Math 331, Fall 2002: Problems 7-10

NOTE: Each exercise is worth 10 points and can be turned in at any time before its ``expiration date''. At the end of the semester, I will expect you to have turned in at least 2/5 of the exercises assigned. If you do more, I will pick your best grades. If you do less, the missing grades will be counted as zeros. Altogether, these will count the same as one project.

(expires 9/30)     Fit the points $(-1.9,-4.7), (-0.8,1.2),
(0.1,2.8), (1.4,-1.2), (1.8,-3.5)$ by means of a quadratic function $f(x)=ax^2+bx+c$, using the least square method. First, do this step by step, as we did in class; then, use the built-in Maple command, described in the notes. Check that the two solutions agree.

(expires 9/30)     Fit the set of points

\begin{displaymath}(1.02,-4.30), (1.00,-2.12), (0.99,0.52), (1.03,2.51), (1.00,3.34),

with a line, using the least square method we used in class. You will see that this is not a good fit. Think of a better way to do the fit and use Maple to do it. Explain in your solution why you think your better way is better.

(expires 10/7)     [In this problem use Maple only as a word processor. If you're more confortable with paper, you can turn in a paper instead of a Maple worksheet.] Let $n$ points of the form $(r_i,r_i^2)$, $i=1,2, \ldots, n$, be given. What is the quadratic function $f(x)=ax^2+bx+c$ that best fits them? Prove your answer. Does it depend on the optimization method (least square or others)?

(expires 10/7)     Once we have calculated the line (or any other curve, for that matter) that best fits a sets of points, we can get an idea how good the fit is by plotting the line together with the points. It is much more scientific, however, to have a measure for this. Come up with a function of the data and parameters of a given best-fit problem that is small when the fit is good and large when the fit is bad, no matter how many points are used. Justify your answer.

MAT 331 2002-09-25