"For the things of the world cannot be made without a knowledge of mathematics." -- (Roger Bacon) Mathematical Drought in Western World (500 - 1200): But mathematics was flourishing in the East. And, in the East appeared perhaps the greatest of all Arab mathematicians, Muhammad ibn Musa al-Khwarizmi (ca 780 - 850), who wrote the first known text in elementary algebra. In algebra, you often "balance" an equation to preserve equality by (for instance) adding a like quantity on both sides of the equality. Today's algebra comes from the al jabr, which is balance in Arabic. The Islamic world was a fertile ground for both mathematical development and the preservation of ancient mathematical knowledge. The city of Baghdad (the capital of what is now Iraq) was established in 766 A.D., and it became a flourishing intellectual center during the early ninth century with the construction of a library and research center.

When Herkimer brought his dirty friend, Hans, home for lunch, what did his mother say? Answer: "Wash your Hans before eating." Herky 's words of wisdom: A day without sunshine is like night. If marriage were outlawed, only outlaws would have inlaws.

Reading: Section 2.7, pages 114-117. Written: Pages 117-119/21-24, 27, 35, 41, 44, 50, 51. NOTE THE SECTION ON GRAPHING PIECEWISE FUNCTIONS ON PAGE 121.

Mathematical word analysis: CIRCUMFERENCE: From the Latin word circumferre, meaning "to carry around."

Reading is important in this section.
Life is much simpler with piecewise functions if you will simply read what is written. Here's an example: A function f is defined by

f(x) = 2x + 1 if x > 4

f(x)= 7x if x = 4

f(x) = x - 3 if x < 4.

OK, f is simply 3-part rule. Here it is in words:

If you are given a number greater than 4, double it, then add 1. If you are given the number 4, multiply it by 7. If you are given a number less than 4, subtract 3 from it.

So, f(9) = 2(9) + 1 = 19, f(4) = 7(4) = 28, and f(-33) = -33 - 3 = -36.

We frequently encounter piecewise rules in real life. A great example appears at the top of page 119 (Postal Rates).

The greatest integer function is also very useful. This is simply a rule that takes an number and "returns" the largest integer that is equal to or less than the number. For instance, INT(4.3) = 4, INT(99.93) = 99, INT(3) = 3, and INT(-4.5) = -5. Note carefully that INT(-4.5) is not -4. The greatest integer that is equal to or less then -4.5 is -5. Think of the real number line.

Your TI-83 has the greatest integer function. Check MATH --> NUM --> 5:int(.

MATH IS POWER.

Problem: To rent a specific item, there is a standard charge of $20. If you rent fewer than 6 items, the weekly rental rate is $12 per item. If you rent at least 6 items, and fewer then 10 items, the weekly rate is $10 per item. If you rent 10 or more items, the weekly rental rate is $8 per item. Create a piecewise function that gives the total weekly cost for a rental of x items. Study the function created and comment on it. Solution (with communication): If C(x) is the weekly cost (in dollars) to rent x items, then C(x) = 20 + 12x if x £ 5 C(x) = 20 + 10x if 6 £ x £ 9 C(x) = 20 + 8x if x > 9 Comment: It can be noted that C(9) = 20 + 10(9) = 20 + 90 = 110 and C(10) = 20 + 10(8) = 20 + 80 = 100. In other words, using this cost function, the weekly cost for renting 9 items is $110, while the weekly cost for renting 10 items is $100. We can also note that C(5) = C(6). A graph of this function would clearly display these characteristics.