Math 472/572 - Assignment 1
Math 472/572 - Assignment 7
Due: Thursday, November 16. Nothing accepted after Tuesday, November 21. 10% off for being late. Please work by yourself. See me if you need help.
The following table gives the mean daily temperature in Chicago for each month of the year.
Month Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
Temp 26 27 36 46 57 69 72 71 67 53 41 30
1. (3 points) Let 0 represent January, 1 – February, …, and 11 represent December. Let m be the month and T the temperature. Make a plot of T vs m. In Mathematica you need to enter the data into a 12(2 matrix d with the m values in the first column and the T values in the second and use ListPlot[d]. In MATLAB you need to enter the m values into one vector m and the T values into another vector T and use Plot(m, T, '*').
2. (3 points) Use mathematical software to get the 11th degree polynomial p(m) that interpolates the (m, T) data. In Mathematica you need to use p = InterpolatingPolynomial[d, m]. In MATLAB you need to use p = polyfit(m, T, 11). The output from MATLAB is a vector with the coefficients of the polynomial from highest to lowest.
3. (3 points) Make a graph of the polynomial. In MATLAB you can use the following to make the graph. m1 = linspace(0, 11, 111); Tpoly = polyval(p, m1); plot(m, T, '*', m1, Tpoly, ':'). Based on this graph do you see any problems in this polynomial in describing the relationship between T and m?
4. (3 points) Extend the data on both ends by six months by repeating July through December at the beginning and January through June at the end. Let –6 represent July at the beginning, –5 - August at the beginning, etc. Let 12 represent January at the end, 13 - February etc. The data is now as follows.
m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
T 72 71 67 53 41 30 26 27 36 46 57 69 72 71 67 53 41 30 26 27 36 46 57 69
Make a plot of the extended data.
5. (3 points) Let q(m) be the piecewise polynomial function that is equal to p(m) for 0 ( m < 12 and is equal to p(m+12) for -6 ( m < 0 and is equal to p(m-12) for 12 ( m ( 17. Make a graph which shows both the extended data and q(m). Based on this graph do you see any problems in q(m) in describing the relationship between T and m that you didn't note in part 3?
6. (5 points) Find some other function f(m) to describe the relationship between T and m that doesn't have the drawbacks you noted in parts 3 and 5.
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