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\bf Math 3310/6310 Homework Assignment 3 Due Feb. 13, 2017
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\begin{enumerate}

\item Write comments for each line of the function \verb+mystery.m+ that explain what each line does.


\item Use Newton's method with $x_0=0.8$ to find a root of the function
\[
f(x)=\frac{1}{2}+\frac{1}{4}x^2-x\sin(x)-\frac{1}{2}\cos(2x)
\]
within $10^{-5}$.  Compute the order of convergence and explain what is happening.  

\item The polynomial $P(x) = x^3 - 7x^2 + 14x - 5$ has one real root and two complex roots.
\begin{enumerate}
\item 
Use Newton's method to find all the roots. You may try an {\em imaginary} initial guess to find the 
complex roots. 

 \item Use the program \verb+muller.m+ to find the root. 

\item Compare the two methods.
\end{enumerate}

\item
Consider the function $f(x) = 2e^{-(x-1)^2} + e^{-3x^2}$.
Plot the function and find the value of $x$ up to 7 decimal
places where this function has a maximum.


\item Plot the polynomial $f(x) = x^2+x+1$.  Find the roots of this 
polynomial to within a tolerance of $10^{-6}$.   Show and explain what happens with some of your
unsuccessful attempts.
\end{enumerate}


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