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.
1. Find the cubic polynomial C(x) = a x3 + b x2 + 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.