Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (76) |
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Assigned Tuesday February 10; due Monday February 23 at the tutorials

This document in PDF: HW.pdf

**Required reading. ** All of Spivak Chapter 19.

**To be handed in. ** From Spivak
Chapter 19: Part (vi) of each of problems 1, 2, 3, 5, 7, 9.

**Recommended for extra practice. ** All else in
problems 1-9 of Chapter 19. Never finish your work!!! Just get to the
point where you are convinced that you know how to continue. In
particular, avoid writing what you can do in your head and don't bother
to simplify your results.

**Just for fun. ** We all know that
is a very
good approximation to ; in fact, it is not difficult to find
people who think that *is*
. Prove them wrong,
and also decide which one is bigger ( or
) by
computing the integral

**Aside. ** More on the irrationality of :

Mathematica 4.1 for IBM AIX Copyright 1988-2000 Wolfram Research, Inc. -- Motif graphics initialized -- In[1]:= p5 = x^5(a - b x)^5 5 5 Out[1]= x (a - b x) In[2]:= Expand[p5]/5! 5 5 4 6 3 2 7 2 3 8 4 9 5 10 a x - 5 a b x + 10 a b x - 10 a b x + 5 a b x - b x Out[2]= ------------------------------------------------------------------ 120 In[3]:= derivatives = Table[D[x^7, {x, n}], {n, 0, 10}] 7 6 5 4 3 2 Out[3]= {x , 7 x , 42 x , 210 x , 840 x , 2520 x , 5040 x, 5040, 0, 0, 0} In[4]:= derivatives /. x -> 0 Out[4]= {0, 0, 0, 0, 0, 0, 0, 5040, 0, 0, 0}

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Dror Bar-Natan 2004-02-09