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\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Lab \#2, Wednesday, October 8} \\
Professor Hwang \\
\end{center}

\subsection*{Purpose} 

We have seen that the derivative of a function and linear
approximations are closely related: The graph of a differentiable
function looks like the tangent line on sufficiently small scales. In
this lab, you will investigate this principle graphically for specific
functions and intervals, by having Maple plot functions over intervals
of the form $[a-h,a+h]$ or $[a,a+h]$ for different choices of $a$~and
$h$. As you make $h$~smaller, you should see the graphs become ``more
and more linear''. However, you will also see that at intermediate
scales there can be complicated behavior, and that mathematical
behavior can be difficult to see graphically.

There is a Maple worksheet at
\begin{center}
\verb+  http://mathcs.holycross.edu/~ahwang/teach/131/lab02.mws+
\end{center}
The worksheet makes a few preliminary definitions for you, but you
will have to do more typing yourself than on the last lab. (In many
cases you can just modify existing lines of the lab rather than typing
commands from scratch.) The plotting option ``scaling=constrained''
makes the horizontal and vertical scales the same, which is essential
for this lab. You are now ready to begin.

\subsection*{Approximate Linearity}

The first lines of the worksheet define constants $a=0.5$~and $h=0.1$,
and the function $f(x)=x^2$. The next line plots~$f$ on the interval
$[a-h,a+h]$. 

\prob{1.} Use Maple to plot~$f$ on the specified interval. Re-plot,
using $h=0.05$; $h=0.01$; and $h=0.001$. What is the slope of the
graph in your plots? Does this agree with what you expect from your
knowledge of the derivative? Explain.

\vspace*{1.5in}

\prob{2.} Set $h=0.000001$ (five zeroes) and plot. Repeat for
$h=0.0000001$ (six zeroes). Are these plots nearly linear? Explain.

\vfil

\eject

Define the constants as $a=1$, $h=0.1$, and define the function
$f(x)=x^3$. 

\prob{3.} Use Maple to plot~$f$ on the interval $[a-h,a+h]$. Re-plot,
using $h=0.01$. Use the axis labels to estimate the slope of the line
you see. What answer do you expect?

\vspace*{0.5in}

\sep

Now that you have a feel for routine functions, let's take a look at
some less obvious examples. 

\prob{4.} Define $a=0$, $h=0.001$, and $f(x)=x^x$, and plot~$f$ on the
interval $[a,a+h]$. (Note the asymmetry.) About how steep is the graph
(on the average) over the interval $[0,h]$? Make~$h$ one-tenth as
large, re-plot, and re-estimate the average slope. Continue until
numerical errors are visible. Tabulate your results below. What do you
think $\lim\limits_{x\to0^+}f'(x)$ is equal to?

\[
\begin{array}{|c||c|c|c|c|c|c|}
\hline
\vp h & 10^{-3} & 10^{-4} & 10^{-5} & 10^{-6} & 10^{-7} & 10^{-8} \\
\hline
\vp \Delta f/\Delta x & & & & & & \\
\hline
\end{array}
\]

\sep

Define $a=0$, $h=0.01$, and $f(x)=x^2\sin(1/x)$.

\prob{5.} Plot this~$f$ over the interval $[-h,h]$. Make a guess as to
the value of~$f'(0)$.

\vspace*{0.25in}

\prob{6.} Set $a=0.01$ and $h=0.005$; plot~$f$ over the interval
$[a-h,a+h]$. Does~$f$ look approximately linear at this scale? If not,
what does the graph of~$f$ resemble? Try making~$h$ smaller to ``zoom
in''. 

\vspace*{1in}

\prob{7.} Set $a=0.001$ and $h=0.00005$; note that this $a$~is ten
times closer to the origin than in the previous question, but~$h$ is
100~times smaller. Again, plot~$f$ over the interval
$[a-h,a+h]$. Does~$f$ look approximately linear at this scale? If not,
what does the graph of~$f$ resemble?

\vspace*{1in}

\prob{8.} On the basis of this evidence, does $\lim\limits_{x\to0}
f'(x)$ exist? Explain.

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