© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: | (95) |
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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:

- Prove that if exists, then
*Schwarz second derivative*of at . Hint: Use the Taylor polynomial with and with . - Let for and for . Show that
- Prove that if has a local maximum at , and the Schwartz second derivative of at exists, then it is .
- 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