Number of People in the US

United States Population as a Function of Time

This data below shows the US population as estimated during the years 1900 through 2000 by the Census Bureau. First, the data is entered into the Graphical Analysis program (GAX) and graphed on the appropriate axes. It was then found that the curve produced by plotting a population versus time graph fit a sixth-degree polynomial function. That is, it fits the general form A + Bx + Cx² + Dx³ + Ex⁴ + Fx⁵ + Gx⁶, where the letters A through G are coefficients determined through the polynomial regression.

Click for a larger graph, larger derivative, or to view the raw data file.
Source: US and Texas populations, 1850-2001 - Texas State Library.

This regression conveys that at the beginning of the 20^th century, the population increased gradually. However, from the obvious polynomial, the population of the US increases more and more throughout the century. However, there is dip, or sharp decrease in the graph around the years 1943-1945. This signifies that the population of the country has decreased during this time period. This is due to the large number of casualties in World War II, which took place around this time. The population the increases at an even greater pace, which shows that the United States recovered from the devastating war.

For this function and its parameters, the derivative dy/dx is defined as the rate at which the population of the United States is changing at a given time. As shown in the graph, after the World War II decrease in population, the derivative, or slope, of the graph increases substantially from that time to the present. Since the slope of the graph is becoming more positive, it is inferred that the rate of population increase is on the rise. This could spell disaster in later years. If the population continues to increase as much and fast as it has been, then the United States, which is the land of opportunity, could face the prospect of severe overcrowding.

When the derivative of this curve is graphed with respect to time, we see that after 1995, the derivative takes on unlikely values. This is due to that fact that the sixth-degree polynomial curve fit may only be valid for the time frame of 1900 to 1995. We must keep in mind that curve fitting can be similar to art, for it is only valuable under certain conditions. As shown in this graph, the sixth-degree polynomial can only be applicable under the certain time conditions. Certain mathematical equations, such as a complex curve like a sixth degree polynomial, have closed intervals in which they are valid. Outside these ranges, the equation ceases to yield reliable data.

Johnny Alicea -- 2002

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