Math113 Cubic and Quartic Functions

Here’s data from the US Energy Information Administration (EIA) on the total US energy supply coming from crude oil products, in quadrillion BTUs (British thermal units).

Year 1955 1960 1965 1970 1975 1985 1990 1995 2000 2010 2015 2020

Quadrillion BTUs 14.4 14.9 16.5 20.4 17.7 19.0 15.6 13.9 12.4 11.6 19.7 23.5
1. Letting your independent variable \(t\) be the number of years since 1950, use technology to find the cubic polynomial function and the quartic polynomial function that best fit this data. Write down the formulas for these functions below, and plot the data along with the graphs of these two functions on these axes below on the domain \(0 \leq t \leq 80.\)
  
  Protip: you don’t have to start the labels for the tics on your \(y\)-axis at zero. Decide what the minimum and maximum values of \(y\) you need to consider to get a good looking plot first, then label the \(y\)-axis.
2. Between the cubic and the quartic model, which appears to fit the data best? Do either of these functions appear to be viable models to predict future crude oil use in the US in the long term?
3. The cubic model achieves a local minimum value and a local maximum value on the domain \(0 \leq t \leq 80.\) Using technology, find the approximate coordinates of the points corresponding to this min and max, and interpret the coordinates of those points in the context of the situation we’re modelling.
4. Using Desmos, fit a polynomial function of degree higher than four to the data. Can you find a high-degree polynomial that appears to fit the data substantially better than the cubic or quartic model?

Year	1955	1960	1965	1970	1975	1985	1990	1995	2000	2010	2015	2020
Quadrillion BTUs	14.4	14.9	16.5	20.4	17.7	19.0	15.6	13.9	12.4	11.6	19.7	23.5

Challenge

What is the maximum number of times that the graph of a cubic polynomial and the graph of a quartic polynomial can possibly intersect each other?

Puzzle

Draw a circle, and draw \(2\) points on the circle. If you connect those two points with a line, it’ll divide the circle’s interior into two regions.

If instead you draw \(3\) points on the circle, and you draw a line connecting each of those points to the other two, it’ll divide the circle’s interior into four regions.

Now draw \(4\) points on the circle, and draw a line connecting each of those points to the others (there should be six lines total). Into how many regions does this divide the circle’s interior? What if you do this starting with \(5\) points on the circle? Do you suspect a pattern? If you do this starting with \(6\) points on the circle, does it adhere to the pattern you suspected? What’s going on here?