Assignment 40

"The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical facilities to the same level of rationality." -- (C. O. Oakley, The American Mathematical Monthly, 56, 1949, p.19)

Math History Tidbit:

Ala Byron Lovelace (1815-1852): She is sometimes referenced as the first computer programmer. Lovelace worked with Charles Babbage, who invented a calculating machine called the Analytical Engine. Lovelies invented the "logic" that made the machine run and "programmed" the machine with this logic. Among other things, she observed that there were frequent situations where repeated calculations of a certain type were needed, so she conceived of the concept of a subroutine with what we today call a loop. Programming techniques developed by Lovelace are used today.

 

Herkimer's Corner

When Herkimer was shy, why did he hesitate to open the refrigerator door?

Answer: He was afraid the mayonnaise was dressing.

Herky's friends:

BRA QUIET...she was always annoyed with people who talked during the showing of a movie.

KENT KETCHUP ...he ran on the track team, but always finished last.

ASSIGNMENT #40

Reading: Section 6.8, pages 373-375.

Written: Page 377/23-33(odds), 35, 36. (Do the things listed below in Items for reflection.)

Items for reflection:

Mathematical word analysis:
NUMERATOR: From the Latin, numerous, meaning "number."

A basic purpose of this exercise is to look at the graphs of functions and do the following:

  • Describe behavior as x -> infinity. (What happens to functional values as x increases indefinitely?)
  • Describe behavior as x -> -infinity. (What happens to functional values as x decreases indefinitely?)
  • Identify x-intercepts. (Points where the functional value, y, is equal to 0.)
  • Identify y-intercepts. (Points where x = 0.)
  • Identify points that represent local maximums.
  • Identify points that represent local minimums.

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Some of what is below is a review of some complex number concepts. It is not directly related to the assignment above.

We know that i = (-1) and i2 = 1. Note the following:

i3 = i2i = -i

i4 = i2i2 = (-1)(-1) = 1

i5 = i4i = (1)i = i.

If you do a bit of experimenting, you will find that

i = i5 = i9 = i13 = ...

-1 = i2 = i8 = i10 = ...

-i = i3 = i7 = i11 = ...

1 = i4 = i8 = i12 = ...

Another thing to notice: If you put your calculator in complex number mode, then go to the home screen and type in the square root of -36, you calculator will produce 6i. In real number mode, you will get an error statement.

Problem: Construct a second-degree polynomial function that has zeros at x = 5i and x =-5i.

Solution (with communication):

The function f(x) = (x - 5i)(x + 5i) will satisfy the request.

This can be written

f(x) = x2 + 25.

This form clearly demonstrates that the function is indeed a second-degree polynomial.

Problem: Describe some characteristics of the cubic polynomial function

y = f(x) = (x+7)(x-2)(x-5).

Solution (with communication):

The function has x-intercepts at (0,-7), (0,2), and (0,5).

f(x) > 0 on the intervals x > 5 and -7 < x < 2.

f(x) < 0 on the intervals 2 < x < 5 and x < -7.