\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose $f(x)=e^{x^2+\sin x}$. Here are values of $f$ and
some of its derivatives at 0:
$$ f(0)=1;\thinspace \ 
f'(0)=1;\thinspace \ 
f''(0)=3;\thinspace \ 
f^{(3)}(0)=6;\thinspace \ 
f^{(4)}(0)=21;\thinspace \ 
f^{(5)}(0)=52\, .$$

\noindent Below are graphs of $f^{(3)}(x)$, $f^{(4)}(x)$, and $f^{(5)}(x)$ on
the interval $[-.5,.5]$. 

\medskip

\centerline{\epsfxsize=1.73in\epsfbox{sg7a.eps}
\quad
\epsfxsize=1.75in\epsfbox{sg7b.eps}\quad
\epsfxsize=1.68in\epsfbox{sg7c.eps}}

\overfullrule=0pt

\line{\ Graph of $f^{(3)}(x)$ on $[-.5,.5]$\qquad
Graph of $f^{(4)}(x)$ on $[-.5,.5]$\ \ \quad
Graph of $f^{(5)}(x)$ on $[-.5,.5]$
}

\medskip

\noindent Assume this information is correct. No additional
computation of the values of $f$ or any of its derivatives is needed
for this problem.

\medskip

\noindent a) What is the second degree Taylor polynomial centered at 0
of $f$?  {\it Do no unnecessary arithmetic!}

\medskip

\noindent b) Find a polynomial $P(x)$ so that $|P(x)-f(x)|<.01$ for
all $x$ in the interval $\left[-{1\over 4},{1\over 4}\right]$. You
should write the polynomial and explain why the error is less than
$.01={1\over {100}}$.



\vfil\eject\end