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\hskip23pc \= Math.\ 270-03\\
\>Spring, 1997\\
\>R. B. Kearfott
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\begin{center}
\bf Fourth Examination \\
\it Friday, April 18, 1997
\end{center}

\noindent {\small
{\bf Instructions:}
This exam should be done on your own paper.  Your
name should be on each sheet and on the back of
the last sheet; the answers should appear written carefully and in
order.  If in doubt, show intermediate steps:  Full credit may not be
given, even for correct answers, unless work is arranged clearly. This
exam is closed book. You may leave after handing in your exam
paper, but be sure to check your answers carefully. Keep your
exam sheet.}

\begin{enumerate}

\item {\it (48 points)} Find $f'(x)$, for $f(x)$ as follows:
$$\begin{array}{rrclrrcl}
      \mbox{(a)} & f(x) &=& 3x^2 - 2x + 1 & 
\qquad\mbox{(b)} & f(x) &=& e^{x^2}       \\
      \mbox{(c)} & f(x) &=& \frac{\sin(x)}{x} &
\qquad\mbox{(d)} & f(x) &=& \sqrt{1-x^2} \\
      \mbox{(e)} & f(x) &=& \arctan(1+x) &
\qquad\mbox{(f)} & f(x) &=& x^2e^x
\end{array}$$

\item {\it (20 points)} Find $dy/dx$ at $x=0$,$y=1$, if $(x-1)^2 + y^2
= 2$.

\item {\it (32 points)} The acceleration $g$ in meters per second per
second
      due to gravity, at a distance of $r$ kilometers from 
      the center of the earth, is given approximately by
      \begin{equation} \label{eq:gravity}
         g(r) = \frac{4.002\times10^8}{r^2}.
      \end{equation}
      \begin{enumerate}
         \item Find $\frac{dg}{dr}$.
         \item Use a linear approximation to $g(r)$ at the
               surface of the earth ($r = 6400$ kilometers)
               to form a table of approximations to $g(r)$ at
               200, 400, and 600 kilometers above the earth.
         \item Compute $g(7000)$ directly from equation 
               (\ref{eq:gravity}), and compare with the
               corresponding value from the linearization?
               Is the actual value less than or greater than
               the value given in the linearization?  How
               could you tell this from the graph of $g(r)$?
               Illustrate with a graph.
         \item Based on the value $\frac{dg}{dr}$ at $r=6400$,
               is it safe to say that $g$ is constant within
               5 kilometers (about the height of Mount Everest)
               of the surface of the earth?
      \end{enumerate}

\end{enumerate}

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