Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: (159) Next: Wang Ying's Solution of Homework Assignment 15
Previous: Class Notes for the Week of January 6 (9 of 9)

Homework Assignment 15

Assigned Tuesday January 14; due Friday January 24, 2PM at SS 1071

this document in PDF: HW15.pdf

Required reading

All of Spivak Chapter 15.

To be handed in

From Spivak Chapter 15: 2 (even parts), 4 (even parts), 9, 15, 18.

Recommended for extra practice

From Spivak Chapter 15: 2 (odd parts), 4 (odd parts), 7, 14, 27.

An aside

Here's a short Mathematica session that computes an approximation of $ 2\int_{-1}^1\sqrt{1-x^2}dx$:
drorbn@coxeter:~/classes/157AnalysisI:1 math
Mathematica 4.1 for IBM AIX
Copyright 1988-2000 Wolfram Research, Inc.

In[1]:= n = 1000; t[i_] := -1. + 2i/1000; f[x_] := Sqrt[1 - x^2]

In[2]:= 2 * Sum[f[t[i]]*(t[i] - t[i - 1]), {i, 1, 1000}]

Out[2]= 3.14149

Just for fun

Compute the limit

$\displaystyle \lim_{N\to\infty}\cos x
\cdot\cos\frac{x}{2}
\cdot\cos\frac{x}{4}
\cdot\cos\frac{x}{8}
\cdots\cos\frac{x}{2^N}
$

The generation of this document was assisted by LATEX2HTML.


Dror Bar-Natan 2003-01-14