Assignment 51

"Angling may be said to be so like mathematics that it cannot fully be learned." -- (The Complete Angler, 1653)

Math History Tidbit:

Trigonometry: An amazing number of useful relationships come from the three simple definitions for sine, cosine, and tangent. The Muslims introduced these ratios and established many of the extremely important trigonometric relationships that we use today. Muslim developers of trigonometry include

Al-Battani (ca. 920)
Abu Wafa (ca. 980)
Ibn Yunus (ca. 1000)
Thabit ibn Qurra (ca. 900)
Nasir al-Din al-Tusi (ca. 1260)

Medieval Islamic mathematicians were expert astronomers and contributed much to the development of spherical trigonometry. They made an effort to preserve, translate, and study works of Greek mathematicians such as Euclid, Apollonius, and Diophantus. The city of Baghdad (the capital of what is now Iraq) was established in 766 A.D. and became a flourishing intellectual center.


Herkimer's Corner

Why did Herkimer cut a hole in the carpet at a Las Vegas nightclub?

Answer: So he could see the floor show.

Things Herky would like to know:

Some folks seem to be luckier than others. Why does an orthopedist get all the breaks?

Is it OK to get angry with a dermatologist who makes rash statements?


Reading: Review Section 7.4, as necessary.

Written: Pages 427-428/57-67. Write these up neatly. Communicate.


Items for reflection:

Mathematical word analysis:
RADICAL: From the Arab word jidr, which means "plant root." In mathematics texts of the Arabs, a number such as 25 was thought to "grow" from a root number 5. When translated into Latin, jidr became radix, which is the Latin term for root.

The concept of a function inverse is important. A characterization (not a definition) of a function is that is is a collection of ordered pairs. The inverse of a function is a collection of ordered pairs obtained simply by interchanging the x and y values.

As a simple example, if the side of a cube is x inches, then the function f(x) = y = x3 would provide the volume (in cubic inches) of the cube. For instance, if a cube has a side of 3.8 inches, then y = f(3.8) = 3.83 = 54.872 (cubic inches) is the volume of the cube. Now, note that

x3 = y ==> x = y1/3

Now think! We have a rule that simply says that if we are given the volume of a cube, we can find the side simply by taking the cube root of the volume. Remember, our original rule was f(x) = x3. OK, the inverse rule is f-1(x) = x1/3. The variable x is a dummy variable. As long as you know what it represents, and you have the correct formula, everything is totally logical. It is the rule that is important, and the rule can be expressed in terms of any variable you choose.

So, if x is the length of a side of a cube, and if x = 7.34 inches, then f(x) = x3 = (7.34)3 = 395.45 (cu. in.) is the volume of the cube.

And, if x is the volume of a cube, and if x = 327.76 cu. inches, then f-1(x) = x1/3 = (327.76)1/3 = 6.895 (in.) is the length of a side of the cube.


OK, here is a financial example. If $1,000 is invested for for 25 years at interest rate x, then

f(x) = y = $1000(1+x)25 will give you the accumulation of the investment for an interest rate x. If x = 8%, for example, then f(.08) = $1,000(1.08)25 = $6,848.48.

Now, if y = 1000(1+x)25 ==> (1+x)25 = y/1000 ==> 1+x = (y/1000)1/25 ==> x = (y/1000)1/25 - 1.

OK, we now have a rule that will tell us the interest rate earned if someone tells us the accumulation at the end of 25 years. Since our original rule was f, this rule would be f-1, and would be defined by

f-1(x) = (x/1000)1/25 - 1.

For instance, if x is the amount of the accumulation after 25 years, and if x = $8,834, then f-1(8834) = (8834/1000)1/25 - 1 = 0.09105, or approximately 9.1%.


Problem: A function w is a rule that says "take a number, add 12, and then divide what you have by 3." Express this in functional notation, find w-1, and state the rule w-1 in words.

Solution (with communication): If x represents a number, then the rule w can be written

w(x) = (x+12)/3.

Hence, w-1 is a rule that says "take a number, multiple it by 3, then subtract 12." We can write

w-1(x) = 3x - 12.

Problem: Use the functions w and w-1 as defined in the problem at the right and show that w(w-1(x)) = x and that w-1(w(x)) = x.

Solution (with communication):

w(w-1(x)) = w(3x-12)

= [(3x-12)+12]/3 = 3x/3 = x, and

w-1(w(x)) = w-1[(x+12)/3]

= 3[(x+12)/3] - 12 = x+12-12 = x.