Assignment 37

"There is no smallest among the small and no largest among the large; but always something still smaller and something still larger." -- (Anaxagoras, ca. 450 B.C.)

Math History Tidbit:

Mary Somerville (1780-1872): Scottish-born Somerville lived during a time when women were not supposed to be educated in mathematics and science. Despite opposition from her own parents, she created her own personal library with books on algebra, geometry, logarithms, calculus, physics, astronomy, and probability theory. She eventually published many outstanding papers and articles...and went on to publish four books, including The Mechanism of the Heavens , which resolved many mathematical problems relating to astronomy. In 1835, Somerville and Caroline Herschel were named as the first honorary female members of the Royal Astronomical Society.


Herkimer's Corner

Why did Herkimer go to the zoo to try to improve his reading comprehension?

Answer: He was told he should practice reading between the lions.

Things Herky would like to know:

Are Santa's elves just a bunch of subordinate Clauses?

If stupidity gets you into a mess, why can't it get you out?


Reading: Section 6.2, pages 329-332.

Written: Page 333/15-26. In each case, find the zeros of the function (the values of x that make f(x) = 0, and describe the end behavior of the function. Indicate whether or not the function represents a polynomial.

Items for reflection:

Mathematical word analysis:
PERPENDICULAR: From the Latin words per (thoroughly) and pendre (to hang). A hanging weight forms a right angle with a horizontal surface.


Consider f(x) = x3 - 7x2 +10x.

If you use your calculator to graph this function, you will find that f(x) = 0 when x = 0 or x = 2 or x = 5. The end behavior can be described as follows: As x --> infinity, then f(x) -->, and as x --> -infinity, then f(x) --> -infinity. Basically, this is saying that as x gets very large, f(x) also gets very large, and as x gets very small (decreased indefinitely), f(x) gets very small (decreases indefinitely). Note that f(x) is a polynomial. Also, note that --> is not ==>.


Consider f(x) = 2 + x-1 = 2 + 1/x.

We have f(x) = 0 when x = -1/2. The end behavior can be described as follows: As x --> infinity, f(x) --> 2, and as x --> -infinity, f(x) --> 2. That is, as x becomes indefinitely large or indefinitely small, the function values approach the number 2. This should make sense. If x = 100,000, for instance, then f(100,000) = 2 + 1/(100,000), which is very close to 2.

We can also note that f(0) is undefined. That is, the function cannot be evaluated at x = 0 since this would involve a division by 0. What about the behavior of this function near x = 0? We can write f(x) --> infinity as x approached 0 from the right, and f(x) --> -infinity as x approaches zero from the left. Note that f(x) is not a polynomial.

Problem: Given the function f defined by

f(x) = 3x3 + 4x -10x-1

Evaluate the function at x = 2. Is f a polynomial function?

Solution (with communication):

f(2) = 3x23 + 4x2 + 10x2-1

= 24 + 8 + 5

= 37.

f is not a polynomial function since the term 10x-1 has an exponent that is not a non-negative integer.

Problem: Show that the sum of two non-polynomial functions can be a polynomial function.

Solution (with communication):

Let f(x) = x3 + x-2 and g(x) = 3x2 - x-2. Note that f and g are not polynomial functions.

However, h(x) = f(x) + g(x) = x3 + 3x2. Note that h is a polynomial function.