\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} The linear approximation for the function $f(x)=x^5$ near $x=2$ is
$32+80(x-2)$. (You should check this!)

\medskip

\noindent a) What number $a$ will give the best quadratic
approximation
$
x^5\approx 32+80(x-2)+a(x-2)^2$ near $x=2$?

\medskip
\font\ssb=cmssbx10.tfm

\noindent b) If this approximation is used for various $x$'s in the interval
$[2,2.1]$, can you be certain that the error is no bigger than $.05$?
Explain, using Taylor's inequality (the {\ssb Error Bound}). 

\medskip 

\noindent c) Graph $x^5-\left(32+80(x-2)+a(x-2)^2\right)$ (using the
value of $a$ previously found) in the interval $[2,2.1]$.

\vfil\eject\end