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\textbf{Math 126A: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}
 
\textbf{Final Exam}\quad \textsl{December 16, 1999}
 
\bigskip
 
There are 12 problems on 12 pages worth of total of 125 points. You start
with 25 points. To receive full credit you must show all your work and
include all important steps.

\medskip You may use a calculator.

\bigskip

\begin{enumerate}
 
\item (10 pts)

\begin{enumerate}

  \item Suppose $f(x)$ is a one-to-one function on the interval $[-3,3]$
  with some of the values of $f(x)$ and $f'(x)$ given in the following
  table.
  \[
  \begin{tabular}{|r|c|c|}
  \hline
  $x$ & $f(x)$ & $f'(x)$ \\
  \hline
  \hline
  $-2$ &    0   &  $1/3$ \\
  \hline
  $-1$ &  $1/2$ &  $1$   \\
  \hline
   $0$ &   $1$  &  $5/3$ \\
  \hline
   $1$ &  $5/2$ &  $5$   \\
  \hline
   $2$ &   $5$  &  $7$   \\
  \hline
  \end{tabular}
  \]
  Compute $(f^{-1})'(0)$, the derivative of the inverse of $f$ at~0.

  \answerspace

  \item Let $f(x) = (x^2-4)^2$. Show that $f$ restricted to the interval
  $[0,2]$ is one-to-one and compute its inverse, $f^{-1}(x)$.

  \answerspace

\end{enumerate}

\problemspace

\item (10 pts) Let $f(x) = x^{\sqrt{x}}$ for $x>0$.

  \begin{enumerate}

  \item Compute $f'(x)$ for $x>0$.

  \answerspace

  \item Compute $\lim_{x\to0^+} f(x)$.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Evaluate the following limits.

  \begin{enumerate}

  \item  $\lim_{x\to0} \frac{x-\tan^{-1}(x)}{x^3}$

  \answerspace

  \item  $\lim_{x\to0} \frac{1-\cosh(x)}{x\sinh(x)}$

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Water drains from a pond at a rate proportional to the
depth of the water. If the depth is 4~ft at noon on Monday and 3.5~ft at
noon on Tuesday, determine the depth of the water at noon on Wednesday.
You may assume that the volume of water in the pond is proportional to
its depth, $y$, so that the rate of change of $y$ is also proportional to~$y$.

\problemspace

\item (10 pts)
Evaluate the following integrals.

  \begin{enumerate}

  \item $\int_8^{\infty}\frac{dx}{x\sqrt{x+1}}$.

  \answerspace

  \item $\int z\,\ln(z)\,dz$.

  \answerspace

  \end{enumerate}

\problemspace

\item (15 pts) Determine whether the following series converge.
 
  \begin{enumerate}

  \item $\sum_{n=1}^\infty \frac{(n!)^2}{(2n)!}$

  \answerspace

  \item $\sum_{n=2}^\infty \frac{1}{n[\ln(n)]^3}$
 
  \answerspace

  \item $\sum_{n=1}^\infty n^2 e^{-n^2}$
 
  \answerspace
 
  \end{enumerate}
 
\problemspace

\item (15 pts)
Find the interval of convergence of each of the following power series.

  \begin{enumerate}

  \item $\sum_{n=1}^\infty n!x^n$

  \answerspace

  \item $\sum_{n=3}^\infty n^2 (x+2)^n$

  \answerspace

  \item $\sum_{n=5}^\infty \frac {(x-3)^n}{n}$

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) 

  \begin{enumerate}

  \item Find the Taylor polynomial $P_n(x)$ for $f(x) = \sin(x)$
  centered at $a = \pi/2$. (\emph{Hint:} The polynomial should
  be written in terms of powers of $(x-\pi/2)$, not $x$.)

  \answerspace

  \item Determine a value of $n$ such that $P_n(x)$ approximates $f(x)
  = \sin(x)$ for $x$ between $\frac{\pi}{2}-0.1$ and $\frac{\pi}{2}+0.1$
  with an error less than $10^{-6}$.

  \answerspace

  \end{enumerate}

\problemspace

  \item (5 pts) Express $\int_{0}^{1} e^{x^2}\, dx$ as an infinite
  series. Be sure to include an expression for the general term of
  this series.

  \answerspace

\problemspace

\item (10 pts) Consider the curve given by the parametric equations
\[
x = t^2-1,\qquad y = t(t^2-2),\qquad -2\le t \le 2
\]

  \begin{enumerate}

  \item Sketch the curve, indicating any intercepts and the direction
  of the curve for increasing values of $t$.

  \answerspace

  \item Find the equation for the line tangent to the curve at the point
  $(2,\sqrt{3})$.

  \answerspace

  \end{enumerate}

\problemspace

\item (10 pts) Determine the foci, vertices, center, asymptotes, and
directrix as appropriate for the conic section given by $x^2 - y^2 -
2x + 8y - 6 = 0$.

\problemspace

\item (10 pts)

  \begin{enumerate}

  \item Find the Cartesian equation for the curve given by the polar
  equation $r=\cos^3(\theta)$.

  \answerspace

  \item Find the length of the curve given by the polar equation
  $r = \cos^2(\theta/2)$ for $0 \le \theta \le \frac{\pi}{3}$.

  \answerspace

  \end{enumerate}


\end{enumerate}

\end{document}
