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\begin{center}

{\bf APPM 1360 \hspace{1.0in}  Make-up FINAL EXAM  \hspace{1.0in} FALL 2000}
% December 20, 2000, ~~~  7:00 - 8:30 p.m.
\end{center}


\noindent {\bf On the front of your bluebook, please write:
a grading key, 
your name, student ID, and
instructor (Dougherty or Biswas). }  This 
exam is worth 200 points and has 9 questions.   A list of 
formulas is given on the back of this exam.  
{\bf Show all work!}   Answers with no justification will
receive no points. (Note: Problems 3, 4, 5 and 7 are worth 25 points.
All other problems are worth 20 points.) 

\be

\item Find the volume created by revolving the region between the
curve $y=4x-x^2$ and the $x$-axis about the (a) $x$-axis and
about the (b) $y$-axis.

\item Let $\ds x(t) = \frac{1}{3} (2t+3)^{3/2}$ and
$\ds y(t) = t + t^2/2$ for $0 \leq t \leq 3$.
\be
\item Find the length of the curve.
\item Find the equation of the tangent line to the curve at $t=1$.
\ee

\item Do the following sequences converge or diverge? If they
converge, find the limit.  Support your answer.
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\; \ds a_n = \left(1+ \frac{1}{n+3} \right)^n & & 
(b) \;\;\; \ds b_n = \frac{\sin^2 n}{n^3}  & &
(c) \;\;\; \ds c_1 = 3$ and $\ds c_{n+1} = 2 c_n
\end{array}
\end{eqnarray*}

\item Evaluate the following integrals. If the integral is improper,
determine
whether if conveges or diverges. If it converges, evaluate it.
If it diverges, justify your answer.
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\; \ds \int \frac{2x^3 + 2x + 1}{x^2(x^2+1)} \; dx  & & 
(b) \;\;\; \ds \int_1^{\infty} \frac{x^3 + 4x + 3}{x^4+2x+1} \; dx  & &
(c) \;\;\; \ds \int \frac{1}{\sqrt{x^2 + 2x}} \; dx
\end{array}
\end{eqnarray*}

\item Do the following series converge absolutely, converge conditionally
or diverge? Be sure to justify your answer and state the test that you
used.
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\; \ds \sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2+1} & &
(b) \;\;\; \ds \sum_{n=1}^{\infty} \frac{n! 2^n}{(2n)!} & &
(c) \;\;\; \ds \sum_{n=1}^{\infty} \tan^{-1} n
\end{array}
\end{eqnarray*}

\item For what values of $x$ do the following series converge
absolutely, converge conditionally or diverge?
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\; \ds \sum_{n=1}^{\infty} \frac{n^3(x+2)^n}{3^n} & & &
(b) \;\;\; \ds \sum_{n=1}^{\infty} \frac{x^n}{2^n \sqrt{n}}
\end{array}
\end{eqnarray*}

\item 
\be
\item Use the Binomial Theorem to calculate the first 4 terms of the
McLaurin series for $\ds f(x)= \frac{1}{\sqrt{1-x^3}}$.
\item Using the McLaurin series for $\sin x$ and your answer to
part (a), find the Taylor polynomial of order four for the function
$\ds g(x) = \frac{\sin x}{\sqrt{1-x^3}}$.
\ee

\item Solve the differential equation $ \ds \frac{dy}{dx} - y = x$
with initial condition $y(0) = 0$.

\item Let $r_1 = \cos \theta$ and $r_2 = 1- \cos \theta$.
\be
\item Graph each equation and find the intersection points.
\item Find the area that is inside the graph of $r_1$ and
outside the graph of $r_2$.
\ee

\ee

\newpage

 
\begin{center}

{\bf APPM 1360 \hspace{1.0in}  Final Exam  Formula Sheet } 
% December 20, 2000, ~~~  4:30 - 7:00 p.m.
\end{center}

\be

\item A short table of integrals.  In the following, $a \neq 0$.
\be
\item $\ds \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}(u/a) + C $ for
$u^2 < a^2$
\item $\ds \int \frac{du}{a^2+u^2} = (1/a) \tan^{-1}(u/a) + C$
\item $\ds \int \frac{du}{u \sqrt{u^2-a^2}} = (1/a) \sec^{-1} |u/a| + C$
for $u^2 > a^2 +C$
\item $\ds \int \frac{du}{\sqrt{a^2 + u^2}} = \sinh^{-1} (u/a) + C$ for
$a>0$
\item $\ds \int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}(u/a) + C $ for
$u> a > 0$
\item $\ds \int \frac{du}{a^2 - u^2} = \left\{ \begin{array}{ll} (1/a)
\tanh^{-1}(u/a) + C  & \mbox{ if }
u^2 < a^2 \\ (1/a) \coth^{-1}(u/a) + C & \mbox{ if } u^2 > a^2
\end{array} \right. $
\item $\ds \int \frac{du}{u \sqrt{a^2 - u^2}} = -(1/a) \, \mbox{sech}^{-1}(u/a) +
C$ for $0 < u < a$
\item $\ds \int \frac{du}{u \sqrt{a^2 + u^2}} = -(1/a) \, \mbox{csch}^{-1} |u/a| +
C $ for $u \neq 0$
\ee

\item Some  trig identities.
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\;\; \ds \sin^2x + cos^2x=1 & & & &
(d) \;\;\;\; \ds \sin^2x= (1- \cos(2x))/2 \\

(b) \;\;\;\; \ds \cos^2x = (1 + \cos (2x))/2 & & & & 
(e) \;\;\;\; \ds \cosh^2x - \sinh^2x=1 \\
(c) \;\;\;\; \ds \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B & & & &
(f) \;\;\;\; \ds \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
\end{array}
\end{eqnarray*}


\item Some useful trig substitutions.

\be
\item $\ds x = a \tan \theta$ replaces $a^2 + x^2$  by $a^2 \sec^2 \theta$
\item $\ds x = a \sin \theta$ replaces $a^2 - x^2$  by $a^2 \cos^2 \theta$
\item $\ds x = a \sec \theta$ replaces $x^2 - a^2$  by $a^2 \tan^2 \theta$
\ee

\item Some useful limits. 
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\;\; \ds \lim_{n \rightarrow \infty} \frac{\ln n}{n} = 0 & & & & 
(d) \;\;\;\; \ds \lim_{n \rightarrow \infty} \sqrt[n]{n} = 1 \\ 
(b) \;\;\;\; \ds \lim_{n \rightarrow \infty} x^{1/n} = 1 \mbox{ for $x > 0$} & & & &
(e) \;\;\;\; \ds \lim_{n \rightarrow \infty} x^{n} = 0 \mbox{ for $|x| < 1$}  \\ 
(c) \;\;\;\; \ds \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} 
= e^x \mbox{ for any $x$} & & & &
(f) \;\;\;\;  \ds \lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0 \mbox{ for any $x$}.
\end{array}
\end{eqnarray*}
Note: In limits (c) through (f), $x$ remains fixed as $n \rightarrow \infty$. 

\item Frequently used Maclaurin series
\begin{eqnarray*}
\begin{array}{llll}
(a) \;\;\; \ds \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n  \mbox{ for $|x| < 1$}
& & &
(e) \;\;\;  \ds e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \mbox{ for $|x| < \infty$  } \\
(b) \;\;\; \ds \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} 
\mbox{ for $|x| < \infty$ }  & & &  
(f) \;\;\; \ds \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}
\mbox{ for $|x| < \infty$  } \\
(c) \;\;\; \ds \ln (1+ x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}, 
\mbox{ for $-1< x \leq 1$ }   & & &
(g) \;\;\; \ds \tan^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)}, 
\mbox{ for $|x| \leq 1$  } \\
(d) \;\;\;  \ds (1+x)^m = 1 + \sum_{k=1}^{\infty} \left( 
\begin{array}{c} m \\ k \end{array} \right) x^k \mbox{ for $|x| < 1$} & & & 
\end{array}
\end{eqnarray*}

\ee


\end{document}