Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:

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Homework Assignment 2

Assigned Tuesday September 16; due Friday September 26, 2PM, at SS 1071

this document in PDF: HW02.pdf

Required reading. All of Spivak Chapter 1.

To be handed in. From Spivak Chapter 1: 11 odd parts, 12 odd parts, 14, and also

1. Show that if $a>0$, then $ax^2+bx+c\ge0$ for all values of $x$ if and only if $b^2-4ac\le0$.

2. Prove the Cauchy-Schwartz inequality
   $$\bigl(a_1b_1+a_2b_2+\cdots a_nb_n\bigr)^2 \le\ \bigl(a_1^{ 2}+\cdots+a_n^{ 2}\bigr) \bigl(b_1^{ 2}+\cdots+b_n^{ 2}\bigr)$$
   in two different ways:
   1. Use $2xy\le x^2+y^2$ (why is this true?), with
      $$x=\frac{\vert a_i\vert}{\sqrt{a_1^{ 2}+\cdots+a_n^{ 2}}}\qquad y=\frac{\vert b_i\vert}{\sqrt{b_1^{ 2}+\cdots+b_n^{ 2}}}$$

   2. Consider the expression
      $$(a_1x+b_1)^2+(a_2x+b_2)^2+\cdots+(a_nx+b_n)^2,$$
      collect terms, and apply the result of Problem 1.

Recommended for extra practice. Spivak Chapter 1: 7, 15, 18, 20, 21, 22, 23.

Just for fun. Seen on the web, source unknown (though see http://www.mrc-cbu.cam.ac.uk/~matt.davis/Cmabrigde/):

Accdronig to a rscheearch at an Elingsh uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is taht frist and lsat ltteer is at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae we do not raed ervey lteter by it slef but the wrod as a wlohe.

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