  Assignment 7

 "I have not been in this country very long, but I do know this. For a 16-year old, low-income Hispanic kid growing up in East Los Angeles, there are a lot of things that are dangerous. Calculus is not one of them." -- (Jaime Escalante) The Greek Age of Reason extended from roughly 600 B.C. to 300 B.C. There was the realization that humans have intellect, and that through observation and experimentation, the human mind can discover truths. Mathematics became a useful tool in this discovery process. Greeks such as Pythagoras, Plato, Aristotle, Euclid, Eratosthenes, Archimedes, and others began to develop this magnificent academic discipline in their attempts to understand the universe as they knew it. Given the limitations of the times in which they lived, the Greeks and some other ancient societies produced and developed an incredible amount of mathematical knowledge. When Herkimer was the captain of a ship, what did he say to the panhandler who tried to come on board? Answer: "Beggars can't be cruisers." Herky's friends: JOE KERR...guy who likes to tell funny stories. DINAH MITE...she has an explosive personality. Reading: Section 2.1, pages 67-71. Written: Pages 72-73/35-49(odds), 50-54. Write neatly. Your homework papers should always represent something you would show to a college admission's representative and be willing to say "this is an example of my work." Written homework problems should be self-explanatory. That is, the reader should understand the PREMISE (information provided in the statement of the problem) and your CONCLUSION. Mathematical word analysis:SLOPE: From the Latin word sleubh (slip), perhaps suggesting the ground slipping away as you go forward.

You need two things to determine the equation of a line: a point and a slope. These are the very basic things that are required. (If you have two points, you still must determine the slope before writing the equation.) If the slope of a line is m, and if it contains the points (x1,y1), then the equation can be written y - y1 = m(x - x1). For instance, if a line containing (5,7) has slope 3, its equation can be written y - 7 = 3(x - 5). If you want the equation in the form y = mx + b, simply solve for y, obtaining y = 3x - 8.

Make sure you can interpret slope, which is change in y divided by change in x. If a line has slope 3.4, this means that for every unit increase in x, y increases by 3.4 units. If a line has slope -2/3, the means that if x increases by 3 units, y decreases by 2 units.

I like to think of a numerical function as a rule that assigns to a number another unique number. For instance, when you write

f(x) = 3x -2

then f is a rule that says "take a number, multiply it by 3, then subtract 2." What does the rule f do to 10? Well, f(10) = 3x10 - 2 = 28. That is, f says "take 10, multiply it be 3, then subtract 2." This yields 28. In other words, the rule f assigns to 10 the number 28.

Math is a language. If you will read it properly, and interpret it properly as you do your assignments, you will gain MATH POWER on a daily basis.

 Problem: Let f be a function defined by f(x) = | x - 7 | + 9 (a) What is f(2)? (b) Find all values of x such that f(x) = 12. Solution (with communication): (a) f(2) = | 2 - 7 | + 9 = | -5 | + 9 = 5 + 9 = 14. (b) | x - 7 | + 9 = 12 ==> | x - 7 | = 3 ==> x - 7 = 3 or x - 7 = -3 ==> x = 10 or x = 4. Problem: If y = 2x - 12, then y is a function of f. That is, given an x value, we double it and subtract 12 to get the corresponding value for y. Express x as a function of y. Solution (with communication): y = 2x - 12 ==> 2x = y + 12 ==> x = (y + 12)/2. If g is a function, we could write x = g(y) = (y + 12)/2. In other words,g is a rule that says "take a number, add 12, then divide by 2." Since (y + 12)/2 = y/2 + 6, the rule g could also be stated "take a number, divide by 2, then add 6."