%Limits and Derivatives
%Math263A

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\begin{center}
{\Large
Limits and Derivatives\footnote{Copyright \copyright 2002 Todd Young and Larry Snyder. 
All rights reserved.
Please address comments to young@math.ohiou.edu}}
\end{center}

\begin{enumerate}

\item Try the following commands:\\
    \verb&  syms  x  h&  \\
    \verb&  f = x^3 + x^2 + x + 1 &  \\
    \verb&  m = (subs(f, x+h)-f)/h &  \\
    \verb&  f1 = limit(m, h, 0) &\\
    Explain what happened.

\item Try the following sequence: \\
    \verb&  syms  x  h& \\
    \verb&  f = exp(sin(x)) &  \\
    \verb&  m = (subs(f, x+h)-f)/h &  \\
    \verb&  f1 = limit(m, h, 0) &   \\
    \verb&  subs(f1, pi) &  \\
    \verb&  X = -10:.05:10;& \dotfill Makes an array of \verb$x$ values.\\
    \verb&  F = subs(f, X);& \dotfill Makes an array of \verb$f(x)$ values.\\
    \verb&  F1 = subs(f1, X);& \\
    \verb&  plot(X, F, 'b', X, F1, 'r')& \\
    Explain what happened.

\item Now repeat the steps above for the function:
$$
    f(x) = (x-1)^2\sqrt{x} 
    \qquad  \verb& (f = (x-1)^2*sqrt(x))& 
$$
    Is the function defined for all real numbers? What about the derivative? 
    How is the graph misleading?

\item Next repeat  this procedure for the function
    $f(x) = {(x-1)^2}{x^{1/3}}$. 
    Are the function
   and its derivative defined for all real
   numbers? How is this graph misleading?

\item Use \verb& ezplot(f) & and \verb& ezplot(f1) & to get another picture
      for \verb$f$ and \verb$f'$ from \#4. In what ways are these graphs misleading?

\item Prepare a brief (\verb$< $1 page) written report answering all
      the questions. Use complete sentences and standard mathematical notation.
      Do {\bf not} get a printout.

\end{enumerate}

\vfill

\textsf{This assignment is intended to reinforce the user's understanding of
the definition of the derivative. They must think about the domains
of the function and its derivative.}

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