Assignment 49

"As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them." -- (Brahmagupta)

Math History Tidbit:

Brahmagupta (c. 598 A.D.): One of the greatest of Indian mathematicians, Brahmagupta was instrumental in the development of algebra for problem solving. Among other things, he wrote an amazing mathematical treatise in which he covered subjects such as square and cube roots, and fractions. He also enjoyed working with irrational numbers, such as the square root of 2, and calculated values of irrational numbers accurate to many decimal places. Brahmagupta's most important work was the Brahmasphutasiddhanta (Correct astronomical system of Brahma, 628 A.D.) In medieval India, most mathematical works were written as chapters of astronomy books, and the mathematical concepts and techniques were applied to astronomical problems. This was true of the Brahmasphutasiddhanta ... and it was written completely in verse.

Herkimer's Corner

What did Herkimer say when he saw a boy and his father being treated to dinner by a hockey goalie?

Answer: "I've seen the father, son, and the goalie host."

Things Herky would like to know:

When a mathematician hits a drive on a golf course, is it OK if he loudly yells "square root of 16"?

Would it be necessary for the president of a janitor's union to call for sweeping reforms?


Reading: Review Section 7.3, as necessary.

Written: The problems in Items for reflection (below).

Items for reflection:

Mathematical word analysis:
AXIOM: From the Greek word axioma, which means "that which is thought fitting." An axiom is a statement accepted without proof. It is something that appears obvious and "fits" the situation.

1. If f(x) = x3 + 2x2 - 7x + 3 and g(x) = x2 - 7, find

(a) f(x) + g(x); (b) f(w) - g(w); (c): g(f(1)); (d) f(g(1)); (e) g(g(g(2)))

2. If F(x) = x3 and G(x) = x-2, find (and simplify, if appropriate)

(a) F(x)G(x); (b) F(x)/G(x); (c) F(G(2)); (d) G(F(2)); (e) F(F(2)); (f) F(F(F(F(F(1))))); (g) F(2)/G(4) =

3. (a) If f(x) = x2, then f(a+b) =

(b) If W(t) = t + 33, then W(x+8) =

(c) If f(x) = 100(1+x)20, then f(.07) =
Can you put an investment interpretation on the function f?

4. A store selling very expensive items will, on February 8, sell any item for $50 less than the listed price. On any day, the store will give a discount of 15% to any customer who can prove he/she contributed to a local charity. Let x be the listed price of an item. If P(x) is the price you will actually pay for an item on Feb. 8, then

P(x) =

If, on a day other than February 8, you prove that you have contributed to a local charity, then if D(x) is the discounted price,

D(x) =

Calculate P(D(x)) and D(P(x)). What does each function represent, and if you are a customer on Feb. 8 who has contributed to a local charity, why should being able to do this be important to you?

5. The surface area S (in square meters) of a hot air balloon is given by S(r) = 4(pi)r2, where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to the formula r(t) = (2/3)t3, find the surface area S of the balloon as a function of time t.

Problem: If F and G are functions defined by F(t) = t2 and G(w) = 9w - 14, find all values of x such that F(x) = G(x).

Solution (with communication):

F(x) = G(x)

==> x2 = 9x - 14

==> x2 - 9x + 14 = 0

==>(x-7)(x-2) = 0

==> x = 7 or x = 2.

CHECK: F(7) = G(7) = 49 and F(2) = G(2) = 4.

Problem: If f and g are functions defined by f(x) = x + 2 and g(x) = x2, find g(f(f(g(w)))).

Solution (with communication):

g(f(f(g(w)))) = g(f(f(w2)))

= g(f(w2+2))

= g(w2+4)

= (w2+4)2

= w4 + 8w2 + 16.