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Topic: The calculus is still about limits?
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Posts: 108
Registered: 12/8/04
The calculus is still about limits?
Posted: Jan 6, 2002 12:11 AM
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At this moment the outstanding issue seems to be that "Calculus is still
about limits."

Some 50 years ago in his book An Introduction to Mathematics Alfred North
Whitehead warns

"The study of mathematics is apt to commence in disappointment ... The reason
for the failure of the science to live up to its expectations is that its
fundamental ideas are not explained to the student disentangled from the
technical procedure which has been invented to facilitate their exact
presentation in particular instances."

The calculus is not about limits. The calculus is about the study of
continuous change - changes in quantity, rates of change, and ultimately
their reciprocal determination. Limits are simply "the technical procedure
which has been invented to facilitate their exact presentation in particular
instances." The utility of a calculus on polynomial functions is that it
allows us to critically examine the calculus on a function domain where a
particular technical procedure, the use of limits, is not necessary. We must
be careful not to conflate the means with the ends that they are intended to
facilitate. In order to evaluate the validity of the claim that limits do not
sneak in the back door let us take another look at the previous response to
the algebraic quadrature of the parabola "without limits" which raised the

"However, one hidden assumption is that the area under a curve whose heights
are k times the heights of a second curve is k times the area under that
second curve. We think this is obvious geometrically because we think of
areas as limits of areas of rectangles. Archimedes would prove such a
statement by using similarity and exhaustion, and we would use limits of
Riemann sums, but it does require a proof of this nature. Thus, the limits
are hidden but there." But this is not the only way to think about area. Here
is another.

The principle of trichotomy tells us that, for all rectangles having the same
base as a given parabolic right triangle, the area of any one of these
rectangles is either greater than, equal to, or less than the area of the
parabolic triangle. The rectangle whose area is equal to the area of the
parabolic right triangle we call the mean value rectangle. Just as we
associate a slope function with the existence of tangent lines to a curve, in
an analogous fashion, we can associate an area function, say A( f ), with the
existence of a mean value rectangle for a curve over a given interval.

There is a "hidden" assumption behind the algebraic quadrature of the
parabola and it revolves around this geometric notion of a mean value
rectangle and its algebraic representation as an area function. The key
property used in this quadrature calculation is the linearity of the area
function. If A( f ) is the area under the curve over a specified interval
then the property we used is A( af+bg ) = aA( f )+bA( g ). Can the linearity
of the area function only be justified by the use of limits? No. The
properties of the area function can also be justified from the geometric
properties of the mean value rectangle. In order to do this we need to
introduce a little terminology. We will call the original rectangle of base b
and height b^2 the principal rectangle. In a sense this principal rectangle
will act as a standard or benchmark. We will measure all other areas in
comparison to it. The area of the mean value rectangle or what is the same
thing, the area of the parabolic right triangle, will then be some fraction
of the area of the principal rectangle.

A uniform vertical scaling of the principal rectangle by a factor r creates
the following changes. ( h -> h' = rh ) The heights of the mean value
rectangle and the curve are scaled by the same factor. While the absolute
vertical dimensions of the figures have changed, the relative heights are
invariant. ( y/h = y'/h' ) A uniform horizontal scaling of the principal
rectangle produces an analogous change. ( b -> b' = sb and x/b = x'/b' ) Now
this uniform scaling of the principal rectangle can be thought of in the
following way. In effect the scaling exchanges a tiling of the principle
rectangle using unit squares with a tiling using r by s rectangles. This
exchange of tiles is one-to-one and in each substitution the relative sizes
of all portions of figures falling within the tile is left invariant.
Consequently the absolute areas are scaled by a factor of rs. ( A -> A' = rsA

The difference between these two approaches to area is this. The construction
of area as approximating rectangles is focused on determining bounds on the
actual area under the curve. And ultimately, limits are required to force
these bounds to deliver an equality. The area as a mean value rectangle is
not being used to determine bounds or measures of the area directly. There
are other algebraic means for accomplishing that task. In this approach it
is necessary to determine the scaling properties of figures with polynomial
boundaries by relating them to the scaling properties of figures with
rectilinear boundaries. It is just these properties of scaling that are
captured in the linearity of the area function and used in the quadrature of
the parabola. The trick here is to recognize the pragmatic difference between
the existence problem (of a mean value rectangle) and the construction
problem (of approximating the area of a figure by rectangles).

One quick, little piece of related geometry: All parabolic triangles of the
same hypotenuse-type are similar. Suppose we start with a p-triangle whose
hypotenuse-type is given by y=ax^2. Let this p-triangle be situated in a
principal rectangle of base b and height b^2. Now scale the rectangle and
consequently the p-triangle by a factor s so that its base goes from b to b'
= sb and its height goes from b^2 to (b')^2 = (sb)^2. We see that the
relative positions of all horizontal and vertical displacements of the
hypotenuse from the origin are invariant. That is, x/b = x'/b' and y/b^2 =
y'/b'^2. The shape and relative size of the p-triangle within the principal
rectangle are scale invariant, and consequently, the area of the p-triangle
scales like s^3. Hence the factor of 8 in the previous proof.

Unfortunately in the conventional presentation of the calculus the mean value
rectangle never acquires the prominence in the integral calculus that the
tangent line does in the differential. It is instructive to reconsider the
relationship between fundamental ideas such as mean value rectangles and
tangent lines and their associated technical procedures, namely limits of
Riemann sums and secant lines and to note the differences and similarities in
their conceptual development as course material. There are striking
differences in the conceptual architecture of the differential calculus and
the integral calculus as it is usually taught. There are striking costs
associated with these differences.




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