\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement}~a) Suppose $f(x)=x^3$ and $g(x)= 4
\cos (7x+5) + 8 \sin (x^2 -9) +6$. Find specific numbers $A$ and $B$
so that all values of $g$ are between $A$ and $B$ (that is, $A \le g(x)
\le B$ for all $x$). The values of $A$ and $B$ don't have to be precise!
Find a value of $x$ (call it $x_A$) so that $f(x_A)<A$ and another value
of $x$ (call it $x_B$) so that $f(x_B)>B$.

\medskip

\noindent b) Make a rough sketch on the same graph of $y=f(x)$ and
$y=g(x)$ and $y=A$ and $y=B$ for $x$ between $x_A$ and $x_B$.

\medskip

\noindent c) Find one root of $f(x)=g(x)$ approximately.

\medskip

\noindent d) Explain why the following result is correct: {\bf if} $F$
and $G$ are continuous functions defined on all real numbers and {\bf
if} $\lim\limits_{x \to +\infty} F(x) = +\infty$ and $\lim\limits_{x
\to -\infty} F(x)= -\infty$ and {\bf if} $G$ is bounded (this means
there are numbers $A$ and $B$ so that $A \le G(x) \le B$ for all $x$) then
the equation $F(x)=G(x)$ {\bf must} have at least one root. (Look up
the Intermediate Value Theorem.)
  

\vfil\eject\end