\documentclass[fleqn, 12pt]{article}
%\pagestyle{empty}
\setlength{\topmargin}{-.5in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\newcommand{\sol}{{\textbf{Solution: }}}


\begin{document}

\begin{center}
{\Large Warm-Up Problems and Lecture Problems

February 10, 2003}
\end{center}

\begin{enumerate}

\item Suppose you are given the following three points:
  \[
  (-1, 1) \qquad (0,4) \qquad (1,3)
  \]
  Your goal is to find the quadratic function that fits this data.
  In other words, you need to find $A$, $B$, and $C$ so that for these three points:
  \[
  y= Ax^2+Bx+C
  \]
  \begin{enumerate}
  \item  Find a system of three equations and three unknowns by plugging the data points into the 
    equation.

  \item  Solve this system of equations and write down the quadratic function that fits the data.

  \end{enumerate}


\newpage


\item  Completing the square.  
  In case you don't remember what this means, here is an example.
  We will complete the square for the expression $x^2+6x+25$:
  \begin{eqnarray*}
    x^2+6x+25 & = & x^2+6x + \left(\frac{6}{2}\right)^2 - \left(\frac{6}{2}\right)^2 + 25 \\
    & = & (x^2+6x + 9) - 9 + 25 \\
    & = & (x^2+6x + 9) + 16 \\
    & = & (x+3)^2 + 16
  \end{eqnarray*}
  Complete the square for the following expressions:
  \begin{enumerate}
  \item  $x^2-4x+5$
  \item  $5-4x-x^2$
  \end{enumerate}

\newpage
\textbf{Lecture Problems}

\item  \label{table}
  Using the table of integrals in the back of your book, evaluate the integral.
  \[
  \int \frac{14x^3}{\sqrt{6+5x^2}} \; dx
  \]
  Hint~1: You might have to do a preliminary substitution or algebraic manipulation.
  \\
  Hint~2: You might want to use integral number~55:
  \[
  \int \frac{u \; du}{\sqrt{a+bu}} = \frac{2}{3b^2}(bu-2a)\sqrt{a+bu} +C 
  \]



\newpage

\item  
  Consider the integral:
  \[
  \int_1^3 \frac{1}{x} \; dx
  \]
  \begin{enumerate}
  \item  Using Simpson's Rule with $n=6$, find an approximation for the definite integral:

  \item  Compute the integral using the Fundamental Theorem of Calculus.

  \item  Compute the error of Simpson's rule (i.e., how far off is Simpson's Rule?).

  \end{enumerate}

\end{enumerate}

\end{document}