\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\usepackage{latexsym,amsmath,amssymb,epsfig,graphicx}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}
\setlength{\topmargin}{-.5in}
\newtheorem{Theorem}{Theorem}
\begin{document}
\centerline{MATH 135 -- Calculus 1}
\centerline{The Intermediate Value Theorem}
\centerline{October 7, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
\noindent
It's our lucky day, because today we get to discuss some beautiful \emph{real mathematics}(!)
The Intermediate Value Theorem (IVT) is the following statement:

\begin{Theorem}[IVT]  Let $f(x)$ be a function that is continuous at every $x$ in a closed
interval $[a,b]$.  Then for every $M$ between and $f(a)$ and $f(b)$, there exists at least
one $x = c$ in the interval where $f(c) = M$.  
\end{Theorem}

What this is saying in intuitive terms is that the graph of a function that is continuous
at every point of a closed interval \emph{really is one unbroken curve} that can be drawn without
lifting your pen or pencil from the paper.  In the process it goes through every $y$-value
between the $y$-coordinates of the endpoints $(a,f(a))$ and $(b,f(b))$.  This statement
has some important and surprising consequences!   
\vskip 10pt
\noindent
{\it Questions}
\begin{enumerate}
\item[(1)] An important consequence:  The IVT implies that equations have solutions(!)
The IVT implies, for instance, that every cubic polynomial
$$P(x) = Ax^3 + Bx^2 + Cx + D$$
(where $A \ne 0$) has some real root.  Here's why:
\begin{enumerate}
\item[(a)]  Since $A \ne 0$ we can divide both sides of the equation $Ax^3 + Bx^2 + Cx + D = 0$
by $A$ to get an equation of the form $x^3 + \beta x^2 + \gamma x + \delta = 0$, where $\beta = B/A$, etc.  Explain
why, no matter what $\beta,\gamma,\delta$ are, you can always find a really big $x = b > 0$ where 
$b^3 + \beta b^2 + \gamma b + \delta > 0$.  On the other hand there must also be really negative $x = a < 0$
where $a^3 + \beta a^2 + \gamma a + \delta < 0$. 
\item[(b)]  Then think about what the IVT says about the cubic polynomial $x^3 + \beta x^2 + \gamma x + \delta$
on the interval $[a,b]$ between the very negative number $a$ and the very positive number $b$.  (Hint: We want 
to say $c^3 + \beta c^2 + \gamma c + \delta = 0$ for some $x = c$ in the interval.  Why does that follow?)  
\end{enumerate}
\item[(2)]  A surprising consequence.  Take any map and draw a circle on it anywhere.  (See Figure 6 on 
page 103 of our textbook, for example.)  I claim that the there are two distinct points on that circle where
the temperature at the current instant is \emph{exactly the same}.  Here's the idea:  If we put in a coordinate
system with the center of the circle at $(0,0)$, then we can measure the counterclockwise angle $\theta$
from the positive $x$-axis to any diameter of the circle.  Let $f(\theta)$ be \emph{the difference between
the temperatures at the two endpoints of the diameter}.  In other words, if $A$ and $B$ are the two endpoints
as in the figure, then $f(\theta) = $ temperature at $A$ minus the temperature at $B$.  It is reasonable 
to assume that $f(\theta)$ is a continuous function of $\theta$ (Why?).  Then consider what happens as 
$\theta$ increases from $0$ to $\pi$ radians.  How are the values $f(0)$ and $f(\pi)$ related?  What
does the IVT tell you about $f$ on the interval $[0,\pi]$?
\end{enumerate}

\end{document}
