Worksheet on Smoothing

Smoothing is the process of choosing a lower- order polynomial and de termining a best -fit according to some criterion such
as least - squares .

Since we already know how to apply the criterion, we need to determine how to choose the polynomial. We will be considering
the following two questions : (1) Should a polynomial be used ? and, if so (2) What order would be appropriate?

Let be the change in the y -values, i.e.
be the change in the -values, i.e.
be the change in the- values i .e.
 

1. Given the data use these definitions to complete the fol lowing table of differences:



2.if then for smallSimilarly,and so on. Explain how the derivatives
can be related to the quantities ,,...

(a). To complete the first table below, look at the columns for  ,,... above and describe any pattern or trend you
observe. For the second table, compute the derivatives of a general quadratic function P(x) = ax² + bx + c and write
down the behavior or trend of each derivative.
 

Difference Coln	Pattern/Trend

P(x) = ax2 + bx + c	Behavior/Trend

(b). Based on the tables in part (a), does it seem reasonable to try and find a quadratic polynomial to pass through these
points? If so, explain why and find the quadratic.

Note: All of the data points in problem 1 should lie on the quadratic found in #2(b).

3. Given the x-values in the table below, use the quadratic found in #2(b) to compute the y-values. Enter them in the table.

(a). Since all of these points lie on the quadratic curve , what do you expect for the ,, and differences?

(b). Compute the differences. Are they what you expected? Do you see any patterns appearing?

(c). What noticable difference is there between this data set and the set given in problem #1?

4. Given the x-values in the table below, use the quadratic found in #2(b) to compute the y-values. Enter them in the table.

(a). Since these points lie on the quadratic curve, what do you expect for the ,, and differences?

(b). Compute the differences. Are they what you expected? Do you see any patterns appearing?

(c). What noticable difference is there between this data set and the set given in problems #1 and 2? Explain how this

observation may be the reason the differences did not give you what you expected.

(d). Can you think of a modification to the differences that might take this observation into account. [Hint: Look at #2]