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The reader who knows something about physics may still be unsatisfied
with the degree-6 polynomial used to fit the NHRA data. To see
why, consider the first and second derivatives of the polynomial fat x
= 0.
f'(0) = 71.682
f''(0) = -60.427
This would imply that not only is the funny car moving at the beginning of the race, but it is also decelerating. We know that this is not true. The speed at x = 0 should be 0, and the acceleration should be positive.
In order to describe the physical problem in the model more accurately,
Professor Tapia added the additional condition that
f'(0) = 0.
In other words, the polynomial f used to approximate the NHRA data will have a c1 term equal to zero. Recall that the seven data points provide seven equations. Assuming that f'(0) = 0 gives an eighth equation: c1 = 0. In order to form a system of equations with eight unknowns, Professor Tapia decided to look for a polynomial of the form
f(x) = c0 + c1x + c2x2 + c3x3 + c4x4 + c5x5 + c6x6 + c7x7,
with the added condition that f'(0) = 0.
This system can be written in matrix form as
Xc = y,
where 
and 
This system gives the following polynomial:
f(x) = 121.9205 x2 - 85.8203 x3 + 50.6903 x4 - 14.8661 x5
+ 2.1261 x6 - 0.1197x7.
It is fortunate and satisfying that f''(0) turned out to be positive because we have no straightforward way to incorporate this inequality condition into our mathematical model.
The head start for the funny car computed from this polynomial
is 140.6714, again within inches of the values determined using
the other polynomials.
Figure 3.3 contains a plot of this polynomial,
along with the NHRA data points.
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Send comments on material to Cynthia Lanius
These pages are maintained by Hilena
Vargas (hvargas@rice.edu)
Updated: February 28, 2001
Copyright © 2001 Richard Tapia and Cynthia Lanius
