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

Solutions

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.
    \\\sol
    \begin{eqnarray*}
      A-B+C & = & 1 \\
      C & = & 4 \\
      A+B+C & = & 3
    \end{eqnarray*}
  \item  Solve this system of equations and write down the quadratic function that fits the data.
    \\\sol
    Solving gives $A=-2$, $B=1$ and $C=4$ so the equation is:
    \[
    y = -2x^2+x+4
    \]
  \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  $x^2-4x+5$
    \\\sol
    $x^2-4x+5 = (x-2)^2+1$
  \item  $5-4x-x^2$
    \\\sol
    $5-4x-x^2 = 9-(x+2)^2$
  \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:
  \[
  \int \frac{u \; du}{\sqrt{a+bu}} = \frac{2}{3b^2}(bu-2a)\sqrt{a+bu} +C 
  \]
  \\\sol
  Make the substitution $u=x^2$ to get (notice that $a=6, b=5$):
  \begin{eqnarray*}
  \int \frac{14x^3}{\sqrt{6+5x^2}} \; dx
    & = & 
    \int \frac{7u}{\sqrt{6+5u}} \; du \\
    & = & 
    7\left(\frac{2}{75}\right)(5x^2-12)\sqrt{6+5x^2} +C 
  \end{eqnarray*}


\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:
    \\\sol
    \begin{eqnarray*}
    S_6 & = & \frac{1}{9} \left( 
      \frac{1}{1} + 4\frac{1}{(4/3)} + 2\frac{1}{(5/3)} + 4\frac{1}{(6/3)} + 2\frac{1}{(7/3)} 
      + 4\frac{1}{(8/3)} + \frac{1}{(9/3)}
      \right) \\
      & \cong & 1.09894179894
    \end{eqnarray*}
  \item  Compute the integral using the Fundamental Theorem of Calculus.
    \\\sol $\ln 3 \cong 1.9861228867$
  \item  Compute the error of Simpson's rule (i.e., how far off is Simpson's Rule?).
    \\\sol $|E| = |S_6 - \ln 3| \cong 0.00032951027$.
  \end{enumerate}

\end{enumerate}

\end{document}