(a) Fit a quadratic polynomial to the data. Do this by setting up and using the normal equations, not just by calling polyfit. You will get an error message about the matrix being extremely badly conditioned, but you can ignore that. You could improve the condition number by rescaling the time axis from 1900, 1910, ..., 2020 to 0,0.1, ..., 1.2, but you get a decent-looking fit even with the original time scale. Print out the normal equations, and the coefficients of the polynomial. Hint: The num- bers are of vastly different orders of magnitude. Exponential notation (format long e) is recommended. Plot the original data and the polynomial in a single picture. (b) Fit a curve of the form p(t) = beat, by fitting a straight line to ln(p(t)). Again, print out the normal equations, the coefficients a and b, and plot the original data and the exponential curve in a single picture.
1900 75994575
1910 91972266
1920 105710620
1930 122775046
1940 131669275
1950 150697361
1960 179323175
1970 203235298
1980 226547082
1990 248709873
2000 281421906
2010 308745538
2020 331449281