Assigned: THU 24 SEP 09
THU 08 OCT 09
Problem 1. Boas Problem #1, Ch.11.5, page 542.
Problem 2. Boas Problem #2, Ch.11.5, page 542.
Problem 3. Boas Problem #4, Ch.11.5, page 542.
Problem 4. Boas Problem #6, Ch.11.7, page 544.
Problem 5. Boas Problem #7, Ch.11.7, page 544.
Problem 6. Boas Problem #3, Ch.11.8, page 546.
This last problem is the solution to the infamous "Brachistochrone
Problem" that led to the invention of the Calculus of Variations.
From Wikipedia: "Johann Bernoulli solved the problem ... before posing
it to readers
of Acta Eruditorum in June
1696. Five mathematicians responded with solutions: Isaac Newton, Jakob
Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von
Tschirnhaus, and Guillaume de l'Hôpital. Four of the solutions
(excluding l'Hôpital's) were published in the May 1697 edition of
the same publication." Newton, who was semi-retired from
mathematics and just taking up a position at the Royal Mint of
his solution anonymously.
Bernoulli immediately recognized that this short and elegant solution
— inventing along the way the calculus of variations as a lemma — could only be that of
Newton, and exclaimed to his brother, "The Old Lion
has again shown his
claws!" (Men of
Mathematics by E. T. Bell.)