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Homework Assignment 8

Assigned Tuesday October 29; due Friday November 8, 2PM at SS 1071

this document in PDF: HW08.pdf

Required reading

All of Spivak Chapter 9.

To be handed in

From Spivak Chapter 9: 1, 9, 15, 23.

Recommended for extra practice

From Spivak Chapter 9: 8, 11, 21, 28.

Also, let $ p(x)$ be the polynomial $ x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$. Now that we know that for $ \vert x\vert>2n\max(\vert a_{n-1}\vert,\dots,\vert a_1\vert,\vert a_0\vert,1)$ we have that

$\displaystyle \frac12 \vert x^n\vert > \vert a_{n-1}x^{n-1}+\dots+a_1x+a_0\vert, $

complete the proof of the following


Just for fun

Write a computer program that will allow you to draw the graph of the function

$\displaystyle f(x)=\sum_{n=0}^\infty \frac{1}{2^n}\sin 3^nx, $

and will allow you to zoom on that graph through various small ``windows''. Use your program to convince yourself that $ f$ is everywhere continuous but nowhere differentiable. The best plots will be posted on this web site! (Send pictures along with window coordinates by email to drorbn@math.toronto.edu).

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Dror Bar-Natan 2002-10-28