%Limits and Derivatives
%Math263A
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{\Large
Limits and Derivatives\footnote{Copyright \copyright 2002 Todd Young and Larry Snyder.
All rights reserved.
Please address comments to young@math.ohiou.edu}}
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\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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