Assignment 43

"Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs is pencil and paper." -- (D. J. Albers and G. L. Alexanderson, Mathematical People )

Math History Tidbit:

Triskaidekaphobia: This is fear of the number 13. Interestingly, no one really knows how this developed. It has been suggested that it might be related to the Biblical Last Supper, but there is no real evidence that this is the source.

Mark 14: 17-21
And it was in the evening he came with the twelve. And as they were at the table eating, Jesus said "Truly, I say to you, one of you will betray me, one who is eating with me." They began to be sorrowful, and to say to him one after another, "Is it I?" He said to them, "It is one of the twelve, one who is dipping in the same dish with me. For the Son of man goes as it is written of him, but woe to that man by whom the Son of man is betrayed! It would have been better for that man if he had not been born."

The inference is that one in the group of 13 is doomed. Famous people who would not dine in a group of 13 include Napoleon, Paul Getty, and Franklin Delano Roosevelt.

Even today, some tall hotels don't have a 13th floor because many people are superstitious about the number. Floors might be numbered 12, 12A, 14 to avoid the number 13.

It is easy to prove that every year must have at least one Friday the 13th. If you are not convinced, start by considering January 1. It any year, it must be one of the seven days of the week. If you will do a bit of calendar research using this fact, you will find the Friday the 13th can't be avoided.

Herkimer's Corner

What does Herkimer call a window that has been installed in a glass building?

Answer: A pane in the glass.

Herky's friends:

WILLIE MAKEIT... everybody wonders if the guy can be successful.

MILLIE METER... she believes the U.S. should convert to the metric system.

ASSIGNMENT #43

Reading: Review text, as necessary.

Written: Review problems (below) for Chapters 1-6.

Items for reflection:

Mathematical word analysis:
OBTUSE: From the Latin ob (against) + tundere (to beat). Basically, when contrasted to an acute angle, an obtuse angle is "not pointed." An object becomes blunt or dull when one beats it against something.

REVIEW PROBLEMS:

1. Expand (a + 2b)2.

2. Convince yourself that (a+b)9 + (a+b)7 = (a+b)7(a2 + 2ab + b2 + 1).

3. Solve for x: x6 - 4x4 = 0.

4. Solve for x: (a) |x + 7| = 10; (b) |x + 7| = -10; (c) |x + 7| = 0.

5. Given the function y = f(x) = (x - 72)2 - 83, (a) What geometric figure is represented by the function: (b) What is the maximum value of y? (c) What is the minimum value of y? (d) What is the domain of f? (e) What is the range of f? (f) For what values of x does f(x) = 17?

6. The function f is a three-part function defined by

f(x) = 7x if x > 5
f(x) = 2x -1 if 0 x 3
f(x) = 5x if x < 0

(a) Sketch a graph of x. (b) f(1) + f(-2) + f(6) = ? (c) Solve f(x) = 10. (d) Solve f(x) = -10. (e) What is the domain of f?

7. In the form y = mx+b, what is the equation of the line containing (2,4) and (4,40)?

8. Solve for x: x2 - 92 = 12.

9. Solve the system x + y + z = 11, 4x - y = 11, and x + y = 9.

10. Solve for x in terms of a, b, and c: a/x + b = c.

11. Solve for x in terms of p and q: 2/x - p = 1/x + q.

12. Here are three constraints: x >2, y < 8 and y > x. (a) Graph the region containing points (x,y) that satisfy the constraints: (b) Given P = -2x + 3y, find the maximum and minimum value of P subject to the given constraints.

13. Fit a cubic polynomial equation to the points (-1,-7), (0,5), (1,15), (2,27).

14. Describe the behavior of the function y = -x4 + x3 + 2x +4. That is, describe behavior as x gets very large, as x gets very small, and identify x-intercepts, y-intercepts, local maximums, and local minimums.

Problem: Solve the system

(a): x+y = 99
(b): x+y+z = 200
(c): x+z = 103

Solution (with communication):

Substituting (a) into (b) yields 99+z = 200 ==> z = 101.

(c) ==> x+101 = 103 ==> x = 2.

(a) ==> 2+y = 99 ==> y = 97.

The three planes intersect at the
point (2,97,101).

Problem: Solve a + w/x = b for x.

Solution (with communication):

a + w/x = b

==> w/x = b-a

==> w = x(b-a)

==> x = w/(b-a) if b-a is not zero.