  Assignment 41

 "The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth." -- (J. B. Shaw, Mathematical Maxims and Minims, 1988) Girolamo Cardano (1501-1576): Born in Italy, Cardano was an unprincipled mathematical genius whose greatest work, Ars Magna , is one of the most important treatises ever written on algebra. During his turbulent life he was imprisoned for heresy because he published a horoscope of Christ's life. A frequent gambler, he was eventually to become known as a founder of probability theory. He became involved in a famous dispute with another Italian mathematician, Niccolo Tartaglia of Brescia (1499-1577) over the discovery of solution methods for cubic equations. Both men contributed much to the development of mathematics. What does Herkimer call a crazy man who inflates balloons? Answer: A balloon-atic. Herky's friends: PETER DOUT...this guy usually gets tired very quickly. NICK O'TEEN ...a nice guy with a bad habit. ASSIGNMENT #41 Reading: Section 6.9, pages 380-383. Written: No problems from book. Mathematical word analysis:MILLION: Related to the Latin mille, meaning thousand. It literally means "great thousand." Italian mathematician Luca Paciola (1445-1515) was the first to use the word million in published writings.

A cubic polynomial has the form y = f(x) = ax3 + bx2 + cx + d.

Note that values of a, b, c, and d are needed to identify a specific polynomial. Basically, four non-collinear points determine a cubic polynomial. Consider the points (-1,0), (0,3), (1,0), and (3,0). If you put the x-values in list L1, and the y-values in list L2, and then do a cubic regression computation, you will find a = 1, b = -3, c = -1, and d = 3, with R2 = 1. The cubic function that contains the four given points is y = x3 - 3x2 - x + 3.

Problem: Fit a polynomial model to the data displayed in the table.

 x 1 2 3 4 5 6 y -4 1 10 26 54 76

Solution (with communication):

If you use cubic regression, you will find that the cubic polynomial function

y = -0.324x3 + 6.278x2 - 13.970x + 4.667

fits the data set quite well, with R2 = 0.99639

For this data set, a quadratic polynomial function

y = 2.875x2 - 3.696x - 3.5

is also a good fit, with R2 = 0.99505.