Precalculus · Problem 3.1.34
Polynomial Functions — Stitz & Zeager
Mathematics — Precalculus · polynomial-regression
Problem
We now revisit the data set from Exercise 6b in Section 2.5. In that exercise, you were given a chart of the number of hours of daylight they get on the 21st of each month in Fairbanks, Alaska based on the 2009 sunrise and sunset data found on the U.S. Naval Observatory website. We let represent January 21, 2009, represent February 21, 2009, and so on. The chart is given again for reference. [table — see PDF] Find cubic (third degree) and quartic (fourth degree) polynomials which model this data and comment on the goodness of fit for each. What can we say about using either model to make predictions about the year 2020? (Hint: Think about the end behavior of polynomials.) Use the models to see how many hours of daylight they got on your birthday and then check the website to see how accurate the models are. Knowing that Sasquatch are largely nocturnal, what days of the year according to your models are going to allow for at least 14 hours of darkness for field research on the elusive creatures?Intuitive explanation
Fitting polynomials to periodic data like daylight hours is about balancing fit quality with long-term reliability. A cubic will capture the smooth rise and fall within one year, while a quartic adds extra flexibility—but here's the key insight: polynomials have end behavior that shoots off to , so they'll eventually fail catastrophically for predictions far from your data (like 2020). You're really asking whether a local fit is worth the global cost, and whether the oscillating pattern repeats predictably enough for a polynomial to work at all.
Intuitive explanation
Fitting polynomials to periodic data like daylight hours is about balancing fit quality with long-term reliability. A cubic will capture the smooth rise and fall within one year, while a quartic adds extra flexibility—but here's the key insight: polynomials have end behavior that shoots off to , so they'll eventually fail catastrophically for predictions far from your data (like 2020). You're really asking whether a local fit is worth the global cost, and whether the oscillating pattern repeats predictably enough for a polynomial to work at all.