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\title{Sample Problems}
\pagestyle{myheadings}
\author{Math 464}
\date{}

\begin{document}
\maketitle
\markboth{Sample Problems}{Sample Problems}


One notebook sized page of notes will be allowed on the final.  You may
write on both sides of the page.  No electronic devices  will be allowed.  This set of
problems only covers material since the midterm.  The final will be
comprehensive.  There will be questions covering the first half and second
half of the course. The exam is at \textbf{8:30 a.m. on Wednesday,
  December 13}.  

\bigskip

\begin{enumerate}

\item The weights of a Gaussian quadrature formula are always
  positive.  Prove that this is true for the Gaussian quadrature
  formula based on three points in the interval $[-1,1]$. Hint: you
  don't need to explicitly find the Legendre polynomial of degree three.

\item  Let ${\displaystyle
    Q(f)=af(\frac{1}{4})+bf(\frac{1}{2})+cf(\frac{3}{4})}$  
    be a quadrature formula on $[0,1]$.
\begin{itemize}
\item[a)]  If $Q$ is to have precision at least 2, what are the values
  of $a,b,c$? 
\item[b)]  What is the exact precision of the resulting quadrature?
\end{itemize}

\bigskip

\item Let $p(x)$ be the linear function that interpolates $\sin (x)$ at $0$ 
and $\pi/2$. Prove that $|p(x)- \sin (x)| \le \frac{1}{2}(\pi/4)^2$ on
$[0,\pi/2]$. 

\bigskip

\item Let $A$ be a nonsingular $2\times 2$ matrix. Prove that the $\infty$-norm
condition number and $1$-norm condition number are equal.

\bigskip 

\item  Find the polynomial $p$ of degree $3$ that satisfies the following 
conditions: $p(1)=1, p'(1)= 2, p''(1)=1, p(2)=3$. Prove that the polynomial 
is unique.


\bigskip

\item 
\begin{itemize}
\item[a)]  What points $x_0,x_1,x_2,x_3$ should be used in $[0,1]$ to
  minimize the error bound in polynomial interpolation at four points?
  Express your answer using the symbols  Hint: Use 
$\displaystyle{\cos x = \sqrt{\frac {1+\cos 2x}{2}}}$.
\item[b)]  What is the largest value attained by
  $(x-x_0)(x-x_1)(x-x_2)(x-x_3)$ in $[0,1]$, where $x_0,x_1,x_2,x_3$
  are the points chosen in part a)?  
\medskip
\item[]  {\color{blue} In this problem, $[0,1]$ is \underline{not} the
  interval $[-1,1]$.}
\end{itemize}

\bigskip

\item Prove that 
\[
2T_n(x) =  \left(x+i\sqrt{1-x^2}\right)^n +\left(x-i\sqrt{1-x^2}\right)^n,
\]
for $n=0,1,2$, where $i=\sqrt{-1}$ and $T_n$ is the $n^{th}$ Chebyshev polynomial. (This is formula is true for all $n$.)

\bigskip

\item The Chebyshev polynomial of degree $3$ is $T_3(x)=4x^3-3x$. 
\begin{enumerate}
\item What is the interpolatory quadrature rule on the interval $[-1,1]$ that uses the roots of $T_3$ as nodes?
\item What is the corresponding rule on any interval $[a,b]$? ("Translate" the interval.) 
\end{enumerate}

\bigskip

\item  
\begin{itemize}
\item[a)]  Let $p(x)=2x^4-3x^3+5x^2-2x+7$.  {\it Use nested
    multiplication} to evaluate $p(2)$.  Show the multiplications.
\item[b)]  Use nested multiplication to evaluate $p'(2)$.  Show the
  multiplications.  {\it Do \underline{not} take the derivative of $p(x)$
    and then use the polynomial $p'(x)$ to evaluate $p'(2)$.}
\item[c)] Use nested multiplication again to evaluate $p''(2)$.
\end{itemize}

\bigskip

\item  Let $p_3(x)$ be the unique polynomial of degree at most $3$ that
  interpolates $f(x)=2^{x+2}$ at the points
  $(x_0,x_1,x_2,x_3)=(-2,-1,1,2)$.  Notice  
  that $0$ is \underline{not} one of the points.
\begin{itemize}
\item[a)]  Form the divided difference table for $f$ at these points and
  write $p_3(x)$ in its Newton form.
\item[b)]  Give the best upper bound for $\Vert f-p_3\Vert _{\infty}$ on
  $[-2,2]$. 
\end{itemize}

\end{enumerate}

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