Assignment 48

 "It is the high privilege and sacred duty of those now living to educate their successors and fit them, by intelligence and virtue, for the inheritance which awaits them." -- (James A. Garfield at his inauguration, March, 1881) James Abram Garfield (1831-1881): Garfield was sworn in as the twentieth United States President on March 4, 1881. His time in office was brief since he was shot by a disappointed office seeker on July 2, 1981, and died from the wounds on September 19, 1981. Garfield is the only U. S. President to develop a proof of the Pythagorean Theorem. History records over 100 different proofs of this Theorem. Garfield's proof is quite eloquent, and easy to understand. Garfield was an interesting character. He was the first president to have his mother present during his inauguration ceremonies, the first to view an inaugural parade from a stand in front of the white house, and the first to be left-handed. In fact, he was ambidextrous and would entertain friends by simultaneously writing a statement in Greek with one hand and in Latin with the other. What did Herkimer say to the monk who was always talking about different religious orders? Answer: "Can't you talk about anything except sects?" Things Herky would like to know: If the price of duck feathers increases, do you say that down is up? When nylon stockings were first sold in the thirties, was there a run on them? ASSIGNMENT #48 Reading: Section 7.3, pages 415-417. Written: Pages 418-419/32-39, 48-51. Write neatly. Communicate.

 Mathematical word analysis:VECTOR: From the Latin word vectus (to carry). A vector has a directional heading and "carries" a force along with it.
OK, some simple ideas here, but you do need to learn the language. If f and g are functions, this section defines what is meant by (f+g)(x), (f/g)(x), etc. This is pretty much common sense. Some simple examples:

If f(x) = 3x3 + 7 and g(x) = x2 + 55, then

(f+g)(x) = f(x)+g(x) = 3x3 + x2 + 62

(f-g)(x) = f(x)-g(x) = 3x3 - x2 - 48

(fg)(x) = f(x)g(x) = (3x3+7)(x2+55) = 3x5+165x3+7x2+385

(f/g)(x) = f(x)/g(x) = (3x3+7)/(x2+55)

The composition of functions is important. Consider two functions f and g defined by f(x) = x+6 and g(x) = x2. Then

f(g(x)) = f(x2) = x2+6, and

g(f(x)) = g(x+6) = (x+6)2 = x2+12x +36.

We can also note the

f(g(3)) = f(9) = 15, and

g(f(3)) = g(9) = 81.

If one thinks of f and g as rules, then an expression such as f(g(x)) is a rule that says "apply rule g, then apply rule f to what you get." Note that in this case, f(g(3)) is not the same as g(f(3)). The composition of functions lacks commutativity.

Here's a real-life application of one rule followed by another rule that will show that the order in which the rules are carried out is significant, since the results can differ depending on the order. Assume that you are facing north. Rule 1 says "take ten steps forward" and Rule 2 says "turn to your right." Imagine yourself doing Rule 1 followed by Rule 2. Now, go back to the starting point and do Rule 2 followed by Rule 1. Did you end up at the same spot as before?

 Problem: If f is a function defined by f(x) = x + 2, evaluate f(f(f(f(1)))). Solution (with communication): f(f(f(f(1)))) = f(f(f(3))) = f(f(5)) = f(7) = 9. Problem: If f is a function, is f(a+b) = f(a) + f(b)? Solution (with communication): It is easy to show that f(a+b) does not always equal f(a) + f(b). For instance, if f(x) = x + 1, then f(a+b) = (a+b)+1 = a + b + 1 and f(a) + f(b) = (a+1) + (b+1) = a + b + 2.