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\Huge{
Cobwebbing Sequences}

\small{Adapted from homework created by Jonny Comes.} 
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\vskip .1 in

\begin{enumerate}
\item Let $f(x)=\frac{x^2}{2}$.  Carefully sketch the graph of $f$ below.

\includegraphics[height=2.2in]{upper_half}

Suppose $a_n=f(a_{n-1})$ for all integers $n>0$.  Use cobwebbing techniques to investigate the sequence $\{ a_n \}_{n=1}^{\infty}$.
\begin{enumerate}
\item For which values of $a_1$ is the sequence $\{ a_n \}$ converging?
\vskip 1in
\item For each value of $a_1$ that makes the sequence $\{ a_n\}$ converge, state the limit.
\end{enumerate}

\newpage
\item Pictured below are the graphs of $y=\sin(x) +x$ and $y=x$.

\includegraphics[height=3in]{sin+x}

Suppose $a_n=\sin(a_{n-1})+a_{n-1}$ for all integers $n>0$.

\begin{enumerate}
\item For which values of $a_1$ does the sequence $\{ a_n\}_{n=1}^{\infty}$ converges to $0$?
\vskip 1in
\item For which values of $a_1$ does the sequence $\{ a_n\}_{n=1}^{\infty}$ converges to $\pi$?
\vskip 1in
\item For which values of $a_1$ does the sequence $\{ a_n\}_{n=1}^{\infty}$ converges to $2\pi$?
\vskip 1in
\item For which values of $a_1$ does the sequence $\{ a_n\}_{n=1}^{\infty}$ converges to $3\pi$?
\end{enumerate}

\newpage
\item Suppose $r$ is a real number and $a_n=ra_{n-1}$ for all integers $n>0$.
\begin{enumerate}
\item Explain what happens to the sequence $\{a_n\}_{n=1}^{\infty}$ when $r=1$ or $r=0$.
\vskip 1in
\item Assume $r>1$.  What functions should you plot to make use of cobwebbing?  Use cobwebbing to find all values of $a_1$ for which the sequence $\{ a_n\}$ converges.  
\vskip 1.7in
\item Repeat part b for $0<r<1$.
\vskip 1.7in
\item Repeat part b for $r=-1$.
\vskip 1.7in
\item Repeat part b for $-1<r<0$.
\end{enumerate}

\end{enumerate}


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