Problem:
Given the following table of U.S. population:

YearPopulation
(millions)
17903.929
18005.308
18107.240
18209.638
183012.861
184017.063
185023.192
186031.443
[Extracted from the Census Bureau WWW site.]

find and graph the exponential model which best approximates the data.

Visualization:
Using Maple:

First, we'll plot the data points:

> with(plots):
> plot([[1790,3.929], [1800,5.308], [1810,7.24], [1820,9.638], [1830,12.861], [1840,17.063], [1850,23.192], [1860,31.443]], style=point, symbol=diamond, color=blue);

We'll store this information using the name dataplot:

> dataplot:=":

Next, we'll plot the data points with ln(population) against year:

> plot([[1790,log(3.929)], [1800,log(5.308)], [1810,log(7.24)], [1820,log(9.638)], [1830,log(12.861)], [1840,log(17.063)], [1850,log(23.192)], [1860,log(31.443)]], style=point, symbol=diamond, color=blue);

Note that this collection of data points almost appears linear. We'll now use the Maple command fit:

> with(stats):
> fit[leastsquare[[x,z]]]([[1790, 1800, 1810, 1820, 1830, 1840, 1850, 1860],[log(3.929), log(5.308), log(7.24), log(9.638), log(12.861), log(17.063), log(23.192), log(31.443)]]);

We now define the exponential function using this linear function:

> f:=x -> exp(-51.45501529+.2951412006e-1*x);

We now plot the graph of the exponential function:

> plot(f(x),x=1790..1860);

We'll store this information using the name fctplot:

> fctplot:=":

Finally, we combine these plots:

> display({dataplot,fctplot}, view=[1790..1860,0..30]);