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.
|
|