Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: (5) Next: Partial Solution of Homework Assignment 1
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Homework Assignment 1

assigned Sep. 10; due Sep. 24, 2PM at SS 1071

this document in PDF: HW01.pdf

Required reading

All of Spivak Chapter 1.

To be handed in

From Spivak Chapter 1:

Hand in Don't hand in
11 even parts 7
12 even parts 15
14 18, 20

And also (to be handed in)

  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

    $\displaystyle \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

      $\displaystyle 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

      $\displaystyle (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: 21, 22, 23

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Dror Bar-Natan 2002-09-20