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\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Final}\quad \textsl{December 15, 2000}
\hfill\makebox[2.5in][r]{Instructor:\hrulefill}

\bigskip

There are 14 problems on 10 pages worth a total of 120 points.  You start
with 30 points. Each part of a problem is worth the same number of points.

\medskip
You may use a calculator if you wish.

\medskip
To receive partial credit on a problem, you must \emph{show your work
and all important steps}. No credit will be given for an answer if no
work is shown.

\bigskip

\begin{enumerate}

\item (12 pts) Consider the following information about functions $f$
and $g$.

\begin{center}
  \begin{tabular}{|r|r|r||r|r|}    
  \hline
  $x$ & $f(x)$ & $f'(x)$ & $g(x)$ & $g'(x)$\\
  \hline
  \hline
  $0$ &  $2$ & $-3$ & $-3$ & $0$\\
  \hline
  $1$ &  $0$ & $-4$ &  $0$ & $5$\\
  \hline
  $2$ & $-1$ & $-7$ &  $8$ & $6$\\
  \hline
  \end{tabular}
\end{center}  

\begin{enumerate}[a)]

\item Suppose in addition $f$ is one-to-one on the interval $[-1,3]$. 
Compute $(f^{-1})'(0)$, the derivative of the inverse of $f$ at $0$.

\answerspace

\item Calculate the derivative of $\sinh (2f(x)+g(x)) $ at $x=0$.

\end{enumerate}
\answerspace

\problemspace

\item (6 pts) Compute $\frac {d\ }{dx}x^{\tan ^{-1}x}$.

\answerspace

\item (6 pts) Einsteinium-253 decays at a rate proportional to the amount
present.  Determine the half-life if this material has 2/3 of its original
mass left after 11.7 days.

\answerspace

\problemspace

\item (12 pts) Evaluate the following limits.

  \begin{enumerate}[a)]

  \item $\lim_{x\to 0}\frac{\sin (x^{10}) - x^{10}}{x^{30}}$

  \answerspace

  \item $\lim_{x\to \infty}x^2 e^{-x}$

  \answerspace

  \end{enumerate}


\problemspace

\item (24 pts) Evaluate the integrals.

  \begin{enumerate}[a)]

  \item $\int \frac {e^x}{\sqrt{1-4e^{2x}}}\,dx$

  \answerspace

  \item $\int x^2 \ln(x)\,dx$

  \answerspace

\problemspace

  \item $\int \frac{1}{x^3-x^2}\,dx$

  \answerspace

  \item $\int \frac{\sqrt{4-x^2}}{x^2}\,dx$

  \answerspace

  \end{enumerate}

\problemspace

\item (12 pts) Determine whether the following series converge:

  \begin{enumerate}[a)]
  \item \begin{math}\sum_{n=1}^\infty \frac{1}{\ln{n} + n}\end{math}

  \answerspace

  \item \begin{math}\sum_{n=2}^\infty \frac{1}{n\ln{n}}\end{math}

  \answerspace

  \end{enumerate}

\problemspace

\item (6 pts) Find the sum of the series $\sum_{n=1}^\infty (-1)^n e^{-n}$.

\answerspace

\item (6 pts) Find the Taylor polynomial of order 4, $P_4(x)$, generated by
$\sin(x)$ centered at $\pi/2$.

\answerspace

\problemspace

\item (6 pts) Find the interval of convergence of the power series
$\sum_{n=1}^\infty \frac{2^n}{n}(x-1)^n$.

\answerspace

\item (6 pts) Estimate the error of approximating $e^{-x}$ by $1 - x +
\frac{1}{2}x^2$ for $|x|<1/2$.

\answerspace

\problemspace

\item (6 pts)  Find the equation of the line tangent to the curve
parameterized by
\begin{eqnarray*}
x &=& t + \cos(\pi t)\\
y &=& t - \sin(\pi t)\\
\end{eqnarray*}
at the point $t=2$.

\answerspace

\item (6 pts) Find the length of the curve parameterized by
\begin{eqnarray*}
x&=&t^3 + 1\\
y&=&t^2 - 1\\
\end{eqnarray*}
for $0 \le t \le 1$.

\answerspace

\problemspace

\item (6 pts) Find the area inside both of the curves $r = \sin(\theta)$
and $r = 1+\cos(\theta)$.

\answerspace

\item (6 pts) Use power series to evaluate the integral
$\int_0^x e^{t^3}\,dt$.

\answerspace

\end{enumerate}

\end{document}
