Many populations -- human, plant, and bacteria -- grow exponentially, at least at first. In time, these populations start to lose their resources (space, food, and so on).
Note 6

Here's an example:

"Whale Numbers up 12% a Year" was a headline in a 1993 Australian newspaper. A 13-year study had found that the humpback whale population off the coast of Australia was increasing significantly. The actual data suggested the increase was closer to 14 percent!

When the study began in 1981, the humpback whale population was 350. Suppose the population has been increasing by about 14 percent each year since then. To find an increase of 14 percent, you could do either of the following (P stands for population):

P_{this year} = P_{last year} + (0.14) * P_{last year}

P_{this year} = (1.14) * P_{last year}

Each of these is a recursive rule. The first rule says that to know this year's population, start with last year's population (which is P_{last year}), then add the population growth. Since the growth is 14 percent of last year's population,
add (0.14) * P_{last year}.

The second rule is used more frequently because it's easier to calculate -- it incorporates the adding in one calculation. This equation shows that the second computation is equivalent to the first:

x + (0.14) * x = (1.14) * x

The second computation fits the format of an exponential function, because successive outputs have the same ratio (in this situation, the ratio is 1.14).

Problem B4

Make a table that shows the estimated whale population for the 5 years after 1981.
Note 7

Years after 1981   Estimated Population 0   350 1   2   3   4   5

Years after 1981   Estimated Population 0   350 1   399 2   455 3   519 4   591 5   674   hide answers

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