% Calculus II exam, prepared by Russell Brown, spring 2002.

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\newcommand{\CourseName}{Calculus II}
\newcommand{\Instructor}{Your name goes here}
\newcommand{\DateofExam}{5 March 2002}
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\newcommand{\score}[1]{\hfill\rule{0.3in}{0.01in}~/#1}
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\CourseName \hfill  \Instructor \\
Exam 2  \hfill  \DateofExam

\vfill
%Boilerplate instructions, modify as necessary.

Answer all of the following questions.  Additional sheets are
available if necessary.  No books or notes may be used. You may use a
calculator. You may not use a calculator which has symbolic
manipulation capabilities.  When answering these questions, please be
sure to 1) check answers when possible, 2) clearly indicate your
answer and the reasoning used to arrive at that answer {\em
(unsupported answers may not receive credit).}

Each question is followed by space to write your answer. Please lay
out your solutions neatly in the space below the question.  You are
not expected to write each solution next to the statement of the
question. 

The total on this test is 105. However, no student will earn more than
100. 

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Name \rule {2in}{0.01in}

\smallskip



Section \rule{ 1in }{0.01in}

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\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\[0.1in]
\hline \hline
1       &       & 40   \\[0.1in]\hline
2       &       & 15   \\[0.1in]\hline
3       &       & 15   \\[0.1in]\hline
4       &       & 15   \\[0.1in]\hline
5       &       & 15   \\[0.1in]\hline
6       &       &  5   \\[0.1in]\hline   
				\hline 
Min(Total,100)   &       & 100  \\[0.1in]
\hline  
\end{tabular}
\end{center}

\vfill

\newpage



\begin{enumerate}
\item Compute 5 of the 6 indefinite integrals below. Write here
\rule{0.5in}{0.01in} the letter of the integral that is not to be
graded. If you do not specify the an integral which is not to be
graded, we will  take the five lowest scores. 

\begin{enumerate}
\item $ \displaystyle \int \sin^2 x\, dx$
\item $ \displaystyle \int \sin^3 x\, dx$
\item $ \displaystyle \int \frac { 1} { (4-x^2 )^{ 3/2}} \, dx $
\item $ \displaystyle \int \frac x { x^2 + 2x + 2} \,dx $
\item $ \displaystyle \int x \cos( 2x) \, dx $
\item $ \displaystyle \int \frac 1 { 1 + \sqrt x } \, dx $
\end{enumerate}


\item \item Find the values of $ \lambda $ for which $y (x) = e^ { \lambda x }$
satisfies the equation 
$$
  y '' - 4 y ' - 5 y = 0.
$$


\item Find the form of the partial fraction decomposition for the
following rational functions.  DO NOT SOLVE FOR THE CONSTANTS. 
\begin{enumerate}
\item $\displaystyle \frac x { (x^2 -4 )(x^2 + 4)} $
\item $ \displaystyle \frac { x^ 2-43x } { ( x+ 1) ^ 3 ( x+ 2) (
x^2-4)} $
\item $ \displaystyle \frac { x+ 1} { x^2 ( x^2+ 1)^2 ( x^2+ 2x +1)} $
\end{enumerate}

\newpage
\item   The trapezoid rule $T_n$  and Simpson's rule $S_n$ for approximating
the integral  $
\int_a^b f(x) \, dx$ are
\begin{eqnarray*}
T_n &=& \frac h 2 ( f( x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) +f(x_n))
\\
S_n& =& \frac h 3 ( f( x_0) + 4f(x_1) + 2f(x_2) + 4 f(x_3) + \dots +
   4f(x_{n-1}) +f(x_n))
\end{eqnarray*}

The errors satisfy 
$$ |E_T| \leq \frac {M_2(b-a)^3 }{ 12n^2} \quad \mbox{and}\quad  |E_S| \leq
\frac {M_4 (b-a)^5 } { 180 n^4}
$$
where $M_k$ is a number which satisfies $|f^{(k)}(x)|\leq M_k$ for all $
x$ with $ a\leq x \leq b$. 

\begin{enumerate}
\item Use the trapezoid rule and Simpson's rule with $n =4$ to
approximate the integral
$$
\int_3^6 \sin(2x) \, dx.
$$
Give your answers correctly rounded to four decimal places. 
\item Find $n$ so that the error in the trapezoid rule is at most $10^{-4}$,
 $$
\left| \int_3^6 \sin(2x) \, dx - T_n \right| \leq 10^{-4}.
$$
\item Find $n$ so that the error in Simpson's rule is at most
$10^{-4}$, 
 $$
\left| \int_3^6 \sin(2x) \, dx - S_n \right| \leq 10^{-4}.
$$
\end{enumerate}

\vfill 

%\score{15}
\newpage

\item \begin{enumerate} \item State the comparison theorem for
improper integrals. 
\item Use the comparison theorem to determine if the following
improper integrals converge. 
\begin{enumerate} \item $\displaystyle \int_1 ^ \infty \sin^2 (x) e^{
-x} \,dx $
\item $ \displaystyle \int_ 1 ^ \infty \frac { 2 + \sin x } x \, dx $ 
      \end{enumerate}
      \end{enumerate}

\newpage

\item \begin{enumerate} \item    Find a function $y= y(x) $ which satisfies
$$
 \frac { dy}{dx}  = y ( 2-y) \quad \mbox{and} \quad y ( 0 ) = 1.
$$
\item
Find 
$$
\lim _{ x\rightarrow \infty } y( x) .
$$
      \end{enumerate}


\newpage

\item There are thirty-two teams in the 2002 FIFA Copa del Mundo.
Thus, it is interesting to know the value of the integral
$$
\int _ 0 ^ \infty x^{32} e^{ -x}\, dx .
$$
Suppose that we know that  
$$
\int_0 ^ \infty x ^{31} e^{ -x} \, dx =A.
$$
Find a simple expression involving $A$ which gives the value of
the integral
$$
\int _ 0 ^ \infty x^{32} e^{ -x}\, dx .
$$

It is known that $A= 8,222,838,654,177,922,817,725,562,880,000,000$,
but I doubt this is very helpful. 

\end{enumerate}



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