\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} If a function $f$ is continuous on
the interval $[a,b)$ and if $\lim\limits_{x \to b^-} f(x) = +\infty$
or $-\infty$ then we say ``$f$ explodes at $b$.''

\medskip

\noindent a) Consider the following functions on an interval $[0,b)$
with $b>0$. For each function, find $b$ so that the function explodes
at $b$. Use your calculator to show graphically what occurs.

$${{{x^2+1}\over {x-1}}}\, ; \quad 
{{{\cos x} \over {x-2}}}\, ; \quad {{{x-2} \over {\cos x}}}\, .$$

\noindent b) Write all solutions to the differential equation $y'=y^2$
subject to the initial condition $y(0)=y_0$. Can you find a solution
that explodes at 10? And another one that explodes at 5? Can you find
a solution that explodes at $x_0$ when $x_0 > 0$? How does the initial
condition at $x=0$ connect with a specified explosion at $x_0$? Graph
one exploding solution.










\vfil\eject\end