\magnification=\magstep2
\everymath={\displaystyle}
\parskip=3pt plus 3pt

\def\bull{$\bullet$\quad}
\def\L{{\cal L}}
\def\F{\relax\ifmmode \Phi \else $\Phi$ \fi}
\def\G{\relax\ifmmode \Gamma \else $\Gamma$ \fi}
\def\f{\relax\ifmmode \phi \else $\phi$ \fi}
\def\and{\hbox{\rm\ and\ }}
\def\ref#1{[{\bf#1}]}

\hyphenation{pa-ram-e-triz-ed}

\font\summaryfont=cmr5 scaled \magstep2

\outer\def\beginsection#1\par{\vskip0pt
plus.08\vsize\penalty-250\vskip0pt plus
-.08\vsize\bigskip\vskip\parskip
\message{#1}\leftline{\bf#1}\nobreak\smallskip}

\def\openbox#1#2{\setbox4=\vbox{\hsize #1
\tolerance=3000\noindent\strut#2\strut}}
\def\boxit#1#2{\openbox{#1}{#2}
\vbox{\hrule\hbox{\vrule\kern3pt
\vbox{\kern3pt\box4\kern3pt}\kern3pt\vrule}\hrule}}

% Function box: input, function, output, width
\def\fbox#1#2#3#4{$
\vcenter{\noindent\hsize#4
\hfill$#1\rightarrow$}\vcenter{\hsize#4
\boxit{#4}{\hfill$#2$\hfill}}
\vcenter{\noindent\hsize#4$\rightarrow#3$}$}

\def\grafbox#1#2{\vtop to 2.5truein{\hsize=2.14truein%
\vfill\special{pcl:#1}\hfill\smallskip%
\line{\noindent\hfil#2\hfil}
}}

\def\2line#1#2#3#4{\line{\noindent\hfill\grafbox{#1}{#2}
\hfill\grafbox{#3}{#4}
\hfill}}

% format: \2line{file1}{title1}{file2}{title2}


% REAL N-SPACE: #1 = DIMENSION
\def\rspace#1{\ifmmode{\sl R}^#1\else${\sl R}^#1$\fi}

% AND FOR USE IN MATH MODE
\def\and{\hbox{ and }}

\def\boxstrut{\vrule height 15pt width 0pt depth 5pt}

\centerline{\bf Dynamic Function Visualization}
\bigskip

\centerline{\bf Abstract}

{\narrower\summaryfont\advance\baselineskip by -2pt
\tolerance=1000
The graph of a function provides a useful but incomplete means
for visualization. Newly developed software which connects
points on a domain line with their function images on a
parallel range line can help to present a function as a
mapping of points. Furthermore, the paths of
points toward their images can be animated in such a way
that speeds are proportional to the derivatives at the
individual points. The resulting movement of an interval of
points produces moving curves whose geometric properties
reflect properties of the function and its derivatives. These
images suggest new types of results, some of which are stated
and proved.\par}
\bigskip

%\head{Introduction}
\beginsection{Introduction}

The problem first appears in 2nd-year Calculus. As soon as the
discussion moves to functions $\rspace1\rightarrow\rspace3$
and $\rspace2\rightarrow\rspace3$ a slightly puzzled look
comes over the faces of many students. This gets worse when
I refer to the ``image'' of such a function and try to draw it
on the board. I can see that all is not well. Finally, a
student may ask: ``Is this (like) the graph?'' Once again it
becomes clear that most students don't know the difference
between a function and its graph, or between the graph and the
image of a function.

As I discovered some time ago, the problem is not simply that
students have never seen these distinctions made, though for
many this is the case. Even after examples are given and
definitions presented, the confusion persists, sometimes
through the entire course. As I have come to understand, the
problem results from a confused or imperfect conception of
{\it function}. This has, of course, been recognized by
others, and studied fairly extensively (see Dubinsky \ref{5},
for example). More precisely, however, I believe that the
difficulty my students have results from their identification
of a function with its graph.

In this article I will examine some of the history of this
identification, which I call the {\it Static} concept of a
function, and suggest a way in which a complementary view, the
{\it Dynamic} view, can be fostered.
\medskip

%\head{Functions as Ordered Pairs}

\beginsection{Functions as Ordered Pairs}

Not so long ago it was fashionable to define a function as a
set of ordered pairs having the property that distinct pairs
have distinct first elements. This definition was inspired by
Peano (\ref{9}) (see also \ref{10}). When Wiener (\ref{12})
showed how to define ordered pairs as sets, namely: $(a,b)
=\bigl\{\{a\}, \{a,b\}\bigr\}$, this notion became part of the
program to reduce mathematics to set theory and, eventually,
to logic. The formal appeal of this program probably reached
its high-point with the influential volumes of Bourbaki
(\ref{1}), and was loosed on the mathematics education
world by the ``New Math'' in the 1960's.

In his influential and classic text {\bf General Topology}
(\ref{8}), John Kelley writes: ``Moreover, there is no
information about the function which cannot be derived from
its graph. In brief, there is no reason why we should attempt
to distinguish between a function and its graph.'' Indeed, the
Bourbaki definition makes a function identical to its graph.

Sometime, a decade or so ago, this approach fell out of favor
in most calculus and precalculus courses. (Though not
entirely: my daughter learned it recently in a ``rigorous''
high school course.) Nevertheless, its influence is still very
strongly felt in several ways. Many of us who teach
mathematics think of functions in this way because we were
taught that it is the only ``correct'' formal definition of
function. Some researchers into the learning of mathematics
have suggested that this level of understanding (implicitly
the highest level) is important in order to achieve
``encapsulation'': the concept of a function as a {\it thing}
or object in itself. Encapsulation enables one to conceive of
spaces of functions and, for example, differentiation as a
function on functions (see \ref{5}).

The ordered pair definition of function has been at least
partially supplanted by some form of the {\it process}
definition: A function $f$ is a rule or process which
associates with each (allowable) element $a$  a well-defined
element $f(a)$. This phraseology, which appears in most
contemporary calculus books, dates back at least to
Dirichlet --- for more information, consult \ref{2}. In spite
of the return to this concept of function, a more recent
development has perpetuated the identification of a function
with a set of ordered pairs. \bigskip

%\head{Graphing or: Confusion Happens}

\beginsection{Graphing or: Confusion Happens}

For many years the trend in elementary mathematics texts has
been to include more and more diagrams and pictures,
especially of function graphs. Personal computers and,
especially, graphing calculators, have made function plotting
easy and widely available. Since calculators and most software
packages draw traditional rectangular plots, that's what is
used and that's what students think of when they think of
functions. Although the Educationally Correct (EC) position
has been that functions are to be approached from (at least) 3
points of view, the simple fact is that the trend is to view
functions as graphs, no matter how functions are actually
defined.

Herein lies the problem. If a function is defined by a
process (all well-defined mathematical functions are) then
visualizing a graph does not in any sense capture that
process, nor does it enable one to see or analyze the process.
A good example is the so-called ``tangent line
approximation.'' My students have absolutely no trouble
accepting the visual image of the tangent to a graph being
close to the graph near the point of tangency. Computer or
black/white board ``zooms'' convince even the slowest of them.
They can repeat back this new local--linearization catechism
instantaneously and in unison. Nevertheless, ask them to
translate this precisely into a statement that the value of
one function is close to the value of another function and
most will become glassy-eyed. Even after a quarter of calculus
from a popular vehicle for ``reform'' they do not see,
without a lot of prompting, that ``the tangent line close to
the graph at $P$'' means that ``$f(x)\approx f(a) +
f'(a)(x-a)$'' for $x$ near $a$. The reason is that they don't
associate the tangent line with the function $x\rightarrow
f(a) + f'(a)(x-a)$. In fact, once the graph of $f$ has been
constructed, they identify it with $f$ and forget the rule
$x\rightarrow f(x)$. The picture has swallowed its creator.

When these same students get to sophomore calculus and have to
cope with curves and surfaces parametrized by functions, they
have no graphs to fall back on. They have not been trained to
think of a function as {\it doing something}, namely
creating an association or mapping: $P\rightarrow Q$, since
they have never thought of their freshman calculus functions
in this way. Well into multivariable calculus, hardly any of
my students could explain why the {\it graph} of
$(x,y)\rightarrow f(x,y)$ is the same as the {\it image} of
$(x,y)\rightarrow (x,y,f(x,y))$, even after we spent a lot of
time with examples of both concepts.

No student or elementary text I have encountered has explained
what ``polar graphs'' have to do with graphing, or what polar
coordinates have to do with plotting functions. Think about
it.
\medskip

\beginsection{Notation Reform: A First Step}

Mathematicians and scientists routinely use phrases such as:
``Consider the function $y=$ [some expression].''  Indeed,
in most texts this is the standard way of presenting a function for
consideration. Although this is nearly universal and quite widely
understood by professionals and relatively advanced students,
I believe that in high school precalculus courses and lower
level undergraduate courses --- the training grounds for
function users --- we would do well to take more care in our
notation.

One of the simplest improvements is to use ``arrow'' notation
for functions: $f:x\longmapsto x^3-5x^2+4x-1$. This notation
is used throughout some recent texts (e.g. \ref{4}),
and has the obvious advantage of emphasizing the dynamic or
process nature of functions. One of its drawbacks is that it
ties the student to the name of a particular variable, in this
case $x$. An alternative, one that I use frequently, is the
``blank'' notation: $f:(\ )\longmapsto (\ )^3-5(\ )^2+4(\
)-1$. Evaluation is done by simply filling in the blank. This
makes composition of functions very easy; for example:
$$f:(9x+7)\longmapsto(9x-7)^3-5(9x-7)^2+4(9x-7)-1.$$


Finally, many of the newer algebra and precalculus texts make
frequent use of ``function boxes'':
\bigskip

\centerline{\hfill
\fbox{x}{f}{f(x)}{.7truein}\hfill
\fbox{t}{\sin(\ )}{\sin(t)}{.75truein}\hfill
}
\medskip

\centerline{\hfill
\fbox{x}{4(\ )^3-2(\ )+7}{4x^3-2x+7}{1.2truein}\hfill
\fbox{{\pi/6}}{\sin(\ )}{{1/2}}{.8truein}\hfill
}

\bigskip

\beginsection{Mapping Diagrams}

The notation $y=f(x)$ and the standard graph, using ordered
pairs, are static representations of a function. On the other
hand, the forms of notation just described capture the dynamic
nature of functions. What should be the dynamic visual image
corresponding to the static graph? One idea that has been used
in various texts and articles is the {\it mapping diagram}.
Basically, such a diagram is formed from two parallel lines,
one representing the domain of a function $f$, the other
representing the range. We shall take them to be vertical
lines, with the domain line to the left of the range line.
Function evaluation is represented by drawing a line from a
number $x$ on the domain line to its image $f(x)$ on the
range line. We call this the {\it mapping line at $x$}. For
the moment, let's use equal scales on the domain and range,
and position them so that the points representing 0 on each
are at the same horizontal level.

Figure 1 shows some mapping lines for a function $f$. Note
that where the function is {\it decreasing}, say between
domain points $x_3\and x_4$, the mapping lines {\it must
cross} somewhere between the domain and range lines, since
$x_4 < x_3\Rightarrow f(x_4) > f(x_3)$.

\medskip
\line{\hfill
\vbox to 2.5truein{\hsize3truein\vfill\special{pcl:fv.pcl}
\hfill}\hfill}
\smallskip
\centerline{\bf Figure 1}
\medskip

From the perspective of mapping diagrams, {\it linear
functions} $x\mapsto ax+ b$ are characterized by the property
that they send equally spaced points into equally spaced
points. It is also an elementary theorem that a function is
linear if and only if it has a ``focal point.'' In other
words, there is some point $P$ not on the domain line such
that for any $x$, $f(x)$ is determined by constructing a line
joining $x$ and $P$ and seeing where this line intersects the
range line. Focal points to the right of the range line
correspond to values of $a$ such that $0<a<1$. For $P$ {\it
on} the range line, we get the {\it constant} linear functions
{$a=0$). As $P$ moves to the left of the range line, we get
the {\it decreasing} linear functions ($a < 0$}, with $P$
halfway between range and domain corresponding to $a=-1$. $P$
lying on the domain line is, of course, a discontinuity. As P
moves to the left of the domain line we get the remaining
positive values of $a$: $a > 1$. The value $a=1$ corresponds
to the ``point at $\infty$'' since, for such linear functions,
the mapping lines are parallel.

For functions in general, the divergence or convergence of the
mapping lines for nearby domain points represents the local
dilation of the function. Representing this as the ratio of
output displacements to input displacements gives an average
rate of change. The limit of these dilations then represents
the derivative. Differentiable functions being locally linear,
they exhibit the behavior of linear function just described
after repeated zooming.

Some of these properties of Mapping Diagrams have been
described in Richmond \ref{11} and Brieske \ref{3}.
More recently, Goldenberg et.~al.~(\ref{7}) have used
a variation of this idea, implemented on a computer, to
examine how students can visualize the dependence of one
variable on another. These authors all used parallel {\it
horizontal} lines to represent the domain and range.
Subsequent testing on students seemed to indicate that drawing
arrows from left to right between two vertical lines is a more
natural setup. Flashman makes extensive use of this kind of
mapping diagram in his forthcoming calculus book \ref{6}.
\bigskip

\beginsection{The Dynamic Function Visualizer}

After using mapping diagrams for a while, it occurred to me
that there was even more one could do to emphasize the dynamic
nature of functions. Over the past year, H.~Hohn
and I have collaborated on a software implementation
of standard mapping diagrams, while also adding several new
features, including automatic zoom, variable scaling of the
domain and range and, especially, an {\it animation mode}.

In animation mode, the domain point $P$ moves toward its
image $f(P)$, from left to right, at a horizontal speed $k +
K\|f'(P)\|$. The constants $k$ and $K$ are adjustable, and
represent, respectively, a {\it minimal} speed and a {\it
spread} between the minimal and maximal speeds. Roughly
speaking (neglecting the minimal speed), a point moves toward
its image at a rate proportional to the absolute value of the
derivative at the point. Furthermore, in animation mode, the
whole domain interval can be made to move, not just a few
individual points. Finally, for clarity, points are
color-coded, depending on whether the derivative is positive
or negative.

The images produced by this scheme are striking and show
characteristic features which one learns by experience to
recognize and interpret. A verbal description can hardly be
adequate, but I will try to describe, with some diagrams, some
of the things we have observed.

1. Linear functions are easily recognizable since the
derivative is constant; thus, all points move at the same
speed, and so one observes a vertical ``wave-front'' moving
from left to right. If domain and range have the same scale,
then the front stretches or contracts vertically depending on
whether the derivative is greater or less than 1 in absolute
value. If the derivative is negative, then the wave front
flips over itself vertically since lower points must move up
and higher points move down. This happens at the focal point,
which lies between the domain and range lines. However, this
flip is invisible since the front is vertical. It's only when
one looks at non-linear functions that this flipping becomes
apparent and striking.

2. Monotone increasing functions show  curvy wavefronts
which bulge toward the right where points have larger
derivatives, and toward the left for smaller derivatives. The
tips of the bulges represent maxima or minima of the
derivative, hence zeros of the second derivative. (Reversing
the animation shows the behavior of the inverse function.) The
diagrams in Figure 2 show some snapshots of the
animation of the functions $x\mapsto(12x+x^3)/13$ and
$x\mapsto(12x-x^3)/13$ on the interval $[-1,1]$.


3. Monotone decreasing functions are even more interesting,
since the mapping lines for smaller arguments must cross the
mapping lines for larger ones, as seen in Figure 1. If the
derivative is itself monotone, then this motion is
accomplished without the wavefront crossing itself, since, for
monotone increasing derivative, points starting higher up on
the domain line move at strictly greater speeds than ones
starting lower down. Thus, they pass through the crossing of
the mapping lines sooner than the lower ones. However, if the
derivative has a maximum or minimum, then there is a leading
or trailing point, and those points starting above or below it must
move in opposite vertical directions as they move rightward.
Eventually, they must cross at the crossing point of the
mapping lines, forming the characteristic loops shown
in the snapshot images of Figure 3 for the functions
$x\mapsto(x^3-12x)/13$ and $x\mapsto(-x^3-12x)/13$. (The
division by 13 in these functions is to make the range have
the same scale as the domain interval $[-1,1]$.) \bigskip

\2line{cubic2.dmp}{$x\mapsto(12x+x^3)/13$}%
{cubic4.dmp}{$x\mapsto(12x-x^3)/13$}%
\bigskip
\centerline{\bf Figure 2}
\bigskip

\2line{cubic1.dmp}{$x\mapsto(-12x+x^3)/13$}
{cubic3.dmp}{$x\mapsto(-12x-x^3)/13$}
\bigskip
\centerline{\bf Figure 3}
\bigskip

In the case where the function is not monotone, the Visualizer
color codes points of increase and decrease. The animation
shows vividly first how the large scale behavior can be broken
down into intervals of monotonicity, and how the function
becomes a several-fold covering of various segments of its
range. Reversing the motion of points and restriction of the
domain shows how inverses can obtained.
\bigskip

\beginsection{Mapping Diagrams: Technicalities}

In order to analyze the Visualizer from a mathematical
standpoint, we make the following conventions.
\medskip

\noindent\quad\bull
The domain axis is the line $x=0$; points on the domain axis
will be denoted $(0,s)$.

\noindent\quad\bull
The range, or image, axis is the line $x=1$; points here are of
the form $\bigl(1,f(s)\bigr)$.

\noindent\quad\bull
The domain point $P_s=(0, s)$ is joined to the image point
$\bigl(1, f(s)\bigr)$ by the mapping line, which we will
denote $\L_s$.
\medskip

We choose two constants: $k$, the minimal speed, and $K$, the
spread between the minimal speed and the maximal speed. A
point $P_s$ moves on $\L_s$ at a horizontal speed $v_s=k+K\|f'(s)\|$.
The following fundamental map
$\F:\rspace2:\rightarrow\rspace2$ determines where the point
$P_s$ is at time $t$ on its journey to its image.
$$\eqalign{
\F:(s,t)\rightarrow&\Bigl(
\bigl(k+K\|f'(s)\|\bigr)\cdot t,\
s+\bigl(f(s)-s\bigr)\bigl(k+K\|f'(s)\|\bigr)\cdot t\Bigr)\cr
=&\Bigl(v_s\cdot t, s+\bigl(f(s)-s\bigr)v_s\cdot t\Bigr)}
$$

Finally, let  $\G_f\subset\rspace2$ denote the classical graph
of the function $f$. There are relations between the points on
the lines $P_s$ and the points of $\G_f$; to find them, the
following observation by T. Gaffney is extremely useful.
\medskip

\proclaim Lemma. Suppose $\F\left(s_1,t_1\right)
=\F\left(s_2,t_2\right)$; then the secant line connecting
the points $\bigl(s_1, f(s_1)\bigr)$ and $\bigl(s_2, f(s_2)
\bigr)$ on $\G_f$ has slope ${C-1\over C}$ where
$C=\bigl(k+K\|f'(s_1)\|\bigr)\cdot t_1 =
   \bigl(k+K\|f'(s_2)\|\bigr)\cdot t_2$.
\medskip

This establishes a ``duality'' between pairs of mapping lines
$\L_{s_1}\and\L_{s_2}$ in the function diagram and pairs of points
$\left(s_1,f(s_1)\right)\and \left(s_2,f(s_2)\right)$ on the
graph $\G_f$. The abscissa $C$ of the intersection of the
mapping lines determines the slope ${C-1\over C}$ of the secant
line to the graph, i.e. the average rate of change of $f$. If
the mapping lines are parallel (intersect at infinity), the
average rate of change is $\lim_{C\rightarrow\infty}{C-1\over
C} = 1$.

If the value $s_1$ is held fixed, and $s_2\rightarrow s_1$,
then the secant line approaches the tangent line and the rates
of change (slopes) approach the derivative. In the mapping
diagram, the mapping line $\L_{s_2}$ approaches the mapping line
$\L_{s_1}$, with their point of intersection approaching a
point whose abscissa $C$ gives ${C-1\over C} = f'(s_1)$. The
proof of this fact is a nice exercise in elementary calculus.
This limiting point lies on the ``envelope'' formed by the
mapping lines: see the top of figure 4. Thus, limiting
points of intersection in the mapping diagram correspond to
tangent lines to the graph.


\medskip
\line{\hfill
\vbox to 3truein{\hsize5truein\vfill\special{pcl:envelope.pcl}
\hfill}\hfill}
\smallskip
\centerline{\bf Figure 4}
\medskip



We can actually find the (parametric) equation of the
envelope formed by the mapping lines. The mapping line $\L_s$
has equation $y = s + \bigl(f(s)-s\bigr)\cdot x$ since it
joins $(0,s)$ with $(1,f(s))$. The envelope point on it has,
as we have seen, abscissa $C$, where ${C-1\over C} = f'(s)$;
i.e. $C = {1\over 1-f'(s)}$.
\medskip

\proclaim Proposition 1. The envelope of the mapping diagram
for $f$ is parametrized by:
$$
s\mapsto \left({1\over 1-f'(s)},\ {f(s)-sf'(s)\over
1-f'(s)}\right).
$$

Because of the way the Visualizer has been set up, it only
displays those parts of the mapping lines which lie between
the domain line ($x=0$) and the range line ($x=1$).
Consequently, the intersection of mapping lines will only be
visible for {\it decreasing} functions. This is not a serious
limitation, since one can replace $f$ by $-f$ if necessary.

Figure 4 shows an example for the function
$s\mapsto -s^2$. The mapping line $\L_{1.6}$ has been
highlighted (done in color by the Visualizer). It is tangent
to the envelope at a point whose abscissa can be easily
estimated (using a mouse-sensitive coordinate display) to be
$C\approx 2.4$. This gives an estimate for the derivative of
$-3.167$ which is fairly close to the correct value of $-3.2$.
Since $f'(s)= -2s$, the parametrization of the envelope
becomes
$$
s\mapsto \left(\underbrace{1\over 1+2s}_x, \underbrace
{s^2\over 1+2s}_y\right).
$$

Here it's possible to eliminate the parameter $s$, which
yields the explicit equation for the envelope: $ y = (1/x - 2
- x)/4$.

Finally, returning to the general case, those critical points
for which $f'(a) = 0$ correspond to points on the envelope
$(1,f(a))$; thus, $f'(a) = 0$ if and only if the mapping line
$\L_a$ is tangent to the envelope at the range line $x=1$.
Thus, the mapping diagram detects these ordinary critical
points.
\bigskip


\beginsection{Wave Fronts and Singularities}

As mentioned above, cusps form at places where the derivative
has a maximum or minimum; i.e. where $f''(s)=0$. One can ask
exactly where on the mapping diagram this occurs. The answer
is that it occurs where the envelope meets the mapping line
for such points $s$. More precisely, define the wave front at
(time) $t$ to be the curve $\f_t:\rspace1\rightarrow \rspace2$
given by
$$\eqalign{\f_t(s) = \F(s,t)&=\Bigl(
\bigl(k+K\|f'(s)\|\bigr)\cdot t,\
s+\bigl(f(s)-s\bigr)\bigl(k+K\|f'(s)\|\bigr)\cdot t\Bigr)\cr
&=\Bigl(v_st, s+\bigl(f(s)-s\bigr)v_st\Bigr)}$$

\noindent (where $v_s=k+K\|f'(s)\|$ is the point's speed).

Because the speed the points move is determined by $\|f'(s)\|$,
there will {\it always} be a cusp at places where $f'(s)=0$
and $f''(s)\ne0$. We shall consider, then, only places where
$f'(s)\ne0$.
\medskip

\proclaim Proposition 2. Suppose $f'\ne0$ in some region of the
domain. Then cusps of the wave front $\f_t$ form at those
places which are the intersection of the envelope for $f$ with
mapping lines $\L_s$, for those $s$ having $f''(s) = 0$.
\medskip

{\bf Proof}  Since the Visualizer only displays the envelope
for decreasing functions, we will assume $f'(s) < 0$, so that
$\|f'(s)\| = -f'(s)$. A nearly identical proof works for
places where $f'(s) > 0$. Since cusps occur at places where
$\f_t'(s)=\vec0$, we calculate this tangent vector:
$$\f_t'(s) = \Bigl(tv_s',\ 1+\bigl(f'(s)-1\bigr)v_st +
\bigl(f(s)-s\bigr)v_s't\Bigr)$$

\noindent where $v_s'= -Kf''(s)$ since $\|f'(s)\|=-f'(s)$.

The starting curve $\f_0$ is a vertical line, so has no cusps,
hence we may assume $t\ne0$; the minimal velocity $K$ is also
non-zero, so $tv_s'=0\iff v_s'=0\iff -Kf''(s)=0\iff f''(s)=0$.
This being the case, we see that
$$
\f_t'(s)=\vec0\iff f''(s)=0\and t={1\over
v_s\bigl(1-f'(s)\bigr)}.$$

Substituting this value of $t$ into the formula for $\f_t(s)$
tell us that the cusp must be the point
$$
\left({1\over1-f'(s)},\ s+{f(s)-s\over1-f'(s)}\right)
=\left({1\over1-f'(s)},\ {f(s)-sf'(s)\over1-f'(s)}\right).$$

Applying the description of the envelope given by Proposition
1 now completes the proof.
\bigskip

\beginsection {Conclusions}

The Dynamic Function Visualizer presents new and illuminating
pictures of functions considered not as static
graphs but as mappings of points. As the above examples show,
this picture is rich enough to provide insight at levels of
mathematical sophistication ranging from high school algebra
to research study of singularities. Like the traditional
graph, the Visualizer can provide an intuitive and geometric
language for describing many aspects of functional behavior.
The work on the Visualizer was mostly supported by the
National Science Foundation and, as all works supported by
the public should be, is in the public domain for all
non-profit purposes. Copies for the Mac and DOS environments
can be downloaded via {\bf ftp} from {\bf
wuarchive.wustl.edu}; write to me (e-mail) for further
information.

\bigskip

\noindent REFERENCES
\smallskip
\hrule
\bigskip

\item{1.} Bourbaki, N., {\it El\'ements de Math\'ematique},
Hermann, Paris, 1958.
\medskip

\item {2.} Boyer, C.~B.~and Merzbach, U.~C., {\it A History of
Mathematics}, Wiley, New York, 1989.
\medskip

\item{3.} Brieske, T., Mapping Diagrams, Continuous Functions
and Derivatives, {\it College Math.~Journal}, 9 (1978) 67-72.
(Reprinted in {\it A Century of Calculus, Part II}, MAA
(1992), 113-118.)
\medskip

\item{4.} Dick, T.~P.~and Patton, C.~M., {\it Calculus}, PWS
Kent, Boston, (1992).
\medskip

\item{5.} Dubinsky, E.~and Harel, G., eds.: {\it The Concept of
Function}, {\it MAA Notes} v.~5, (1992).
\medskip

\item{6.} Flashman, M., {\it Sensible Calculus} (to appear).
\medskip

\item{7.} Goldenberg, P.~et.~al., Representation and the
Development of a Process Understanding of Function, in {\bf
[5]}, 235-260.
\medskip

\item{8.} Kelley, J., {\it General Topology}, Van Nostrand,
Princeton N.~J.~, (1955).
\medskip

\item{9.} Peano,  G., Sulla definizione di funzione, {\it Atti
della Reale Accad.~dei Lincei}, v.~20 (1911)
\medskip

\item{10} Quine, W., {\it Mathematical Logic}, Harper \& Row,
New York (1951).
\medskip

\item{11} Richmond, D.~E., Calculus --- A new look, {\it
Amer.~Math.~Monthly}, 70 (1963) 415-423. (Reprinted in {\it A
Century of Calculus, Part I}, MAA (1992), 84-92.)

\medskip

\item{12} Wiener, N., A Simplification of the Logic of
Relations, {\it Proc.~Camb. Phil.~Soc.}, 17 (1914).

\bigskip

\leftline{Mark Bridger}
\leftline{\it Mathematics Department}
\leftline{\it Northeastern University}
\leftline{\it Boston, MA 02115}
\leftline{\it bridger@neu.edu}


\bye

