MAT131 - Project 1

*due in lecture on Monday 9/28 or Tuesday 9/29*

Several years ago, a population of 470 kafit birds were resettled into the Greater Boomchalaka game preserve. The number of kafit birds found in the preserve in the spring of a number of years is given in the table below.

year | 0 | 1 | 5 | 6 |

# birds | 470 | 940 | 25,220 | 93,590 |

**1.**
Find the cubic polynomial
*C*(*x*) = *a x*^{3} + *b x*^{2} + *c x* + *d* that
goes through all four of the data points. Use this to get an estimate
of the population of kafit birds in years 2, 3, and 4. Does your
answer seem reasonable? If not, can you think of a way to get a
polynomial that gives more reasonable estimates?

**2.**
Find an exponential function
that models the data
well, and use it to estimate the population in years 2, 3, and 4.
Does your answer seem reasonable? If not, can you think of a way to
get better estimates?
(Note that it is impossible for *E*(*x*) to go through all four data
points exactly. Instead, you will have to use some judgement and
ingenuity to find one that ``fits pretty well''.)

In both parts, make a good graph of the functions you find, and
explain clearly your methodology. For the graph, you may either use a
computer or use a sheet of graph paper and sketch what you see on your
graphing calculator to scale. Indicate the actual data points on your
graph. In your explanation, state clearly which steps
you took and why. You do not need to show every minor computation-
instead, do your work on another piece of paper, then rewrite it
leaving out trivial calculations and false starts. Neatness and
clarity of presentation counts in the grading.

You may (in fact, you are encouraged to) work in groups of 4 or 5 to
do this project, and may use any resources you care to (graphing
calculators, computers, math professors, ...), provided you state what
they are. If you work with other students from your recitation section,
you may turn in a single write-up, listing everyone who participated.

Suggestions for good group results: Pick one person to be the coordinator, and another to be the scribe. The main task of the coordinator is just to ensure that everyone does his or her jobs, and to arrange times to meet. The scribe should be responsible only for transcribing the first draft into a final write-up. Of course, everyone should participate in the discussions.