\documentclass[11pt]{article} 

\usepackage{amsmath}    
\usepackage{amsfonts}  
\usepackage{amssymb}
\usepackage{graphicx}  
\usepackage{subfig}
\usepackage{tabularx}
\usepackage{url}
\setlength{\parskip}{3pt plus 2pt}
\setlength{\parindent}{20pt}
\setlength{\oddsidemargin}{0.5cm}
\setlength{\evensidemargin}{0.5cm}
\setlength{\marginparsep}{0.75cm}
\setlength{\marginparwidth}{2cm}
\setlength{\marginparpush}{1.0cm}
\setlength{\textwidth}{165mm}
\setlength{\textheight}{9 true in}
\setlength{\topmargin}{-2cm}
\setlength{\footskip}{10pt}

\linespread{1.5}

\begin{document}

\pagestyle{empty}

%\begin{flushright}
%Name: \line(1,0){200}
%\end{flushright}
%\vskip1ex
\begin{center}{
\large{ \bf{Numerical Analysis, Spring 2019 \\ Homework 1 due January 28th}}
}\end{center}

Please show your work. Exercises in 1-5 should be done by hand (or using \LaTeX). For the programming exercises in Chapter 4,
hand in your MATLAB codes and outputs. (You can use the MATLAB function \emph{diary} to copy your command window output - read about it in Chapter 2.)
Page \url{http://academics.davidson.edu/math/chartier/Numerical/matlab.html} contains MATLAB m-files mentioned in the assignments,
but you can (and are encouraged to) write your own routines.

\begin{enumerate}
 
\item Show that the following equations have at least one solution in the given intervals.
  \begin{enumerate}
   \item $2x \cos (2x) - (x-2)^2 = 0$, intervals $[2, 3]$ and $[3, 4]$ 
   \item $x-(\ln x)^x = 0$, interval $[4, 5]$
  \end{enumerate}
\vskip2ex
\item Find $\text{max}_{0 \leq x \leq 1} |f(x)|$ for $f(x) = (2-e^x+2x)/3$.
\vskip2ex
\item Given $f(x) = 1-e^x +(e-1) \sin ((\pi/2)x)$, show that $f'(x)$ is $0$ at least once in the interval $[0, 1]$.
\vskip2ex
\item Let $f(x) = x^3$.
  \begin{enumerate}
    \item Find the second Taylor polynomial $P_2(x)$ about $x_0 = 0$.
    \item Find $R_2(0.5)$ and the error in using $P_2(x)$ to approximate $f(0.5)$.
    \item Repeat part (a) using $x_0 = 1$.
    \item Repeat part (b) using the polynomial from part (c).     
  \end{enumerate}
\vskip2ex
\item Use three iterations of the Bisection method to find approximation of the root of $f(x) = \sqrt{x} - \cos x = 0$ on $[0, 1]$. 
\end{enumerate}
\vskip2ex
\emph{Programming part:} \vskip1ex
NOTE: Hand in all your MATLAB codes and outputs. Answer all qeustions.
When you submit any graphs (generated in MATLAB), make sure you label axes and title and annotate your plots.  
\begin{itemize}
\item Chapter 4, Exercise 2(a).
\vskip2ex
\item Chapter 4, Exercise 18.
\end{itemize}

\end{document}







