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\begin{center}
{\bf APPM 1360 ~~~~~~ FALL 1996}   \\
\medskip

\underline{\bf  Final Exam} \\
December 14, 1996, ~~~  11:30 - 2:30
\end{center}

\noindent
ON THE EXAM BOOKLET PLEASE:~~  (1) Write your name,  (2) Student ID,
(3) Lecture Number, and (4) Recitation Instructor.

\underline {Show all work}.  A correct answer with no relevant work may
receive no credit, while an incorrect answer accompanied by some correct
work may receive partial credit.
\begin{center}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{center}
\begin{itemize}

\item [1] ~
  Compute the following integrals.  You must show (neatly) all of your
  work.
  \[ \begin{array}{lll}
      \ds         \mbox{\rm (a) } \int_{3}^8 \frac{dx}{x - 4}
    & \ds ~~~~~~  \mbox{\rm (b) } \int \frac{dx}{x^2 \sqrt{1 - x^2}}
    & \ds ~~~~~~  \mbox{\rm (c) } \int \frac{dx}{x^2 (1 - x)}
  \end{array} \]
  
\item [2] ~ Determine whether the following sequences converge or
  diverge.  If the sequence converges find the limit.  State why you
  are answering as you are.
  \[ \begin{array}{ll}
    \ds \mbox{\rm (a) } a_n = n 2^{1/n} & ~~~~~~~~~~~~
    \ds \mbox{\rm (b) } a_n = \frac{n}{\ln ( n^2)}
    \end{array} \]

\item [3] ~
  Decide whether the following series converge absolutely, converge
  conditionally, or diverge.  As usual show your work.
  \[ \begin{array}{ll}
    \ds \mbox{\rm (a) } \sum_{n=1}^\infty  \ln \left ( \frac{n}{n+1}
      \right ) & ~~~~~~~~~~~~
    \ds \mbox{\rm (b) } \sum_{n=0}^\infty \frac{(-1)^n}{n \sqrt{n^2 + 1}}
    \end{array} \]

\item [4] ~
  Determine the interval of convergence of the following power
  series.  Be sure to check the endpoints if necessary.
  \[ \begin{array}{ll}
    \ds \mbox{\rm (a) } \sum_{n=1}^\infty \frac{x^n}{\sqrt{n}} & ~~~~~~~~~~~~
    \ds \mbox{\rm (b) } \sum_{n=0}^\infty \frac{(-1)^{n-1} (x-2)^n}{n 2^n}
    \end{array} \]

\item [5] ~
  Compute the area enclosed by the polar equation curve $\ds r =
  \sqrt{\theta} \sec \theta $ on the interval $\ds 0 \leq \theta \leq
  \pi/4$.  Be careful!

\item [6] ~
  Compute the tangent lines to the following curves at the indicated
  points.
  \begin{itemize}
    \item [(a)] ~
      $\ds y = (x+1)(x+2) \cdots (x+10) $ at the point $\ds x = 0$.
    \item [(b)] ~
      $\ds y = (x + \cos^{-1} x)^{\tan^{-1} x}$ at the point $\ds x = 0$.
    \item [(c)] ~
      $\ds x = t^2$ and $\ds y = \frac{1}{3} t^3 -t$ at $\ds t = \sqrt{3}$.
    \item [(d)]
      In polar-coordinates $\ds r = 2 \cos \theta + 1$ at $\ds \theta = \frac{3 \pi}{4}$.
  \end{itemize}
  
\item [7] ~ Let $\ds f(x) = \int_0^x \sin (t^2) \,dt$.  Use the
  Maclaurin series for $\sin(t)$ to compute the Maclaurin series for
  $f(x)$.  What is the minimum number of terms of the series for
  $f(x)$ would be needed to compute $f(0.1)$ accurately to 4 decimal
  places?

\item [8] ~
  Decide whether the following are true or false.  As usual, show all
  of your work.
    \begin{itemize}
    \item [(a)] ~
      $ \ds x = {\cal O}(e^x) $ as $\ds x \rightarrow \infty$.
    \item [(b)] ~
      $x$ and $\ds \ln ( \int e^{x^2} \, dx)$ grow at the same rate as $x \rightarrow \infty$.
    \item [(c)] ~
      $f(x) = o(g(x))$ implies that $g(x) = o(f(x))$.
    \item [(d)] ~
      $\ds \ln x = o(\ln (x^2 + 1))$ as $x \rightarrow \infty$.
  \end{itemize}

\item [9]
  Compute the length of the curve $\ds x = t^3$, $\ds y = 3 t^2 /2$
  for $\ds 0 \leq t \leq \sqrt{3}$.

\end{itemize}

A short table of integrals:
\renewcommand{\arraystretch}{1.7}
\[ \begin{array}{lll} \ds
  \int \, du = u + C & & \ds \\ \ds
  \int a \, du = a u + C & & \ds \\ \ds
  \int (du + dv) = \int \, du + \int \, dv & & \ds \\ \ds
  \int u^n \, du = \frac{u^{n+1}}{n+1} + C, ~~ n \neq 1 & & \ds \int
    \frac{du}{u} = \ln |u| + C \\ \ds
  \int \cos u \, du = \sin u + C & & \ds \int \sin u \, du = - \cos
    u + C \\ \ds
  \int \tan u \, du = - \ln |\cos u| + C & & \ds \int \cot u \,
    du = \ln |\sin u| + C \\ \ds
  \int \sec u \, du = \ln | \sec u + \tan u| + C & & \ds \int \csc u \, du
    = - \ln |\csc u + \cot u| + C \\ \ds
  \int e^u \, du = e^u + C & & \ds \int a^u \, du = \frac{1}{\ln a} a^u +
    C, ~~ a > 0, a \neq 1 \\ \ds
  \int \frac{du}{1 + u^2} = \tan ^{-1} u + C & & \ds \int
    \frac{du}{\sqrt{1-u^2}} = \sin ^{-1} u + C \\ \ds
  \int \frac{du}{u \sqrt{u^2 - 1}} = \sec ^{-1} |u| + C & & \ds 
  \int \sec^2 x \, dx = \tan x + C\\ \ds
  \int u \, dv = uv - \int v \, du & & \ds 
\end{array} \]

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