\input epsf
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\noindent {\bf Problem statement} In this problem,
$f(x)=\displaystyle{1\over{1+x}}-\cos x$. 

\medskip

\noindent a) Graph $f(x)$ in the window $0\le x\le 6$ and $-1\le y\le
1.5$.

\medskip

\noindent b) Write an equation showing how $x_n$, an approximation for
a root of $f(x)=0$, is changed to an improved approximation,
$x_{n+1}$, using Newton's method. Your equation should use the
specific function in this problem.

\medskip

\noindent c) Suppose $x_0=2$. Compute the next two approximations
$x_1$ and $x_2$. Explain what happens to the sequence of
approximations $\{x_n\}$ as $n$ gets large. You should use both
numerical and graphical evidence to support your assertion.

\medskip

\noindent d) Suppose $x_0=4$. Compute the next two approximations
$x_1$ and $x_2$. Explain what happens to the sequence of
approximations $\{x_n\}$ as $n$ gets large. You should use both
numerical and graphical evidence to support your assertion.

\vfil\eject\end

