# Homework Assignment 20

Assigned Tuesday March 8; not to be submitted.

this document in PDF: HW.pdf

Required reading. All of Spivak's Chapter 20.

In class review problem(s) (to be solved in class on Tuesday March 15). Chapter 20 problem 16:

1. Prove that if exists, then

The limit on the right is called the Schwarz second derivative of at . Hint: Use the Taylor polynomial with and with .
2. Let for and for . Show that

exists, even though does not.
3. Prove that if has a local maximum at , and the Schwartz second derivative of at exists, then it is .
4. Prove that if exists, then

Recommended for extra practice. From Spivak's Chapter 20: Problems 3, 4, 5, 6, 9, 18 and 20.

Just for fun. According to your trustworthy professor, should approach when goes to infinity. Here are the first few values of :

 0 157.0 1 -644825.1666 2 794263446.1416 3 -465722259874.7894 4 159244913619814.5429 5 -35629004757275297.7787 6 5619143855101017161.3172 7 -658116552443218272478.0047 8 59490490719826164706638.3418 9 -4275606060900548165855463.4918 10 250142953226934230105633222.4574 100

In widths of hydrogen atoms that last value is way more than the diameter of the observable universe. Yet surely you remember that ; in fact, my computer tells me that is approximately -0.0795485. In the light of that and in the light of the above table, do you still trust your professor?

The Small Print. For we get .

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Dror Bar-Natan 2005-03-09