%% Math 131-01 Midterm 3 Review Problems, Fall 2007

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%\markright{Math 131-01, Midterm}

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\begin{document}


\be
\item{} Compute $y'$ and simplify.
\begin{enumerate}
\item[(a)] $\ds y = \ln(x)(x^7 - \frac{4}{\sqrt{x}})$
\item[(b)] $\ds y = (3^{2x} + 2)^3$
\item[(c)] $\ds y = \frac{x + 1}{3x^4 - 1}$
\item[(d)] $\ds y = \frac{\sin(x)}{1 + \cos(x)}$
\item[(e)] $\ds y = \arctan(e^{5x} - 2^x)$
\item[(f)] $\ds xy^2 - 3y^3 + 2x^4 = 2$
\item[(g)] $\ds y = \cos(x^2)^{x^3}$
\end{enumerate}
\be

\item{} Evaluate the following limits:

\bd

\item{(a)} $\ds\lim_{x\to0}\frac{1-\cos x}{x^2}$

\vfil

\item{(b)} $\ds\lim_{x\to\infty}\frac{\ln(e^x+1)}{x}$

\vfil

\ed


\item{} Both parts of this question refer to the
  function $f(x)=4x^3-x^4$.

  \bd

  \item{(a)} Find and classify the critical point(s) of~$f$.

    \vfil

    \hrulefill

  \item{(b)} Find the absolute maximum and minimum values of~$f$ on
  the interval~$[1,4]$.

  \ed

\item{} A spherical balloon is being inflated at
  $20~\textrm{in}^3/\textrm{sec}$. When the radius is 6~inches, at
  what rate is the radius increasing?  (The volume of a sphere of
  radius~$r$ is $V=\frac{4}{3}\pi r^3$.)

\vfil

\item{} Let $a>0$. A rectangle is inscribed under the
  curve $y=\sqrt{x}$:
  \begin{center}
    \input{midterm3-01b.eepic}
  \end{center}
  Find the value of~$x$, with $0<x<a$, for which the area of the
  rectangle is largest. Show all work for full credit.

\item{} [10~points]\quad Find the inflection point(s) of the function
  $f(x)=x^{2/3}-\frac{1}{5}x^{5/3}$. 


\ee

\end{document}

